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The narrow escape problem : a matched asymptotic expansion approach 2008

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The Narrow Escape Problem: A Matched Asymptotic Expansion Approach by Samara Pillay B.Sc.(Hons.), The University of the Witwatersrand, 2005 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Mathematics) The University Of British Columbia (Vancouver) August, 2008 c© Samara Pillay 2008 Abstract We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Narrow Escape Problem . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . 4 1.2.2 Derivation of the Model Equation . . . . . . . . . . . . . 5 1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Literature Review - Biological Context . . . . . . . . . . . . . . 10 1.4.1 Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.2 The Narrow Escape Problem in Three Dimensions . . . . 15 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 The Narrow Escape Problem in Two Dimensions . . . . . . . 18 2.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . 19 2.2 Two-Dimensional Domain with One Hole on the Boundary . . . 21 2.2.1 Calculation of φ0 and λ0 . . . . . . . . . . . . . . . . . . 21 2.2.2 Calculation of the Mean First Passage Time . . . . . . . 28 2.3 Two-Dimensional Domain with N Holes on the Boundary . . . . 29 2.3.1 Calculation of φ0 and λ0 . . . . . . . . . . . . . . . . . . 30 2.3.2 Calculation of the Mean First Passage Time . . . . . . . 38 iii Table of Contents 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3 Numerical Realizations: The Neumann Green’s Function . . 41 3.1 The Unit Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.1 The Neumann Green’s Function in a Unit Disk . . . . . . 41 3.1.2 The Mean First Passage Time in a Unit Disk with One Hole on the Boundary . . . . . . . . . . . . . . . . . . . . 43 3.1.3 The Mean First Passage Time in a Unit Disk with Two Holes on the Boundary . . . . . . . . . . . . . . . . . . . 46 3.1.4 Equally spaced Points on a Unit Disk . . . . . . . . . . . 48 3.2 The Unit Square . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.1 The Neumann Green’s Function in a Unit Square . . . . 51 3.2.2 Solution Method . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.3 The Mean First Passage Time in a Unit Square with One Hole on the Boundary . . . . . . . . . . . . . . . . . . . . 57 3.2.4 The Mean First Passage Time in a Unit Square with Two Holes on the Boundary . . . . . . . . . . . . . . . . . . . 62 3.3 Arbitrary Domains - A Numerical Approach . . . . . . . . . . . 66 3.3.1 The Boundary Element Method . . . . . . . . . . . . . . 67 3.3.2 The Unit Disk . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3.3 The Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4 Perturbed Circular Domains - Analytical Approach . . . . . . . 79 3.4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4 The Narrow Escape Problem in Three Dimensions - The Sphere 92 4.1 Derivation of the Neumann Green’s function for a Sphere . . . . 92 4.2 Three-Dimensional Sphere with N Absorbing Patches on the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2.1 Calculation of λ0 and φ0 . . . . . . . . . . . . . . . . . . 98 4.2.2 Calculation of the Mean First Passage Time . . . . . . . 108 4.3 Three-Dimensional Sphere with N Absorbing Patches on the Boundary - An Alternate Derivation . . . . . . . . . . . . . . . . 111 4.3.1 Calculation of the MFPT Directly . . . . . . . . . . . . . 111 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 iv Table of Contents 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Appendices A Derivation of d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 B Far-Field Behaviour of ψc . . . . . . . . . . . . . . . . . . . . . . . 130 C Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 v List of Tables 3.1 R(x,x0) for the unit disk with N mesh points on the boundary . 71 3.2 R(x0,x0) for the unit disk with N mesh points on the boundary 72 3.3 R(x,x0) for the ellipse with N mesh points on the boundary . . 74 3.4 R(x0,x0) for the ellipse with N mesh points on the boundary . . 76 4.1 Comparison of v̄2, v̄3 and v̄n for various values of ε, N = 1, 2 and 4.117 vi List of Figures 1.1 Structure of a dendritic spine and the trajectory of a receptor . . 12 1.2 A circular confinement domain . . . . . . . . . . . . . . . . . . . 13 3.1 MFPT for the unit disk as the initial position moves from (−1, 0) towards the absorbing arc at (1, 0) with D = 1, d = 12 and ε = 10 −5. 45 3.2 MFPT from the centre of the unit disk with two holes on the boundary with D = 1, d = 12 and ε = 10 −5. . . . . . . . . . . . . 47 3.3 MFPT, v(x), from the centre of the unit disk with N symmet- rically located holes on the boundary with D = 1, d = 12 and ε = 10−5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 MFPT for the unit square as the initial position moves from (0, 0.5) towards the exit window at (1, 0.5) with D = 1, d = 12 and ε = 10−5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.5 MFPT from the centre of the unit square as the exit window moves from (1, 0.0125) to (1, 0.9875) with D = 1, d = 12 and ε = 10−5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.6 MFPT from the centre of a unit square with two exit windows, one fixed at (0,0.5) and the other moving from (0,0.48) to (1,0.5), sides 1 and 2 with D = 1, d = 12 and ε = 10 −5. . . . . . . . . . . 63 3.7 MFPT from the centre of a unit square with two exit windows, one fixed at (0,0.5) and the other moving from (0,0.48) to (1,0.5), side 3 and all sides with D = 1, d = 12 and ε = 10 −5. . . . . . . 64 3.8 Plot of the Neumann Green’s function for the ellipse, Gm − C, versus θ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.9 Plot of the MFPT, v(x), from the centre of the ellipse versus θ0 with D = 1, d = 12 and ε = 10 −5 . . . . . . . . . . . . . . . . . . 78 3.10 Plot of the MFPT, v(x), from x = (1, 0) for the ellipse versus θ0 with D = 1, d = 12 and ε = 10 −5 . . . . . . . . . . . . . . . . . . 79 vii List of Figures 3.11 Plot of the perturbed unit disk, the curvature and ρ with ε = 0.1 and a = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.12 Plot of Rm(x0,x0) for N = 600 (red) and N = 2400 (blue) with ε = 0.1 and a = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . 89 viii Acknowledgements I would like to thank my supervisors, Michael Ward and Anthony Peirce, for introducing me to the field of asymptotics and allowing me to work on this project. I appreciate all the help they have provided. I would like to thank Sydney Pachmann, a Mathematics graduate student, for her support during my degree. To all the graduate students, thank you for making my time at UBC unforgettable. I also express my sincere appreciation to all the helpful staff of the Mathematics Department and the IAM. Financial support was provided by my supervisors and a University Graduate Fellowship. ix Dedication To my family and Ryan. x Chapter 1 Introduction Brownian motion describes the perpetual irregular motion of small particles, such as the random motion of smoke particles. This phenomenon was first studied by the British botanist Robert Brown, who noticed the chaotic mo- tions of small grains of pollen, immersed in water under a microscope in 1827. Though Brown was never able to explain what he observed, he was able to dis- pel the notion that the random movements were exclusive to pollen particles by observing the similar behaviour of dust particles. A mathematical description of Brownian motion was first formulated by Thorvald N. Thiele in 1880 in his paper on the method of least squares. This mathematical formalism was con- tinued independently by Albert Einstein in 1905 and Marian Smoluchowski in 1906. Initially, the term Brownian motion was reserved for the description of the random movement of particles immersed in a liquid or gas. However, since Brown’s discovery, the study of Brownian motion has been extensive, finding applications in various fields such as finance, biology, and statistical mechanics. This thesis is devoted to a specific aspect of Brownian motion, the mean first passage time, abbreviated as MFPT. We employ both analytic and numerical techniques to solve this problem for the mean first passage time and compare our findings to relevant empirical results found in the literature. 1.1 Brownian Motion We will give a brief mathematical description of Brownian particles immersed in a fluid in order to introduce certain key aspects of the motion. This is the simplest physical model of Brownian motion and is adapted from [20]. We assume that the size of the colloidal particles are much larger than the molecules of the surrounding fluid. With this assumption it is clear that each collision has a negligible effect, however, the collective effect of multiple collisions with the surrounding fluid alters the path of the colloidal particle. We expect 1 Chapter 1. Introduction that these collisions happen in rapid succession and in very high numbers. For example there are 1021 collisions per second for gold particles of radius 50µm immersed in a fluid under standard conditions [20]. Since it is the collective effect of many collisions that cause an observable effect, we must describe the path of a Brownian particle statistically. In the case under consideration, there are two main forces that act on the Brownian particle. Firstly, by Stokes law, which assumes low Reynolds number, there is a drag force exerted on a spherical colloidal particle by the fluid. The drag force per unit force per unit mass acting on a spherical colloidal particle is −βv, where β = 6piaη/m. Here, v is the particle’s velocity, a is the radius of the particle, η is the coefficient of dynamic viscosity of the fluid and m is the mass of the particle. The second force acting on the Brownian particle is due to the individual collisions with the molecules of the surrounding fluid. These individual collisions produce instantaneous changes to the Brownian particle’s acceleration. These changes are random both in magnitude and direction. We denote this fluctuating force by f(t) which satisfies the following properties [20]: • f(t) is statistically independent of v(t), • the variations in f(t) are more frequent than the variations in v(t), • the average of f(t) is zero. Newton’s equations of motion lead us to dv(t) dt = −βv(t) + f(t). (1.1) This equation is also known as Langevin’s equation. This is a stochastic differ- ential equation, which motivates the fact that Brownian motion is among the simplest continuous-time stochastic processes. Typically, a stochastic differen- tial equation describing a stochastic or Brownian process will be driven by two terms, a locally deterministic velocity or drift term and a volatility or Gaus- sian disturbance term. The Gaussian disturbance term or white noise term in equation (1.1) is the fluctuating force f(t). The solution of a stochastic differential equation determines the transition probability density of the random process it describes. In our case, the solution to equation (1.1) will determine the transition probability density p(v(t), t | v0) of the random process v(t), where v(0) = v0. We denote the probability density 2 Chapter 1. Introduction function from an initial position v0 to v at time t as p(v(t), t | v0). We will show later that a diffusion equation describes the time evolution of the probability density function. The transition probability density function, or pdf as it usually called, can be used to determine the probability. For example we can find the probability that v(t) ∈ A given that the v(t) at t = 0 is v0 by integrating over the pdf. That is, P (v(t) ∈ A | v0(0) = v0) = ∫ A p(v(t), t,v0)dV. (1.2) We assume that v0 is given, and thus p(v(t), t | v0)→ δ(v − v0) as t→ 0. (1.3) The situation under consideration can also be described by using statistical mechanics. In fact, we can deduce the statistical properties of f(t) by comparing the solution to equation (1.1) to known physical laws. From statistical physics we know that p(v(t), t | v0) must tend towards the Maxwellian density for the temperature T of the surrounding fluid independently of v0 as t→∞. Hence, p(v(t), t | v0)→ ( m 2pikT )3/2 exp ( −m |v| 2 2kT ) as t→∞. (1.4) The conditions on p(v(t), t | v0) impose conditions on f(t). The solution to (1.1), derived upon using an integrating factor, is v(t) = v0e−βt + ∫ t 0 e−β(t−s)f(s)ds. (1.5) Since, for large t, v(t) ≈ ∫ t 0 e−β(t−s)f(s)ds, we can conclude that the integral in this equation must have the same properties as v(t) and satisfy the Gaussian properties prescribed by the pdf (1.4). It it worth noting that the integral in (1.5) is not a standard Riemann integral. The integral is a stochastic integral. The white noise term f(t) is of locally unbounded variation, it is everywhere continuous but no-where differentiable and thus the integral cannot be defined in straightforward manner. In fact, a new framework has to be used to deal with this integral, namely the Itô calculus. 3 Chapter 1. Introduction We can describe the displacement of the Brownian particle, x(t), by x(t) = x0 + ∫ t 0 v(s)ds. (1.6) This is also a stochastic process driven by a locally deterministic velocity along with a Gaussian disturbance term. We state the probability density of x(t) for large t, p(x(t), t | x(0) = x0) ≈ (4piDt)−3/2 exp ( − |x− x0|2 4Dt ) , D = kT mβ = kT 6piaη . (1.7) The interested reader is directed to [20] for further details. It can be shown that p satisfies the diffusion equation ∂p(x(t), t | x(0) = x0) ∂t = D∆xp(x(t), t | x(0) = x0), (1.8) where D is the diffusion coefficient. We have found the pdf for the process x. In addition, we can state the following properties of the process x [20]: • The increments x(t+s)−x(t) and x(t)−x(t−u) are independent of each other and are independent of t for s ≥ 0 and u ≥ 0. • The paths of x(t) are continuous. • For s < t, x(t) − x(s) has a Gaussian distribution with mean zero and variance t− s. These are properties of a standard stochastic process known as a Wiener process. It is equation (1.8), which describes the time evolution of the probability density function for the process x that is of particular importance to this thesis. 1.2 The Narrow Escape Problem We have described certain key aspects of Brownian motion. We now state the specific research problem considered in this thesis. 1.2.1 Statement of the Problem We consider the motion of a Brownian particle trapped in an arbitrary bounded domain, Ω ∈ Rd d = 2, 3, whose boundary is reflecting, ∂Ωr, except for a small 4 Chapter 1. Introduction absorbing window, ∂Ωa. We assume that ∂Ω = ∂Ωa+∂Ωr is a d−1 dimensional analytic surface and that ∂Ω is sufficiently smooth. Furthermore, we assume that the size of the absorbing window, centred at x0, is small in comparison to the reflecting portion of the boundary. We define the small parameter ε as ε = |∂Ωa| |∂Ωr| ¿ 1. (1.9) Alternatively, we may define ε = |∂Ωa|. However, it must be understood that the absorbing part of the boundary is asymptotically small in comparison to the reflecting part. The trajectory of the Brownian particle is denoted by x(t). The mean first passage time or exit time is defined as the time taken for the Brownian particle to escape from the domain, Ω, through the absorbing arc, ∂Ωa, centred at x0, from some initial position x(0) ∈ Ω. The mean first passage time, from a fixed starting position x(0) = x is defined in [20] as v(x) = E [τ | x(0) = x] . (1.10) The focus of this thesis is to find the mean first passage time for a variety of two and three dimensional domains, Ω, with N absorbing windows on the bound- ary. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, or as ε → 0, the calculation is a singular per- turbation problem. A plethora of work has already been done on this problem. In this thesis we use an alternative method based on matched asymptotic ex- pansions, a singular perturbation technique, to calculate the mean first passage time. 1.2.2 Derivation of the Model Equation The narrow escape problem is equivalent to solving an inhomogeneous mixed Neumann-Dirichlet boundary value problem for the Poisson equation. We will derive this equation. Our starting point is equation (1.8), which is a diffusion equation describing the time evolution of the probability density function asso- ciated with the process y. From [5], the probability density function satisfies 5 Chapter 1. Introduction the Fokker-Planck equation ∂p(y, t | x) ∂t = D∆yp, y,x ∈ Ω, ∂p(y, t, | x) ∂n(y) = 0, y ∈ ∂Ωr,x ∈ Ω, p(y, t, | x) = 0, y ∈ ∂Ωa,x ∈ Ω, p(y, 0 | x) = δ(y − x). (1.11) where ∂n = n̂·∇ and n̂ is the unit outward normal. The boundary conditions are easily explainable in the context of the physical problem at hand. The Neumann boundary condition on the reflecting part of the boundary represents the no flux boundary condition. The Dirichlet boundary condition on the absorbing part of the boundary represents the fact that the Brownian particle is absorbed at the boundary. The first passage time to the absorbing boundary is defined in [5] as τ = inf{t > 0 : y(t) ∈ ∂Ωa}. (1.12) We are interested in the mean first passage time to ∂Ωa given that the particle starts at some initial position x, that is y(0) = x. We define the survival probability as S(x, t) = ∫ Ω p(y, t | x)dy, (1.13) as in [18]. The nth moments of the first passage time are given by τ̄n = − ∫ ∞ 0 tn ∂ ∂t S(x, t)dt (1.14) = −tnS(x, t) |∞0 +n ∫ ∞ 0 tn−1S(x, t)dt, (1.15) in [18]. The first line is obtained using integration by parts. In the second line, it is important to note that we can set the first term to zero in both the upper and lower limit. This is a consequence of the fact that the pdf p(y, t | x) given by (1.11) tends to zero as t → ∞. Thus, in the limit as t → ∞, S(x, t) → 0 much faster than tn →∞ [18]. We are interested in the mean first passage time, which is the first moment of the equation above [18]. Thus, we find that the mean first passage time is the integral of the survival probability over t. Thus, 6 Chapter 1. Introduction the mean first passage time satisfies the conditional expectation τ̄x = E [τ | y(0) = x] = ∫ ∞ 0 ∫ Ω p(y, t | x)dydt, (1.16) as in [20], [18] and [5]. We return to the pdf, (1.11). We integrate this equation over t from 0 to ∞ and make use of the boundary conditions to find −δ(y − x) = D∆y ∫ ∞ 0 p(y, t | x)dt. (1.17) We let g(y | x) = ∫∞ 0 p(y, t | x)dt. Using this we find that ∆yg(y | x) = −δ(y − x) D , y,x ∈ Ω, (1.18) ∂g(y | x) ∂n(y) = 0, y ∈ ∂Ωr,x ∈ Ω, (1.19) g(y | x) = 0, y ∈ ∂Ωa,x ∈ Ω. (1.20) Thus, we can find the MFPT by solving for g from the equation above and integrating the final solution over Ω. That is τ̄x = ∫ Ω g(y | x)dy = ∫ ∞ 0 ∫ Ω p(y, t | x)dydt. (1.21) There is a simpler way to find the MFPT. The pde given by (1.11) is the Fokker-Planck equation. That means that the differentiation with respect to the spatial variable and the boundary and the initial conditions are given in terms of the forward variable, x, not the initial position, y. One can state a similar equation for the pdf in terms of backward variables, known as the Kolmogorov backward equation ∂p(y, t | x) ∂t = −D∆xp, y,x ∈ Ω, (1.22) ∂p(y, t | x) ∂n(x) = 0, x ∈ ∂Ωr,y ∈ Ω, (1.23) p(y, t | x) = 0, x ∈ ∂Ωa,y ∈ Ω, (1.24) p(y, t | x) = δ(y − x). (1.25) There are several differences between the Kolmogorov backward equation and 7 Chapter 1. Introduction the Fokker-Planck equation. Firstly, the spatial operator in the Fokker-Planck equation is the adjoint of the operator in the Kolmogorov equation, which acts on the forward variable, y. In the case under consideration the operator is the Laplacian, which is a self-adjoint operator. However, there is a sign difference in the coefficient of the Laplacian. In the Fokker-Planck equation, (1.11), it is a +1 while in the Kolmogorov equation, (1.22), it is a −1. Furthermore, if one compares the boundary conditions (1.23) and (1.24) to the boundary conditions of (1.11), we see that the initial position x starts on the boundary in the Kolmogorov backward equation, while the final position y is within Ω. Lastly, there is no initial condition but a final condition in the Kolmogorov backward equation. Thus, in the limit as t → ∞, the pdf p(y, t | x) tends to δ(y − x) and the pdf p(y, t | x)→ 0 as t → 0. This is also different to the pdf given by (1.11) We know the definition of the MFPT, given by (1.16). Thus, we must integrate (1.22) over Ω with respect to y and from 0 to ∞ with respect to t. We let v(x) = ∫∞ 0 ∫ Ω p(y, t | x)dydt. We find that v(x) = τ̄x satisfies ∆v(x) = − 1 D , x ∈ Ω, (1.26) v(x) = 0, x ∈ ∂Ωa, (1.27) ∂v(x) ∂n(x) = 0, x ∈ ∂Ωr. (1.28) The reader is directed to [20] for further reading and an alternate derivation of the MFPT, v(x) satisfying (1.26)-(1.28). We also define the average MFPT, v̄, based on an assumed uniform distri- bution of starting points, v̄ = 1 |Ω| ∫ Ω v(x)dx. (1.29) 1.3 Literature Review The mean first passage time is a first-passage probability, which is the proba- bility that a diffusing particle or random walker first hits a specified site. The first instances of such a study came with Lord Rayleigh, who considered the flux through a small hole in the context of acoustics [17]. Today, there are many applications of first-passage probabilities, for example fluorescence quenching, integrate-and-fire neurons and the execution of buy/sell orders when a stock 8 Chapter 1. Introduction price reaches a certain threshold [18]. Moreover, the narrow escape problem often appears in the context of electrostatics and biology. As a result of the far reaching applications of the MFPT, the approach to solving the problem has been varied. Statistical, numerical and analytical techniques have all been employed to solve the narrow escape problem. We present a brief overview of a few subject areas where the narrow escape problem arises. As we have shown, the narrow escape problem reduces to a mixed boundary value problem for the Poisson equation. This scenario comes up in a variety of contexts. In particular these problems arise in classical electrostatics, for example, the electrified disk problem [6]. Furthermore, the problem arises in elasticity theory, diffusion and conductance theory and acoustics [17]. These problems are generally solved using a separation of variables technique, with a subsequent summing of the resulting series. This approach to obtain analyti- cal results has proved to be challenging, especially when considering complex domains, and ultimately, numerical techniques are employed. The first passage probability and mean first passage time play fundamental roles in electrostatics as a result of their physical interpretation. In fact, the probability of a particle, initially at a position x(0), escaping at the point x0 on the boundary is equal to the electric field at the point x0, when a point charge is located at x(0) and the boundaries are grounded conductors [18]. This analogy is powerful, and can be applied to calculations in many areas. For example, one can determine the ‘break-even’ probabilities in the stock market more easily by considering this analogy [18]. One can obtain digitized representations of composite materials via Brow- nian motion. The first passage equations are adapted to digitized media and are solved numerically. The first passage time plays a role in determining the conductivity, dielectric constant, and diffusion coefficient of digitized composite media [32]. In biology, the motion of ions, molecules or receptors are modelled as free Brownian particles. In this context, the mean first passage time represents the mean time for a receptor to hit a target binding site [25], [23], [21], [24], [5], [2], [1]. The applications are far reaching including, receptor trafficking in a synaptic membrane, calcium diffusion in dendritic spines and vesicle trafficking in cells to name a few. Bressloff et. al. in [2] calculated the MFPT in a rectangular planar domain with absorbing portions within the domain. The statement of the problem is 9 Chapter 1. Introduction slightly different to equation (1.11) in that the pdf has periodic boundary condi- tions in addition to the mixed boundary conditions. The method of solution is a combination of statistical properties and an asymptotic approach. This model describes protein receptor trafficking within the membrane of a cylindrical den- drite. If one transforms the problem for the MFPT into an eigenvalue problem we find that the MFPT is inversely proportional to the principal eigenvalue, in the limit of small windows. We shall explore this connection in detail in following sections. In fact, it is the key idea used in this thesis to estimate the MFPT. The problem for the principal eigenvalue and eigenfunction has been extensively studied, for example in [8], and the references therein. The Neumann Green’s function for the domain under consideration is required to solve this problem. As a result, the form of the mean first passage time is different in two and three dimensions because of the difference in the singular behaviour in the Neumann Green’s function. In two dimensions it is a logarithmic singularity and in three dimensions it is a simple pole. We will devote a portion of this thesis to finding the Neumann Green’s function in various domains. There are other closely related problems to the narrow escape problem, such as Kolomogorov’s exit problem and the problem for the Dwell time, which will be discussed later. However, an extensive listing of all instances where the narrow escape problem arises is not relevant to this thesis. Instead, we review the relevant work of a few authors in the area who have produced results for escape from two and three-dimensional domains. In this thesis we intend to expand upon these results, and compare our findings with the work of these authors. 1.4 Literature Review - Biological Context We review the work of certain authors who have determined results for the MFPT in the context of biology. We intend to compare our findings to the results of these authors. The calculation of the mean first passage time is a typical problem in cellular biochemistry. In particular, the function of neurobiological microstructures, such as dendrites, is largely unknown since experimental methods have failed to produce a complete understanding of the neuron. An understanding of the mechanism by which newly synthesized proteins from the soma reach distant 10 Chapter 1. Introduction locations on axons or dendrites is still elusive. Hence, a mathematical framework has been developed to shed light where experimental methods have failed. 1.4.1 Two Dimensions We consider an example to emphasize the importance of the two-dimensional narrow escape problem in a neurobiological context as outlined in [5]. The axon contains presynaptic active zones for neurotransmitter release. Dendrites, on the other hand, contain postsynaptic densities where receptors that bind neurotransmitters cluster. At most excitatory synapses in the brain, it is the dendritic spines that contain the postsynaptic densities (PSD). Dendritic spines are tiny membranous protrusions, on the order of micrometers, on the surface of a dendrite. Generally, the spine head is bulbous in shape and is connected to the stalk of the dendrite via a thin neck. Holcman and Schuss in [5] consider the motion of a receptor inserted into a dendritic spine. The physical question they address is, how long does it take the receptor to reach its final destination on the postsynaptic density (PSD) from its point of insertion? The motion of the diffusing receptor is considered to be free Brownian motion in the plane. Firstly, we describe the geometry of the surface of the dendritic spine as in [5]. We consider the surface of the domain to be planar, thus neglecting the curvature. The surface contains many confinement domains known as cor- rals. These corrals are smooth two-dimensional domains, Ω, whose boundary is reflecting, ∂Ωr, except for a small absorbing portion, ∂Ωa. The reflecting boundary may represent a physical barrier or a potential barrier. We expect that the random walker, the receptor in this case, gets trapped in these con- finement domains during its journey towards its final anchoring position on the PSD. Holcman and Schuss aim to find the mean confinement time, the time a receptor spends in a corral. Figure 1.1 depicts the geometry of the dendritic spine. 11 Chapter 1. Introduction Figure 1.1: Structure of a dendritic spine and the trajectory of a receptor The placement of the corrals, or confinement domains, are also illustrated in Figure 1.1. The confinement domain may be an arbitrary two-dimensional shape. Figure 1.2 depicts a circular confinement domain along with the trajec- tory of a random walker trapped within it. 12 Chapter 1. Introduction Figure 1.2: A circular confinement domain Typically, the trajectory of the Brownian particle occupies a much larger area within the domain than what is depicted in Figure 1.2. Figures 1.1 and 1.2 are based respectively on Figures 1 and 2 in [5]. Physically, the confinement domain represents sites where the diffusing re- ceptor can bind to scaffolding proteins. Thus, the mean time a protein spends in a confinement domain provides information on the type of bonds a receptor can make with scaffolding proteins. Binding increases the mean first passage time. In fact, a diffusing receptor may bind to a scaffolding protein and never leave the confinement domain. This is in alignment with the fact that a receptor inserted into a dendritic spine far from the PSD may remain in the spine with- out reaching the PSD. Holcman and Schuss, in [5], are concerned with the mean time it takes a receptor to leave a confinement domain. In other words, they did not consider the possibility that receptors may be anchored in the confinement domain. 13 Chapter 1. Introduction Holcman and Schuss considered a circular confinement domain, depicted in Figure 1.2. The narrow escape problem for a circular domain with an absorbing arc on the boundary was studied in both [23] and [5]. In [23] and [5], the mean first passage time from the centre of a disk of radius R with a small absorbing arc of length 2ε on the boundary was shown to be E [τ | x(0) = 0] = R 2 D [ log 1 ε + log 2 + 1 4 +O(ε) ] . (1.30) The mean first passage time averaged with respect to an initial uniform distri- bution of starting points in the disk, from [23], is Eτ = R2 D [ log 1 ε + log 2 + 1 8 +O(ε) ] . (1.31) The maximum mean first passage time is attained when the initial position is antipodal to the centre of the absorbing arc. Singer et al., in [25], placed the initial position at x = (r = 1, θ = 0) and the centre of the absorbing arc at x0 = (r = 1, θ = pi). For this situation Singer et al. calculated that E [τ | x = (r = 1, θ = 0)] = R 2 D [ log 1 ε + 2 log 2 +O(ε) ] . (1.32) Holcman and Schuss in [5] were able to calculate the mean time to anchoring in the PSD using the confinement time. This process underlies synaptic plasticity, the change in the number of receptors at a synapse. Synaptic plasticity is conjectured to underlie processes as complex as learning and memory. The results above conform to a general result derived by Singer et al. in [25], in which they showed that the leading order term and error estimate for a two-dimensional Riemannian manifold with metric g is Eτ = |Ω|g piD [ log 1 ε +O(1) ] , (1.33) where |Ω|g is the Riemannian area of the domain with the assumption that ε = |∂Ωa|g |Ω|g ¿ 1. We see that the leading order term here is the same as that obtained for the circle of radius R. Notice here, that to this order in ε, that the mean escape time Eτ is not a function of the initial position, x(0). Hence, it is denoted Eτ , not E[τ | x(0) = x]. In [24], Singer et al. consider two-dimensional domains with cusps and cor- 14 Chapter 1. Introduction ners. In the other words, they modify their domains to include non-smooth boundaries. If the absorbing window is located at a corner of angle α, the mean lifetime is Eτ = |Ω|g αD [ log 1 ε +O(1) ] . (1.34) Singer et al., in [24], find that the MFPT in a rectangle with sides a and b with the absorbing hole of size ε at the corner is Eτ = 2 |Ω| Dpi [ log a ε + log 2 pi + pi 6 b a + 2β2 +O ( ε a , β4 )] , (1.35) where β = e−pib/a. They also consider an annulus and the domain bounded between two tangent circles in [24]. The annulus has applications to a closely related problem for the Dwell time, described in [30]. The Dwell time is the time a Brownian particle spends within a microdomain, a confinement domain, including binding time, before it escapes through ∂Ωa. The confinement domain is a circular annulus of outer radius R and inner radius δ. The boundary of the outer disk, D(R) is reflecting except for a small absorbing portion, ∂Ωa. The boundary of the inner disk, D(δ) is absorbing. A Brownian particle is free to diffuse in D(R)−D(δ). If it is absorbed at D(δ), it is eventually released within the annulus. If it is absorbed at ∂Ωa, it does not return to the annulus. The inner disk represents an immobile trap, a domain of chemical reactions, where it reacts with binding molecules. The results for a two-dimensional domain, given by (1.30)-(1.35), were de- rived using a separation of variables technique and the solution of certain dual integral equations. We intend to reproduce the results obtained for the circu- lar disk with radius one. Furthermore, we intend to determine an asymptotic expression for the MFPT in an arbitrary two-dimensional domain. 1.4.2 The Narrow Escape Problem in Three Dimensions The narrow escape problem has been studied for an arbitrary three-dimensional domain with an arbitrary exit window in [25]. The leading order term for the MFPT is expressed in terms of an integral equation, which can be solved for specific geometries. Singer et al. were able to explicitly determine the leading order term for an absorbing ellipse on the boundary. The additional assumption was that the semi-major axis of the elliptic window, a, was to be much smaller 15 Chapter 1. Introduction than the cube root of the volume, |Ω|1/3. The MFPT was found to be Eτ ∼ |Ω|2piDaK(e), (1.36) where e is the eccentricity of the elliptic hole, and K(.) is the complete elliptic integral of the first kind. For a circular hole, the result above reduces to Eτ ∼ |Ω|4aD . (1.37) These results were obtained by solving for the MFPT using the properties of the Neumann Green’s function in three-dimensions. It is clear from these results that the MFPT depends on the geometry of the exit window, as well as on the domain Ω. These results do not contain any error estimates, and since these are only leading order approximations, the dependence of the MFPT on the initial position and location of the absorbing window is lost. These results were known to Lord Rayleigh in the context of acoustics [17]. For a spherical domain of radius R with a circular hole of radius a on the boundary Singer et al., in [25], derived the second term and error estimate of the MFPT, Eτ = |Ω| 4aD [ 1 + ε log 1 ε +O (ε) ] . (1.38) This result was derived using Collins’ method for solving dual series of integral equations and a subsequent expansion of these results in orders of ε = aR ¿ 1. We intend to expand on the results for the spherical domain with a circular exit window by deriving the next term in the asymptotic expansion for the MFPT, and to extend the results to include N small windows on the boundary of the sphere. 1.5 Outline In Chapter 2 we consider the narrow escape problem in two dimensions. We derive an asymptotic expression for the MFPT in an arbitrary two-dimensional domain with one absorbing arc on the boundary. We extend this framework to include N absorbing arcs on the boundary. We use a matched asymptotic expansions approach, a singular perturbation technique, to find the MFPT. In Chapter 3, we use the findings from Chapter 2 to obtain specific results for the unit disk and unit square. We also consider methods to solve for the 16 Chapter 1. Introduction Neumann Green’s function in arbitrary two-dimensional domains. We consider both an analytical approach and a numerical approach. In Chapter 4 we solve the narrow escape problem for a three-dimensional sphere. This involves finding the Neumann Green’s function for a sphere with a singularity on the boundary of the sphere. Wherever possible, we compare our results to relevant findings in the literature. 17 Chapter 2 The Narrow Escape Problem in Two Dimensions In this chapter, we consider the motion of a Brownian particle trapped in an arbitrary bounded two-dimensional domain, Ω, whose boundary is reflecting, ∂Ωr, except for a small absorbing window, ∂Ωa, through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean escape time. The concepts and method of solution introduced in this chapter are important as we use similar techniques in Chapter 4, where we consider three-dimensional domains. Initially, we begin our analysis with an arbitrary two-dimensional domain with one absorbing arc or hole on the boundary. We then extend our results to include N absorbing arcs of varying length. In addition, we derive results for N identical holes. To recap, the length of the absorbing arc is asymptotically small in compar- ison to the length of the reflecting boundary. In two-dimensions we define ε as lε = |∂Ωa| ¿ 1, (2.1) where l is O(1). The MFPT v(x), from a fixed starting position x is given by the probabilistic formula [20], [5], [23] and [25] v(x) = E[τ | x(0) = x]. (2.2) As the absorbing window shrinks, the mean free path time diverges. We expect the behaviour v(x)→∞ as ε→ 0 and v(x)→ 0 as x→ x0. Note that x(0) = x is the initial position of the Brownian particle and x0 denotes the centre of the absorbing arc. These are not to be confused. The MFPT v(x) satisfies the 18 Chapter 2. The Narrow Escape Problem in Two Dimensions mixed boundary value problem [20], [5], [23] and [25] ∆v(x) = − 1 D , x ∈ Ω, (2.3) v(x) = 0, x ∈ ∂Ωa, (2.4) ∂v(x) ∂n(x) = 0, x ∈ ∂Ωr, (2.5) where D is the diffusion coefficient. Our task is to calculate an asymptotic solution for v(x) in the limit ε→ 0 that satisfies (2.3)-(2.5). 2.1 Formulation of the Problem We know from the literature, [23] and [8], and we will show that the mean first passage time is asymptotically inversely proportional to the principal eigenvalue. Motivated by this fact, we transform our problem into an eigenvalue problem for the principal eigenvalue. We solve equations (2.3) to (2.5) by writing v(x) in terms of a complete set of eigenfunctions φj(x), that is v(x) = ∞∑ j=0 cjφj(x), (2.6) where φj(x) satisfies the eigenvalue problem ∆φj + λjφj = 0, x ∈ Ω, (2.7) φj(x) = 0, x ∈ ∂Ωa, (2.8) ∂φj(x) ∂n(x) = 0, x ∈ ∂Ωr. (2.9) Recall that the eigenfunctions φj satisfy the following properties (φj(x), φi(x)) = { 0 if i 6= j, 1 if i = j, (2.10) where (u, v) denotes ∫ Ω uvdx. To find cj in (2.6) we use the orthogonality properties of the eigenfunctions φj . We substitute the expression for v(x) from (2.6) into (2.3) and use (2.7) to 19 Chapter 2. The Narrow Escape Problem in Two Dimensions obtain 1 D = ∞∑ j=0 cjλjφj(x). (2.11) By multiplying both sides of (2.11) by φi and integrating over the domain, and using (2.10), we find cj = (1, φj(x)) Dλj(φj(x), φj(x)) . (2.12) Substituting the result (2.12) for cj into (2.6) we find v(x) = ∞∑ j=0 (1, φj(x)) Dλj(φj(x), φj(x)) φj(x). (2.13) where (φj(x), φj(x)) = 1. Note that we have an expression for the mean escape time, v(x), in terms of an infinite series, which is not simple to work with. Luckily, we can make some approximations that simplify the expression for the escape time. Notice that as ε → 0, or in other words as ∂Ωa → x0, where x0 denotes the centre of the absorbing arc, the eigenvalue problem given by equations (2.7) to (2.9) reduces to the unperturbed problem ∆φj + λjφj = 0, x ∈ Ω, (2.14) ∂φj(x) ∂n(x) = 0, x ∈ ∂Ω. (2.15) We know that the first eigenvalue and eigenfunction for this problem are λ0 = 0 and φ0 = |Ω|− 1 2 . Thus, in the limit as ε→ 0, we know λ0 → 0, while the other λj for j ≥ 1 remain bounded. We can expect λ0 ∼ O ( 1 log ε ) as ε → 0, as will become more apparent in the calculations that follow. This implies that 1λ0 →∞ and that 1λj << 1 λ0 for j ≥ 1 as ε → 0. Furthermore, by integrating (2.7) over the domain and using the Divergence Theorem along with the boundary conditions (2.8) and (2.9) we find∫ Ω ∆φjdx = −λj ∫ Ω φjdx, = ∫ ∂Ωr ∂nφjdS + ∫ ∂Ωa ∂nφjdS, = ∫ ∂Ωa ∂nφjdS. (2.16) 20 Chapter 2. The Narrow Escape Problem in Two Dimensions We know λj ∼ O (1) as ε→ 0 for j ≥ 1 and we can assume that |∂Ωa| ∼ O (ε). By considering the first and third line of (2.16), we can deduce that ∫ Ω φj ∼ O (ε) for j ≥ 1. Thus, we can see that the most significant contribution to (2.13) comes from the first term, j = 0. Thus to leading order v(x) ∼ (1, φ0(x)) Dλ0 φ0(x), (2.17) with (φ0(x), φ0(x)) = 1. Therefore, the expected lifetime or exit time of a Brownian particle is proportional to 1λ0(ε) in accordance with theory, [8] and[23]. Our task now is to find φ0 and λ0 using strong localized perturbation theory. The absorbing arc ∂Ωa provides a perturbation that is large in magnitude but small in extent, hence it is known as a strong localized perturbation. Gen- erally strong localized perturbations produce large changes to the solution in a localized region. We use the method of matched asymptotic expansions to con- struct the solution in a similar way as outlined in [8], [33] and [34]. In particular there will be expansions for the region in the vicinity of the absorbing arc, known as the inner region, and expansions in the outer region, the region away from the hole. By matching these expansions we can accurately describe the large local changes that occur and the relatively small changes to the solution that occur away from the absorbing arc. This is how we will proceed. 2.2 Two-Dimensional Domain with One Hole on the Boundary We start our analysis with one hole or absorbing arc on the boundary. In the next section we will extend this framework to include N holes on the boundary of varying length. We must first calculate the principal eigenvalue λ0 and the principal eigenfunction φ0 before we can calculate the mean escape time v(x) given by equation (2.17). 2.2.1 Calculation of φ0 and λ0 We need to solve the system of (2.7)-(2.9) for the principal eigenvalue λ0(ε) and the corresponding eigenfunction φ0. We start by expanding the eigenvalue λ0(ε) = λ00 + ν(ε)λ01 + (ν(ε))2λ02 + · · · , (2.18) 21 Chapter 2. The Narrow Escape Problem in Two Dimensions with the condition that ν(ε) → 0 as ε → 0, where ν(ε) = − 1log(εd) . Here d is the logarithmic capacitance, a constant, which is dependent on the length of the perturbing arc. This expansion was used for a similar problem, where the holes were located within the domain instead of on the boundary. It was shown by [34], that λ01 is independent of the position of the hole, x0. Though, the problem studied in [34] is a slight variation of the problem at hand, the result is analogous to our problem. Thus, we must go to higher orders to find terms that do depend on the location of the hole. This is important so that we obtain a formula for the MFPT that depends on the location of the hole as well as on the initial position. We can write λ0(ε) in terms of an infinite logarithmic expansion λ0(ε) = λ∗(ν) +O ( ε log ε ) where ν(ε) = − 1 log(εd) . (2.19) In [33] it was shown how to formulate this expansion. We expand λ0(ε) = λ∗(ν) + µλ1 + · · · , (2.20) where µ¿ νm for m > 0. In the outer region, away from the absorbing arc, we expand the global solution as φ0(x, ε) = u∗(x, ν) + µu1(x, ν) + · · · . (2.21) Substituting (2.20) and (2.21) into (2.7) and (2.9) we obtain for O(µ0) that ∆u∗ + λ∗u∗ = 0, x ∈ Ω \ {x0}, ∂nu ∗ = 0, x ∈ ∂Ωr,∫ Ω (u∗)2 dx = 1, (2.22) with some singularity condition as x → x0 to be found from matching inner and outer solutions. For order O(µ1) we obtain ∆u1 + λ∗u1 = −λ1u∗, x ∈ Ω \ {x0}, ∂nu1 = 0, x ∈ ∂Ωr,∫ Ω u∗u1dx = 0. (2.23) The integral conditions on u∗ and u1 in the last line of the system of (2.22) and 22 Chapter 2. The Narrow Escape Problem in Two Dimensions (2.23) are an implementation of the normalization condition on the eigenfunc- tions. Now in the inner region near the absorbing portion of the boundary ∂Ωa we let y = (x− x0) ε , (2.24) so that the arc is rescaled as ∂Ω0 = ∂Ωaε ∼ O(1). In the inner region we let v(y, ε) = φ0(x0 + εy, ε). Then we expand v(y, ε) = ν(ε)v0(y) + · · · . (2.25) Substituting (2.25) into (2.7) and (2.8), we find to order O(ν(ε)0) that ∆yv0 = 0, y /∈ ∂Ω0, v0 = 0, y ∈ ∂Ω0. (2.26) We write v0(y) = A(ν)vc(y), (2.27) with A(ν) ∼ O(1) as ε→ 0 and vc defined later. We will need to match v(y, ε) to φ0(x, ε) in the limit as x→ x0. The limit x → x0 is equivalent to ε → 0, which implies that |y| → ∞. We will need an O(1) term at leading order. Notice that v0 is multiplied by a factor of ν(ε) in the expansion (2.25). To ensure that we obtain an O(1) term in the expansion for v(y, ε) in the limit as |y| → ∞ we would need v0(y) ∼ log |y| as |y| → ∞. We find that vc(y) satisfies ∆yvc = 0, y /∈ ∂Ω0, vc = 0, y ∈ ∂Ω0, vc → log |y| as |y| → ∞. (2.28) This ensures that the inner (local) solution has logarithmic behaviour at infinity. The problem for vc(y), (2.28), has a unique solution and the behaviour at ∞ is [8] vc(y) ∼ log |y| − log d+ p.y|y|2 . (2.29) 23 Chapter 2. The Narrow Escape Problem in Two Dimensions where d, the logarithmic capacitance and p = (p1, p2), the dipole vector, are determined from the length of the hole, ∂Ωa. In our case, where the length of the hole is |∂Ωa| = 2ε, d = 12 . In general, for an arc of length l, d = l4 . This is obtained by a special solution in terms of elliptic cylindrical coordinates. We present a derivation in Appendix A. The inner and outer solutions must match to obtain a consistent solution. Thus, from matching in the vicinity of the arc, in the limit as x→ x0 or |y| → ∞ we must have u∗(x, ν) + µu1(x, ν) + · · · = ν(ε)A(ν) [ log |y| − log d+ p.y|y|2 ] + · · · . Writing the inner variables in terms of outer variables using (2.24) u∗(x, ν) + µu1(x, ν) + · · · = ν(ε)A(ν) log |x− x0| − ν(ε)A(ν) log(εd) + · · · . We recall that ν(ε) = − 1log(εd) . Thus, from matching we find that u∗ → A+Aν(ε) log |x− x0| as x→ x0. (2.30) This gives us the missing singularity condition as x→ x0. Thus u∗ must satisfy the system (2.22)-(2.30). From the matching, we see that the next order in the inner expansion is O(εν). To ensure matching of the inner and outer solutions µ = O(εν). To find u∗ and λ∗ we introduce the Helmholtz Green’s function G(x,x0, λ∗) and its regular part R(x,x0, λ∗). This Green’s function satisfies ∆G+ λ∗G = 0, x ∈ Ω, ∂nG = 0, x ∈ ∂Ω \ {x0}, (2.31) G(x,x0, λ∗) = − 1 pi log |x− x0|+R(x,x0, λ∗). (2.32) Notice that if we did not have an absorbing arc, but instead a hole in the interior of the domain, so that the singular point x0 is in the interior of Ω, the behaviour of the Green’s function would be G(x,x0, λ∗) = − 12pi log |x− x0|+R(x,x0, λ ∗). However the singularity of the Green’s function at the boundary is twice as large 24 Chapter 2. The Narrow Escape Problem in Two Dimensions as it is in the interior of the domain, thus we have a factor of − 1pi in front of log |x− x0|. The solution to (2.22) with the correct factor for the logarithmic singularity is u∗ = −piAνG(x,x0, λ∗). (2.33) The normalization condition in (2.12) determines A as pi2A2ν2 ∫ Ω G2(x,x0, λ∗)dx = 1. (2.34) By using (2.32) and (2.33) we calculate u∗ in the limit as x→ x0 as u∗(x, ε) = Aν log |x− x0| − piAνR(x0,x0, λ∗). (2.35) Comparing this with (2.30), we find that u∗ has the desired behaviour as x→ x0 provided that R(x0,x0, λ∗) = − 1 piν . (2.36) Equation (2.36) is a transcendental equation for λ∗. To proceed we expand G in powers of λ∗ for λ∗ << 1 as G(x,x0, λ∗) = 1 λ∗ G0(x,x0) +G1(x,x0) + λ∗G2(x,x0) + · · · . (2.37) We substitute (2.37) into (2.31) to obtain for O ( 1 λ∗ ) that ∆G0 = 0, x ∈ Ω, ∂nG0 = 0, x ∈ ∂Ω \ {x0}. (2.38) For O ( (λ∗)0 ) we have ∆G1 = −G0, x ∈ Ω, ∂nG1 = 0, x ∈ ∂Ω \ {x0},∫ Ω G1dx = 0. (2.39) 25 Chapter 2. The Narrow Escape Problem in Two Dimensions For O ( (λ∗)j−1 ) with j = 1, 2, 3, · · · ∆Gj = −Gj−1, x ∈ Ω, ∂nGj = 0, x ∈ ∂Ω \ {x0},∫ Ω Gjdx = 0. (2.40) We see that G0 is a constant from (2.38). The condition that the integral of Gj over the domain must be zero for j ≥ 1 is a consequence of the normalization condition. It also ensures that we obtain a unique solution, as the Green’s function is only unique up to a constant as a result of the Neumann boundary condition. We can solve for the G′js recursively. We can solve for G0 from (2.38) and (2.39). We make a semi-circular cut out of radius σ, σ << 1, near x0. We integrate (2.39) over Ω and using the Divergence Theorem and the boundary condition for G1 we find that lim ε→0 ∫ Ω\Ωσ ∆G1dx = lim ε→0 ∫ pi 0 ( −∂G1 ∂ρ |ρ=ε ) εdϕ, = − lim ε→0 ∫ Ω\Ωσ G0dx, (2.41) where ρ = |x− x0|. We know that G1 ∼ − 1pi log ρ as x→ x0, thus ∂G1∂ρ = − 1piρ . Substituting this into (2.41) we find that G0 = − 1|Ω| . (2.42) Thus G1 satisfies ∆G1 = 1 |Ω| , x ∈ Ω, ∂nG1 = 0, x ∈ ∂Ω \ {x0},∫ Ω G1dx = 0. (2.43) We call the Green’s function satisfying (2.43) the Neumann Green’s function or modified Green’s function Gm(x,x0) with regular part Rm(x,x0). We will refer to G1 as Gm. We know that as x→ x0 Gm(x,x0) = − 1 pi log |x− x0|+Rm(x0,x0). (2.44) 26 Chapter 2. The Narrow Escape Problem in Two Dimensions Using our expressions for G0 and G1 given by (2.42) and (2.44) respectively, we can rewrite expansion (2.37) as G(x,x0, λ∗) = − 1 λ∗ |Ω| +Gm(x,x0) +O(λ ∗). (2.45) Comparing the behaviour of G(x,x0.λ∗) from (2.35) and the behaviour of G(x,x0) from (2.44) we see that R(x,x0, λ∗) = − 1 λ∗ |Ω| +Rm(x,x0) +O(λ ∗). (2.46) Substituting the expression for R(x,x0, λ∗), given by (2.46), in the limit as x→ x0 into (2.36), we find that λ∗ = piν |Ω| (1 + piνRm(x0,x0)) +O(ν 3). (2.47) We have found λ0 and φ0 to first order in the outer region, from (2.20) and (2.21) we see that λ0(ν) = piν |Ω| − pi2ν2 |Ω| Rm(x0,x0) +O(ν 3), (2.48) φ0(x, ν) = piAν ( 1 λ0 |Ω| −Gm(x,x0) ) +O(λ∗), (2.49) where Gm(x,x0) satisfies (2.43). It remains to find the constant A = A(ν). To find the constant we must impose the normalization condition, (φ0, φ0) = 1, which gives A(ν) = 1 |Ω| 12 (1− piνRm(x0,x0)) +O(ν2), (2.50) where we have used the fact that ∫ Ω Gmdx = 0 and the expression for λ0 given by (2.48). Substituting this into the expression for φ0(x, ν), (2.49), and using (2.48) we have the following main result: Proposition 2.1: (One Hole) For ε→ 0, the first eigenvalue, λ0, and the first eigenfunction, φ0, of the system (2.7)-(2.9), have the two-term asymptotic 27 Chapter 2. The Narrow Escape Problem in Two Dimensions behaviour λ0(ν) = piν |Ω| − pi2ν2 |Ω| Rm(x0,x0) +O(ν 3), φ0(x, ν) = |Ω|− 1 2 − piν |Ω|− 12 Gm(x,x0) +O(ν2). (2.51) 2.2.2 Calculation of the Mean First Passage Time Using (2.17) we can find an expression for v(x) that holds in a general two- dimensional domain with one hole on the boundary. Substituting the leading order expressions for λ0 and φ0, given by (2.48) and (2.51) respectively, into (2.17) gives v(x) = |Ω| piD ( ν−1 + pi (Rm(x0,x0)−Gm(x,x0)) ) +O(ν), (2.52) where ν = − log(εd), with d the logarithmic capacitance. Hence we have the following result: Proposition 2.2: (One Hole) For ε → 0, the mean first passage time, v(x), given by (2.17), has the two-term asymptotic behaviour v(x) = |Ω| piD (− log(εd) + pi (Rm(x0,x0)−Gm(x,x0))) +O(ν). (2.53) The average MFPT, v̄, is v̄ = |Ω| piD (− log(εd) + piRm(x0,x0)) +O(ν). (2.54) Recall that v̄ = 1|Ω| ∫ Ω v(x)dx. This expression for the mean first passage time v(x) holds in a general two- dimensional domain. The modified Green’s function remains to be found for the geometry under consideration. As ε→ 0, we see that v(x)→∞, in accordance with the behaviour that we had anticipated. Also note that we have used the outer expansion for the eigenfunction to construct the mean first passage time. In other words (2.53) holds in the outer region. To find the behaviour in the local region, we must use the local expansion for the eigenfunction v(y). In this region, we do obtain the behaviour v(x)→ 0 as ε→ 0 for |x− x0| = O(ε). 28 Chapter 2. The Narrow Escape Problem in Two Dimensions We compare our result to that of Singer et al., [25], in which they obtained the one-term expansion for the behaviour of v(x) in a two-dimensional domain with one hole given by Eτ = |Ω| piD [ log 1 ε +O(1) ] . Our result, (2.53), significantly improves upon theirs. We have a two-term expansion for the mean first passage time. Consequently, our result depends on the location of the hole, x0, the initial position, x, as well as on the length of the hole, through d, and the shape of the domain, through the Green’s function. Note that if |∂Ωa| = εl, then d = l4 . 2.3 Two-Dimensional Domain with N Holes on the Boundary We can extend the expression for v(x) to include N absorbing arcs. The bulk of the analysis in the previous section remains the same, except that we replace the single arc with N arcs, ∂Ωi for i = 1, 2, .., N , of length O(ε), where xi can be interpreted as the centre of the arc. Here the length of each arc is |∂Ωi| = εli, (2.55) for some li = O(1). We assume that the arcs are non-overlapping. We restate the problem for the mean first passage time v(x) with N absorbing arcs on the boundary as ∆v(x) = − 1 D , x ∈ Ω, (2.56) v(x) = 0, x ∈ ∂Ωi, i = 1, 2, ..., N, (2.57) ∂v(x) ∂n(x) = 0, x ∈ ∂Ωr, (2.58) where ∂Ωi denotes the ith absorbing arc on the boundary. We proceed as before, by writing v(x) in terms of a complete set of eigen- 29 Chapter 2. The Narrow Escape Problem in Two Dimensions functions φj(x) which satisfy ∆φj + λjφj = 0, x ∈ Ω, (2.59) φj(x) = 0, x ∈ ∂Ωi, i = 1, 2, ..., N, (2.60) ∂φj(x) ∂n(x) = 0, x ∈ ∂Ωr. (2.61) After imposing the orthogonality and orthonormal properties, we arrive at the same expression for the MFPT v(x) as in equation (2.17). The difference is that our eigenfunctions satisfy a slightly different problem to that for one hole. We must solve solve the system (2.59)-(2.61) for the leading order eigenfunction and eigenvalue. 2.3.1 Calculation of φ0 and λ0 For ε → 0 we expand the eigenvalue λ0 and the eigenfunction φ0(x) as given by (2.20) and (2.21). The subtlety to notice here, is that we no longer have one parameter ν, instead, we have N parameters νi for i = 1, 2, ...N where νi = − 1log(εdi) , (2.62) where we incorporate the logarithmic capacitance, di of each arc. In this manner we capture the length of each arc. In particular, if |∂Ωi| = εli, then d = li4 . Now we proceed as in the case for one hole. The problem in the outer region away from the hole at order O(µ0) reads as ∆u∗ + λ∗u∗ = 0, x ∈ Ω \ {x1,x2, ...,xN}, ∂nu ∗ = 0, x ∈ ∂Ωr. (2.63) Now in the inner region near the absorbing portions of the boundary ∂Ωi for i = 1, ...N we let yi = (x− xi) ε , so that the arc is rescaled as |∂Ω0i| = |∂Ωi|ε = O(1). In the inner region we let v(yi, ε) = φ0(xi + εy, ε). 30 Chapter 2. The Narrow Escape Problem in Two Dimensions and we expand v(yi, ε) = ν(ε)v0(yi) + · · · . (2.64) Proceeding as before we arrive at the order O(ν(ε)0) equation for each hole ∆yv0 = 0, yi /∈ ∂Ω0i, v0 = 0, yi ∈ ∂Ω0i. (2.65) According to the same reasoning outlined for one hole, we want v0(yi) ∼ log |yi| as |yi| → ∞. This ensures that we can eventually match the inner and outer solutions. We write v0(y) = Ai(νi)vc(yi), (2.66) with Ai(νi) ∼ O(1) as ε→ 0. So we find that vc(yi) satisfies 4yvc = 0, yi /∈ ∂Ω0i, vc = 0, y ∈ ∂Ω0i, vc → log |yi| as |yi| → ∞. (2.67) Recall that if |∂Ωi| = εli, then di = li4 . Matching the inner and outer solutions as before we find that u∗ must satisfy ∆u∗ + λ∗u∗ = 0, x ∈ Ω \ {x1,x2, ...,xN}, ∂nu ∗ = 0, x ∈ ∂Ωr, ∫ Ω (u∗)2dΩ = 1, (2.68) u∗ → Ai +Aiνi(ε) log |x− xi| as x→ xi. (2.69) This is similar to (2.22) and (2.30) for u∗ for one hole. However, notice that we have N unknowns Ai for i = 1, ..., N , with only one normalization condi- tion. Thus, we will have a single relation between each of the Ai’s set by the normalization condition. We write the solution for u∗ in terms of the Helmholtz Green’s function u∗ = −piΣNk=1AkνkG(x,xk, λ∗) (2.70) where k is the index of the N absorbing arcs. The Green’s function G(x,xk, λ∗) 31 Chapter 2. The Narrow Escape Problem in Two Dimensions must satisfy a system similar to (2.31). The Green’s function satisfies ∆G+ λ∗G = 0, x ∈ Ω, ∂nG = 0, x ∈ ∂Ω \ {x1,x2, ...,xN}, (2.71) G(x,xk, λ∗) ∼ − 1 pi log |x− xk|+R(xk,xk, λ∗) as x→ xk. (2.72) Substituting (2.72) into (2.70) will give an expression for u∗ in the limit as x → xi. This expression must match to (2.69). After some simplification, in the limit as x→ xi, we require that Ai(1 + piνiR(xi,xi, λ∗)) + pi N∑ k=1,k 6=i AkνkG(xi,xk, λ∗) = 0, i = 1, 2, ..., N, (2.73) to satisfy the matching condition to the inner solution. This is different from the result for one hole, and is a consequence of the fact that there are N holes. System (2.73) is an NXN homogeneous linear system for the N unknowns Ai, i = 1, ..., N , which we write in matrix form as MA = 0, where A = (A1, A2, ..., AN )T is a vector and M is the matrix shown below, whose determinant must be zero in order for nontrivial solution to exist. This allows us to solve for the Ai numerically. Here we have denoted R(xi,xi, λ∗) by Rii(λ∗) and G(xi,xj , λ∗) by Gij(λ∗) for i 6= j. The matrix M is 1 + piν1R11(λ∗) piν2G12(λ∗) piν3G13(λ∗) · · · piνNG1N (λ∗) piν1G21(λ∗) 1 + piν2R22(λ∗) piν3G23(λ∗) · · · piνNG2N (λ∗) piν1G31(λ∗) piν2G32(λ∗) 1 + piν3R33(λ∗) · · · piνNG3N (λ∗) ... ... ... . . . ... piν1GN1(λ∗) piν1GN1(λ∗) · · · · · · 1 + piνNRNN (λ∗)  We expand each of the Gij in orders of λ∗ as G(xi,xj , λ∗) = 1 λ∗ G0(xi,xj) +G1(xi,xj) + λ∗G2(xi,xj) + · · · . (2.74) Consequently, each of the Gij must satisfy equivalent forms of (2.38), (2.39) 32 Chapter 2. The Narrow Escape Problem in Two Dimensions and (2.40). Applying the Divergence Theorem as we previously did in the case of one hole, we find that for each Gij we have G0(xi,xj) = − 1|Ω| , (2.75) and thus G1 satisfies ∆G1(xi,xj) = 1 |Ω| , x ∈ Ω, ∂nG1(xi,xj) = 0, x ∈ ∂Ω \ {x1,x2, ...,xN}, (2.76)∫ Ω G1(xi,xj)dx = 0, G1 ∼ − 1 pi log |x− xj |+R1(xj ,xj) as x→ xj . (2.77) This is the problem for the modified Green’s function or Neumann’s Green function. We shall henceforth refer to G1 as Gm. Now we can rewrite (2.74) for λ∗ << 1 using Gm as G(xi,xj , λ∗) = − 1 λ∗ |Ω| +Gm(xi,xj) +O(λ ∗). (2.78) Comparing the behaviour of this expression as xi → xj with (2.72), where we set x = xi, we find that R(xi,xi, λ∗) = − 1 λ∗ |Ω| +Rm(xi,xi). (2.79) We substitute (2.78) and (2.79) into (2.73) to obtain Ai ( 1 + piνiRm(xi,xi)− piνi|Ω|λ∗ ) + pi N∑ k=1,k 6=i Akνk ( Gm(xi,xk)− 1 λ∗ |Ω| ) = 0, (2.80) for i = 1, ..., N . It is convenient to rewrite this in matrix form as an eigenvalue problem. 33 Chapter 2. The Narrow Escape Problem in Two Dimensions Firstly we do some rearranging Ai(1 + piνiRm(xi,xi)) + pi N∑ k=1,k 6=i AkνkGm(xi,xk) = pi |Ω|λ∗ Aiνi + N∑ k=1,k 6=i Akνk  , (2.81) for i = 1, ..., N . We can write this equation in matrix form as CA = pi |Ω|λ∗BVA, (2.82) where C = I + piΓV, (2.83) and V =  ν1 0 · · · · · · 0 0 ν2 0 · · · 0 0 0 ν3 · · · 0 ... ... ... . . . ... 0 · · · · · · 0 νN  , B =  1 1 · · · · · · 1 1 1 · · · · · · 1 1 1 · · · · · · 1 ... ... ... . . . ... 1 1 · · · · · · 1  , A = ( A1 A2 · · · · · · AN )T , Γ =  Rm(1, 1) Gm(1, 2) Gm(1, 3) · · · Gm(1, N) Gm(2, 1) Rm(2, 2) Gm(2, 3) · · · Gm(2, N) Gm(3, 1) Gm(3, 2) Rm(3, 3) · · · Gm(3, N) ... ... ... . . . ... Gm(N, 1) Gm(N, 2) · · · · · · Rm(N,N)  . The Green’s function is symmetric since Gm(i, j) = Gm(j, i) where Gm(i, j) de- notes Gm(xi,xj) and Rm(i, i) denotes Rm(xi,xi). As a result, Γ is a symmetric matrix. Let νm = max νk for k = 1, ..., N . For νm sufficiently small, we can invert C so that we obtain a matrix eigenvalue problem for the eigenvalue λ∗ pi |Ω|C −1BVA = λ∗A. (2.84) 34 Chapter 2. The Narrow Escape Problem in Two Dimensions We let Υ = pi|Ω|C −1BV , this gives us ΥA = λ∗A, Υ = pi |Ω|C −1BV. (2.85) We can find λ∗ by finding the eigenvalue of the system above. We follow the procedure as outlined in [8]. Notice that each row of BV is the same [ν1, ν2, ..., νn], thus there is only one linearly independent row. Hence, BV has rank one. As a result, we can conclude that Υ has rank one. Thus, we can conclude that λ∗ is the only unique non-zero eigenvalue of Υ. Hence, λ∗ = Trace(Υ). By using the structure of Υ as defined in (2.85), we find that λ∗ = Trace(Υ) = pi |Ω| N∑ j=1 νj ( N∑ k=1 cjk ) , cjk = ( C−1 ) jk . (2.86) We are left to find C−1. We have assumed that νm << 1, thus the asymp- totic inverse of C is C−1 = I − piΓV + · · · . Substituting this into equation (2.86) we find that λ∗ = pi |Ω| N∑ j=1 νj ( N∑ k=1 Ijk − pi(ΓV )jk ) , (2.87) where ΓV = ∑N j=1 ∑N k=1 νkΓjk and ∑N k=1 Ijk = 1 for each j. Thus λ∗ = pi |Ω|  N∑ j=1 νj − pi N∑ j=1 νj N∑ k=1 νkΓjk +O(ν3m). (2.88) To find u∗ we use (2.70) and (2.74). We have found a two-term expansion for λ0 and φ0. We summarize our result as follows: Proposition 2.3: (N Holes) For ε→ 0, the first eigenvalue, λ0, and first eigenfunction, φ0, of the system (2.59)-(2.61), have the two-term asymptotic 35 Chapter 2. The Narrow Escape Problem in Two Dimensions expansions λ0(ε) = pi |Ω|  N∑ j=1 νj − pi N∑ j=1 N∑ k=1 νjνkΓjk +O(ν3m), (2.89) φ0(x, ε) ∼ pi λ∗ |Ω| N∑ k=1 Akνk − pi N∑ k=1 AkνkGm(x,xk). (2.90) This gives us a general expression for a two-dimensional domain withN holes of differing length on the boundary. It must be noted that we have not explicitly imposed the normalization condition here to find the relationship between the Ai. The Ai can be found by finding the eigenvector of the matrix system (2.85). Notice that in the case of N identical holes, there is simplification in the expression for the eigenvalue λ0 and the eigenvalue φ0. Suppose that we have N identical holes. Thus, each of the νi are identical. The expression for φ0 given by (2.90) reduces to φ0(x, ε) ∼ piν λ0 |Ω| N∑ k=1 Ak − piν N∑ k=1 AkGm(x,xk). (2.91) Note that ∑N j=1 ∑N k=1 Γjk = ∑N k=1 ( Rm(xk,xk) + ∑N j=1,j 6=kGm(xj ,xk) ) . To simplify the notation, we define p(x1,x2, ...,xN ) by p(x1,x2, ...,xN ) = N∑ k=1 Rm(xk,xk) + N∑ j=1,j 6=k Gm(xj ,xk)  (2.92) In the case of N identical holes we can impose the normalization condition explicitly to find the relationship between the Ai. We calculate (φ0, φ0) using (2.91) and (2.89) to find that N∑ k=1 Ak = N |Ω| 12 ( 1− piν N p(x1,x2, ...,xN ) ) +O(ν2), (2.93) where p is given by (2.92). As a Corollary to Proposition 2.3, we obtain the following result for N iden- tical holes: Corollary 2.4 (N Identical Holes) Suppose that the N holes are identical, 36 Chapter 2. The Narrow Escape Problem in Two Dimensions that is νj = ν. The expressions for λ0 and φ0, given by (2.89) and (2.90), simplify to λ0(x, ε) = piν |Ω| (N − piνp(x1,x2, ...,xN )) +O(ν 3), (2.94) φ0(x, ε) = |Ω|− 1 2 − piνΣNk=1AkGm(x,xk) +O(ν2). (2.95) For ν ¿ 1, we observe that the eigenvalue, λ0, given by (2.94) is largest when the hole locations x1,x2, ...,xN are chosen so as to minimize p(x1,x2, ...,xN ). The mean escape time, which is inversely proportional to the principal eigen- value, will be a minimum in this case. There is in fact one more special case that we can consider. In the case of N identical holes, there arises the possibility of attaining a cyclic matrix. Recall from the matrix equation (2.85) where we have ΥA = λ∗A, we can find A as the eigenvector associated with the eigenvalue λ∗. If we have an even number of equally spaced identical points in certain geometries, we find that the matrix Υ is cyclic. For example, if we have two holes placed at the corners of a rectangle or square, we obtain a cyclic matrix. In addition, we find that this property holds for any two points located on the circumference of the unit circle. Furthermore, we find that the property holds for four points, located along lines of symmetry. These will be illustrated in following sections. The cause of this special property of the matrix Υ is that the matrices Γ and C and C−1 are cyclic. As a result, we find that the eigenvector associated with the eigenvalue λ∗ has identical entries. Thus, A1 = A2 = ... = AN . Using this unique property, we can find a simple expression for v(x) analogous to the expression we found for one hole in (2.53). Notice that, for identical A′s, φ0 becomes φ0(x, ε) ∼ piNAν λ∗ |Ω| − piνA N∑ k=1 Gm(x,xk). (2.96) We can simplify the expression for ∑N k=1Ak given by (2.93) to find A = 1 |Ω| 12 ( 1− piν N p(x1,x2, ...,xN ) ) +O(ν2). (2.97) We can now state the new expression for φ0 in the case of a cyclic matrix 37 Chapter 2. The Narrow Escape Problem in Two Dimensions Corollary 2.5: (N Identical Holes, Υ Cyclic) Suppose that the N holes are identical, that is νj = ν and the holes are placed in such a manner that the matrix Υ in system (2.85) is cyclic. The expression for φ0, given by (2.90), simplifies to φ0(x, ε) = |Ω|− 1 2 − piν |Ω|− 12 N∑ k=1 Gm(x,xk) +O(ν2). (2.98) 2.3.2 Calculation of the Mean First Passage Time Next we use (2.17) to find the MFPT in an arbitrary two-dimensional domain with N holes on the boundary. We will have three different expressions for the MFPT for the following cases, N holes of differing length on the boundary, N identical holes on the boundary, and N identical holes where we have a cyclic matrix. For N holes of differing length we substitute the expressions for φ0 and λ0 from (2.89) and (2.90) into (2.17). In terms of λ0, we find a two-term expression for v(x): Proposition 2.6: (N Holes) For ε → 0, the two-term asymptotic be- haviour of the mean first passage time, v(x), is v(x) ∼ pi ∑N k=1Akνk D(λ0)2 ( pi λ0 |Ω| N∑ k=1 Akνk − pi N∑ k=1 AkνkGm(x,xk) ) . (2.99) The average MFPT, v̄, is v̄ ∼ pi ∑N k=1Akνk D(λ0)2 ( pi λ0 |Ω| N∑ k=1 Akνk ) . (2.100) Notice that the normalization condition has not been imposed in equation (2.99) for the MFPT. Therefore, we must divide expression (2.99) by (φ0, φ0), once we have found the Ai numerically by solving the eigenvalue problem (2.85). For N identical holes, we use expressions (2.94) and (2.95) for the eigen- function and eigenvalue. We have already imposed the normalization condition in this case. We obtain that the mean first passage time for a two-dimensional 38 Chapter 2. The Narrow Escape Problem in Two Dimensions domain with N identical holes on the boundary is given by the following: Corollary 2.7: (N Identical Holes) For ε→ 0, the two-term asymptotic behaviour of the mean first passage time, v(x), is v(x) = |Ω| piND ( − log(εd)− |Ω| 12 pi N∑ k=1 AkGm(x,xk) + pi N p(x1,x2, ...,xN ) ) +O(ν), (2.101) where p(x1,x2, ...,xN ) is defined by (2.92). The average MFPT, v̄, is v̄ = |Ω| piND ( − log(εd) + pi N N∑ k=1 p(x1,x2, ...,xN ) ) +O(ν). (2.102) In the case of the matrix Υ being cyclic we find: Corollary 2.8: (N Identical Holes, Υ Cyclic) For ε→ 0, the two-term asymptotic behaviour of the mean first passage time, v(x), is v(x) = |Ω| piND ( − log(εd) + pi ( 1 N p(x1,x2, ...,xN )− N∑ k=1 Gm(x,xk) )) + O(ν). (2.103) The average MFPT, v̄, is v̄ = |Ω| piND ( − log(εd) + pi N p(x1,x2, ...,xN ) ) +O(ν). (2.104) Notice that by setting N = 1 in equations (2.98) and (2.103), we recover (2.51) and (2.53). Also notice that in (2.102) and (2.104) that the average MFPT is minimized for a configuration of arcs that minimize p(x1,x2, ...,xN ). Holcman and Schuss, in [5], consider the case of N absorbing arcs of differing length. We cannot directly compare our results with those of Holcman and Schuss, since their results are in terms of an infinite series. In the case of N identical arcs, Holcman and Schuss find that the leading order term for the mean first passage time is proportional to 1N . The leading order term in our results, (2.101) and (2.103), are also proportional to 1N . Our results for the mean first passage time depend on x, xk, for k = 1, ..., N , the length of the arcs through d 39 Chapter 2. The Narrow Escape Problem in Two Dimensions and the shape of the domain, Ω. This is an improvement on the results derived in [5]. 2.4 Discussion In this chapter we solved for the MFPT in an arbitrary two-dimensional domain. We transformed the problem at hand to an eigenvalue problem. We used the method of matched asymptotic expansions to solve for the principal eigenvalue, λ0, and eigenfunction, φ0. In the case of one hole on the boundary, our expres- sion for the MFPT, (2.53), agrees with the leading order term derived by Singer et al. in [25]. Furthermore, our result is an improvement on the result derived by Singer et al. since we have the next term in the asymptotic expansion for the MFPT, which depends on the position of the hole at x0 and the initial position at x. We were able to extend this framework to find general expressions for the MFPT with N holes on the boundary. We found expressions for the MFPT with N holes of differing size on the boundary, (2.99) and N identical holes on the boundary, (2.101). Furthermore, we found the MFPT, equation (2.103), in the special case of a cyclic matrix in the system (2.85). In the next chapter we will apply these results to calculate the MFPT in a few domains. 40 Chapter 3 Numerical Realizations: The Neumann Green’s Function In this chapter, we apply the results derived in Chapter 2 to a few special domains. It is clear from our expressions for the MFPT in Chapter 2, that the Neumann Green’s function plays an important role. In fact, to find the MFPT in a given domain, one must find the Neumann Green’s function. For the unit disk and unit square we can calculate the Neumann Green’s function, Gm(x,x0), and its regular part, Rm(x,x0), analytically. For more general domains, we present and implement a boundary element method to numerically calculate Gm(x,x0) and Rm(x,x0). We use these results to find the mean first passage time. 3.1 The Unit Disk We have found an expression for the mean first passage time, (2.53), for a general two-dimensional domain with one hole on the boundary. In addition, we have found three differnt formulas for the mean first passage time with N holes on the boundary, namely (2.99), (2.101) and (2.103). We would now like to investigate the behaviour of the MFPT in specific geometries. We begin with the unit disk. In the next section we will consider the unit square. 3.1.1 The Neumann Green’s Function in a Unit Disk To investigate the mean free path time in the unit disk, we must find the Neu- mann Green’s function Gm for the unit disk. For x0 ∈ Ω, so that x0 is in the 41 Chapter 3. Numerical Realizations: The Neumann Green’s Function interior of Ω, the modified Green’s function satisfies ∆Gm = 1 |Ω| − δ(x− x0), x ∈ Ω, (3.1) ∂nGm = 0, x ∈ ∂Ω, (3.2)∫ Ω Gm(x,x0)dx = 0, (3.3) where the integral condition ensures that we obtain a unique solution to the problem. The solution to the problem above for the unit disk is well known, and takes the form [8] Gm(x,x0) = − 12pi log |x− x0|− 1 2pi log ∣∣∣∣x |x0| − x0|x0| ∣∣∣∣+ 14pi (|x|2+ |x0|2)+C(x) (3.4) where the singularity, x0, is located in the interior of the disk and C(x) is to be determined from the integral condition. Writing the expression above in the form Gm(x,x0) = − 12pi log |x− x0|+Rm(x,x0), we see that the regular part Rm is given by Rm(x,x0) = − 12pi log ∣∣∣∣x0 |x0| − x0|x0| ∣∣∣∣+ 14pi (|x|2 + |x0|2) + C(x). (3.5) For the theory in Chapter 2, the singularity is located on the boundary of the circle, so that |x0| = 1. However, we first find the constant C before we take the limit as x0 → ∂Ω. To determine C, we multiply (3.4) for Gm(x,x′0) by (3.1) and integrate over Ω. We make use of the integral property (3.3) to obtain Gm(x0,x′0) = − ∫ Ω Gm(x,x′0)∆Gm(x,x0)dx. (3.6) We now use integration by parts and the Neumann boundary condition (3.2) to find Gm(x0,x′0) = ∫ Ω ∇Gm(x,x′0) · ∇Gm(x,x0)dx. (3.7) We can see from this equation that Gm(x0,x′0) = Gm(x ′ 0,x0). This is just a proof of the symmetry property of the Green’s function, which follows from the fact that the Laplacian is a self-adjoint operator. It does allow us to deduce that 42 Chapter 3. Numerical Realizations: The Neumann Green’s Function C(x0) = C(x′0) = C. Before we find C, we show another property. Following the same argument as outlined above, along with the symmetry property of the Green’s function, we find that 1 |Ω| ∫ Ω G(x,x′0)dx = Gm(x0,x ′ 0)− ∫ Ω ∇Gm(x,x′0) · ∇Gm(x,x0)dx. (3.8) Imposing the symmetry property, we see that the ∫ Ω Gm(x,x0)dx is indepen- dent of x0. We will need this result later on when we consider a numerical approach to finding the Neumann Green’s function. To determine C, we integrate (3.4) over Ω. Since we know that the integral over G is independent of x0, we pick x0 = 0. We know that ∫ Ω log |x| dx = −pi/2, ∫ Ω |x|2 dx = pi/2 and ∫ Ω Gm(x,0)dx = 0. Using this, we find that C = − 38pi . Using this, the expression for the Neumann Green’s function in a unit disk with a singularity on the boundary is obtained by letting |x0| → 1. We obtain Gm(x,x0) = − 1 pi log |x− x0|+ 14pi |x| 2 − 1 8pi . (3.9) Note that this expression is consistent with the remark regarding (2.31) for a surface Green’s function, in that a singularity on the boundary is twice as large as one in the interior. Now we are ready to use expression (2.53) to find the mean free path time for a Brownian particle trapped in a unit circle with one hole on the boundary. 3.1.2 The Mean First Passage Time in a Unit Disk with One Hole on the Boundary We set our initial position to be the origin, x = (0, 0) and we place the singu- larity at x0 = (1, 0). Note that Rm(x0,x0) = 18pi . We substitute for Rm(x0,x0) and Gm(x,x0) with the starting point x at the centre of the disk and the sin- gularity x0 at (1, 0) into (2.53) to obtain v(x) = |Ω| piD ( − log(εd) + 1 4 ) +O(ν). (3.10) where d is the logarithmic capacitance. For a length |∂Ω| = 2ε, d = 12 and |Ω| = pi we find that: 43 Chapter 3. Numerical Realizations: The Neumann Green’s Function Proposition 3.1: (One Hole in the Unit Disk) In the limit that ε→ 0, we find the two-term asymptotic behaviour of the MFPT from the centre of the disk to be E[τ | x(0) = (0, 0)] = v(x) = 1 D ( log 1 ε + log 2 + 1 4 ) +O(ν). (3.11) Notice that with our initial position, x, at the origin, and an x0 located anywhere on the boundary, we will always arrive at the same result, (3.11). This is a consequence of the symmetry of the circle, and that all points on the boundary are located an equal distance r = 1 from the origin. This is comparable to the result derived by Singer et al., equation (1.2) of [23] Next we find the MFPT with the initial position at the antipodal point to the absorbing arc, x = (−1, 0): Proposition 3.2: (One Hole in the Unit Disk) In the limit that ε→ 0, we find the two-term asymptotic behaviour of the MFPT from the antipodal point to the absorbing arc to be E[τ | x(0) = (−1, 0)] = v(x) = 1 D ( log 1 ε + 2 log 2 ) +O(ν). (3.12) This result is also comparable to that derived by Singer et al. in equation (1.4) of [23]. Lastly, we investigate the behaviour of moving the initial position x from the antipodal point at x = (−1, 0) towards the singular point, at (1,0), in fixed increments of 140 in the x direction. We set D = 1, d = 1 2 and ε = 10 −5. We expect that the mean free path time decreases as the initial position moves towards the hole. The configuration and results are shown in Figure 3.1 below. 44 Chapter 3. Numerical Realizations: The Neumann Green’s Function −1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y initial position moves towards the absorbing arc absorbing arc (a) The initial position moves from (−1, 0) (blue) towards the hole at (1, 0) (red) −1 −0.5 0 0.5 1 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 x − initial position M FP T fo r t he  u ni t c irc le (b) Plot of the MFPT, v(x), as a function of the x coordinate of the initial position Figure 3.1: MFPT for the unit disk as the initial position moves from (−1, 0) towards the absorbing arc at (1, 0) with D = 1, d = 12 and ε = 10 −5. 45 Chapter 3. Numerical Realizations: The Neumann Green’s Function It is important to note that the initial position does not actually reach the absorbing arc, at (1, 0), since our expression for the MFPT is not valid in the inner region. The MFPT does indeed decrease as the initial position moves towards the hole. 3.1.3 The Mean First Passage Time in a Unit Disk with Two Holes on the Boundary We will now make use of formula (2.103) to find the MFPT in a unit circle with two holes on the boundary. As mentioned, in this case the placement of the holes results in the matrix Υ in the system (2.85) being cyclic. To illustrate the behaviour of the MFPT, we hold one absorbing arc fixed at x0 = (1, 0) while we move another hole in fixed angular increments of θ = pi32 from a position an increment away from (1, 0), at position 1, to the antipodal point at (−1, 0), at position 32. We keep the initial position fixed at the origin. We set D = 1, d = 12 and ε = 10 −5. The simplified expression for the MFPT from the origin is E[τ | x(0) = (0, 0)] = 1 2D ( log 1 ε + 3 8 + 1 2 log 2− 1 2 log(1− cos θ) ) +O(ν). (3.13) The configuration and the behaviour of the MFPT as a function of positions 1 to 32 are shown in Figure 3.2. 46 Chapter 3. Numerical Realizations: The Neumann Green’s Function −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y position 32 initial position moved position 1 fixed (a) Configuration of the absorbing arcs (red) and initial position (blue) 0 5 10 15 20 25 30 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 Absorbing arcs 1−32 M FP T fo r t he  u ni t d isk (b) Plot of the MFPT, v(x), versus the configuration number Figure 3.2: MFPT from the centre of the unit disk with two holes on the boundary with D = 1, d = 12 and ε = 10 −5. 47 Chapter 3. Numerical Realizations: The Neumann Green’s Function We see that the MFPT decreases as the two holes on the boundary move further apart. The MFPT reaches a minimum when the two holes are antipodal to each other, which is what we would expect. 3.1.4 Equally spaced Points on a Unit Disk We investigate one more case in the unit disk. By placing N absorbing arcs of equal length at equally spaced points along the circumference of a unit circle, we can find a closed form expression for the eigenvalue λ0 and the mean free path time v(x). For a pattern of N identical holes located symmetrically on the circumference of the unit disk, we have xj = e2piij/N j = 1, ..., N, (3.14) with N > 1 and i = √−1. We start by simplifying p(x1,x2, ...,xN ) = N∑ k=1 Rm(xk,xk) + N∑ j=1,j 6=k Gm(xj ,xk)  . (3.15) Recall that Gm(xj ,xk) = − 1pi log |xj − xk|+ 18pi as stated in (3.9). We first state the following lemma [8]: Lemma 3.3: Let N > 0 and n be integers, and i = √−1. Then, for y > 0, we have N∏ j=1 ( x− ye2pii(j−n)/N ) = xN − yN . (3.16) Proof: The interested reader is directed to the proof in [8]. Now, we consider each term of Gm separately and along with the previous Lemma we formulate the following Lemma: Lemma 3.4: Let N > 1 be an integer and let xj for j = 1, 2, ..., N satisfy (3.14). Then we have N∑ j=1 ( |xj |2 + |xk|2 ) = 2N, (3.17) N∑ j=1,j 6=k log |xj − xk| = logN. (3.18) 48 Chapter 3. Numerical Realizations: The Neumann Green’s Function Proof: The first result, (3.17), is immediate. To prove the second result (3.18), we start with N∑ j=1,j 6=k log |xj − xk| = N∑ j=1,j 6=k log ∣∣∣e2piij/N − e2piik/N ∣∣∣ , = log ∣∣∣∣∣∣ N∏ j=1,j 6=k ( 1− e2pii(j−k)/N )∣∣∣∣∣∣ . (3.19) Now using Lemma 3.3 we find log ∣∣∣∣∣∣ N∏ j=1,j 6=k ( x− ye2pii(j−k)/N )∣∣∣∣∣∣ = log ∣∣∣∣xN − yNx− y ∣∣∣∣ = log ∣∣∣∣xN−1 [1 + (yx)+ · · ·+ (yx)N−1 ]∣∣∣∣ (3.20) By using (3.20) with x = y = 1, and substituting this into (3.19), we obtain (3.18).¥ Using the Lemma above along with the Green’s function (3.9) we can rewrite p as p = N∑ k=1  N∑ j=1 ( 1 4pi ( |xj |2 + |xk|2 ) − 3 8pi ) − 1 pi N∑ j=1,j 6=k log |xj − xk|  , = N∑ k=1 [ 2N 4pi − 3N 8pi − 1 pi logN ] . (3.21) Proposition 3.5: (Unit Disk) Let N > 1 be an integer, and let xj satisfy (3.14). Then p = p(N), given by (2.93) reduces to p(N) = 1 pi [ N2 8 −N logN ] . (3.22) We can use the simplification above in (2.94) for λ0 to give us: Proposition 3.6: (Unit Disk) For ε → 0, the two-term asymptotic be- 49 Chapter 3. Numerical Realizations: The Neumann Green’s Function haviour of the principal eigenvalue, λ0, in a unit disk with N symmetrically located holes on the boundary is λ0(ε) = piν |Ω| ( N − ν ( N2 8 −N logN )) +O(ν3). (3.23) From (2.101) we determine the mean first passage time by using (3.22) for p. This leads to the following result: Proposition 3.7: (Unit Disk) For ε → 0, the two-term asymptotic be- haviour of the mean first passage time, v(x), in a unit disk with N symmetrically located holes on the boundary is v(x) = |Ω| piND ( ν−1 + N 8 − logN − pi N∑ k=1 Gm(x,xk) ) +O(ν). (3.24) We now find an explicit expression for the mean escape time from the centre of the disk with N absorbing arcs each with length 2ε, for which d = 12 : Proposition 3.8: (Unit Disk) For ε → 0, the two-term asymptotic be- haviour of the mean first passage time, v(x), in a unit disk with N symmetrically located holes on the boundary, from the centre of the disk is E[τ | x(0) = (0, 0)] = v(x) = 1 ND ( log 1 ε + log 2 + N 4 − logN ) . (3.25) Notice that for N = 1 we recover the result obtained for one hole in the unit disk given in Proposition 2.3. Lastly, we expect that as N gets larger, the MFPT tends to get smaller and smaller. In fact as N → ∞, the MFPT given by (3.25), tends to 14D when log 1ε >> N . This is depicted in Figure 3.3 below for D = 1, d = 1 2 , ε = 10 −5 and N = 100. 50 Chapter 3. Numerical Realizations: The Neumann Green’s Function 0 20 40 60 80 100 0 2 4 6 8 10 12 14 N − number of holes M FP T fo r t he  u ni t d isk Figure 3.3: MFPT, v(x), from the centre of the unit disk with N symmetrically located holes on the boundary with D = 1, d = 12 and ε = 10 −5. 3.2 The Unit Square We have found an expression for the mean first passage time, given by (2.53), for a general two-dimensional domain with one hole on the boundary. In addition, we have found three formulas for the mean first passage time with N holes on the boundary, namely (2.99), (2.101) and (2.103). We would now like to investigate the behaviour of the MFPT in the unit square. 3.2.1 The Neumann Green’s Function in a Unit Square To investigate the mean first passage time in the unit square we must first derive the Neumann Green’s function for the unit square. We start with a derivation of the Neumann Green’s function for a rectangle. We will calculate the Neumann Green’s function in a rectangle, where Ω = [0, L] × [0, d] with |Ω| = Ld. For the moment we place the singular point x0 within the domain Ω. We will eventually take the limit as x0 → ∂Ω so that we can use the result to find the mean escape time. We understand x = (x, y) so 51 Chapter 3. Numerical Realizations: The Neumann Green’s Function that Gm(x,x0) = Gm(x, y, x0, y0), where x and y are the Cartesian coordinates of x, and x0 and y0 are the Cartesian coordinates of x0. We need to solve (3.1)-(3.3) for the Neumann Green’s function. That is, we need to solve ∆Gm(x,x0) = 1 |Ω| − δ(x− x0), x ∈ Ω, ∂xGm(0, y,x0) = ∂xGm(L, y,x0) = 0, (3.26) ∂yGm(x, 0,x0) = ∂yGm(x, d,x0) = 0. We impose the following integral condition to ensure uniqueness of the Neu- mann Green’s function, ∫ Ω Gm(x,x0)dx = 0. (3.27) 3.2.2 Solution Method We expand Gm in terms of a complete set of eigenfunctions, and we use a separation of variables technique to solve the resulting eigenvalue problem. In other words, we use a Fourier series representation. We find that Gm(x,x0) = ∞∑ n=0 ∞∑ m=0 cm,n cos (npix L ) cos (mpiy d ) . (3.28) To enforce the integral condition, we must set c0,0 = 0. That is, we set the coefficient of the n = 0, m = 0 mode to zero. This ensures that the integral condition is satisfied. We then define φ0,n = cos (npix L ) , φm,0 = cos (mpiy d ) , φm,n = cos (npix L ) cos (mpix d ) . (3.29) We decompose the n = 0, m = 0 modes as follows Gm(x,x0) = ∞∑ n=1 c0,nφ0,n + ∞∑ m=1 cm,0φm,0 + ∞∑ n=1 ∞∑ m=1 cm,nφm,n. (3.30) 52 Chapter 3. Numerical Realizations: The Neumann Green’s Function We find that c0,n = 2L2 |Ω| cos ( npix0 L ) n2pi2 , cm,0 = 2d2 |Ω| cos ( mpiy0 L ) m2pi2 , cm,n = 4 |Ω| cos ( npix0 L ) n2pi2 L2 + m2pi2 d2 . (3.31) We will make use of the following identities ∞∑ k=1 cos (kpix) k2 + α2 = pi 2α cosh (αpi(1− x)) sinh(αpi) − 1 2α2 , 0 ≤ x ≤ 2, (3.32) ∞∑ k=1 cos (kpix) k2 = pi2 ( 1 6 − x 2 + x2 4 ) , 0 ≤ x ≤ 2. (3.33) To simplify the first term in (3.30) we make use of the identity (3.33). We find that the first term reduces to H(x, x0) = L2 12 |Ω| ( 4− 6 ( x+ x0 L ) + 3 ( x+ x0 L )2 − 6 ( |x− x0| L ) + 3 ( x− x0 L )2) (3.34) To simplify the third term in (3.31) we make use of identity (3.32). The second term in (3.31) cancels with a term in this simplified expression. Making all the necessary simplifications we find that (3.31) reduces to Gm(x,x0) = H(x, x0) + 1 2pi ∞∑ m=1 cosh ( mLpi d ( 1− ( |x+x0| L ))) + cosh ( mLpi d ( 1− ( |x−x0| L ))) m sinh ( mLpi d ) × ( cos ( mpi (y + y0) d ) + cos ( mpi (y − y0) d )) . (3.35) We now make use of the following identity cosh(a− b) + cosh(a− c) sinh a = 1 1− e−2a ( e−b + e−c + eb−2a + ec−2a ) , (3.36) 53 Chapter 3. Numerical Realizations: The Neumann Green’s Function with a = mLpid , b = mLpi d |x+x0| L , c = mLpi d |x−x0| L and q = e − 2Lpid . We expand the second term in (3.35) into exponentials. We define the fol- lowing complex constants: r+,+ = − |x+ x0|+ i(y + y0), r+,− = − |x+ x0|+ i(y − y0), r−,+ = − |x− x0|+ i(y + y0), r−,− = − |x− x0|+ i(y − y0), ρ+,+ = |x+ x0|+ i(y + y0)− 2L, ρ+,− = |x+ x0|+ i(y − y0)− 2L, ρ−,+ = |x− x0|+ i(y + y0)− 2L, ρ−,− = |x− x0|+ i(y − y0)− 2L. (3.37) Then we introduce the new variables z+,+ = er+,+pi/d, z+,− = er+,−pi/d, z−,+ = er−,+pi/d, z−,− = er−,−pi/d, ξ+,+ = eρ+,+pi/d, ξ+,− = eρ+,−pi/d, ξ−,+ = eρ−,+pi/d, ξ−,− = eρ−,−pi/d. (3.38) Using (3.37) and (3.38) in (3.35) and recalling that 11−qm = ∑∞ n=0 (q n)m, we find that Gm(x,x0) = H(x, x0) + 1 4pi ∞∑ m=1 ∞∑ n=0 (qn)m ( zm+,+ + z̄ m +,+ ) + 1 4pi ∞∑ m=1 ∞∑ n=0 (qn)m ( zm+,− + z̄ m +,− + z m −,+ + z̄ m −,+ + z m −,− + z̄ m −,− ) + 1 4pi ∞∑ m=1 ∞∑ n=0 (qn)m ( ξm+,+ + ξ̄ m +,+ + ξ m +,− + ξ̄ m +,− ) + 1 4pi ∞∑ m=1 ∞∑ n=0 (qn)m ( ξm−,+ + ξ̄ m −,+ + ξ m −,− + ξ̄ m −,− ) . (3.39) Furthermore, we recall that Re [∑∞ m=1 wm m ] = − log |1− w|. Provided that each of the z’s and ξ’s do not equal one, the double sum is absolutely convergent, and we can change the order of summation in (3.39). We perform the summation 54 Chapter 3. Numerical Realizations: The Neumann Green’s Function over m to obtain Gm(x,x0) = H(x, x0) − 1 2pi ∞∑ n=0 log |1− qnz+,+| |1− qnz+,−| |1− qnz−,+| |1− qnz−,−| − 1 2pi ∞∑ n=0 log |1− qnξ+,+| |1− qnξ+,−| |1− qnξ−,+| |1− qnξ−,−| . (3.40) At this point we must extract the regular and singular part of the Green’s function as x → x0. We observe that r−,− = 0 as x → x0 causing the log |1− z−,−| term to diverge when n = 0. We simplify as follows log |1− z−,−| = log |r−,−|+ log ∣∣∣∣1− z−,−r−,− ∣∣∣∣ . (3.41) Then we write the Green’s function as Gm(x,x0) = − 12pi log |x− x0|+Rm(x,x0), (3.42) with Rm(x,x0) = H(x, x0)− 12pi log ∣∣∣∣1− z−,−r−,− ∣∣∣∣− 12pi ∞∑ n=1 log |1− qnz−,−| − − 1 2pi ∞∑ n=0 log |1− qnz+,+| |1− qnz+,−| |1− qnz−,+| − 1 2pi ∞∑ n=0 log |1− qnξ+,+| |1− qnξ+,−| |1− qnξ−,+| |1− qnξ−,−| . (3.43) We have derived the Neumann Green’s function for a rectangle. We now consider the specific case with L = d = 1, the unit square. Now we determine expressions for a singular point on the boundary, x0 ∈ ∂Ω. Using the symmetry of the square, we see that we need only find the Green’s function on one side and on one corner. The method of extracting the regular and singular parts is identical to that used above. We state the results here. For a unit square, L = d = 1, with a singular point along the bottom side, that is y0 = 0 and 55 Chapter 3. Numerical Realizations: The Neumann Green’s Function x0 ∈ (0, 1) we find Gm(x,x0) = − 12pi log |x− x0| − 1 2pi log √ |x− x0|2 + |y + y0|2 +Rm(x,x0), (3.44) with Rm(x,x0) = H(x, x0)− 12pi log ∣∣∣∣1− z−,−r−,− ∣∣∣∣− 12pi log ∣∣∣∣1− z−,+r−,+ ∣∣∣∣ − 1 2pi ∞∑ n=1 log |1− qnz−,−| |1− qnz−,+| − 12pi ∞∑ n=0 log |1− qnz+,−| |1− qnz+,+| − 1 2pi ∞∑ n=0 log |1− qnξ+,+| |1− qnξ+,−| |1− qnξ−,+| |1− qnξ−,−| . (3.45) As x→ x0 we find Gm = − 1 pi log |x− x0|+Rm(x0,x0), (3.46) with Rm(x0,x0) = H(x0, x0)− 1 pi log pi − 1 pi ∞∑ n=1 log |1− qn| − 1 2pi ∞∑ n=0 log ∣∣1− qnz0+,+∣∣ ∣∣1− qnz0+,−∣∣ − 1 2pi ∞∑ n=0 log ∣∣1− qnξ0+,+∣∣ ∣∣1− qnξ0+,−∣∣ ∣∣1− qnξ0−,+∣∣ ∣∣1− qnξ0−,−∣∣ , (3.47) where z and ξ are defined as in (3.38) with r0+,+ = −2x0, r0+,− = −2x0, r0−,+ = 0, r 0 −,− = 0, ρ0+,+ = 2x0 − 2, ρ0+,− = 2x0 − 2, ρ0−,+ = −2, ρ0−,− = −2. (3.48) Notice that the singularity is now twice what it was with the singular point within the unit square. For a unit square, L = d = 1, with a singular point at the left end corner, 56 Chapter 3. Numerical Realizations: The Neumann Green’s Function that is x0 = 0 and y0 = 0 we find Gm(x,0) = − 12pi log |x− x0| − 1 2pi log √ |x+ x0|2 + |y + y0|2 − 1 2pi log √ |x+ x0|2 + |y − y0|2 − 12pi log √ |x− x0|2 + |y + y0|2 +R(x,x0), = − 2 pi log |x|+Rm(x,0), (3.49) with Rm(x,0) = H(x, 0)− 12pi log ∣∣∣∣1− z+,+r+,+ ∣∣∣∣ ∣∣∣∣1− z+,−r+,− ∣∣∣∣ ∣∣∣∣1− z−,+r−,+ ∣∣∣∣ ∣∣∣∣1− z−,−r−,− ∣∣∣∣ − 1 2pi ∞∑ n=1 log |1− qnz+,+| |1− qnz+,−| |1− qnz−,+| |1− qnz−,−| − 1 2pi ∞∑ n=0 log |1− qnξ+,+| |1− qnξ+,−| |1− qnξ−,+| |1− qnξ−,−| . (3.50) As x→ 0 we find Gm(0,0)→ − 2 pi log |x|+Rm(0,0), (3.51) with Rm(0,0) = H(0, 0)− 2 pi log pi − 4 pi ∞∑ n=1 log |1− qn| , (3.52) with H(0, 0) = 13 . Notice that the log |x− x0| term in (3.51) is multiplied by 2 pi . This means that singularity at a corner is four times larger than a singular point in the interior. Now that we have expressions for the Neumann Green’s function in a unit square, we can proceed to the calculation of the MFPT using (2.17). 3.2.3 The Mean First Passage Time in a Unit Square with One Hole on the Boundary We can use (2.53) for the MFPT for one hole along with the expressions for the Green’s function for a unit square given above. Since all of the expressions for the Green’s function involve an infinite series, it is more instructive to plot the results until a certain tolerance is reached. That is, until the terms in the series are negligible in comparison to some specified tolerance. We use a tolerance 57 Chapter 3. Numerical Realizations: The Neumann Green’s Function 10−12. D = 1 and d = 12 in all cases investigated. Firstly, we consider one hole in the unit square located at the position (1, 0.5). The initial position starts at (0, 0.5) and moves closer to the hole in fixed incre- ments of 180 . The initial position does not reach the hole. We investigate the behaviour of the MFPT in this case with D = 1, d = 12 and ε = 10 −5. The configuration and the MFPT are shown in Figure 3.4. 58 Chapter 3. Numerical Realizations: The Neumann Green’s Function 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y initial position moves towards the exit window exit window (a) Plot of the initial position (blue), as it moves from the antipodal point at (0,0.5) towards the singular point (red) at (1,0.5) 0 0.2 0.4 0.6 0.8 1 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 x − initial position M FP T fo r t he  u ni t s qu ar e (b) Plot of the MFPT, v(x), versus the x-coordinate of the initial position Figure 3.4: MFPT for the unit square as the initial position moves from (0, 0.5) towards the exit window at (1, 0.5) with D = 1, d = 12 and ε = 10 −5. 59 Chapter 3. Numerical Realizations: The Neumann Green’s Function We see that the MFPT decreases as the initial position moves towards the exit window. Now we consider the case where we hold the initial position fixed at the centre of the unit square, at x = (0.5, 0.5). We place the exit window a small increment, 180 , away from the corner at (1, 0). We then move the exit window in increments of 180 from (1, 0.0125) to (1, 0.9875) and calculate the MFPT with D = 1, d = 12 and ε = 10 −5. The configuration is shown below in Figure 3.5 along with the MFPT plotted against the y-coordinate of the moving exit window. 60 Chapter 3. Numerical Realizations: The Neumann Green’s Function 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y initial position exit window moves from one end towards the other (a) Plot of the initial position (blue), and the singular point (red) as it moves along the side 0 0.2 0.4 0.6 0.8 1 11.5 12 12.5 13 13.5 14 14.5 15 y − exit window M FP T fo r t he  u ni t s qu ar e (b) Plot of the MFPT, v(x), versus the y-coordinate of the exit window Figure 3.5: MFPT from the centre of the unit square as the exit window moves from (1, 0.0125) to (1, 0.9875) with D = 1, d = 12 and ε = 10 −5. 61 Chapter 3. Numerical Realizations: The Neumann Green’s Function We observe symmetric behaviour in the MFPT, with a minimum where the distance from the exit window to the initial position is shortest. Our results are consistent with what we would expect. 3.2.4 The Mean First Passage Time in a Unit Square with Two Holes on the Boundary Now we consider two holes in a unit square. We fix the initial position at the centre of the unit square at x = (0.5, 0.5). One exit window is fixed at (0, 0.5) while the other moves from (0, 0.48) towards the corner at (0, 0). Then it moves from (0, 0) to (1, 0) and finally from (1, 0) to (1, 0.5). We plot the behaviour of the MFPT as a function of the moving coordinate of the moving exit window on each side, excluding the corner point. We then plot the total behaviour, as the exit window moves from (0, 0.48) to (1, 0.5), including the corner points, with the points numbered from 1 to 100. We set D = 1, d = 12 and ε = 10 −5. The results are plotted shown in Figures 3.6 and 3.7 below. 62 Chapter 3. Numerical Realizations: The Neumann Green’s Function 00.10.20.30.40.5 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 Increments along Y axis M FP T fo r t he  u ni t s qu ar e (a) Plot of the MFPT as a function of the y-coordinate of the moving exit window along side 1, from (0,0.48) to (0,0.02) 0 0.2 0.4 0.6 0.8 1 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 Increments along the X axis M FP T fo r t he  u ni t s qu ar e (b) Plot of the MFPT as a function of the x-coordinate of the moving exit window along side 2, from (0.02,0) to (0.98,0) Figure 3.6: MFPT from the centre of a unit square with two exit windows, one fixed at (0,0.5) and the other moving from (0,0.48) to (1,0.5), sides 1 and 2 with D = 1, d = 12 and ε = 10 −5. 63 Chapter 3. Numerical Realizations: The Neumann Green’s Function 0 0.1 0.2 0.3 0.4 0.5 1.75 1.8 1.85 1.9 1.95 2 Increments along the Y axis M FP T fo r t he  u ni t s qu ar e (a) Plot of the MFPT as a function of the y-coordinate of the moving exit window along side 3, from (1,0.02) to (1,0.5) 0 20 40 60 80 100 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 Absorbing Arcs 1−100 M FP T fo r t he  u ni t s qu ar e (b) Plot of the MFPT as a function of the moving exit window from (0,048) to (1,0.5), including the corner points. The position of the moving exit window is labelled from 1-100 Figure 3.7: MFPT from the centre of a unit square with two exit windows, one fixed at (0,0.5) and the other moving from (0,0.48) to (1,0.5), side 3 and all sides with D = 1, d = 12 and ε = 10 −5. 64 Chapter 3. Numerical Realizations: The Neumann Green’s Function Notice that the local minima in Figures 3.6 (a) and (b) are not where the distance of the moving exit window from the initial position is a minimum. This is consequence of the fact that there are two holes, and there is an interaction between the two holes, as well as with the initial position. We see that the minimum is shifted to the right slightly. In Figure 3.7 (a), we do find a minimum for the MFPT in the expected position. This is a result of the symmetry of the placement of the two holes. In fact, if we continue to move the second exit window from (1, 0.02) to (1, 0.98), the plot of the MFPT is symmetric in this case. Also notice the discontinuities in Figure 3.7 (b) at the corner points. This is expected, since the boundary is non-smooth at these points. We remark that our formula for the MFPT is not valid when the arc is at the corner of the square. The local geometry at the corner of a square is different. Therefore, we need a different form for the outer solution. If we look at the expression for the Neumann Green’s function at the corner, given by (3.49), (3.50), (3.51), (3.52), we see that the singularity is twice as large at the corner than a singularity on the smooth portion of the boundary. This agrees with the result for a rectangle obtained by Singer et al. in [24]. Here, Singer et al. state that the leading order term for the MFPT is twice as large as the case with the singular on the smooth portion of the boundary. We proceed to derive an expression for the MFPT from the centre of the unit square with a single arc at the corner, x0 = (0, 0). We alter the method outlined in Chapter 2, Section 2.2. The outer solution for the principal eigenfunction, u∗, is written in terms of the Helmholtz Green’s function given by (2.31) and (2.32). We alter (2.32) so that G(x,x0, λ∗) = − 2 pi log |x− x0|+R(x,x0, λ∗). (3.53) Thus, u∗ = −piAν2 G(x,x0, λ∗). Matching to the inner solution, we find that R(x0,x0, λ∗) = − 2piν . We expand G(x,x0, λ∗) as in (2.37). We find that Gm satisfies (2.43) with Gm(x,x0) = − 2pi log |x− x0| + Rm(x,x0). We find the principal eigenvalue and eigenfunction to be λ0(ν) = piν 2 |Ω| − pi2ν2 4 |Ω|Rm(x0,x0) +O(ν 3) φ0(x, ν) = |Ω|− 1 2 − piν |Ω| − 12 2 Gm(x,x0) +O(ν3). (3.54) 65 Chapter 3. Numerical Realizations: The Neumann Green’s Function We calculate the MFPT to be v(x) = 2 |Ω| piD ( − log(εd) + pi 2 (Rm(x0,x0)−Gm(x,x0)) ) +O(ν), (3.55) with the average MFPT given by v̄ = 2 |Ω| piD ( − log(εd) + pi 2 Rm(x0,x0) ) +O(ν). (3.56) Now, for the corner point x0 = (0, 0), Rm(0,0) is given by (3.52) with H(0, 0) = 13 and Gm(x,0) is given by (3.49) and (3.50). Finally, the constant d in (3.55) is determined from the far-field behaviour of the inner problem. In particular, d depends on the manner in which the absorbing arc of length 2ε is placed at the corner. If the arc is placed along one side so that v = 0 on 0 < x < 2ε with y = 0, then d = 1. If the arc is placed along two sides such that v = 0 on y = 0 with 0 < x < ε and on x = 0 with 0 < y < ε, then d = 14 . Taking an arc of length ε placed along one side such that v = 0 on y = 0 with 0 < x < ε, we find that d = 12 . Approximating Rm(0,0) by the first term in (3.56) we find that v̄ ∼ 2 |Ω| Dpi ( log 1 ε + log 2 pi + pi 6 + 2e−2pi ) . (3.57) This agrees with equation (2.8) of [24], which was found by solving certain integral equations asymptotically. 3.3 Arbitrary Domains - A Numerical Approach As we have seen, to investigate the MFPT in a particular domain we must find the Neumann Green’s function and its regular part. We have succeeded in doing this for a unit circle and square. However, other geometries are more challenging as an analytical solution for the Green’s function is unattainable in most cases. In this section we introduce a numerical procedure to calculate the Neumann Green’s function and its regular part in an arbitrary domain. 66 Chapter 3. Numerical Realizations: The Neumann Green’s Function 3.3.1 The Boundary Element Method We introduce the boundary element method, the method we will use to cal- culate the Neumann Green’s function in an arbitrary domain. We alter the statement of the problem for the Neumann Green’s function slightly, by putting the singularity directly on the boundary as ∆Gm(x,x0) = 1 |Ω| , x ∈ Ω, (3.58) ∂nGm(x,x0) = δ(s− s0), x ∈ ∂Ω, (3.59)∫ Ω Gm(x,x0)dx = 0. (3.60) where s is an arclength parameter and the normal derivative is taken with respect to the unit outward normal. As mentioned, the condition that the integral of the Green’s function over the domain is zero ensures that the solution to the problem is unique. We decompose Gm(x,x0) as Gm(x,x0) = − 1 pi log |x− x0|+R(x,x0) + 14 |Ω| ( |x|2 + |x0|2 ) (3.61) We will define w as w(x,x0) = 1 4 |Ω| ( |x|2 + |x0|2 ) . (3.62) The Neumann Green’s function is given by Gm(x,x0) = − 1pi log |x− x0| + Rm(x,x0). Notice that the regular part of the Neumann Green’s function in (3.61) is given by Rm(x,x0) = R(x,x0) + 14 |Ω| ( |x|2 + |x0|2 ) . (3.63) We solve for R(x,x0) numerically. Substituting (3.61) into (3.58)-(3.60) we find that R(x,x0) must satisfy ∆R(x,x0) = 0, x ∈ Ω, ∂nR = −∂nw, x ∈ ∂Ω,∫ Ω Rdx = − ∫ Ω ( 1 pi log |x− x0| − w ) dx. (3.64) 67 Chapter 3. Numerical Realizations: The Neumann Green’s Function Note that ∆w = 1|Ω| . We now apply Green’s identity given by (u1,∆u2)− (u2,∆u1) = ∫ ∂Ω u1∂nu2dS − ∫ ∂Ω u2∂nu1dS. (3.65) We let u1 = R(x,x0) and u2 = g(x0, ξ), where g(x, ξ) is the free-space Green’s function satisfying ∆g(x, ξ) = −δ(x− ξ), x ∈ Ω, which is given by g(x, ξ) = − 1 2pi log |x− ξ| . (3.66) Substituting u1 and u2 into (3.65) we find that the integral equation for R at x ∈ Ω is R(x, ξ) = − ∫ ∂Ω R(x,η)∂ng(η, ξ)dS(η)− ∫ ∂Ω g(x,η)∂nw(η, ξ)dS(η). (3.67) Now we discretize the boundary ∂Ω into n arcs ∂Ω1, ∂Ω2, ..., ∂Ωn. We can then use the midpoint rule to approximate the integrals in (3.67). For example, we can approximate R(x,η) by R(x, ξi) where ξi is the midpoint of each arc ∂Ωi. We let Rj = R(x, ξj), using this we find Rj = −Σni=1 (aijRi + bij) , (3.68) where aij = ∫ ∂Ωi ∂ng(η, ξj)dS(η), bij = ∫ ∂Ωi g(x,η)∂nw(η, ξj)dS(η) and Ri = R(x, ξi). Now we compute aij and bij by the midpoint rule to find aij = li∂ng(ξi, ξj), bij = g(x, ξi)li∂nw(ξi, ξj), (3.69) where li is the length of the arc ∂Ωi. Using this we find that we can represent R by R(x, ξj) + n∑ i=1 liR(x, ξi)∂ng(ξi, ξj) = − n∑ i=1 lig(x, ξi)∂nw(ξi, ξj). (3.70) Firstly, we calculate ∂nw(ξi, ξj) in (3.70). We denote the x and y component 68 Chapter 3. Numerical Realizations: The Neumann Green’s Function of ξi as ξix and ξiy respectively. We obtain that ∇w(ξi, ξj) = 1 2 |Ω| (ξix, ξiy), ∂nw(ξi, ξj) = 1 2 |Ω| (ξix, ξiy).n̂. (3.71) Hence ∂nw is independent of its second argument ξj . Now we calculate ∂ng(ξi, ξj), ∇g(ξi, ξj) · n̂ = − 1 2pi ξi − ξj∣∣ξi − ξj∣∣2 · n̂. (3.72) When calculating ∂ng(ξi, ξj), the case i = j requires special care because of the singularity of the Green’s function. We must calculate the first integral in (3.67) differently. We introduce a local coordinate system. We let 1κi be the radius of curvature of ∂Ωi at ξi where κi is the curvature. Since we have assumed each arc ∂Ωi to be very small, we may assume that ∂Ωi is parametrized for t << 1 as η(t) = 1 κi (cos t, sin t), − li 2κi ≤ t ≤ li 2κi , (3.73) with ξi = ( 1 κi , 0 ) . We compute ∂ng(η, ξ) = ∇g.n̂ where n̂ = (cos t, sin t) is the unit outward normal in the local coordinate system. We find ∇g = − 1 2pi η − ξi |η − ξi|2 , η − ξi = ( 1 κi cos t− 1 κi , 1 κi sin t ) , n̂.(η − ξi) = 1 κi (1− cos t), |η − ξi|2 = 2 κ2i (1− cos t). (3.74) Therefore ∂ng(η, ξi) = − 1 2pi 1 κi (1− cos t) 2 1 κ2i (1− cos t) = − κi 4pi . (3.75) Thus aii = − liκi4pi . (3.76) We do not encounter any problems when i = j for bij . 69 Chapter 3. Numerical Realizations: The Neumann Green’s Function We can represent (3.70) as a matrix system AR = −B as follows: R = ( R(x, ξ1) R(x, ξ2) · · · · · · R(x, ξn) )T , (3.77) A =  ( 1− l14pir1 ) l2∂ng(ξ2, ξ1) l3∂ng(ξ3, ξ1) · · · ln∂ng(ξn, ξ1) l1∂ng(ξ1, ξ2) ( 1− l24pir2 ) l3∂ng(ξ3, ξ2) · · · ln∂ng(ξn, ξ2) l1∂ng(ξ1, ξ3) l2∂ng(ξ2, ξ3) ( 1− l34pir3 ) · · · ln∂ng(ξn, ξ3) ... ... ... . . . ... l1∂ng(ξ1, ξn) l2∂ng(ξ2, ξn)) · · · · · · ( 1− ln4pirn )  , B = l1g(x, ξ1)∂nw(ξ1, ξ1) + l2g(x, ξ2)∂nw(ξ2, ξ1) + · · ·+ lng(x, ξn)∂nw(ξn, ξ1) l1g(x, ξ1)∂nw(ξ1, ξ2) + l2g(x, ξ2)∂nw(ξ2, ξ2) + · · ·+ lng(x, ξn)∂nw(ξn, ξ2) l1g(x, ξ1)∂nw(ξ1, ξ3) + l2g(x, ξ2)∂nw(ξ2, ξ3) + · · ·+ lng(x, ξn)∂nw(ξn, ξ3) ... l1g(x, ξ1)∂nw(ξ1, ξn) + l2g(x, ξ2)∂nw(ξ2, ξn) + · · ·+ lng(x, ξn)∂nw(ξn, ξn)  , We solve for R using Gaussian elimination. Our goal is to calculate the MFPT. To this end, we must determineR(x0,x0). Using this formulation, this means that we must find R(ξj , ξj). Thus, we pick x = ξj for some j = 1, 2, ..., N in (3.70). This introduces another case where the possibility of i = j arises. We have to recalculate ∫ ∂Ω g(ξ,η)∂nw(η, ξ)dS(η) in (3.67). We use the same local coordinate system as outlined above. Thus, we have g(ξ,η) = − 12pi log |x− ξ| = − 14pi log ( 2 κ2i (1− cos t) ) and ∂nw(η, ξ) = 1 2|Ω|κi . Once again, 1 κi is the radius of curvature in the local coordinate system in segment i. The integral becomes − 1 8pi |Ω|κ2i ∫ liκi/2 −liκi/2 log ( 2 κ2i (1− cos t) ) dt. (3.78) Notice that when t = 0 we have a singularity, so we can use gaussian quadrature to get around this by using an even number of sample points. However, for certain domains we remove the singularity by rewriting the integral, as will be discussed in later sections. 70 Chapter 3. Numerical Realizations: The Neumann Green’s Function Notice that we have not imposed the integral condition. We must still deter- mine the constant C so that (3.64) is satisfied. To do this, we must integrate our numerical result for R over the domain Ω, which proves to be a computationally intensive process. We now illustrate the use of the method for a circle and an ellipse. 3.3.2 The Unit Disk We make use of the boundary element method described in the previous section to determine the Neumann Green’s function for a unit disk. This allows us to compare our numerical method to the analytical result given by (3.9). According to the formulation above, we let Gm(x,x0) = − 1 pi log |x− x0|+ 14 |Ω| ( |x|2 + |x0|2 | ) +R(x,x0) + C (3.79) Thus, the unknown is R(x,x0) and C is to be determined from the integral condition once R has been found. If we compare the expression above (3.79) with (3.9), we see that we already have the expression for the unit circle. Our solution forR in this case should be zero for all x. In this section we confirm that the boundary element formulation does in fact give us this result. We illustrate the results in terms of N , the number of segments used for the calculation. We aim to attain zero to six decimal places. Note that since we are essentially looking for a constant, each entry of the vector R, (3.77), will be the same. We begin with the calculation of R(x,x0) for x ∈ Ω. We notice, by consid- ering Table 3.1 below, that as x moves towards the boundary of the unit disk, we require more segments to attain the desired accuracy. N 40 80 100 x = (0, 0) −0.3534× 10−17 −0.6184× 10−17 −0.5624× 10−17 x = (0.5, 0) 0.7237× 10−14 0.1084× 10−16 0.2602× 10−16 x = (0.75, 0) 0.8003× 10−7 0.4024× 10−12 0.1016× 10−14 x = (0.99, 0) 0.0041 0.0015 0.9928× 10−3 Table 3.1: R(x,x0) for the unit disk with N mesh points on the boundary Table 3.1 shows the value of R(x,x0) for each x shown above, and for all N 71 Chapter 3. Numerical Realizations: The Neumann Green’s Function x0’s on the boundary. The x0’s on the boundary are the ξi’s in the boundary element formulation. There is only one value shown for the vector R since each entry of the vector is the same, since we are looking for a constant. It is clear that as x moves towards the boundary more mesh points are re- quired to attain the desired result of zero to six decimal places, whereas with x at the centre of the disk we attain convergence immediately. This is a consequence of the fact that as x tends towards the boundary, the self effect starts to play an important role. For N = 800, with x = (0.99, 0) we obtain R = 0.1282× 10−6. To calculate R(x0,x0) we need to recalculate the second integral in equation (3.67) as outlined in the previous section. We use integral (3.78) instead. We calculate the integral (3.78) using a four point Gaussian quadrature rule. We do not rewrite the integral to remove the singularity in this case. As long as we do not use an odd number of sample points in the Gaussian quadrature procedure, we are able to avoid singular contributions to the integral. We also anticipate that we will require many mesh points to attain the desired accuracy. Additionally, as a result of the formulation of the boundary element method, we are able to check the symmetry of R in this case. In other words, for x = ξ1 we require R(ξ1, ξj) = R(ξj , ξ1) for each j. The symmetry of the problem ensures that R(xj ,xj) are identical for each j. We tabulate the results in Table 3.2 below N 50 100 200 400 800 R(x0,x0) 0.0025 0.0012 0.6193× 10−3 0.3097× 10−3 0.1548× 10−3 Table 3.2: R(x0,x0) for the unit disk with N mesh points on the boundary We see that as we double N , the value of R divides by 2. Thus, we can conclude that R(x0,x0) is tending to zero. We are left to calculate the integral over Gm, equation (3.79). It is simple to calculate the integral over the domain of each term in equation (3.79) with the exception of the integral over R. We use the composite trapezoidal rule to numerically compute the integral over R and log |x− x0|. As shown in (3.8), the integral of Gm over the domain is independent of x0. We calculate the desired integrals for several x0’s. We expect each integral to return the same value. This is another way to check that the numerical method is working correctly. We describe the integration method 72 Chapter 3. Numerical Realizations: The Neumann Green’s Function over R. We need to calculate∫ 2pi 0 ∫ 1 0 R(x,x0)rdrdθ (3.80) We first split the integral over θ using the composite trapezoidal rule with n intervals. There will be n remaining integrals over r. We evaluate the remaining n integrals using the midpoint rule. We run our boundary element method for the values of x that are required to compute the integral. The integration over log |x− x0| is similar. Notice that we compute the integrals over r using the midpoint rule to avoid any singularities. We find that the integral over log |x− x0| is zero for all x0 and we find the integral over R to be zero for all x0. We are now ready to calculate the constant C. Since the integral over R and log |x− x0| are zero, we are left to calculate the integral over w = 1 4|Ω| ( |x|2 + |x0|2 ) . For any x0 on the boundary, |x0| = 1. We find C = − 38pi , which agrees with what we find analytically. We have confirmed that the boundary element method does in fact give us the desired results for a unit circle. 3.3.3 The Ellipse We are now ready to use the boundary element method to determine the regular part of the Neumann Green’s function for an ellipse. We state the properties that will be required for the boundary element method. For a curve defined in polar coordinates, we have r = r(θ), x = r cos(θ), y = r sin(θ). For an ellipse r = ab√ a2 sin2(θ) + b2 cos2(θ) , (3.81) r′ = − ab sin(θ) cos(θ)(a 2 − b2) a2 sin2(θ) + b2 cos2(θ)(3/2) , (3.82) r′′ = −ab(a2 − b2)×( cos2(θ)− sin2(θ) (a2 sin2(θ) + b2 cos2(θ))3/2 − 3(a 2 − b2) sin2(θ) cos2(θ) (a2 sin2(θ) + b2 cos2(θ))5/2 ) , (3.83) where a is the semi-major axis and b is the semi-minor axis. 73 Chapter 3. Numerical Realizations: The Neumann Green’s Function The outward normal vector, the arclength and curvature are given by n̂ = 1√ r′2 + r2 (r cos(θ) + r′ sin(θ), r sin(θ)− r′ cos(θ)) , (3.84) s = ∫ θ2 θ1 √ r′2 + r2dθ, (3.85) κ = r2 + 2r′2 − rr′′ (r′2 + r2)3/2 . (3.86) To determine the length of a segment li we will need to calculate the integral (3.85) for the arclength. We compute this numerically using the trapezoidal rule. The algorithm outlined for the boundary element method holds for a general domain. We consider an ellipse with a = 2 and b = 1. Once again, we begin with the calculation for R(x,x0) for x ∈ Ω. We use more mesh points as x moves towards the boundary of the ellipse. The results are tabulated below in Table 3.3. N 80 160 320 640 x = (0, 0) 0.1290 0.1291 0.1291 0.1291 x = (1, 0) 0.1291 0.1291 0.1291 0.1291 x = (1.75, 0) 0.1291 0.1291 0.1291 0.1291 x = (1.99, 0) 0.1331 0.1305 0.1294 0.1291 x = (0, 0.5) 0.1290 0.1291 0.1291 0.1291 x = (0, 0.99) 0.1298 0.1292 0.1291 0.1291 x = (1, 0.5) 0.1291 0.1291 0.1291 0.1291 Table 3.3: R(x,x0) for the ellipse with N mesh points on the boundary Table 3.3 shows the value of R(x,x0) for each x within the ellipse, and for all x0 on the boundary. We find that each entry of the vector R is identical, regardless of where we place x or x0. This implies that R is a constant, R = 0.1291. The behaviour of the Green’s function for the ellipse is thus contained within log |x− x0| and w(x,x0). We have found that Gm(x,x0) = − 1 pi log |x− x0|+ 14 |Ω| ( |x|2 + |x0|2 ) + 0.1291 (3.87) Notice that we have not imposed ∫ Ω G(x,x0)dx = 0, and thus the expression for 74 Chapter 3. Numerical Realizations: The Neumann Green’s Function the Neumann Green’s function, (3.87), for the ellipse is unique up to a constant, C. Below, in Figure 3.8 we show the behaviour of the Neumann Green’s function for the ellipse, Gm − C. 0 pi/2 pi 3pi/2 2pi 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Θ G m − C fo r t he  e llip se Figure 3.8: Plot of the Neumann Green’s function for the ellipse, Gm−C, versus θ0 In Figure 3.8 we have x at the origin and x0 at points along the boundary in increments of pi100 . We observe the symmetric behaviour of the Green’s function for the ellipse, which is consistent with our expectations. We proceed to the calculation of R(x0,x0). We compute the integral (3.78) as follows∫ liκi/2 −liκi/2 log ( 2 κ2i (1− cos t) ) dt = liκi log ( 2 κ2i ) + ∫ liκi/2 −liκi/2 log(1− cos t)dt. (3.88) We rewrite the last integral in (3.88) as = ∫ liκi/2 −liκi/2 log(1− cos t)− log(t2/2)dt+ ∫ liκi/2 −liκi/2 log(t2/2)dt, (3.89) = ∫ liκi/2 −liκi/2 log(2(1− cos t)/t2)dt+ 2liκi log liκi − 2liκi − 3liκi log 2. 75 Chapter 3. Numerical Realizations: The Neumann Green’s Function We compute the remaining integral using Gaussian quadrature, with an even number of sample points. Initially we use weight one with sample points ± 1√ 3 , two point Gaussian quadrature. Notice that since we have observed that R is a constant, R(x0,x0) should be the same for each x0. We encounter extremely slow convergence of the numerical method in this case. We require many mesh points in order for the method to converge to an answer that we expect, that is, all the entries of the vector R(x0,x0) should be identical and each vector R should be the same regardless of x0. In Table 3.4 below, we tabulate the maximum difference between the entries in any vector R(x0,x0). N 150 300 600 1200 2400 min 0.1136 0.1202 0.1240 0.1263 0.1275 max 0.1429 0.1371 0.1336 0.1316 0.1305 |max−min| 0.0293 0.0169 0.0096 0.0054 0.0030 R(x0,x0) 0.1209 0.1244 0.1264 0.1276 0.1282 Table 3.4: R(x0,x0) for the ellipse with N mesh points on the boundary The last row in Table 3.4 shows the average value of R(x0,x0). We find that the vector R satisfies the expected properties when the mesh points are clustered very closely together, at θ = pi/2, 3pi/2. At θ = 0, pi the points are far apart along the boundary due to the curvature of the ellipse. It is here that we obtain the largest discrepancy between values of R. We have been unable to get the method to converge for N = 2400. We propose altering the manner in which we have computed the integrals for the boundary element method by using Gaussian quadrature instead of the midpoint rule to compute the integrals in (3.67). In addition, we did attempt to use four point Gaussian quadrature to compute the last integral in (3.90). However, there was no improvement in the convergence. In fact, we obtained exactly the same results as we did with two point Gaussian quadrature. The reason for this is that the integrand is not sufficiently differentiable in order for four point Gaussian quadrature to make a difference. We expect that R(x0,x0) = 0.1291. We see, by considering the average values of R, that R is tending towards this value. We are left to calculate the integral over Gm, (3.87), to determine the constant C. We use the same procedure as we used for the unit circle. We find the same value for the integral over R regardless of the x0 we choose. 76 Chapter 3. Numerical Realizations: The Neumann Green’s Function This agrees with the fact that integral over Gm should be independent of x0. We find that ∫ Ω R(x,x0)rdrdθ = 0.8108. We know that since R is a con- stant, that ∫ Ω R(x,x0)dx = 2pi × 0.1291 = 0.8112. Our numerical answer is correct up to three decimal places. It is simple to calculate the integral∫ Ω 1 4|Ω| ( |x|2 + |x0|2 ) dx = 14|Ω| ( 7.85398 + 2pi |x0|2 ) . However, the integral over − 1pi log |x− x0| is not simple to calculate as a consequence of the singular- ity when x = x0. We are only able to estimate this integral up to one decimal place accurately. As a result, we find that C = − 12pi (0.8108 + 0.1) = −0.1450 We can now find the MFPT for the ellipse. We know the Neumann Green’s function for the ellipse, (3.87). In addition, we can assume that R(x0,x0) = 0.1291. To calculate the MFPT for the ellipse with one hole on the boundary we use (2.53). It is worth noting that the constant, C, does not play a role directly in this equation since the constant term from Rm(x0,x0) and Gm(x,x0) cancel each other. However, it is assumed that we are able to find the constant C such that ∫ Ω Gm(x,x0)dx = 0 since it is an implementation of the normalization condition. It is only in the case of normalized eigenfunctions that (2.53) holds. We place the initial position at the origin. We place one hole on the bound- ary, initially at an angle pi100 , which moves from this position at equal angular increments of pi50 until it reaches − pi100 . We set D = 1, d = 12 and ε = 10−5. The MFPT from the centre of an ellipse with semi-major axis 2 and semi-minor axis 1 is depicted in Figure 3.9. 77 Chapter 3. Numerical Realizations: The Neumann Green’s Function 0 pi/2 pi 3pi/2 2pi 24.5 25 25.5 26 26.5 27 Θ M FP T fo r t he  e llip se Figure 3.9: Plot of the MFPT, v(x), from the centre of the ellipse versus θ0 with D = 1, d = 12 and ε = 10 −5 We observe the expected symmetric behaviour in the MFPT. The MFPT is a minimum at θ = pi/2, where the distance from the centre is a minimum. We see a maximum in the MFPT at θ = 0, pi, where the distance from the centre is a maximum. Now, we move the initial position from the origin to x = (1, 0). The MFPT in this case is depicted in Figure 3.10. 78 Chapter 3. Numerical Realizations: The Neumann Green’s Function 0 pi/2 pi 3pi/2 2pi 24 24.5 25 25.5 26 26.5 27 27.5 Θ M FP T fo r t he  e llip se Figure 3.10: Plot of the MFPT, v(x), from x = (1, 0) for the ellipse versus θ0 with D = 1, d = 12 and ε = 10 −5 Since the initial position is shifted in this case, we find that the MFPT has a minimum at θ = ±19pi/100 and a maximum at θ = pi, where the hole is the furthest from the initial position. 3.4 Perturbed Circular Domains - Analytical Approach We now derive expressions for the regular part for the Neumann Green’s function in a perturbed circular domain. This work was completed by T. Kolokolnikov, [7]. We present an overview of his work. 3.4.1 Derivation We want to solve system (2.43) for the Neumann Green’s function, with the exception of the integral condition. That is, we solve for the Green’s function 79 Chapter 3. Numerical Realizations: The Neumann Green’s Function up to an arbitrary constant. We have the general expression Gm(x,x0) = − 1 pi log |x− x0|+R(x,x0), x0 ∈ ∂Ω. (3.90) We want to determine Rm(x,x0) and Rm(x,x0) in the limit as x→ x0 for do- mains that are near the unit disk. We parametrize the boundary of the domain, Ω, in polar coordinates. We propose the following theorem: Theorem 3.9: Suppose that the domain Ω is defined in polar coordinates by r = r(θ) = 1 + εσ(θ), ε¿ 1. (3.91) Suppose that σ(θ) is given by σ(θ) = ∞∑ n=1 (an cosnθ + bn sinnθ) . (3.92) Let x0 = (r0 cos θ0, r0 sin θ0) be a point on the boundary where r0 = 1 + εσ(θ0). We define ρ(θ) = Rm(x,x0) and ρ(θ0) = Rm(x0,x0). (3.93) We have the following expression for ρ′(θ0) ρ′(θ0) = ε pi ∑ n≥1 ( n2 + n− 2) (bn cosnθ0 − an sinnθ0) (3.94) Proof: We find that Rm(x,x0) satisfies ∆Rm(x,x0) = 1 |Ω| , x ∈ Ω, (3.95) ∇Rm(x,x0) · n̂ = 1 pi (x− x0) · n̂(x) |x− x0|2 , x ∈ ∂Ω, (3.96) where the unit outward normal n̂ is defined by (3.84). We compute the expres- sion on the right-hand side of (3.96) for ε << 1 as 1 pi (x− x0) · n̂(x) |x− x0|2 = 1 2pi ( 1 + ε [ σ cos(θ − θ0)− σ0 − σ′ sin(θ − θ0) 1− cos(θ − θ0) ]) +O(ε2). (3.97) 80 Chapter 3. Numerical Realizations: The Neumann Green’s Function The expression in square brackets is bounded for θ → θ0. Thus the expression is uniformly valid for all θ ∈ [0, 2pi). We define f(θ) as f(θ) = σ cos(θ − θ0)− σ0 − σ′ sin(θ − θ0) 1− cos(θ − θ0) . (3.98) Then we express f(θ) in terms of a Fourier series as f(θ) = ∞∑ n=1 (An cosn(θ − θ0) +Bn sinn(θ − θ0)) , (3.99) where An and Bn are given by An = 1 pi ∫ 2pi 0 f(θ) cosm(θ − θ0)dθ, Bn = 1 pi ∫ 2pi 0 f(θ) sinm(θ − θ0)dθ. (3.100) We denote I1 and I2 by I1 = piAn and I2 = piBn. Firstly, we consider the case where σ = cosnθ = Re ( einθ ) . We need to calculate I1 = Re ∫ 2pi 0 cos(θ − θ0)einθ − einθ0 − ineinθ sin(θ − θ0) 1− cos(θ − θ0) cosm(θ − θ0)dθ. (3.101) Let z = eiθ and z0 = eiθ0 . The equation above becomes I = ∫ zn( zz0+ zz0 −12 )− zn0 − nzn zz0− zz0 −12 2− zz0 + zz0 −1 (( z z0 )m + ( z z0 )−m) dz iz . (3.102) Let w = zz0 . The integral becomes I = ∫ 1−n 2 w n+1 + 1+n2 w n−1 − 1 w2 − 2w + 1 ( wm + w−m ) dw. (3.103) The integration is performed over the boundary of the unit disk. Note that we can write 1w2−2w+1 as 1 (1− w)2 = d dw 1 1− w = d dw ∞∑ n=0 wn. (3.104) Notice that |w| < 1 since w is within the unit circle. After some simplification 81 Chapter 3. Numerical Realizations: The Neumann Green’s Function we find 1−n 2 w n+1 + 1+n2 w n−1 − 1 w2 − 2w + 1 = − ( 1 + 2w + 3w2 + · · ·+ (n− 1)wn−2 + (n− 1) 2 wn−1 ) . (3.105) We use the residue theorem to calculate the integral. We find that I = zn0 { 2pim, 1 ≤ m < n, pi(n− 1), m = n, 0, m > n. (3.106) We can now state the following results for I1 and I2. For σ0 = cosnθ0: I1 = cosnθ0 { 2pim, 1 ≤ m < n, pi(n− 1), m = n, 0, m > n, (3.107) and I2 = − sinnθ0 { 2pim, 1 ≤ m < n, pi(n− 1), m = n, 0, m > n. (3.108) For σ0 = sin(nθ0): I1 = sinnθ0 { 2pim, 1 ≤ m < n, pi(n− 1), m = n, 0, m > n, (3.109) and I2 = cosnθ0 { 2pim, 1 ≤ m < n, pi(n− 1), m = n, 0, m > n. (3.110) We have now found An = 1pi I1 and Bn = 1 pi I2. We have the following Fourier expansions for f(θ). For σ0 = cosnθ0: f(θ) = n−1∑ m=1 2m [cosnθ0 cosm(θ − θ0)− sinnθ0 sinm(θ − θ0)] + (n− 1) (cosnθ0 cosn(θ − θ0)− sinnθ0 sinn(θ − θ0)) . (3.111) 82 Chapter 3. Numerical Realizations: The Neumann Green’s Function For σ0 = sinnθ0: f(θ) = n−1∑ m=1 2m [cosnθ0 sinm(θ − θ0) + sinnθ0 cosm(θ − θ0)] + (n− 1) (cosnθ0 sinn(θ − θ0) + sinnθ0 cosn(θ − θ0)) . (3.112) Recall that σ = ∑∞ n=1 (an cosnθ + bn sinnθ) from (3.92). We can find f(θ) for σ given by (3.92) by superposition. In this way we find f(θ) = ∞∑ n=1 ( an(n− 1)γcn + n−1∑ m=1 an2mγcm ) + ∞∑ n=1 ( bn(n− 1)γsn + n−1∑ m=1 bn2mγsm ) , (3.113) where γcm = cosnθ0 cosm(θ − θ0)− sinnθ0 sinm(θ − θ0), (3.114) γsm = cosnθ0 sinm(θ − θ0) + sinnθ0 cosm(θ − θ0). (3.115) We can rearrange f(θ) and rewrite it as f(θ) = ∞∑ n=1 [(n− 1) (an cosnθ0 + bn sinnθ0) cosn(θ − θ0)] + ∞∑ n=1 n−1∑ m=1 [2m (an cosnθ0 + bn sinnθ0) cosm(θ − θ0)] + ∞∑ n=1 [(n− 1) (bn cosnθ0 − an sinnθ0) sinn(θ − θ0)] + ∞∑ n=1 n−1∑ m=1 [2m (bn cosnθ0 − an sinnθ0) sinm(θ − θ0)] . (3.116) Then we interchange the order of summation as follows: ∞∑ n=1 n−1∑ m=1 χmn = ∞∑ m=1 ∞∑ n>m χmn = ∞∑ n=1 ∞∑ m>n χnm. (3.117) 83 Chapter 3. Numerical Realizations: The Neumann Green’s Function Using this we find that f(θ) = ∞∑ n=1 An cosn(θ − θ0) +Bn sinn(θ − θ0), (3.118) An = (n− 1) (an cosnθ0 + bn sinnθ0) + 2n ∞∑ m>n (am cosmθ0 + bm sinmθ0) , Bn = (n− 1) (bn cosnθ0 − an sinnθ0) + 2n ∞∑ m>n (bm cosmθ0 − am sinmθ0) . (3.119) To proceed we define S(x,x0) by Rm(x,x0) = S(x,x0) + |x|2 4 |Ω| . (3.120) Substituting this into the system for R, (3.95) and (3.96), we find that ∆S(x,x0) = 0, x ∈ Ω, (3.121) ∂nS(x,x0) = ∂nRm(x,x0)− 14 |Ω|∂n |x| 2 , x ∈ ∂Ω. (3.122) We calculate ∂n |x|2 = 2r√ 1+r′2/r2 . Using this along with (3.96), (3.97) and (3.98) we find that ∂nS(x,x0) = ε 2pi (f(θ)− σ(θ)) +O(ε)2, x ∈ ∂Ω. (3.123) Note that |Ω| ≈ pi. Now we introduce S0(x,x0) by S(x,x0) = ε 2pi S0(x,x0). (3.124) To leading order we can write ∂nS0 = ∂rS0 |r=1 +O(ε). We obtain the following leading order problem: ∆S0(x,x0) = 0, 0 < r < 1, (3.125) ∂rS0(x,x0) |r=1 = f(θ)− σ(θ), r = 1. (3.126) 84 Chapter 3. Numerical Realizations: The Neumann Green’s Function We can solve this system using separation of variables. We find that S0 = D0 + ∞∑ n=1 rn (Dn cosn(θ − θ0) + En sinn(θ − θ0)) . (3.127) To solve for the coefficients Dn and En we must use the boundary condition (3.126). To this end, we must rewrite σ, given by (3.92), in terms of cosn(θ−θ0) and sinn(θ− θ0). Notice that f(θ) is already in this form, (3.99). We calculate for σ that σ = ∞∑ n=1 ( ãn cosn(θ − θ0) + b̃n sinn(θ − θ0) ) , (3.128) ãn = an cosnθ0 + bn sinnθ0, (3.129) b̃n = bn cosnθ0 − an sinnθ0. (3.130) By imposing the boundary condition (3.126) we find that ∂rS0 |r=1= ∞∑ n=1 n (Dn cosn(θ − θ0) + En sinn(θ − θ0)) . (3.131) Equating coefficients with the right-hand side of (3.126) we obtain nDn = An − ãn, nEn = Bn − b̃n. (3.132) To summarize, we have, for x ∈ ∂Ω, that Rm(x,x0) = S(x,x0) + |x|2 4pi = ε 2pi S0(x,x0) + 1 4pi + εσ 2pi +O(ε2). (3.133) Using definition (3.93), we calculate the derivative of ρ′(θ0) as follows ρ′(θ0) = d dθ0 Rm(x0(θ0),x0(θ0)) = 2 d dθ Rm(x(θ),x0(θ0)) |θ=θ0 . (3.134) Substituting (3.133) into (3.134) we find ρ′(θ0) = ε pi [ d dθ S0(x(θ),x0(θ0)) |θ=θ0 +σ′(θ0) ] . (3.135) We differentiate S0(x,x0) given by (3.127) and σ given by (3.92) with respect 85 Chapter 3. Numerical Realizations: The Neumann Green’s Function to θ. We substitute these into the equation above to find that ρ′(θ0) = ε pi ∞∑ n=1 ( nEn + nb̃n ) . (3.136) Then, using equations (3.130) and (3.132) we find ρ′(θ0) = ε pi ∞∑ n=1 [Bn + (n− 1)bn cosnθ0 − (n− 1)an sinnθ0] . (3.137) Next, we use the expression for Bn Bn = (n− 1) (bn cosnθ0 − an sinnθ0) + 2n ∑ m>n (bm cosmθ0 − am sinmθ0) , (3.138) so that ρ′(θ) = ε pi ∞∑ n=1 ( 2(n− 1)γn + 2n ∞∑ m>n γm ) , (3.139) γm = bm cosmθ0 − am sinmθ0. (3.140) We calculate χ = ∑∞ n=1 ∑∞ m>n nγm. We can change the order of summation such that χ = ∑∞ m=2 γm ∑m−1 n=1 n. We find that χ = ∑∞ n=1 n(n − 1)γn. Thus our final result for ρ′(θ0) is ρ′(θ0) = ε pi ∞∑ n=1 (n+ 2)(n− 1) (bn cosnθ0 − an sinnθ0) . (3.141) ¥ We have found R′m(x0,x0). 3.4.2 Example We present an example here that makes use of (3.141). We determine whether there is a relationship between the curvature of ∂Ω and where the regular part of the Green’s function, Rm, has its extrema. In addition, we compare these predictions with Rm from our boundary element method. Consider the shape with σ defined by σ(θ) = cos 2θ + a cos 3θ (3.142) 86 Chapter 3. Numerical Realizations: The Neumann Green’s Function This implies that a1 = 0, a2 = 1, a3 = a, bn = 0. Recall r(θ) = 1 + εσ(θ). The curvature in polar coordinates is given by (3.86). Substituting the expression for r into the expression for the curvature we find κ(θ) = 1 + ε (3 cos 2θ + 8a cos 3θ) +O(ε)2 (3.143) κ′(θ) = −6ε sin 2θ − 24εa sin 3θ (3.144) Using equation (3.141) we find that ρ′(θ) = −4ε pi [ sin 2θ + 5a 2 sin 3θ ] (3.145) Using (3.143) and (3.144), we can find conditions on the parameter a that involves the minima and maxima of κ and ρ. Note that at θ = npi , the first derivative of κ and ρ are zero. Thus, we have extrema at θ = npi. We calculate the second derivatives when θ = pi κ′′(pi) = −6ε (2− 12a) (3.146) ρ′′(pi) = 4ε pi ( 2− 15a 2 ) (3.147) We see that at θ = pi , there is a maximum for κ when a < 16 and there is a maximum for ρ when a < 415 . Thus for a ∈ ( 16 , 415 ), ρ has a maximum where κ has a minimum. We have the reverse result if σ is negative. Thus, the principle eigenvalue, λ0, given by (2.51), (2.89), (2.94) for one hole, N absorbing arcs and N identical absorbing arcs are not in general minimized at the maximum of the domain curvature. Now at θ = 0, the second derivatives are κ′′(0) = −12ε (1 + 36a) (3.148) ρ′′(0) = −4ε pi ( 2 + 18a 2 ) (3.149) We see that κ′′(0) and ρ′′(0) are both negative for a > 0. Thus, both κ and ρ have maximums at θ = 0. In fact, it is where both attain their global maximums. We can relate these findings to our boundary element method by plotting Rm(x0,x0) versus x0(θ0). The shape of the domain is shown in Figure 3.11 (a), while the curvature, κ− 1, and ρ−C, where C is a constant of integration, are plotted in Figure 3.11 (b) for a = 0.2 and ε = 0.1. 87 Chapter 3. Numerical Realizations: The Neumann Green’s Function −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y (a) Plot of the perturbed unit disk with boundary r = 1+ε (cos 2θ + a cos 3θ). 0 pi/2 pi 3pi/2 2pi −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Θ (b) Plot of k(θ)− 1 (red) from (3.143) and ρ(θ)− C (blue) from (3.145). Figure 3.11: Plot of the perturbed unit disk, the curvature and ρ with ε = 0.1 and a = 0.2. We see from Figure 3.11 (a) that at θ = pi the curvature has a local minimum 88 Chapter 3. Numerical Realizations: The Neumann Green’s Function and ρ has a local maximum. At θ = 0 and θ = 2pi, we see that ρ and κ have global maxima here. Now we plot ρ = Rm(x0,x0) from the numerical method for N = 600 and N = 2400 mesh points in Figure 3.12 with a = 0.2 and ε = 0.1. 0 pi/2 pi 3pi/2 2pi 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 Θ R m (x 0 , x 0 ) Figure 3.12: Plot of Rm(x0,x0) for N = 600 (red) and N = 2400 (blue) with ε = 0.1 and a = 0.2. We see that ρ attains a local maximum at θ = pi, while at θ0 = 0, ρ attains a global maximum. Thus, our numerical results using the boundary element method agree with the analytical predictions qualitatively. However, the scaling of our numerical results does not match the analytical results. We see from considering the curve for N = 600 and N = 2400 that the numerical method is converging extremely slowly since the curves are almost identical. To attain accurate results that are quantitatively comparable to the analytical result we require many more mesh points. We are not able to run the method for more mesh points due to computational limitations. 3.5 Discussion In this chapter, we applied the results from Chapter 2 to calculate the MFPT in a few domains. We calculated the MFPT in the unit disk. In particular, we 89 Chapter 3. Numerical Realizations: The Neumann Green’s Function considered one hole on the boundary of the unit disk with the initial position at the centre of the disk and the initial position at the antipodal point to the hole. Our results are comparable to that derived by Singer et al. in [23]. In addition, we investigate the behaviour of the MFPT as the initial position moved towards the hole at the antipodal point. These are depicted in Figure 3.1. With two holes on the boundary, and the initial position at the centre, we found that the MFPT reached a minimum when the holes were furthest apart. This is depicted in Figure 3.2. We also derived results for N equally spaced points on the boundary of the unit disk. We found that v(x)→ 14D when N →∞, when log 1ε >> N We derived the Neumann Green’s function in the unit square in order to calculate the MFPT. We considered the behaviour of the MFPT as the ini- tial position moves towards the hole, and the case where the initial position is fixed, but the hole moves. These are depicted in Figures 3.4 and 3.5. We also considered two holes on the boundary of the unit square. The behaviour in this case was not as predictable as the other cases. We held one hole fixed at (0, 0.5), while the other hole moved from (0, 0.48) to (1, 0.5) along the boundary. We found that the minimum of the MFPT on the first two sides were slightly shifted as a result of the interaction between the two holes. In other words, the minimum was not where the distance between the initial position and moving exit window was a minimum. However, on the third side, this was the case as a result of the symmetry of the configuration. These results are shown in Figures 3.6 and 3.7. It is worth noting that our expression for the MFPT is not valid at the corners of the unit square. We derived an expression for the MFPT that is valid at the corner x0 = (0, 0) given by 3.55. In the last two sections of this Chapter, we considered an analytical and a nu- merical approach to solving for the Green’s function in more arbitrary domains. The boundary element method was introduced as a numerical technique to solve for the Green’s function. We showed that the boundary element method did indeed give the correct results in the case of a unit disk. We then proceeded to use the boundary element method to find the Neumann Green’s function in an ellipse with semi-major axis, a = 2 and semi-minor axis, b = 1. More precisely, we used our numerical technique to find R(x,x0). We found that the numerical method converged for N = 640 mesh points. We found that R(x,x0) = 0.1291 for all x within the Ω and for all x0 on the boundary. Thus, R(x,x0) is a constant. We expect then that R(x0,x0) = 0.1291. To calculate R(x0,x0) we must rewrite integral (3.78) as we did in (3.88) 90 Chapter 3. Numerical Realizations: The Neumann Green’s Function and (3.90). The calculation of R(x0,x0) proved to be a challenge, and requires many mesh points for convergence. By considering Table 3.4 we see that the difference between entries in the vector R decreases by approximately 0.57 as the number of mesh points doubles. The convergence is thus extremely slow. We propose that the integral (3.78) be calculated a more accurate way in this case. We propose using gaussian quadrature in stead of the midpoint rule to compute this integral. This is left for further work. We plotted the behaviour of the Neumann Green’s function for an ellipse with x at the origin and x0 at various points along the boundary in Figure 3.8. We found that the Green’s function satisfied the symmetry properties that we expected. We used the numerical results for the Green’s function to find the MFPT for the ellipse with x at the origin and x0 moving along the boundary. The results are depicted in Figure 3.9. We then moved the initial position to x = (1, 0) with x0 moving along the boundary as before. The results are depicted in Figure 3.10. We see that the MFPT is a minimum when the exit window is closest to the initial position, and a maximum when the exit is furthest from the initial position. To find the Neumann Green’s function in a perturbed circular domain we considered an analytical approach by T. Kolokolnikov [7]. Here, we derived an expression for the regular part of the Green’s function, Rm(x0,x0). One must integrate this expression and impose the integral condition to obtain a unique solution. We considered an example of a particular domain with σ given by (3.142), which is depicted in Figure 3.11 (a). We found that the regular part of the Green’s function, ρ, (3.145), has a local maximum while the curvature, κ, (3.143) has a local minimum at θ = pi for a particular parameter regime, a ∈ ( 16 , 415 ). We plotted the the curvature and ρ in Figure 3.11 (b) for a = 0.2 and ε = 0.1. We then calculatedRm(x0,x0) using our numerical boundary element method and plotted this against θ0 in Figure 3.12 for N = 600 and N = 2400 mesh points. By comparing these curves we see that the numerical method is con- verging extremely slowly since the two curves are similar. We see that our nu- merical results qualitatively agree with the analytical result in that R(x0,x0) has a maximum at θ = pi. However, the scaling is incorrect in comparison to the analytical result. To obtain a more accurate numerical result we must calculate the integral (3.78) more accurately. 91 Chapter 4 The Narrow Escape Problem in Three Dimensions - The Sphere We now proceed to the narrow escape problem for the mean first passage time for a Brownian particle trapped within a three-dimensional domain. The math- ematical statement of the problem remains the same, however, the method of approach and results are different from those in two dimensions. We will derive results for a sphere as the confinement domain. It is an open problem to extend this framework to arbitrary domains in three dimensions. The extension is not as straightforward as in two dimensions, as the local geometry and curvature of the three-dimensional domain plays a significant role in the final solution. It is, thus, necessary to solve for the Green’s function for each three-dimensional domain before one can proceed to find the mean first passage time. 4.1 Derivation of the Neumann Green’s function for a Sphere We begin by deriving the Neumann Green’s function, Gs(x,x0), for a sphere, with a singularity on the boundary, which satisfies ∆Gs = 1 |Ω| , x ∈ Ω, (4.1) ∂rGs = 1 R2 δ(cosϕ− cosϕ0)δ(φ− φ0), x ∈ ∂Ω, (4.2)∫ Ω Gsdx = 0. (4.3) 92 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere where Ω = {x | |x| ≤ 1} and |Ω| = 4piR33 . The angles are defined with 0 < ϕ < pi, the latitude, and 0 ≤ φ < 2pi, the longitude. We have the property that if g(ϕ0) = 0 and g(ϕ) is a monotone function, then δ(g(ϕ)) = δ(ϕ−ϕ0) |g′(ϕ0)| . Thus δ(cosϕ− cosϕ0) = δ(ϕ− ϕ0)sinϕ0 . (4.4) The singularity of the Green’s function in three dimensions is a simple pole. Since the singular point is on the boundary, the singularity is twice as large. Thus, we expect Gs(x,x0) = 1 2pi |x− x0| +R(x,x0), (4.5) where R(x,x0) has a milder singularity than a simple pole as x → ∞. In the three-dimensional case we show that R(x,x0) has a logarithmic singularity as x→ x0. Now we define Gp Gp = 1 6 |Ω| ( |x|2 + |x0|2 ) . (4.6) Note that |x|2 = r2 and |x0|2 = R2. To solve (4.1) and (4.2) we let Gs = Gp + Ḡs. Substituting this into (4.1) and (4.2) we find ∆Ḡs = 0, x ∈ Ω, (4.7) ∂rḠs = 1 R2 δ(cosϕ− cosϕ′)δ(φ− φ′)− 1 4piR2 , x ∈ ∂Ω. (4.8) We will look for a solution written in terms of Legendre Polynomials. To this end, we must rewrite the Neumann boundary condition in terms of Legendre polynomials. This leads to the following Lemma: Lemma 4.1: We claim that δ(cosϕ− cosϕ0)δ(φ− φ0) = 14pi ∞∑ m=0 (2m+ 1)Pm(cos γ), (4.9) 93 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere where cos γ = x · x0 = cosϕ cosϕ0 + sinϕ sinϕ0 cos(φ− φ0), x = (cosφ sinϕ, sinφ sinϕ, cosϕ), x0 = (cosφ0 sinϕ0, sinφ0 sinϕ0, cosϕ0). Here x and x0 are written in spherical coordinates in the unit sphere, R = 1. Proof : We recall the completeness formula ∞∑ m=0 m∑ n=−m Y ∗mn(ϕ0, φ0)Ymn(ϕ, φ) = δ(φ− φ0)δ(cosϕ− cosϕ0), (4.10) where the Ymn are the spherical harmonics. The addition theorem for Legendre polynomials states that (2m+ 1) 4pi Pm(cos γ) = m∑ n=−m Y ∗mn(ϕ0, φ0)Ymn(ϕ, φ). (4.11) Summing (4.11) from m = 0 to ∞, we obtain the desired result (4.9).¥ Using this Lemma, the boundary condition (4.2) becomes ∂rḠs = 1 4piR2 ∞∑ m=1 (2m+ 1)Pm(cos γ), onr = 1. (4.12) We look for a solution for Ḡs in the form Ḡs = ∞∑ m=1 am ( r R )m Pm(cos γ), (4.13) which satisfies Laplace’s equation by construction. Applying the boundary con- dition (4.12) we find that am = 1 4piR (2m+ 1) m . (4.14) 94 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere We can write the solution for Ḡs as Ḡs = 1 4piR ∞∑ m=1 (2m+ 1) m ( r R )m Pm(cos γ) (4.15) = 1 2piR ∞∑ m=1 ( r R )m Pm(cos γ) + 1 4piR ∞∑ m=1 1 m ( r R )m Pm(cos γ). (4.16) We deal with these two terms separately. We start with the first term. Recall the generating function for Legendre polynomials 1√ 1− 2xt+ t2 = ∞∑ m=0 Pm(x)tn. (4.17) Thus, 1 2piR ∞∑ m=1 ( r R )m Pm(cos γ) = 1 2pi 1√ r2 +R2 − 2rR cos γ − 1 2piR . (4.18) Now we consider the second term in (4.16). Let β = rR and I = ∑∞ m=1 1 mβ mPm(cos γ). Using this we find I ′(β) = 1 β ∞∑ m=1 βmPm(cos γ) = 1 β [ 1√ 1− 2β cos γ + β2 − 1 ] (4.19) Also note, that I(0) = 0. We integrate the equation above I = ∫ β 0 ( 1 s 1√ 1− 2s cos γ + s2 − 1 s ) ds = log ( 2 1− β cos γ + √ 1 + β2 − 2β cos γ ) (4.20) Then with Gs = Gp + Ḡs, and substituting for β we find that Gs = 1 6 |Ω| ( |x|2 + |x0|2 ) + 1 2pi 1 |x− x0| − 1 2piR + 1 4piR log ( 2R R− r cos γ + |x− x0| ) + C. (4.21) 95 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere We are ready to use the integral condition (4.3), ∫ 2pi 0 ∫ pi 0 ∫ R 0 Gs sin γr2dγdφdr = 0, to find the constant C. We know that log ( 2R R− r cos γ + |x− x0| ) = ∞∑ m=1 1 m ( r R )m Pm(cos γ). (4.22) We also have that∫ pi 0 Pm(cos γ) sin γdγ = 2 sinmpi mpi(m+ 1) = { 0 if m 6= 0, 2 if m = 0. (4.23) Thus, the integral over γ of (4.22) is zero. Next we note that 1 2pi |x− x0| = 1 2piR ∞∑ m=0 Pm(cos γ) ( r R )m . (4.24) We can perform the integration over γ by using property (4.23). We find that 1 2piR ∞∑ m=0 ∫ 2pi 0 ∫ pi 0 ∫ R 0 Pm(cos γ) ( r R )m sin γr2dγdφdr = 2R2 3 . (4.25) Using (4.22) and (4.25) we find that C = − 15piR for the integral condition, (4.3) to be satisfied. Our final expression for Gs is Gs(x,x0) = 1 2pi |x− x0| + 1 6 |Ω| ( |x|2 +R2 ) + 1 4piR log ( 2R R− |x| cos γ + |x− x0| ) − 7 10piR (4.26) We will need the expression for the Green’s function, (4.26) in the limit that x→ x0. We let y = x− x0 ε , Λ = R− r ε . (4.27) We then calculate using the law of cosines that R− |x| cos γ = 1 2R ( |x− x0|2 − (|x|2 −R2) ) ∼ 1 2R ( O(ε)2 − ((R− εΛ)2 −R2)) ∼ εΛ +O(ε)2. (4.28) 96 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere Therefore, by substituting (4.27) and (4.28) into (4.26), we obtain in the limit as x→ x0 that Gs ∼ 12piε |y| + 1 4piR log ( 2R ε(|y|+ Λ) ) − 9 20piR . (4.29) Now we calculate |x− x0| in the limit as x → x0 in spherical coordinates. We use a Taylor expansion about x0. We find that |x− x0| = √ (r −R)2 +R2(ϕ− ϕ0)2 +R2 sin2 ϕ(φ− φ0)2 as x→ x0 (4.30) We let Λ = R− r ε , s1 = R sinϕ(φ− φ0) ε , s2 = R(ϕ− ϕ0) ε . (4.31) Using this, we find that |x− x0| = ε ( Λ2 + s21 + s 2 2 )1/2 as x→ x0 (4.32) This implies that |y| = (Λ2 + s21 + s22)1/2 . (4.33) Combining (4.29) with (4.33) we obtain the far field behaviour of the Neu- mann Green’s function. 4.2 Three-Dimensional Sphere with N Absorbing Patches on the Boundary We want to find the MFPT in a unit sphere, Ω = {x | |x| ≤ 1}. ∂Ωr is the reflecting part of the boundary, ∂Ω, while ∂Ωa is the absorbing part consisting of N non-overlapping circular patches defined by ∂Ωεj = {(ϕ, φ) | (ϕ− ϕj)2 + sin2 ϕj(φ− φj)2 ≤ ε2a2j = r2ε} (4.34) Thus, the area of the circular patch centred at (1, ϕj , φj) is ∣∣∂Ωεj ∣∣ = piε2a2j for j = 1, 2, ..N . Without loss of generality, there is no absorbing patch centred at a pole of the coordinate system. In Cartesian coordinates, the location of the 97 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere centre of the jth patch is given by xj = cosφj sinϕj , yj = sinφj sinϕj , zj = cosϕj , (4.35) where |xj | = 1. The Laplacian in spherical coordinates is ∆u = 1 r2 ( r2ur ) r + 1 r2 sin2 ϕ ∂φφu+ 1 r2 sinϕ ∂ϕ (sinϕ∂ϕu) , = urr + 2 r ur + 1 r2 sin2 ϕ uφφ + cotϕ r2 uϕ + 1 r2 uϕϕ. (4.36) We write the mean escape time in terms of eigenfunctions as in (2.6), as ∆φ0 + λ0φ0 = 0, x ∈ Ω, φ0(x) = 0, x ∈ ∂Ωa = N⋃ j=1 ∂Ωεj , ∂rφ0 = 0, x ∈ ∂Ωr. (4.37) 4.2.1 Calculation of λ0 and φ0 We need to solve the system (4.37) to find the principal eigenvalue, λ0, and the principal eigenfunction, φ0. The eigenvalue is expanded as λ0(ε) = ελ1 + ε2 log (ε 2 ) λ2 + ε2λ3 + · · · . (4.38) In the outer region, away from the absorbing arcs, we expand the principal eigenfunction as φ0(x, ε) = u0 + εu1 + ε2 log (ε 2 ) u2 + ε2u3 + · · · . (4.39) Here, u0 = 1|Ω| 12 , the solution to the leading order problem. Substituting these expansions into (4.37), we find to order O(ε) that ∆u1 = −λ1u0, x ∈ Ω, ∂ru1 = 0, x ∈ ∂Ωr,∫ Ω u1dx = 0. (4.40) 98 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere The integral condition enforces the normalization condition. At the next order, O ( ε2 log ( ε 2 )) , we find ∆u2 = −λ2u0, x ∈ Ω, ∂ru2 = 0, x ∈ ∂Ωr,∫ Ω u2dx = 0. (4.41) At order O(ε2) we find ∆u3 = −λ1u1 − λ3u0, x ∈ Ω, ∂ru3 = 0, x ∈ ∂Ωr,∫ Ω ( 2u0u3 + u21 ) dx = 0. (4.42) In the absorbing region we introduce the following change of variables, (4.31), near the jth absorbing patch Λ = 1− r ε , s1 = sinϕj(φ− φj) ε , s2 = ϕ− ϕj ε . (4.43) We let v(Λ, s1, s2, ε) = u ( 1− εΛ, εs1 sinϕ + φj , εs2 + ϕj ) . (4.44) To proceed, we must transform the Laplacian, (4.36), into an expression involving the inner variables using the chain rule. We find that ∆w = ε−2 (wΛΛ + ws1s1 + ws2s2) + ε −1 (−2wΛ + 2Λ (ws1s1 + ws2s2)) + ε−1 cotϕj (ws2 − s22ws1s1) +O(1). (4.45) In the inner region we expand the eigenfunction as v = v0 + ε log (ε 2 ) v1 + εv2 + · · · . (4.46) We project all the patches onto a circular disk of radius a2j in the plane. At 99 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere O(ε0) we find Lv0 = v0ΛΛ + v0s1s1 + v0s2s2 = 0, Λ ≥ 0, −∞ < s1, s2 <∞, ∂Λv0 = 0, Λ = 0, s21 + s 2 2 ≥ a2j , v0 = 0, Λ = 0, s21 + s 2 2 ≤ a2j . (4.47) At O ( ε log ( ε 2 )) we find Lv1 = 0, Λ ≥ 0, −∞ < s1, s2 <∞, ∂Λv1 = 0, Λ = 0, s21 + s 2 2 ≥ a2j , v1 = 0, Λ = 0, s21 + s 2 2 ≤ a2j . (4.48) At order O(ε) we have Lv2 = 2 (v0Λ − Λ (v0s1s1 + v0s2s2))− cotϕj (v0s2 − 2s2v0s1s1) ∂Λv2 = 0, Λ = 0, s21 + s 2 2 ≥ a2j , v2 = 0, Λ = 0, s21 + s 2 2 ≤ a2j . (4.49) To proceed, we must match inner and outer solutions. We compare (4.39) to (4.46) in the limit as x → xj or |y| = ( Λ2 + s21 + s 2 2 )1/2 → ∞. We see that v0 = u0 = |Ω|− 1 2 in this limit. We write the solution as v0 = |Ω|− 1 2 (1− vc), (4.50) where vc satisfies Lvc = 0, Λ ≥ 0, −∞ < s1, s2 <∞, ∂Λvc = 0, Λ = 0, s21 + s 2 2 ≥ a2j , vc = 1, Λ = 0, s21 + s 2 2 ≤ a2j . (4.51) This is the well-known electrified disk problem in electrostatics, [6], which has a solution given in terms of the Bessel Function J0(z) by vc = 2 pi ∫ ∞ 0 sinµ µ e−µΛ/ajJ0 ( µσ aj ) dµ, (4.52) 100 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere where σ = ( s21 + s 2 2 )1/2. This solution has the asymptotic behaviour vc ∼ cj|y| +O ( 1 |y|3 ) as |y| = (Λ2 + s21 + s22)1/2 →∞. (4.53) Here, cj is the capacitance of a disk of radius aj in an infinite plane given by cj = 2aj pi . (4.54) The result above is obtained using Laplace’s method and is shown in Appendix B. Thus v0 ∼ 1|Ω| 12 ( 1− cj|y| ) . Writing y in inner variables, as in (4.27) and (4.33), we find from matching in the limit as x→ xj that 1 |Ω| 12 + εu1 + ε2 log (ε 2 ) u2 + · · · ∼ 1|Ω| 12 − εcj |Ω|1/2 |x− xj | + · · · . This gives the missing singular condition on u1 u1 ∼ − cj|Ω| 12 |x− xj | as x→ xj j = 1, .., N. (4.55) We must solve for u1 from (4.40) and (4.55). Recall, from (4.1) and (4.2), that a singularity on the boundary can be written as ∂ru1 = δ(ϕ− ϕj)δ(φ− φj) sinϕj on r = 1. (4.56) This gives rise to a term in the solution of the form u1 ∼ 12pi|x−xj | as x→ xj . Thus, we can write the problem for u1 as ∆u1 = −λ1u0, x ∈ Ω, (4.57) ∂ru1 = − 2pi|Ω| 12 N∑ j=1 cj δ(ϕ− ϕj)δ(φ− φj) sinϕj on r = 1, (4.58)∫ Ω u1dx = 0. (4.59) Before we find the solution for u1 we impose a solvability condition that will allow us to find the eigenvalue λ1. 101 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere We integrate (4.57) over Ω and apply the Divergence Theorem to obtain∫ Ω ∆u1dx = ∫ ∂Ω ∂u1 ∂r |r=1 dS, −λ1u0 |Ω| = − N∑ j=1 2picj |Ω| 12 . Thus, λ1 = 2pi |Ω| ∞∑ j=1 cj , cj = 2aj pi . (4.60) We can write the solution for u1 as u1 = − 2pi|Ω| 12 ∞∑ i=1 ciGs(x,xi), (4.61) where Gs satisfies (4.1)-(4.3) with the Neumann boundary condition written using (4.4). That is, Gs satisfies ∆Gs = 1 |Ω| , x ∈ Ω, (4.62) ∂rGs = δ(ϕ− ϕi)δ(φ− φi) sinϕi on r = 1, (4.63)∫ Ω Gsdx = 0. (4.64) We proceed by matching. In the limit as x→ xj , u1 becomes u1 ∼ − 2pi|Ω| 12 [cjGs(x,xj)]− 2pi|Ω| 12 N∑ i=1,i6=j ciGs(xj ,xi), (4.65) where the near-field behaviour of Gs(x,xj) as x → xj is given by (4.29) with y given by (4.33). The outer expansion in terms of inner variables for x→ xj is 1 |Ω| 12 − cj |Ω| 12 |y| + 2pi |Ω|1/2 ε cj 4pi log (ε 2 ) + εcj 4pi log (|y|+ Λ) + ε9cj 20pi − ε N∑ i=1,i6=j ciGs(xj ,xi)  + ε2 log (ε 2 ) u2 + ε2u3 + · · · . (4.66) 102 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere The inner expansion is 1 |Ω| 12 − cj |Ω| 12 |y| + ε log (ε 2 ) v1 + εv2 + · · · . (4.67) Comparing these expressions we find that v1 ∼ cj 2 |Ω| 12 as |y| → ∞. (4.68) To proceed, we let v1 = cj 2 |Ω| 12 (1− vc), (4.69) where vc satisfies (4.51). Using the behaviour of vc as |y| → ∞ given by (4.53), we find that v1 ∼ cj 2 |Ω| 12 ( 1− cj|y| +O ( 1 |y|3 )) as |y| → ∞. (4.70) Substituting this into (4.67) and comparing to (4.66) we find from matching that u2 ∼ − c2j 2 |Ω| 12 |x− xj | as x→ xj , j = 1, ..., N. (4.71) Thus, we must solve for u2 from (4.41) and (4.71). Proceeding as we did when finding u1, we rewrite the boundary condition of (4.41) as ∂ru2 = − pi|Ω| 12 N∑ j=1 c2j δ(ϕ− ϕj)δ(φ− φj) sinϕj on r = 1. (4.72) Before solving for u2 we use a solvability condition to find λ2. We integrate (4.41) over Ω and use the Divergence Theorem. We find that λ2 = pi |Ω| N∑ j=1 c2j . (4.73) At this stage, as ε→ 0, we have the following two-term asymptotic behaviour λ ∼ 2pi|Ω|ε N∑ j=1 cj + pi |Ω|ε 2 log (ε 2 ) N∑ j=1 c2j +O(ε 2), (4.74) where cj = 2aj pi . 103 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere The solution for u2 is written in terms of the Green’s function Gs as u2 = − pi|Ω| 12 N∑ i=1 c2iGs(x,xi), (4.75) where Gs satisfies (4.62)-(4.64). It is key to note that there is no need to expand u2 as x → xj . This is a consequence of the fact that any unmatched terms in u2 are dominated by the O(ε) terms in the expansion (4.66). The O(ε) terms in u as x→ xj written in the variable y are − 2pi |Ω| 12 Aj + cj 2 |Ω| 12 log(|y|+ Λ), (4.76) where Aj = − 9cj20pi + N∑ i=1,i6=j ciGs(xj ,xi). (4.77) From (4.67) we see that v2 ∼ − 2pi|Ω| 12 Aj + cj 2 |Ω| 12 log(|y|+ Λ) as |y| → ∞. (4.78) Thus v2 must satisfy (4.49) and (4.78). We can rewrite the right-hand side of the first equation of (4.49) by using v0s1s1 + v0s2s2 + v0ΛΛ = 0 as 2 (Λv0Λ)Λ − cotϕj (v0s2 − 2s2v0s1s1) . (4.79) We use an asymptotic decomposition to solve for v2 to obtain v2 = − 2pi|Ω| 12 Aj(1− vc) + cj 2 |Ω| 12 v2p, (4.80) where v2p ∼ log (|y|+ Λ) as |y| → ∞. (4.81) Here, vc is homogeneous solution, satisfying (4.51) with the behaviour as |y| → ∞, given by (4.53). That is vc ∼ cj|y| + O ( 1 |y|3 ) as |y| → ∞. Furthermore, v2p is the particular solution satisfying (4.49) with the far-field behaviour given by (4.81). By using the far-field behaviour of vc, (4.53), combined with a few 104 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere simple estimates, we find that the far-field behaviour of v2 is v2 ∼ − 2pi|Ω| 12 Aj ( 1− cj|y| ) + cj 2 |Ω| 12 log(|y|+ Λ), as |y| → ∞. (4.82) From matching, we must then conclude that u3 ∼ 2pi|Ω| 12 Ajcj |x− xj | as |y| → ∞. (4.83) This matches the unmatched term in the inner expansion v2 ∼ 2pi|Ω| 12 Ajcj |y| . The problem for u3 becomes ∆u3 = −λ1u1 − λ3u0, x ∈ Ω, (4.84) ∂ru3 = 4pi2 |Ω| 12 N∑ j=1 Ajcj δ(ϕ− ϕj)δ(φ− φj) sinϕj on r = 1, (4.85)∫ Ω u3dx = 0. (4.86) We can find λ3 using a solvability condition. We integrate (4.84) over Ω, recall- ing that ∫ Ω u1dx = 0. We use the Divergence Theorem to obtain λ3 = −4pi 2 |Ω| N∑ j=1 Ajcj . (4.87) where Aj is given by (4.77). We have the following proposition: Proposition 4.2: (N Holes) In the limit as ε → 0, λ0 and φ0 have the following three-term asymptotic behaviour λ0(ε) ∼ 2piε|Ω| N∑ j=1 cj + ε2 log (ε 2 ) pi |Ω| N∑ j=1 c2j − 4pi2ε2 |Ω| N∑ j=1 Ajcj , (4.88) φ0(x, ε) = 1 |Ω| 12 − 2pi |Ω| 12 ε N∑ i=1 ciGs(x,xi)− ε2 log (ε 2 ) pi |Ω| 12 N∑ i=1 c2iGs(x,xi) + O(ε2). (4.89) 105 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere We can simplify the notation by introducing the Green’s matrix, Γ, and capacitance vector C by Γ =  R Gs(1, 2) Gs(1, 3) · · · Gs(1, N) Gs(2, 1) R Gs(2, 3) · · · Gs(2, N) Gs(3, 1) Gs(3, 2) R · · · Gs(3, N) ... ... ... . . . ... Gs(N, 1) Gs(N, 2) · · · · · · R  , C = ( c1 c2 c3 · · · cN )T , e = ( 1 1 1 · · · 1 )T . Note that the matrix Γ is symmetric since Gs(j, i) = Gs(i, j), where Gs(j, i) denotes Gs(xj ,xi). Also, we define R by R = − 9 20pi . (4.90) Now, we can write λ as λ0(ε) ∼ 2piε|Ω| ( CTe+ ε 2 log (ε 2 ) CTC − 2piεCTΓC ) . (4.91) We can simplify this further by simplifying the expression for the Green’s func- tion Gs given by (4.26). We know that |xi| = |xj | = 1. Furthermore, by the law of cosines, 1− cos γij = |xi−xj | 2 2 . Using this, the Green’s function reduces to Gs(xj ,xi) = R + 1 2pi ( 1 |xj − xi| + log 2− 1 2 log |xj − xi| − 12 log (2 + |xj − xi|) ) .(4.92) We define Hij by Hij = ( 1 |xj − xi| − 1 2 log |xj − xi| − 12 log (2 + |xj − xi|) ) , (4.93) for i 6= j. Thus, Γ can be written as Γij = R+ log 2 2pi + 1 2pi Hij , (4.94) 106 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere for i 6= j. For i = j we see that Γii = R. We define p(x1,x2, ...,xN ) by p(x1,x2, ...,xN ) = CTΓC = N∑ i=1 N∑ j=1 cicjΓij , = R N∑ i=1 c2i + N∑ i=1 N∑ j=1,j 6=i cicjΓij . (4.95) We now write λ as λ0(ε) ∼ 2piε|Ω| ( CTe+ ε 2 log (ε 2 ) CTC − 2piεp(x1,x2, ...,xN ) ) . (4.96) Now we consider the special case when each of the absorbing patches are of the same radius, that is aj = a. In this case, the capacitance, cj , becomes c = 2api . We have the following proposition: Proposition 4.3: (N Identical Holes) In the limit as ε → 0, λ0 and φ0 have the three-term asymptotic behaviour λ0(ε) ∼ 2piεNc|Ω| ( 1 + ε 2 log (ε 2 ) c ) + 2piεNc |Ω| εc 9N 10 − (N − 1) log 2− 1 N N∑ i=1 N∑ j=1,j 6=i Hij  , (4.97) φ0(x, ε) = 1 |Ω| 12 − 2piεc |Ω| 12 N∑ i=1 Gs(x,xi)− ε2 log (ε 2 ) pic2 |Ω| 12 N∑ i=1 Gs(x,xi) + O(ε2), (4.98) where Hij is given by (4.93). For the case of equally sized absorbing patches, λ0(ε), is maximized with respect to {x1,x2, ...,xN} when we find {x1,x2, ...,xN} with |xj | = 1, for all j that satisfies the following discrete variational problem HN = min{x1,x2,...,xN} H(x1,x2, ...,xN ) = min{x1,x2,...,xN} N∑ i=1 N∑ j=1,j 6=i Hij , (4.99) where Hij is given by (4.93). By using HN in (4.97) for λ0, we obtain a max- 107 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere imum for λ0 up to O(ε2). The first term in H(x1,x2, ...,xN ) is the usual Coulomb singularity in three-dimensions while the other two terms represent a contribution from surface diffusion on the boundary of the sphere. The discrete variational problem (4.99) is a modification of the classical discrete variational problems for finding optimal configurations of N points on the surface of a sphere that minimize the Coulomb energy min {x1,x2,...,xN} N∑ i=1 N∑ j=1,j 6=i 1 |xi − xj | , |xi| = 1, (4.100) or the logarithmic energy − min {x1,x2,...,xN} N∑ i=1 N∑ j=1,j 6=i log |xi − xj | . (4.101) The problem of minimizing Hij appears to be a generalization of these two classic discrete variational problems. An overview of these variational problems are given in [19] and references therein. 4.2.2 Calculation of the Mean First Passage Time We calculate the MFPT in the case where we have N holes on the boundary of a unit sphere, of radius aj . We use expansions (4.88) and (4.89) to compute the MFPT. Recall, that the MFPT is given by v(x) ∼ (1, φ0) Dλ0 φ0, (4.102) where x is the initial position within the sphere. This assumes, that (φ0, φ0) = 1, which is satisfied by (4.89) since the integrals of Gs over Ω are zero. Recall that Gs is the Green’s function in a sphere of radius one. The MFPT for N holes of differing radius in terms of λ0 is given by v(x) = 1 Dλ0(ε) ( 1− 2piε N∑ i=1 ciGs(x,xi)− ε2 log (ε 2 ) pi N∑ i=1 c2iGs(x,xi) +O(ε 2) ) , (4.103) where λ0 is given by (4.88). By substituting for λ0 we have the following proposition for the MFPT with N holes of differing radius ai on the boundary of the unit sphere: 108 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere Proposition 4.4: (N Holes) For ε → 0, the three-term asymptotic be- haviour of the mean first passage time, v(x), is v(x) = |Ω| 2piεD ∑N i=1 ci ( 1 + ε 2 log ( 2 ε ) ∑N i=1 c 2 i∑N i=1 ci − 2piε N∑ i=1 ciG(x,xi) ) + |Ω| 2piεD ∑N i=1 ci ( 2piε∑N i=1 ci p(x1,x2, ...,xN ) +O(ε2 log ε) ) , (4.104) where ci = 2aipi , p(x1,x2, ...,xN ) is given by (4.95), Gs is given by (4.92) and |Ω| = 4pi3 . The average MFPT, v̄, is v̄ = |Ω| 2piεD ∑N i=1 ci ( 1 + ε 2 log ( 2 ε ) ∑N i=1 c 2 i∑N i=1 ci ) + |Ω| 2piεD ∑N i=1 ci ( 2piε∑N i=1 ci p(x1,x2, ...,xN ) +O(ε2 log ε) ) . (4.105) In the case of N identical holes of radius a, so that ci = c = 2api , we find: Corollary 4.5: (N Identical Holes) For ε→ 0, the three-term asymptotic behaviour of the mean first passage time is v(x) = |Ω| 2piεDNc ( 1 + ε 2 log ( 2 ε ) c− 2piεc N∑ i=1 Gs(x,xi) ) + |Ω| 2piεDNc εc −9N 10 + (N − 1) log 2 + 1 N N∑ i=1 N∑ j=1,j 6=i Hij  + O ( ε2 log ε ) , (4.106) where Hij is given by (4.93). The average MFPT, v̄, is v̄ = |Ω| 2piεDNc ( 1 + ε 2 log ( 2 ε ) c ) + |Ω| 2piεDNc εc −9N 10 + (N − 1) log 2 + 1 N N∑ i=1 N∑ j=1,j 6=i Hij  + O ( ε2 log ε ) . (4.107) 109 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere Notice that if we minimize Hij as in (4.99), we minimize the MFPT. We can find an expression for one hole by setting N = 1 in (4.106) to obtain the follow- ing corollary: Corollary 4.6: (One Hole) For ε → 0, the three-term asymptotic be- haviour of the mean first passage time, v(x), is v(x) = |Ω| 2piεDc ( 1 + ε 2 log 2 ε c− 2piεcGs(x,x0)− 9εc10 +O(ε 2 log ε) ) . (4.108) The average MFPT, v̄, is v̄ = |Ω| 2piεDc ( 1 + ε 2 log 2 ε − 9εc 10 +O(ε2 log ε) ) . (4.109) In the case of one hole with a = 1, the MFPT from the centre of the sphere is: Corollary 4.7: (One Hole) For ε → 0, the three-term asymptotic be- haviour of the mean first passage time from the centre of a unit sphere, v(x), with one absorbing patch of radius a = 1 on the boundary is E[τ | x = (0, 0)] = |Ω| 4εD ( 1 + ε pi log 2 ε − 3ε 2pi +O(ε2 log ε) ) . (4.110) Furthermore, for the case of one absorbing circular window of radius ε, the average MFPT is v̄ ∼ |Ω|4εD ( 1 + εpi log 1 ε +O(ε) ) . This is comparable to the result derived by Singer et al in [25] for the MFPT for one absorbing circular window of radius ε. They found the MFPT, and the average MFPT, to be E[τ | x = (0, 0)] = |Ω| 4εD [ 1 + ε pi log 1 ε +O(ε) ] (4.111) The original result in equation (3.52) of [25] omits the pi term in (4.111) due to an omission of an extra factor of pi on the left-hand side of the equation above (3.52) in [25]. Our result, (4.110), agrees asymptotically with that of (4.111) and determines explicitly the O(ε) term to v(x). Our results, (4.104) and (4.106), generalizes the result in [25] to N circular absorbing windows of different radii on the unit sphere. 110 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere 4.3 Three-Dimensional Sphere with N Absorbing Patches on the Boundary - An Alternate Derivation Here, we present an alternate derivation of the MFPT. Instead of writing the MFPT in terms of a complete set of eigenfunctions, we solve for the MFPT directly. That is, we want to solve ∆v(x) = − 1 D , x ∈ Ω, v(x) = 0, x ∈ ∂Ωa = N⋃ j=1 ∂Ωεj , ∂nv(x) = 0, x ∈ Ωr, (4.112) where ∂Ωεj is given by (4.34). The Laplacian is given (4.36). As before, there is no absorbing patch centred at a pole of the coordinate system. In Carte- sian coordinates, the location of the centre of the jth is given (4.35). We will now proceed to solve (4.112) directly using the method of matched asymptotic expansions. 4.3.1 Calculation of the MFPT Directly We want to solve (4.112). In the outer region, away from absorbing patches, we expand v(x) as v(x, ε) = v0 ε + v1 + ε log (ε 2 ) + εv3 + · · · . (4.113) We obtain the following problems: At O(ε−1) ∆v0 = 0, x ∈ Ω, ∂nv0 = 0, x ∈ Ωr, (4.114) at O(ε0) ∆v1 = − 1 D , x ∈ Ω, ∂nv1 = 0, x ∈ Ωr, (4.115) 111 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere at O ( ε log ( ε 2 )) ∆v2 = 0, x ∈ Ω, ∂nv2 = 0, x ∈ Ωr, (4.116) and at O(ε) ∆v3 = 0, x ∈ Ω, ∂nv3 = 0, x ∈ Ωr. (4.117) In the inner region we introduce the change of variables given by (4.31) and (4.43). In the inner region we expand w(Λ, s1, s2, ε) = w0 ε + log (ε 2 ) w1 + w2 + · · · . (4.118) We must transform the Laplacian, given by (4.36) into an expression in terms of local variables by using the chain rule. As before, we obtain (4.45). We obtain to O(ε−1) Lw0 = w0ΛΛ + w0s1s1 + w0s2s2 = 0, Λ ≥ 0,−∞ < s1, s2 <∞, ∂Λw0 = 0, Λ = 0, s21 + s 2 2 ≥ a2j , w0 = 0, Λ = 0, s21 + s 2 2 ≤ a2j . (4.119) At O ( log ( ε 2 )) we have Lw1 = 0, Λ ≥ 0, −∞ < s1, s2 <∞, ∂Λw1 = 0, Λ = 0 s21 + s 2 2 ≥ a2j , w1 = 0, Λ = 0 s21 + s 2 2 ≤ a2j . (4.120) At O(ε0) we have Lw2 = 2 (Λw0ΛΛ + w0Λ)− cotϕj (w0s2 − 2s2w0s1s1) , ∂Λw2 = 0, Λ = 0, s21 + s 2 2 ≥ a2j , w2 = 0, Λ = 0 s21 + s 2 2 ≤ a2j . (4.121) We proceed by matching, which implies that w0 → v0 as |y| → ( Λ2 + s21 + s 2 2 )1/2 112 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere or as x→ xj . We write the solution for w0 as w0 = v0(1− wc), (4.122) where wc satisfies (4.51). The solution to this is given by (4.52). The solution has the asymptotic behaviour (4.53) where cj is the capacitance of a disk of radius aj in an infinite plane given by (4.54). Writing y in outer variables as y = x−xjε and matching w0 to the outer solution we find that v1 ∼ − v0cj|x− xj | as x→ xj . (4.123) Thus, we must solve for v1 from (4.115), and (4.123). Recall from (4.1) and (4.2), that a singularity on the boundary can be written as ∂ru = δ(ϕ− ϕj)δ(φ− φj) sinϕj on r = 1. (4.124) This gives rise to a term in the solution of the form u ∼ 12pi|x−xj | as x → xj . Thus, we can write the problem for v1 as ∆v1 = − 1 D x ∈ Ω, (4.125) ∂rv1 = −2piv0 N∑ j=1 cj δ(ϕ− ϕj)δ(φ− φj) sinϕj on r = 1. (4.126) Before we find v1, we apply a solvability condition that will allow us to find v0. We apply the Divergence Theorem by integrating (4.125) and using the boundary condition (4.126). We find that v0 = |Ω| 2piD ∑N j=1 cj . (4.127) Now, the solution to v1 is written in terms of the surface Green’s function v1 = −2piv0 N∑ i=1 ciGs(x,xi) + χ, (4.128) where Gs satisfies (4.62)-(4.64). Using this, notices that ∫ Ω v1dx = χ |Ω| so that 113 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere χ = v̄. We proceed by matching. In the limit as x→ xj v1 ∼ −2piv0cjGs(x,xj)− 2piv0 N∑ i=1,i6=j ciGs(xj ,xi) + χ, (4.129) where the far-field behaviour of Gs is given by (4.29) along with (4.33). We let χ = χ0 log (ε 2 ) + χ1. (4.130) We write the outer expansion in terms of inner variables and match to the inner solution. That is, in the limit as |y| → ∞ we have v0 ε − v0cj ε |y| + v0cj 2 log (ε 2 ) +Bj + v0cj 2 log (Λ + |y|) + χ0 (ε 2 ) + χ1 + ε (ε 2 ) v2 + εv3 + · · · ∼ v0 ε ( 1− cj|y| ) + (ε 2 ) w1 + w2 + · · · , (4.131) where Bj = v0 9cj 10 − 2pi N∑ i=1,i6=j ciG(xj ,xi)  . (4.132) If we consider the expression for Aj , (4.77), we see that Bj = −2piv0Aj . By matching we see that w1 ∼ v0cj2 + χ0 as |y| → ∞. (4.133) The solution is w1 = (v0cj 2 + χ0 ) (1− wc), (4.134) where wc satisfies (4.51). Using the far-field behaviour of wc given by (4.53) we find that w1 ∼ (v0cj 2 + χ0 )( 1− cj|y| ) as |y| → ∞. (4.135) Substituting this into the matching condition (4.131) we find that v2 ∼ − (v0cj 2 + χ0 ) cj |x− xj | as x→ xj . (4.136) Thus, we must solve for v2 from (4.116) and (4.136). Proceeding as we did when 114 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere finding v1, we rewrite the boundary condition of (4.116) as ∂rv1 = −2pi N∑ j=1 (v0cj 2 + χ0 ) cj δ(ϕ− ϕj)δ(φ− φj) sinϕj on r = 1. (4.137) We use a solvability condition to find χ0. Applying the Divergence Theorem to (4.116) and using the boundary condition (4.137) we find that χ0 = − v0 ∑N j=1 c 2 j 2 ∑N j=1 cj . (4.138) We can write the solution to v2 as v2 ∼ −2pi N∑ i=1 (v0ci 2 + χ0 ) ciGs(x,xi) + χ2. (4.139) At this stage, we have the following outer expansion for the MFPT v ∼ v0 ε + χ0 log (ε 2 ) + χ1 − 2piv0 N∑ i=1 ciGs(x,xi) + ε log (ε 2 ) v2 + εv3, (4.140) where v0 is given by (4.127), χ0 is given by (4.138) and v2 is given by (4.139). The inner solution is w ∼ v0 ε (1− wc) + log (ε 2 )(v0cj 2 + χ0 ) (1− wc) + w2 + · · · , (4.141) where wc satisfies (4.51). Now, if we consider the matching condition, (4.131), with (4.140) and (4.141) substituted, we find that v2 contributes an unmatched term of O ( ε2 log2 ( ε 2 )) . This term can be ignored since the O(1) unmatched terms dominate. Thus, we must have that w2 ∼ Bj + χ1 + v0cj2 log (Λ + |y|) as |y| → ∞, (4.142) so that all the necessary terms are matched. Recall that w2 satisfies (4.121). We solve this problem for w2 by superposition. We let w2 ∼ (Bj + χ1) (1− wc) + v0cj2 w2p, . (4.143) 115 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere Here wc is the homogeneous solution satisfying (4.51) with the behaviour as |y| → ∞, given by (4.53). That is wc ∼ cj|y| + O ( 1 |y|3 ) as |y| → ∞. Further- more, w2p is the particular solution satisfying (4.121) with the far-field behaviour given by (4.81). By using the far-field behaviour of wc, (4.53), combined with a few simple estimates, we find that the far-field behaviour of w2 is w2 ∼ (Bj + χ1) ( 1− cj|y| ) + v0cj 2 log(|y|+ Λ). (4.144) From matching we must then conclude that v3 ∼ −cj (Bj + χ1)|x− xj | as x→ xj . (4.145) We must solve for v3 from (4.117) and (4.145). We rewrite the boundary con- dition of (4.117) as ∂rv3 = −2pi N∑ j=1 cj (Bj + χ1) δ(ϕ− ϕj)δ(φ− φj) sinϕj on r = 1. (4.146) We apply the Divergence Theorem to (4.117) and use the boundary condition (4.146). We find that χ1 = − ∑N j=1 cjBj∑N j=1 cj , (4.147) where Bj is given by (4.132). We let R = − 920pi as in (4.90). Substituting for v0 from (4.127) we find that χ1 = 2piv0∑N j=1 cj  N∑ j=1 c2jR+ cj N∑ i=1,i6=j ciGs(xj ,xi)  . (4.148) Substituting for χ1 and v2 into (4.140), we find that the outer solution for the MFPT is 116 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere v(x) = |Ω| 2piεD ∑N j=1 cj 1− ε 2 log (ε 2 ) (∑N j=1 c 2 j ) ∑N j=1 cj − 2piε N∑ i=1 ciG(x,xi)  + |Ω| 2piεD ∑N j=1 cj  2piε∑N j=1 cj  N∑ j=1 c2jR+ cj N∑ i=1,i 6=j ciGs(xj ,xi)  + O(ε2 log ε). (4.149) Notice that (∑N j=1 ( c2jR+ cj ∑N i=1,i6=j ciGs(xj ,xi) )) is the same expression as p(x1,x2, ...,xN ) given by (4.95). Thus, we have recovered the result that was derived in the previous section for the MFPT with N holes on the boundary, Proposition 4.4, (4.104). Thus, by simplifying (4.149) in the same manner as we did (4.104), we recover Corollary 4.5, 4.6 and 4.7 for N identical holes, one hole and one hole of radius a = 1. Thus, the two methods for deriving the MFPT agree exactly. Lastly, we compare the asymptotic results with the numerical results solving (4.112) computed using COMSOL [15] by R. Straube [28]. We use the expression for the average MFPT for N identical holes, (4.107). We set a = 1, so that we have N identical circular windows of radius ε equidistantly placed on the surface of the sphere. We let v̄2 be the two-term asymptotic result obtained be omitting the O(ε) terms in (4.107), v̄3 be the full three-term asymptotic result (4.107) and v̄n the full numerical result. We tabulate the results for N = 1, 2 and 4 in Table 4.1 below. N = 1 N = 2 N = 4 ε v̄2 v̄3 v̄n v̄2 v̄3 v̄n v̄2 v̄3 v̄n 0.02 53.89 53.29 52.81 26.95 26.40 26.12 13.47 13.10 12.99 0.05 22.17 21.57 21.35 11.09 10.54 10.43 5.54 5.17 5.12 0.10 11.47 10.87 10.78 5.74 5.19 5.14 2.87 2.50 2.47 0.20 6.00 5.40 5.36 3.00 2.45 2.44 1.50 1.13 1.13 0.50 2.56 1.95 1.96 1.28 0.73 0.70 0.64 0.27 0.30 Table 4.1: Comparison of v̄2, v̄3 and v̄n for various values of ε, N = 1, 2 and 4. We see good agreement between the numerical results and the asymptotic results, especially when using the full three-term asymptotic result, v̄3. 117 Chapter 4. The Narrow Escape Problem in Three Dimensions - The Sphere 4.4 Discussion In this Chapter we aimed to find the MFPT in a unit sphere. To this end, we began with a derivation of the Neumann Green’s function for a sphere of radius R. As in two dimensions, we transformed the problem for the MFPT into an eigenvalue problem. We solved for the principle eigenvalue, λ0, and the principle eigenfunction, φ0 for N holes of differing radius, using the method of matched asymptotic expansions. These results are given by (4.88) and (4.89). We then simplified the expressions for λ0 and φ0 for the case ofN identical holes, given by (4.97) and (4.98). We then found the MFPT in a unit sphere with N absorbing patches of differing radii on the boundary given by (4.104). We found a simpler expression in the case of N identical holes, given by (4.106). In the case of one hole in a unit sphere we were able to compare our result, (4.110), to that derived by Singer et al. in [25], (4.111). Our result agrees asymptotically with (4.111) and determines explicitly the O(ε) term. Our result, (4.104) generalizes the result obtained in [25]. Furthermore, the result contains more terms in the asymptotic expansion for the MFPT, which depend on the initial position x, the location of the patches, xj , j = 1, .., N , and the radii of the holes. Next, we derived the MFPT directly using the method of matched asymp- totic expansions. We found that the result for the MFPT derived directly, (4.149), agrees exactly with (4.104). In addition, we compared our asymptotic result for the average MFPT for identical holes, (4.107), with numerical results solving (4.112). We considered N equidistantly placed identical holes of radius ε for N = 1, 2 and 4. The numerical results were computed using COMSOL [15] by R. Straube [28]. We found good agreement with the numerical results when using the three-term expression for the average MFPT, (4.107). The results are tabulated in Table 4.1. It is worth noting that in the case of N circular absorbing windows of com- mon radius ε, that the MFPT is minimized in the limit as ε → 0 at the con- figuration {x1,x2, ...,xN} that minimizes the discrete sum ∑N i=1 ∑N j=1,j 6=iHij . This also maximizes the eigenvalue λ0, thus minimizing the MFPT. Recall that our expressions for the MFPT are valid in the outer region and are not valid in the vicinity of the absorbing windows. That is, our results for the MFPT hold when |x− xj | >> O(ε), for each absorbing window j. 118 Chapter 5 Conclusion In this thesis we focused on the mean first passage time, which is the mean time a Brownian particle takes to escape from a domain Ω ∈ Rd, d = 2, 3, whose boundary is reflecting, ∂Ωr, except for a small absorbing window, ∂Ωa. We let ε = |∂Ωa|. This is known as the narrow escape problem, a singular perturbation problem. As we have discussed, the mean first passage time has applications in many fields including electrostatics, biology, and finance to name a few. We discussed specific applications in biology and relevant results found in [25], [5], [23], [24]. We compared our results with the work of these authors. As it was shown in Chapter 1, the MFPT is defined in terms of a conditional expectation, (1.10), dependent on the initial position x, which satisfies a mixed boundary value problem for the Poisson equation, (1.26)-(1.28). The focus of this thesis was to find the MFPT in various two-dimensional domains and in a unit sphere. In Chapter 2, we transformed the problem for the MFPT, (2.3)-(2.5), into an eigenvalue problem by writing the MFPT in terms of a complete set of eigen- functions. We found that the MFPT to leading order is inversely proportional to the principal eigenvalue in the limit that ε → 0. This agrees with the lit- erature. Recall that the absorbing arc ∂Ωa is a perturbation that is small in extent but large in magnitude in a localized region. Thus, our methodology was to find the principal eigenvalue and eigenfunction, λ0 and φ0, using the method of matched asymptotic expansions, a singular perturbation technique. We found the two-term asymptotic behaviour of the principal eigenvalue and eigenfunction, in the limit that ε → 0. We then proceeded to find the MFPT. We determined the two-term asymptotic behaviour of the MFPT in an arbitrary two-dimensional domain with one hole on the boundary in the limit as ε → 0, given by (2.53). The leading order term agrees with that derived by Singer et. al in [25]. Our result provides more information since it is dependent on the initial position, x, the location of the hole, x0, the length of the hole through d and the shape of the domain. 119 Chapter 5. Conclusion We extended our results to include N absorbing holes on the boundary. We derived results for the principal eigenvalue and eigenfunction with N holes on the boundary and N identical holes on the boundary. In the case of N identical holes, we derived another result that exploited the special property of a cyclic matrix in the matrix system (2.85). We proceeded to derive results for the MFPT in these cases, given by (2.99), (2.101) and (2.103). We found in the case of N identical holes that the leading order term was proportional to 1N , which agrees with a result found by Holcman and Schuss in [5]. With these results in hand, we proceeded to investigate the MFPT in specific geometries, namely the unit disk and unit square in Chapter 3. We compared our results for a unit disk with one hole on the boundary with those derived by Singer et al. in [23]. We found good agreement with their results. Furthermore, we investigated the behaviour of the MFPT as the initial position moves towards the hole on the boundary. We found that the MFPT decreases in this case. We also investigated two holes on the boundary, and found that the MFPT reaches a minimum when the holes are furthest apart. These results are shown in Figures 3.1 and 3.2. With N symmetrically located holes on the boundary of the unit disk, we found a special result. In the limit as N → ∞, v(x) → 14D when x is placed at the centre of the disk, with log 1ε >> N . We derived the Green’s function for a unit square. We used this result to find the MFPT. In the case where the initial position moves towards the hole, the MFPT decreases. When the initial position is fixed and the hole moves along one side of the unit square, we find that the MFPT is a minimum where the distance between the hole and the initial position is a minimum. These results are depicted in Figures 3.4 and 3.5. We then considered two holes on the boundary, one fixed, and one moved. The results are illustrated in Figure 3.6 and 3.7. We found that the MFPT is not a minimum where the distance between the initial position and the moving arc is a minimum along sides 1 and 2. This is a result of the interaction with the second arc which is fixed at the midpoint of side 1. However, along side 3, the MFPT is a minimum when the two holes are furthest apart, and the distance from the moving arc and the initial position is a minimum. This is a consequence of the symmetry of the configuration with the moving exit window along side 3. Notice that our expression for the MFPT is not uniformly valid at the corners of the unit square, since the boundary is non-smooth at these points. By considering the expression for the Green’s function at the corner, we see that the singularity at the corner is twice as large as a singularity along the smooth portion of the 120 Chapter 5. Conclusion boundary. We derived an expression for the MFPT that is valid at the corner x0 = (0, 0) given by (3.55). As we have seen from the general expressions for v(x), the Neumann Green’s function plays an important role. To investigate different geometries, we must find the Neumann Green’s function. We proposed a numerical technique, the boundary element method, to find the Neumann Green’s function. We showed that the method gave accurate results in the case of the unit circle. We then attempted to use the method to find the Green’s function in an ellipse. We found that R(x,x0) = 0.1291 for all x ∈ Ω and x0 ∈ ∂Ω. Thus, we concluded that R was a constant. This implies that R(x0,x0) = 0.1291. We plotted the behaviour of the Green’s function for x at the origin and x0 moving along the boundary in Figure 3.8. We attempted to find R(x0,x0) numerically. However, the numerical method did not converge in this case. We require many more mesh points to attain the desired accuracy. At the present time, we do not have the computational ability to reach the desired accuracy of four decimal places. We propose using an integration technique that converges quicker, or Richardson’s extrapolation to attain a more accurate answer. In the case of perturbed circular domains, we used an analytical approach, derived by T. Kolokolnikov [7], to find the Green’s function. The result, given by (3.141), is unique up to a constant. To find the constant, we must impose the integral condition ∫ Ω G(x,x0)dx = 0. This analytic approach allows for a comparison with the numerical approach. We considered an example of a particular domain with σ given by (3.142), which is depicted in Figure 3.11 (a). We found that the regular part of the Green’s function, ρ, (3.145), has a local maximum where the curvature, κ, (3.143), has a local minimum at θ = pi for a particular parameter regime, a ∈ ( 16 , 415 ). We plotted the the curvature and ρ in Figure 3.11 (b) for a = 0.2 and ε = 0.1. We then used our numerical method to plot Rm(x0,x0) versus θ0 in Figure 3.12, for the shape given by (3.142), for N = 600 and N = 2400 mesh points along the boundary. By comparing these curves we see that the numerical method is converging extremely slowly since the two curves are similar. The numerical results qualitatively agree with the analytical result in that Rm(x0,x0) has a maximum at θ = pi. However, the scaling is incorrect in comparison to the analytical result. To obtain a more accurate numerical result, or at least one that converges faster, we must calculate the integral (3.78) more accurately instead of using the midpoint rule. In three-dimensions, the statement of the narrow escape problem is no dif- ferent from the statement in two dimensions. Thus, the MFPT must satisfy 121 Chapter 5. Conclusion (1.26)-(1.28). As in two-dimensions, we transform the problem into an eigen- value problem, for the principal eigenvalue and eigenfunction. To proceed by the method of matched asymptotic expansions, we solved for the Green’s function first. This is different to the approach in two dimensions. In three-dimensions, one must find the Green’s function first, before finding the MFPT. We found the Neumann Green’s function for a sphere of radius R. We found the principal eigenvalue and eigenfunction in a unit sphere with N holes on the boundary given by (4.88) and (4.89). We derived simplified results in the case of N iden- tical holes. We found the MFPT for N absorbing patches of differing radii on the boundary, (4.104). We simplified this result for the case of N identical patches. The result is given by (4.106). For one hole on the boundary of radius a the MFPT is given by (4.108). In the case of one hole with radius a = 1 in a unit sphere we were able to compare our result, (4.110), to that derived by Singer et al. in [25], (4.111). Our result agrees asymptotically with (4.111) and determines explicitly the O(ε) term. Our result, (4.104), generalizes the result obtained in [25]. Furthermore, the result contains more terms in the asymptotic expansion for the MFPT, which depend on the initial position x, the location of the patches, xj , j = 1, .., N , and the radii of the holes. Next, we derived the MFPT directly using the method of matched asymp- totic expansions. We found that the result for the MFPT derived directly, (4.149), agrees exactly with (4.104). We also compared our asymptotic result for the average MFPT for N identical holes of radius ε on the boundary, (4.107), with numerical results solving (4.112). These numerical results were computed for N = 1, 2 and 4 identical holes placed equidistantly on the surface of the sphere using COMSOL [15] by R. Straube [28]. We found good agreement with the numerical results when comparing them to the three-term asymptotic result (4.107). The results are tabulated in Table 4.1. It is worth noting that in the case of N circular absorbing windows of com- mon radius ε, that the MFPT is minimized in the limit as ε → 0 at the con- figuration {x1,x2, ...,xN} that minimizes the discrete sum ∑N i=1 ∑N j=1,j 6=iHij . This also maximizes the eigenvalue λ0, thus minimizing the MFPT. We note that our expressions for the MFPT are valid in the outer region and are not valid in the vicinity of the absorbing windows. That is, our results for the MFPT hold when |x− xj | >> O(ε), for each absorbing window j. We would like to extend the framework introduced in Chapter 4 to include an arbitrary three-dimensional domain. It is an open problem to find the MFPT and the Green’s function in an arbitrary three-dimensional domain. 122 Chapter 5. Conclusion We would like to improve the convergence of the boundary element method introduced in Chapter 3 when finding R(x0,x0). As mentioned, we could cal- culate the integrals more accurately using gaussian quadrature. Alternatively, we could find the error term for the technique and attempt Richardson extrapo- lation to find more accurate answers for R(x0,x0). Lastly, we could change the elements for the method by using line segments instead of circular arcs. With improved convergence, we could use the technique to find the Neumann Green’s function for domains with smooth boundaries more accurately. Finally, the related problem for the Dwell time would be an extension of the work presented in this thesis. 123 Bibliography [1] P.C. Bressloff and B.A. Earnshaw. Diffusion-trapping model of receptor trafficking in dendrites. Physical Review E, 75:041915–1 – 041915–7, 2007. [2] P.C. Bressloff, B.A. Earnshaw, and M.J. Ward. Diffusion of protein recep- tors on a cylindrical dendritic membrane with partially absorbing traps. SIAM Journal on Applied Mathematics, 68(5):1223–1246, 2008. [3] R.L. Burden and J. Douglas Faires. Numerical Analysis. Brooks/Cole Publishing Company, 7th edition, 2001. [4] I.V. Grigoriev, Y.A. Makhnovskii, A.M. Berezhkovskii, and V.Y. Zitser- man. Kinetics of escape through a small hole. Journal of Chemical Physics, 116(22):9574–9577, 2002. [5] D. Holcman and Z. Schuss. 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SIAM Journal of Applied Mathematics, 60(5):1767–1788, 2000. [32] S. Torquato, I.C. Kim, and D. Cule. Effective conductivity, dielectric constant, and diffusion coefficient of digitized composite media via first- passage-time equations. Journal of Applied Physics, 85(3):1560–1571, 1999. [33] M.J. Ward, W.D. Henshaw, and J.B. Keller. Summing logarithmic ex- pansions for singularly perturbed eigenvalue problems. SIAM Journal on Applied Mathematics, 53(3):799–828, 1993. 126 Bibliography [34] M.J. Ward and J.B. Keller. Strong localized perturbations of eigenvalue problems. SIAM Journal on Applied Mathematics, 53(3):770–798, 1993. 127 Appendix A Derivation of d We want to solve for vc(y), where vc satisfies ∆yvc = 0, y /∈ ∂Ω0, vc = 0, y ∈ ∂Ω0, vc → log |y| as |y| → ∞. (A.1) The problem for vc(y), (2.28), has a unique solution and the behaviour at ∞ is [8] vc(y) ∼ log |y| − log d+ p.y|y|2 . (A.2) where d, the logarithmic capacitance and p = (p1, p2), the dipole vector, are determined from the length of the hole, ∂Ωa. In our case, where the length of the hole is |∂Ωa| = 2ε, d = 12 . This is obtained by a special solution in terms of elliptic cylindrical coordinates. We present a derivation here. In elliptic cylindrical polar coordinates y = (x, y) transforms as x = R cosh ξ cosΛ, y = R sinh ξ sinΛ, (A.3) with 0 ≤ Λ < 2pi. Furthermore we find for a function f(x, y) that fxx+ fyy = fξξ + fΛΛ = 0. Under these coordinates we have that x2 R2 cosh2 ξ + y2 R2 sinh2 ξ = 1. (A.4) Thus, curves of constant ξ form ellipses. We choose R cosh ξ0 = a, R sin ξ0 = b. (A.5) 128 Appendix A. Derivation of d Therefore, R = (a2 − b2)1/2, ξ0 = 12 log ( 1 + b/a 1− b/a ) for a > b. (A.6) We have mapped the ellipse onto an infinite sheet in the Λ− ξ plane. To solve the problem (A.1) we must solve fξξ + fΛΛ = 0, ξ > ξ0, 0 ≤ Λ < 2pi, f = 0 on ξ = ξ0, f ∼ log |r| as ξ →∞, (A.7) where r = √ x2 + y2 and f is 2pi periodic in Λ. The solution is f = C(ξ − ξ0), where C is a constant. As ξ →∞ the coordinates (A.3) become x = Reξ 2 cosΛ, y = Reξ 2 sinΛ. (A.8) In this limit r = Re ξ 2 and therefore ξ = log r + log 2 R . We find the behaviour of f as ξ → ∞ by substituting for ξ in this limit and ξ0 and R from (A.6). We find that f = C ( log r + log 2 R − 1 2 log ( 1 + b/a 1− b/a )) , = C ( log r − log ( a+ b 2 )) . (A.9) Thus, to obtain f ∼ log r at ∞, C = 1. By setting d = a+b2 we obtain the behaviour (A.2) at∞. In our case b = 0, thus for an arc of length 2a, d = a2 . In general, for an arc of length l, d = l4 . The solution for vc is given by C(ξ − ξ0), where one must map back to Cartesian coordinates. The far-field behavior of vc is given by (A.2). 129 Appendix B Far-Field Behaviour of ψc We consider ψcΛΛ + ψcs1s1 + ψcs2s2 = 0,Λ ≥ 0, −∞ < s1, s2 <∞, (B.1) ∂Λψc = 0, Λ = 0, s21 + s 2 2 ≥ a2j , (B.2) ψc = 1, Λ = 0, s21 + s 2 2 ≤ a2j . (B.3) The exact solution from [6] is ψc = 2 pi ∫ ∞ 0 sinµ µ e−µΛ/ajJ0 ( µσ aj ) dµ, (B.4) where σ = (s21+s 2 2) 1/2. Now, we want to derive the asymptotics of this solution as |y| = (Λ2 + s21 + s22)1/2 →∞. We introduce r = µσaj . Then ψc = 2aj piσ ∫ ∞ 0 sin (ajr/σ) ajr/σ e−γrJ0(r)dr, (B.5) where γ = Λ/σ. We would like to determine the asymptotic behavoiur as σ →∞, but with γ fixed. As x→ 0 we know that sin xx ∼ 1− x 2 6 + · · · . Thus, ψc ∼ 2aj piσ ( I0 − a2j 6σ2 I1 + · · · ) , (B.6) where I0 = ∫ ∞ 0 e−γrJ0(r)dr, (B.7) I1 = ∫ ∞ 0 r2e−γrJ0(r)dr. (B.8) 130 Appendix B. Far-Field Behaviour of ψc It follows that I0 is the Laplace transform of J0(r). Hence I0 = 1√ 1 + γ2 . (B.9) Also I1 = d 2 dγ2 I0(γ). We find that ψc ∼ 2aj pi [ I0 σ − a 2 j 6σ3 I1 + · · · ] , (B.10) where I0 σ = 1√ Λ2 + σ2 , (B.11) I1 σ3 = − 1 (Λ2 + σ2)3/2 + 3Λ2 (Λ2 + σ2)5/2 . (B.12) Defining |y| = (Λ2 + s21 + s22)1/2 we find that ψc ∼ 2aj pi ( 1 |y| + a2j 6 ( 1 |y|3 − 3Λ2 |y|5 )) as |y| → ∞. (B.13) This is the asymptotic behaviour at ∞ uniform in Λ, s1 and s2. 131 Appendix C Matlab Code Here we give the code for the boundary element method for an ellipse. We first give the code for finding R(x,x0) and then the code for finding R(x0,x0). Lastly we give the code for finding the integral of R(x,x0) over Ω. clear all; N=640; dth = 2*pi/N; th = 0:dth:2*pi; s = zeros(N,1); %arclength vector m = 5000; %number of steps in the trapezoidal rule for the arclength a = 2; b = 1; sum2 = 0; area = pi*a*b; for j = 1:N lowerth = th(j); upperth = th(j+1); ht = (upperth-lowerth)/m; sum2 = 0; for k = 0:m thk = lowerth + k*ht; if k == 0 sum2 = ht/2*sqrt((a^6*b^2*(sin(thk))^2+a^2*b^ 6*(cos(thk))^2)/((a^2*(sin(thk))^2 + b^2*(cos(thk))^2)^3)) + sum2; elseif k == m sum2 = ht/2*sqrt((a^6*b^2*(sin(thk))^2+a^2*b^ 6*(cos(thk))^2)/((a^2*(sin(thk))^2 + b^2*(cos(thk))^2)^3)) + sum2; else sum2 = ht*sqrt((a^6*b^2*(sin(thk))^2+a^2*b^6* (cos(thk))^2)/((a^2*(sin(thk))^2 + b^2*(cos(thk))^2)^3)) + sum2; end; end; 132 Appendix C. Matlab Code s(j) = sum2; end; % s is the vector of lengths of each segment % Setting up the midpoints for j = 1:N nth(j) = (0.5*(th(j)+th(j+1))); %midpoints end; r = (a*b)./(a^2.*sin(nth).^2 + b^2.*cos(nth).^2).^(1/2); xm =r.*cos(nth); ym = r.*sin(nth); plot(xm,ym,’*r’); %plot of meshpoints x = 1; y = 0.5; c = -a*b*(a^2-b^2); d = a^2.*sin(nth).^2 + b^2.*cos(nth).^2; rdr = c*sin(nth).*cos(nth).*(a^2.*sin(nth).^2+b^2.*cos(nth).^2). ^(-3/2); %first derivative rddr = c.*((cos(nth).^2-sin(nth).^2).*(d.^(-3/2))-3*(a^2-b^2).* (sin(nth).^2).*(cos(nth).^2).*d.^(-5/2)); %second derivative k = (r.^2+2*rdr.^2-r.*rddr)./((r.^2+ rdr.^2).^(3/2)); %curvature nx = (1./sqrt(rdr.^2+r.^2)).*(xm+rdr.*sin(nth)); ny = (1./sqrt(rdr.^2+r.^2)).*(ym-rdr.*cos(nth)); f = inline(’sqrt((x1-x2)^2+(y1-y2)^2)’); for j = 1:N bb(j) = 0; for i = 1:N if i == j A(i,j) = 1-(s(i)*k(i))/(4*pi); bb(j) = s(i)/(4*pi*area)*log(f(x,xm(i),y,ym(i))) *(xm(i)*nx(i)+ym(i)*ny(i))+bb(j); else A(i,j) = -s(i)/(2*pi*f(xm(i),xm(j),ym(i),ym(j))^2)* (nx(i)*(xm(i)-xm(j))+ny(i)*(ym(i)-ym(j))); bb(j) = s(i)/(4*pi*area)*log(f(x,xm(i),y,ym(i))) *(xm(i)*nx(i)+ym(i)*ny(i))+bb(j); end; end; end; R = A’\bb’; 133 Appendix C. Matlab Code Now we give the code for the calculation of R(x0,x0). clear all; clf N=2400; dth = 2*pi/N; th = 0:dth:2*pi; s = zeros(N,1); m = 5000; a = 2; b = 1; area = pi*a*b; sum2 = 0; delta = 0; for j = 1:N lowerth = th(j); upperth = th(j+1); ht = (upperth-lowerth)/m; sum2 = 0; for k = 0:m thk = lowerth + k*ht; if k == 0 sum2 = ht/2*sqrt((a^6*b^2*(sin(thk))^2+a^2* b^6*(cos(thk))^2)/((a^2*(sin(thk))^2 + b^2*(cos(thk))^2)^3)) + sum2; elseif k == m sum2 = ht/2*sqrt((a^6*b^2*(sin(thk))^2+a^2* b^6*(cos(thk))^2)/((a^2*(sin(thk))^2 + b^2*(cos(thk))^2)^3)) + sum2; else sum2 = ht*sqrt((a^6*b^2*(sin(thk))^2+a^2* b^6*(cos(thk))^2)/((a^2*(sin(thk))^2 + b^2*(cos(thk))^2)^3)) + sum2; end; end; s(j) = sum2; end; % s is the vector of lengths of each segment for j = 1:N nth(j) = (0.5*(th(j)+th(j+1))); %midpoints end; r = (a*b)./(a^2.*sin(nth).^2 + b^2.*cos(nth).^2).^(1/2); xm =r.*cos(nth); ym = r.*sin(nth); plot(xm,ym,’.’); c = -a*b*(a^2-b^2); d = a^2.*sin(nth).^2 + b^2.*cos(nth).^2; rdr = c*sin(nth).*cos(nth).*(a^2.*sin(nth).^2+b^2.*cos(nth).^2).^(-3/2); rddr = c.*((cos(nth).^2-sin(nth).^2).*d.^(-3/2)-3*(a^2-b^2). *(sin(nth).^2).*(cos(nth).^2).*d.^(-5/2)); 134 Appendix C. Matlab Code k = (r.^2+2*rdr.^2-r.*rddr)./((r.^2+ rdr.^2).^(3/2)); %curvature nx = (1./sqrt(rdr.^2+r.^2)).*(xm+rdr.*sin(nth)); ny = (1./sqrt(rdr.^2+r.^2)).*(ym-rdr.*cos(nth)); f = inline(’sqrt((x1-x2)^2+(y1-y2)^2)’); Ns = N; for ks = 1:Ns x = xm(ks); y = ym(ks); for j = 1:N bb(j) = 0; for i = 1:N if i == j A(j,i) = 1-(s(i)*k(i))/(4*pi); if (x == xm(i))& (y == ym(i)) c = 0; m = (k(i)*s(i)-delta)/2; bb(j) = 1/(8*pi*area*k(i)^2)*(s(i)*k(i)*log(2/k(i)^2 )+2*(s(i)*k(i))*log(s(i)*k(i)/2) - s(i)*k(i)*log(2) -2*s(i)*k(i)+m*(log(2*(1-cos(c+m/sqrt(3)))/ ((c+m/sqrt(3))^2))+log(2*(1-cos(c-m/sqrt(3)))/ ((c-m/sqrt(3))^2)))) + bb(j); else %THIS INTEGRAL IS CALCULATED IN THE GLOBAL COORD %SYSTEM WITH THE MIDPOINT RULE bb(j) = s(i)/(4*pi*area)*0.5*log((x-xm(i))^2+ (y-ym(i))^2)*(xm(i)*nx(i)+ym(i)*ny(i))+bb(j); end; else %EXPRESSION USING GLOBAL COORDINATE SYSTEM AND MIDPOINT %RULE A(j,i) = -s(i)/(2*pi*((xm(i)-xm(j))^2+(ym(i)-ym(j))^2) )*(nx(i)*(xm(i)-xm(j))+ny(i)*(ym(i)-ym(j))); if (x == xm(i))& (y == ym(i)) c=0; m = (k(i)*s(i)-delta)/2; bb(j) = 1/(8*pi*area*k(i)^2)*(s(i)*k(i)*log(2/k(i)^2 135 Appendix C. Matlab Code )+2*(s(i)*k(i))*log(s(i)*k(i)/2) - s(i)*k(i)*log(2) -2*s(i)*k(i)+m*(log(2*(1-cos(c+m/sqrt(3)))/ ((c+m/sqrt(3))^2))+log(2*(1-cos(c-m/sqrt(3)))/ ((c-m/sqrt(3))^2)))) + bb(j); else %THIS INTEGRAL IS CALCULATED IN THE GLOBAL COORD %SYSTEM WITH THE MIDPOINT RULE bb(j) = s(i)/(4*pi*area)*0.5*log((x-xm(i))^2 +(y-ym(i))^2)*(xm(i)*nx(i)+ym(i)*ny(i))+bb(j); end; end; end; end; R = A\bb’; finalR(1,ks) = R(1,1); %finalR(:,ks) = R; end; %ks min = 1; max = 0; for k = 1:N if finalR(1,k) < min min = finalR(1,k); end; if finalR(1,k) > max max = finalR(1,k); end; end; Finally, we give the code for calculating ∫ Ω R(x0,x0)dx using the midpoint rule. clear all; %changed to use the midpoint rule N=80; %x_0’s, the integral should be independent of this m=5000; %integration steps in theta for finding the arclength dth = 2*pi/N; th = 0:dth:2*pi; s = zeros(N,1); a = 2; b = 1; sum2 = 0; area = pi*a*b; for j = 1:N lowerth = th(j); upperth = th(j+1); ht = (upperth-lowerth)/m; 136 Appendix C. Matlab Code sum2 = 0; for k = 0:m thk = lowerth + k*ht; if k == 0 sum2 = ht/2*sqrt((a^6*b^2*(sin(thk))^2+a^2* b^6*(cos(thk))^2)/((a^2*(sin(thk))^2 + b^2*(cos(thk))^2)^3)) + sum2; elseif k == m sum2 = ht/2*sqrt((a^6*b^2*(sin(thk))^2+a^2* b^6*(cos(thk))^2)/((a^2*(sin(thk))^2 + b^2*(cos(thk))^2)^3)) + sum2; else sum2 = ht*sqrt((a^6*b^2*(sin(thk))^2+a^2* b^6*(cos(thk))^2)/((a^2*(sin(thk))^2 + b^2*(cos(thk))^2)^3)) + sum2; end; end; s(j) = sum2; end; % s is the vector of lengths of each segment for j = 1:N nth(j) = (0.5*(th(j)+th(j+1))); %midpoints/singular points end; r = (a*b)./(a^2.*sin(nth).^2 + b^2.*cos(nth).^2).^(1/2); xm =r.*cos(nth); ym = r.*sin(nth); plot(xm,ym,’*’); hold on; c = -a*b*(a^2-b^2); d = a^2.*sin(nth).^2 + b^2.*cos(nth).^2; rdr = c*sin(nth).*cos(nth).*(a^2.*sin(nth).^2+b^2.*cos(nth).^2).^(-3/2); rddr = c.*((cos(nth).^2-sin(nth).^2).*d.^(-3/2)-3*(a^2-b^2). *(sin(nth).^2).*(cos(nth).^2).*d.^(-5/2)); kr = (r.^2+2*rdr.^2-r.*rddr)./((r.^2+ rdr.^2).^(3/2)); %curvature nx = (1./sqrt(rdr.^2+r.^2)).*(xm+rdr.*sin(nth)); ny = (1./sqrt(rdr.^2+r.^2)).*(ym-rdr.*cos(nth)); %this sets up the integration variables over x n = 100; ht = (2*pi)/n; k = 1:1:n; tk = k.*ht; uppert = (a*b)./((a^2.*(sin(tk)).^2 + b^2.*(cos(tk)).^2).^(1/2)); hr =(uppert); f = inline(’sqrt((x1-x2)^2+(y1-y2)^2)’); %we have separated everything out into integrals 137 Appendix C. Matlab Code for k = 1:n thk = tk(k); h = hr(k); % this is the length of the interval, b-a x = 0.5*h*cos(thk); y = 0.5*h*sin(thk); plot(x,y,’.r’); hold on; for j = 1:N bb(j) = 0; for i = 1:N if i == j A(i,j) = 1-(s(i)*kr(i))/(4*pi); bb(j) = s(i)/(4*pi*area)*log(f(x,xm(i),y,ym(i))) *(xm(i)*nx(i)+ym(i)*ny(i))+bb(j); else A(i,j) = -s(i)/(2*pi*f(xm(i),xm(j),ym(i),ym(j))^2) *(nx(i)*(xm(i)-xm(j))+ny(i)*(ym(i)-ym(j))); bb(j) = s(i)/(4*pi*area)*log(f(x,xm(i),y,ym(i))) *(xm(i)*nx(i)+ym(i)*ny(i))+bb(j); end; end; end; R(:,k) = A’\bb’; end; for ks = 1:N %this is the loop over the x0,y0’s %numerical integration sum3 = 0; sumt = 0; sumw = 0; for k = 1:n %(t index) h = hr(k); thk = tk(k); sumt = sumt + 0.5*ht*0.5*h^2*log((0.5*h*cos(thk)-xm(ks))^2 +(0.5*h*sin(thk)-ym(ks))^2); sum2 = 0.5*h^2*R(ks,k); %r index, midpoint rule sum3 = sum3 + ht*sum2; 138 Appendix C. Matlab Code sumw = sumw + 0.5*ht*h^2*(h^2/4+xm(ks)^2+ym(ks)^2); end; sumfinal(ks) = sum3; sum2final(ks) = sumt; sum3final(ks) = sumw; end; %for ks 139

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