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UBC Theses and Dissertations

Similarity solution of a Fokker-Planck equation with a moving, absorbing boundary Lee, Richard Tsan Ming 1980

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c. SIMILARITY SOLUTION OF A FOKKER-PLANCK EQUATION WITH A MOVING, ABSORBING BOUNDARY by RICHARD TSAN MING LEE B . S c , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES The I n s t i t u t e of A p p l i e d Mathematics and S t a t i s t i c s We a c c e p t t h i s t h e s i s as conforming t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September 1980 (c) R i c h a r d Tsan Ming Lee, 1980 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Mathematics The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date September 30> 1980, i i ABSTRACT A one parameter L i e group of t r a n s f o r m a t i o n s i s used t o d e r i v e a c l o s e d form s i m i l a r i t y s o l u t i o n of t h e e q u a t i o n : u t = u x x + (xu) x , - « > < x < r { t ) # t > 0 , ~ w i t h a ( - o»,t) = u ( r ( t ) , t ) = 0 , u(x,0) = <j>(x) , \ (*) where r (t) = Y 0 e* + Yt e * . ' A s i m p l e e x p r e s s i o n f o r t h e f i r s t passage t i m e d i s t r i b u t i o n of the g e n e r a l F o k k e r - P l a n c k (Kolraogorov forward) e q u a t i o n i n an i r r e g u l a r domain i s d e r i v e d , which i s s u b s e q u e n t l y used t o f i n d the f i r s t passage time d i s t r i b u t i o n s of f o u r e q u i v a l e n t problems. A new d i s t r i b u t i o n i s found. A s m a l l time a s y m p t o t i c e x p a n s i o n of the i n t e g r a l s o l u t i o n of {*) i s c a l c u l a t e d i n order t o be p i e c e d t o g e t h e r t o form the s o l u t i o n f o r a more g e n e r a l r (t) . Examining the f e a s i b i l i t y of t h i s method, we f i n d t h a t i t i s e q u i v a l e n t to a s i m p l e a p p l i c a t i o n of T a y l o r e x p a n s i o n s , and i t i s not b e t t e r t h a n t h e p o w e r f u l method of t r a n s f o r m i n g the i r r e g u l a r domain to a r e g u l a r one and a p p l y i n g some e x p l i c i t schemes. Convergence and S t a b i l i t y c r i t e r i a a re d e r i v e d f o r an e x p l i c i t method which admits an a r b i t r a y r (t) . i i i TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES •••••• • v LIST OF FIGURES v i ACKNOWLEDGEWfWT'. v i i i CHAPTER 0 INTRODUCTION 1 CHAPTER I SIMILARITY SOLUTIONS 6 I.U INTRODUCTION 6 1.1 THE PROBLEM 6 1.2 INVARIANCE OF THE EQUATION 7 1.3 INVARIANCE OF ALL BOUNDARY CONDITIONS 10 1.4 RELATION TO THE HEAT EQUATION .. 15 CHAPTER I I FIRST PASSAGE TIME DISTRIBUTION 22 I I . 0 INTRODUCTION 22 I I . 1 EQUIVALENCE TO u/^-«Jt = 0 WITH FIXED BOUNDARY 23 I I . 1 FIRST PASSAGE TIME PROBLEM 31 CHAPTER I I I ASYMPTOTIC AND NUMERICAL PROCEDURES 42 I I I . O INTRODUCTION 42 111.1 ASYMPTOTIC EXPANSION FOR SMALL TIME 45 111.2 A FINITE DIFFERENCE METHOD FOR ARBITRARY MOVING BOUNDARY 55 I I I . 3 COMPUTATIONS 58 CHAPTER IV DISCUSSION AND CONCLUSIONS 75 BIBLIOGRAPHY 82 APPENDIX V LIST OF TABLES Table 2. 1 C o n n e c t i o n s Between The F o k k e r - P l a n c k E q u a t i o n With Moving Boundary r U ) = Y. e * + Yi £ * And The D i f f u s i o n E q u a t i o n .26 T a b l e 3.1 a) Case Of Y „ = - 1 , Yx = 2, &t=1/200, r.-x,, = 1, Ay=0.1: Approximate S o l u t i o n ...69 T a b l e 3.1 b) Case Of Y» =-1, Yi = 2, At=1/200, r.-x#=1, Ay=0. 1: Exact S o l u t i o n . . . 7 0 Table 3.2 Case Of Y « = - 1 , Yx = 2, At=1/200, r e - x e = 1, Ay=0.2: Approximate S o l u t i o n ....71 T a b l e 3.3 Case Of Y 0 =-1, Yt = 2, At=1/200, r „ - X o = 1 , Ay=0.5: Approximate S o l u t i o n 71 Table 3.4 a) Case Of Y, =-9 r YA=10, At=1/200, r , - x . = 1, ay=0. 1: Approximate S o l u t i o n .72 Table 3.4 b) Case Of I. =-9, Yi=10, 6t=1/200, r o - x 0 = 1, Ay=0.1: Exact S o l u t i o n 73 T a b l e 3.4 c) Case Of Y„ =-9, Y i = 1 0 r At=1/200, r 0 - X o = 1 r Ay=0. 1: E r r o r • 74 v i LIST OF FIGURES F i g u r e 0,1 V a r i o u s Forms Of V a r y i n g T h r e s h o l d s 3 F i y u r e 2.1 a) Green's S o l u t i o n F o r r (t) = YC e* + Y, e"1 . Y»=0, Yi =1 (see Table 2.1) 29 b) Green's S o l u t i o n For r ( t ) = e " t In A Transformed Domain (see Table 2.1) 30 F i g u r e 2.2 a) Graphs Of X = nt> - Y, + Yi £ * , H I s The S i m i l a r i t y V a r i a b l e (see Chapter I) 37 b) Graphs Of 3 = fl«) ~ *Y.$ + Y» , ^ I s The S i m i l a r i t y V a r i a b l e (see Chapter I) 38 F i g u r e 3.1 Graphs Of f(ot,z) Showing Where The Major C o n t r i b u t i o n s Are L o c a t e d 47 F i g u r e 3.2 I l l u s t r a t i n g I n s t a b i l i t y Due To fit/ ( A x) 2> 1/2: a) Ay=0.1, 6t= 1/200: S t a b l e 60 b) Ay=0.1, At=1/195: U n s t a b l e , But I t I s Not A p p r a r e n t From The N u m e r i c a l R e s u l t I n T h i s Time I n t e r v a l 61 c) Ay=0.1, At=1/190: U n s t a b l e 62 F i g u r e 3.3 I l l u s t r a t i n g The P r o p a g a t i o n Of E r r o r : a) A y=0. 1, At=1/200 63 b) 6y=0.2, At= 1/200 64 c) Ay=0.5, At=1/200 65 F i g u r e 3.4 S o l u t i o n With Approximate Boundary C o n d i t i o n 4 * =0 6 7 F i g u r e 3.5 S o l u t i o n With Exact Boundary C o n d i t i o n s 68 Figure 4. 1 Graphs Of g t (t) As Y0 / A Parameter In The Threshold, Changes •••• 76 Figure 4.2 Graphs Of qi (t) As r 4 - x» , The Distance Between The I n i t i a l Membrane Potential And The Threshold At t = 0, changes 77 Figure 4.3 Graphs Of g 3 (?) As Y 0 Changes .78 v i i i ACKNOWLEDGEMENT I am g r a t e f u l t o my s u p e r v i s o r , Dr. George Bluman, f o r i n t r o d u c i n g me t o t h i s t o p i c and g u i d i n g me d u r i n g the c o u r s e of the r e s e a r c h . I would l i k e t o thank Drs. U r i Ascher and John Petkau f o r s u g g e s t i o n s which l e d t o the improvement of t h i s t h e s i s . Thanks a r e a l s o due t o Dr. Henry T u c k w e l l f o r d i r e c t i n g my a t t e n t i o n t o some u s e f u l r e f e r e n c e s . I am t h a n k f u l t o the Department o f Mathematics and t h e I n s t i t u t e o f A p p l i e d Mathematics and S t a t i s t i c s f o r the f i n a n c i a l s u p p o r t s , and t o a l l t h o s e whose encouragements make th e c o m p l e t i o n o f t h i s work p o s s i b l e . 1 CHAPTER 0 INTRODUCTION T h i s t h e s i s i s concerned w i t h s i m i l a r i t y s o l u t i o n s and the n u m e r i c a l c o m p u t a t i o n o f the s o l u t i o n t o a p a r t i c u l a r F o k k e r -Planck e g u a t i o n w i t h a moving, a b s o r b i n g boundary. The F o k k e r - P l a n c k e g u a t i o n a r i s e s v e r y o f t e n i n the p h y s i c a l and b i o l o g i c a l s c i e n c e s . I n g e n e r a l , i t governs t h e time e v o l u t i o n of a p r o b a b i l i t y d e n s i t y f u n c t i o n which a r i s e s from a d i f f u s i o n a p p r o x i m a t i o n o f some c o n t i n u o u s Markovian p r o c e s s . . E i n s t e i n £ 1 ] , Smoluchowski £ 2 ] , Uhlenbeck and O r n s t e i n [ 3 ] , Wang and Uhlenbeck [ 4 ] , Chandrasekhar [ 5 ] , Doob [ 6 ] and o t h e r s have c o n t r i b u t e d t o the d e r i v a t i o n and development of a F o k k e r - P l a n c k e g u a t i o n associated;:- w i t h t h e Brownian motion o f p a r t i c l e s i n a f o r c e f i e l d . The e g u a t i o n t a k e s the f o l l o w i n g form: Bluman [ 7 ] , Bluman and C o l e [ 8 ] o b t a i n e d s i m i l a r i t y s o l u t i o n s of t h i s e g u a t i o n f o r some f i x e d boundary c o n d i t i o n s . O r t h o g o n a l p o l y n o m i a l s o l u t i o n s can be d e r i v e d f o r some forms o f f{x ) £ 9 ] . . A F o k k e r - P l a n c k e g u a t i o n w i t h moving boundary (b=b(t)) appears i n a model f o r t h e f i r i n g of neurons £ 1 0 , 1 1 ] . A f t e r b e i n g s i m p l i f i e d , the e g u a t i o n has t h e form: VP _ V_p ^(*P> -oo < x < rct> , t >o t w i t h boundary c o n d i t i o n s : P ( - o o,t) = P ( r ( t ) , t ) = 0 and i n i t i a l c o n d i t i o n : P ( x , 0 ) = ^ ( x ) , 2 where P ( x , t ) = t r a n s i t i o n p r o b a b i l i t y d e n s i t y f o r the membrane p o t e n t i a l , which s a t i s f i e s t h e Chapman-Kolmogorov e q u a t i o n ; t = ti m e ; x = a measure of t h e membrane p o t e n t i a l ; r (t) = a measure of the t h r e s h o l d (see F i g u r e 0.1); t ( x ) = an a r b i t r a r y d e n s i t y f u n c t i o n , f o r example, i f x = x, a t t = 0, f (x) = £(x-x 0). The s t u d y o f t h i s e g u a t i o n and i t s a s s o c i a t e d f i r s t passage time problem w i l l c o n s t i t u t e t h e major p a r t s o f t h i s t h e s i s . . & number o f forms f o r r ( t ) have been suggested [ 1 2 , pp76-81, t h e c o r r e s p o n d i n g r e f e r e n c e s are a l s o g i v e n h e r e . ] . In t h e f o l l o w i n g l i s t , the s u b s c r i p t s i n 1rs It) i s r e f e r r e d t o the l a b e l S o f t h e graph i n F i g u r e 0.1.. OL ( i ) Hagiwara (1954), f h It) = C T , * > » [ a l s o i n 11, e q u a t i o n 4 ] ; ( i i ) B u l l e r , N i c h o l l s and Strom (1953), rfc(t> = ; ( i i i ) C a l v i n and St e v e n s (1965), Vt It) ~ ; (i v ) G e i s l e r and Goldb e r g (1966), 1-j Li.) = / oo i f o^t<«., V& e T i i f t > * • (v) Weiss (1964), r^(t) * <x! € In Chapter I we w i l l o b t a i n s i m i l a r i t y s o l u t i o n s f o r a two parameter f a m i l y of r (t) which i n c l u d e s (v) as a s p e c i a l case. r It) • . o — j i i i i i i i i i | i i i i i i i i i | i i i i i i i i i | i i i i i i i i i | i i i i i i i i i [ i i i i i i i i > | l i i i i i i i i | i i i i i 0 1 2 3 4 5 6 7 F i g . 0.1 Various Forms of Varying Thresholds 4 Then we t r a n s f o r m the e q u a t i o n t o the s i m p l e h e a t e q u a t i o n and c o n s i d e r the t r a n s f o r m e d boundary c o n d i t i o n s . C o n s e q u e n t l y , s e v e r a l c l a s s e s of moving b o u n d a r i e s d e r i v e d i n [ 8 ] can be used t o c o n s t r u c t s i m i l a r i t y s o l u t i o n s . We i l l u s t r a t e t h i s method by u s i n g the two parameter f a m i l y of r ( t ) . I n Chapter I I we e s t a b l i s h the c o n n e c t i o n between the O r n s t e i n - U h l e n b e c k p r o c e s s and the Wiener p r o c e s s , and s u b s e q u e n t l y r e l a t e t h e Green's f u n c t i o n s t o each o t h e r . The f i r s t passage time problem [13, 14] i s f o r m u l a t e d g e n e r a l l y and i t s p h y s i c a l i n t e r p r e t a t i o n g i v e n . T h i s f o r m u l a t i o n i s then used to f i n d the f i r s t passage t i m e d i s t r i b u t i o n of f o u r e q u i v a l e n t problems. The r e s u l t s of t h i s c h a p t e r demonstrate t h a t some of the models are a c t u a l l y e q u i v a l e n t when viewed i n the proper c o o r d i n a t e s . To our knowledge the form of the f i r s t passage t i m e d i s t r i b u t i o n i n the o r i g i n a l v a r i a b l e s (x, t) i s new. On t h e o t h e r hand the s p e c i a l c a s e , which c o r r e s p o n d s t o a one parameter f a m i l y of r (t) , i s w e l l known [11, 15, 16].. Chapter I I I i s devoted t o examining t h e f e a s i b i l i t y o f computing the s o l u t i o n u ( x , t ) f o r a r b i t r a r y moving b o u n d a r i e s by p i e c i n g t o g e t h e r t h e s m a l l time a s y m p t o t i c e x p a n s i o n s o f the i n t e g r a l s o l u t i o n s f o r t h e two parameter moving b o u n d a r i e s . We d e f i n e f o u r d i s t i n c t a l g o r i t h m s and show t h a t t h e method of t r a n s f o r m i n g the moving boundary problem t o a f i x e d boundary problem [10] i s t h e most p o w e r f u l one. We demonstrate the s e t t i n g up of an e x p l i c i t c o m p u t a t i o n scheme based on the t r a n s f o r m a t i o n method and d e r i v e the a s s o c i a t e d convergence and s t a b i l i t y c r i t e r i a . F i n a l l y we g i v e some n u m e r i c a l examples t o 5 i l l u s t r a t e the s t a b i l i t y problem. 6 CHAPTER I SIMILARITY SOLUTIONS Ls.9. INTRODUCTION In s e c t i o n (1) we d e f i n e the problem. In s e c t i o n (2) we f i n d s u f f i c i e n t c o n d i t i o n s which t h e i n f i n i t e s i m a l s of t h e one parameter L i e group of t r a n s f o r m a t i o n s must s a t i s f y i n o r d e r t o l e a v e i n v a r i a n t t h e e g u a t i o n i t s e l f w i t h o u t c o n s i d e r a t i o n of i n i t i a l and boundary c o n d i t i o n s . . In s e c t i o n (3) we f u r t h e r r e s t r i c t t h e i n f i n i t e s i m a l s i n o r d e r t h a t i n i t i a l and boundary c o n d i t i o n s are a l s o i n v a r i a n t under t h e t r a n s f o r m a t i o n s . S u b s e q u e n t l y t h e r e emerges a s i m i l a r i t y s o l u t i o n which admits a two parameter f a m i l y of moving b o u n d a r i e s . . I n s e c t i o n (4) we t r a n s f o r m the problem i n t o one f o r the heat e q u a t i o n and i l l u s t r a t e t h a t t h e group of the s i m p l e heat e q u a t i o n can be used t o c o n s t r u c t s i m i l a r i t y s o l u t i o n s of our o r i g i n a l problem. THE PROBLEM C o n s i d e r the f o l l o w i n g l i n e a r p a r t i a l d i f f e r e n t i a l e g u a t i o n w i t h o n e - s i d e d moving boundary, - u t + c x u )x = o (1 • 1) where - «*> < x <• r i t ) , t > o , s u b j e c t t o t h e boundary c o n d i t i o n s u (-«>-+.> « u cirtt), t ) = o f o r t >• o , (1- la) u i * , o ) : U x - , X o < r c > ) . (1.1b) We s h a l l f i n d the forms of r (t) a d m i t t e d by the 7 s i m i l a r i t y method. . I_2 INVARIANCE OF THE EQUATION We s h a l l f o l l o w t h e g e n e r a l set-up of Bluman and C o l e [ 8 , p a r t I I s e c t i o n 2.0 and 2 .1 ] , C o n s i d e r a one parameter L i e group of t r a n s f o r m a t i o n s : u* = U * ( x , t . u ; £) = u + £•? ( * , t , t 0 ( £ l ) , X* = X * U , t , R ; t ) = X + £ * Cx,t,U> + Oil*) I ( U 2 ) i * = T * U , i , u ; Z) = t •» i T C x , t , + 0 ( t v ) . J I f u = 9 (*, -t ) i s an i n v a r i a n t s o l u t i o n of (1.1), i t s a t i s f i e s Expanding <*x.*x* , U^* and U x* i n T a y l o r s e r i e s about t = o, one has from (1.3) * U + 11**' » flt + + lfi + CM + <*+ c*>(8*+i>T) + 0 ( £ l ) * 0 *K - e t + (x©) x = o , (1.4) where t h e f i r s t e x t e n s i o n s : and the second e x t e n s i o n *2.x*u e x 6* - fix* - ?uu exl e t fix* - 2 . e x t - 3 8 « fix - &«x e t - 2 Tu e*t 0x . (1.4) i m p l i e s the c o e f f i c i e n t of 2 must be z e r o . A f t e r making the s u b s t i t u t i o n 9„y r g t - (x©)* r we have the 8 f o l l o w i n g equation.: t <M*u - • St + + S ) 8* + < *U« - * Sxu + a- * \» ) + l" * !?« " * T** > 6« 0± * ° ' (1.5) The group of (1. 1) i s found by e q u a t i n g t o z e r o the c o e f f i c i e n t s of the terras 0 X , 8 t » 6 0* r S S t r 8 X X , $x 6t , 0 X 3 , 0<0tr e * t a n d fl*t 6x • The r e s u l t i n g l i n e a r p a r t i a l d i f f e r e n t i a l e g u a t i o n s i n ^ , f and T a r e c a l l e d the d e t e r m i n i n g e q u a t i o n s . S o l v i n g t h i s s e t of d e t e r m i n i n g e g u a t i o n s we f i n d the i n f i n i t e s i m a l s which s a t i s f y (1.5). These e q u a t i o n s are l i s t e d below. M*« - * It + » + ! - • ( 1 " 7 ) S „ , o n-9) T „ = * (1. 10) * U « - * + = 0 (1.11) ! H + ? » « = O (1. 12) f « H - 0 (1. 13) T H M = o (1. 14) ? x - ° (1. 15) 7 * r 0 (1. 16) 9 ( 1 . . 1 6 ) , ( 1 . 9 ) and ( 1 . 1 1 ) g i v e T = ?<x,t> ( 1 . 1 7 ) 5 r $ (x,t ) ( 1 . 1 8 ) f r i <x,t> u + 3 <x,-fc) ( 1 . 1 9) where f , 3 are a r b i t r a r y . ( 1 . 1 0 ) , ( 1 . 1 2 ) , ( 1 . 1 3 ) and (1 - 1 4 ) are s a t i s f i e d c o n s e q u e n t l y . ( 1 . 1 5 ) and ( 1 . 1 7 ) i m p l y ? - 7 < t ) . ( 1 . 2 0 ) R e c a l l i n g t h a t u t 0 ( « , t ) , from ( 1 . 6 ) and ( 1 . 1 9 ) we have <7** * + * i < + ( 3 x X - 3 t • <*3>x ) * 0 ( 1 . 2 1 ) N o t i c i n g t h a t w= J U , * ) i s a s o l u t i o n of ( 1 . 1 ) , we have the f o l l o w i n g s u f f i c i e n t c o n d i t i o n s : 9 * o and f » K - * t + * a f« * o ( 1 . 2 2 ) ( 1 . 7 ) => 2f„ . |„ K + $ t + + $ * 0 ( 1 . 2 3 ) ( 1 . 8 ) x ' ^ z \ £ x * „ ( 1 . 2 4 ) S o l v i n g ( 1 . 2 4 ) and ( 1 . 2 3 ) we have I (x,t) = t A l t ) , ( 1 . 2 5 ) i Ix,*) = - — ( A + A') - *. ( t ' + 1.") + 3 (t) ( 1 . 2 6 ) where A(t) and 5<t) are a r b i t r a r y f o r the time b e i n g . The s u b s t i t u t i o n of ( 1 . 2 4 ) and ( 1 . 2 6 ) i n t o ( 1 . 2 2 ) g i v e s X \ V " + 4 . T ' > + 1 I A " - A ) - ( ( 3 ' t l " - 2)*o . 8 2. 4 J. Hence t h e f o l l o w i n g s u f f i c i e n t c o n d i t i o n s a r e o b t a i n e d : T " ' - 4 ? = 0 ( 1 . 2 7 ) 10 (1. 29) N o t i c e t h a t t h e r e a r e a l l t o g e t h e r s i x f r e e parameters i n the s o l u t i o n o f (1.27), (1.28) and ( 1 . 2 9 ) , and hence i n the i n f i n i t e s i m a l s 7 , _j and \ . We s h a l l show the f o l l o w i n g : ( i ) I f a l l t h e boundary c o n d i t i o n s a r e a l s o i n v a r i a n t , t h e s i m i l a r i t y s o l u t i o n o f ( 1 . 1 ) , (1.1a) and (1.1b) admits a two parameter r i t ) . . ( i i ) I f o n l y (1.1) and (1.1a) a r e i n v a r i a n t , the s i m i l a r i t y s o l u t i o n of (1.1) and (1.1a) admits a f i v e parameter r ( t ) . The c o n d i t i o n a t t = 0 can be s a t i s f i e d by s u p e r p o s i t i o n . . We s h a l l show case ( i ) i n s e c t i o n (3) , and i n v e s t i g a t e case ( i i ) i n s e c t i o n (4) by mapping t h e F o k k e r - P l a n c k e g u a t i o n i n t o the heat e g u a t i o n . I_3 INVARIANCE OF ALL BOUNDARY CONDITIONS In t h i s s e c t i o n we s h a l l show t h a t i f a l l t h e boundary c o n d i t i o n s a r e i n v a r i a n t under the group t r a n s f o r m a t i o n s we have a s i m i l a r i t y s o l u t i o n a d m i t t i n g a two parameter r t t ) . (a) I n v a r i a n c e o f (1.1a) i s e q u i v a l e n t t o the c o n d i t i o n u * i m > , t ) e w ( r i t * ) , t*) r o . Expanding about l ? 0 g i v e s M r ( t i , t ) + € j <•* (ti . t )u : u t n t i . t ) t u«( r c t i . t ) t 7 l t ) r t t i + u t ( rtt),t )£Tlt) +-0.(O) = « <r(t), t ) + U * ( r ( t ) , t ) l \ ( r<0, t ) + H t ( r ( t ) , t ) % - r i m o u l ) . Thus by (1. 25) 11 T<t)r'(t^ = $ < m i , t ) = Yli£T + A(t>. (1.30) (b) I n v a r i a n c e of (1.1b) means i n v a r i a n c e of the s o u r c e at x * * 0 f o r t = o „ Thus X*= X a t t i [ (X,,0) r X, , t * = o + £ T(-o) ^ 0 and u * ( x , 0 = W ( x * , o ) r S ( X*- X, ) . (1.31) I t f o l l o w s t h a t ]f OC f /o) r T (o) r a (1.32) and by expanding (1.31) about 6 t o , we have f (X, ; o) ? - ( x„. 0 ) , (1.33) (1.25) and (1.32) g i v e X» Vco) + A ( 0 > - 0 I r (1.30) and (1.32) g i v e r° ^ ( a ) + A to) = o . ft* S i n c e x» ^ r 8 , i t f o l l o w s t h a t T ' I O r ACO = o . Thus we have the i n i t i a l c o n d i t i o n f o r (1.27) and (1.28). Moreover (1.26) and (1.33) g i v e (c) The s o l u t i o n o f the system of l i n e a r o r d i n a r y e g u a t i o n s (1.27-29) with i n i t i a l c o n d i t i o n s TCo> = T'<») = A(») ^ 3(o) - |? ( A'fo)+ I J ^ ) r o i s found t o be T It) r Z0. S^t A (t) r b t & (t) = -| ( C*»Ut - J ^ A 2 t ) -r £ ( a x * + |>xe -a) where o., b are a r b i t r a r y c o n s t a n t s . S i n c e tit) must s a t i s f y ( 1 .30), we have t h e f o l l o w i n g s o l u t i o n : 12 ret) = -c Q**k t + i s^k t , where C = — = -r to) = -f 0 , £ i s a r b i t r a r y . Hence r(t) 2. a. contains two free parameters. The s i m i l a r i t y solution K.- 9(x,t) i s obtained by solving the invariant surface condition (see [8, section 2.1]); | QK + T 6t = t and therefore the c h a r a c t e r i s t i c eguations jJ(M) ~ T(t>> - f ( ^ u ' (1.34) In t h i s problem T I t ) = i » . 5 " J ^ K * t + § C Co* 2t - S^k2t) + £ ( Oi x, + t> x 0 - a. ) t The f i r s t eguation of (1.34) gives the s i m i l a r i t y varible ZCS^t (1.35) where c i s an arbitrary constant, assuming tx% o . The second eguation of (1.34) gives the following: (1. 36) where A : ^ • ( X o+c)^, F ( ? ) i s a function of £ only. For convenience l e t > f = c ( ^ - ^ ) . Then we have 13 u c x , t ) r i — ex e e 0 . 3 7 ) S u b s t i t u t i n g (1.37) i n t o (1.1) we have an o r d i n a r y d i f f e r e n t i a l e g u a t i o n g o v e r n i n g 3 ( Y ) • A f t e r some a l g e b r a i t i s found t h a t 3" _ A-i 3 = o • (1.38) The c o n d i t i o n u l n t > , - t ) s o g i v e s 3 ( Y.) = 0 (1.39) where y<> " £ ( * - ~ c > . The s o l u t i o n o f (1.38) and (1.39) i s 3(Y)= A * U J ? ( Y - Y 0 ) ) , where 1/T r -c -x. >o. Thus 3 ( Y ) = A* SUk X„tc) c Y - YoO (1.40) To f i n d A * such t h a t the i n i t i a l c o n d i t i o n u . ( x , o ) = S i * - * © ) i s s a t i s f i e d we c o n d i d e r t h e b e h a v i o u r of « ( x , t ) as t o • A * rx 0 +c> < Y-Yo) Let 3 = 3, t 3 i w i * e r e 3, = ~ e the n u : R + P, where 1 , / V l ± -£c*kt i<«-e l t)Y v p,. M i € e e P> = hSS. e * e e I t i s easy t o show the f o l l o w i n g : Y r * ! £ . - £ + 0 ( t ) , 14 J - Q i - fl* ^ t O \ C f X y t Q , Y Q ( X 0 4 C ) • • e ~ ~ ^ * 1 . T h U S /x H)Y i-rtx+e^OCoto) -0i+O V r = - A J ? e e e and ii I K o\ - £ f v- X-S A = — (> ^ t - » o T /TT + u ( x , o ) = S ( x - x 0 > => A = 7= e Hence «( *,t) = (x, X0, t) 47 Suit e -where e e i - e 15 s o l v e s ( 1 . 1 ) , (1.2a) ana (1.2b),. As a check f o r the answer, p u t t i n g Y = Yo w e have u ( r(t>, t ) = o I i . 4 . E I M I I O N TO THE HEAT EQUATION Doob [ 6 ] used a t r a n s f o r m a t i o n e g u i v a l e n t t o _ = x e* , s = •£ e l t . tr = u s ' 4 (1.42) t o reduce (1. 1) t o the heat e g u a t i o n - v s = o . I n t h i s case the c o r r e s p o n d i n g boundary c o n d i t i o n s can be t r a n s f o r m e d as f o l l o w s ; ( i ) For t = o , u ( x, o) s J( x-x 6) For S r £ , v( j . t ^ s J T J i y - U0 ^  where «j„= x„ . ( i i ) For x : r | t ) , u ( n t i . t ) = o 4=* For 3 = & ( S ) r t ( i n ^ s ) /TT > V ( r t t t > , s ) » o , where -oo<|j<ft(S); S > i s t h e domain. Thus the f o l l o w i n g a r e e g u i v a l e n t t o ( 1 . 1 ) , (1.1a) and (1.1b) r e s p e c t i v e l y : V„ -V. . • < 1- 4 3> where -«o < j < S > i . , s u b j e c t t o t h e boundary c o n d i t i o n s V(»M , S ) s V(A < < ) , S ) = o f o r £ > _ , (1.43a) O s t f S C M , ) . (1.43b) The group as w e l l as g e n e r a l s i m i l a r i t y s o l u t i o n s of t h e heat e g u a t i o n can be found i n [ 8 ] . . The group i s 16 ) where ^ , ? and f V c o r r e s p o n d t o t h e i n f i n i t e s i m a l s of M,S and V r e s p e c t i v e l y . L e t ^ be the s i m i l a r i t y v a r i a b l e , t h e r e a r e f o u r d i s t i n c t c a ses [ 8 ] : e a s e l . | J x ^ * r . J T * O V - 3 - ( A ? t 6) w h e r e A.litlie. f B s J f L J s f . The moving boundary has t h e form: s j ^ + i f s + rj 1 + As + s ^ Case II . , T * © ^ = ( W + J + J<JlIf } — L _ where the parameter 6" i s t h e assumed v a l u e of the s i m i l a r i t y v a r i a b l e )> • Case I I I . |3vr«<.r , f = |1 = o , ' a * o t = H - K S T where t h e parameter <T i s the assumed value o f t h e s i m i l a r i t y v a r i a b l e p Case IV. u = (I = f = o , S~ ^ o * - * ^ = S , which means t h a t t h e r e i s no moving boundary. I n the f o l l o w i n g example we s h a l l c o n s i d e r the moving boundary c o r r e s p o n d i n g t o a two parameter l * < t ) . 17 _______ _ _ 1 ret) r -c + -fe s^ k t = e* + e then ft<s) - i*s r (J3«777) s tfi { l£±J-/Ts + S^=M = *r,s 4 Y, where y _ -c + , v _ >c» ft . ft _— '« " z Thus i t i s i n the form of Case II i f Invariance of S = £ => 7 f •£ ) = + ^ + * * " ( :»> . .'. o(+ (I -t i t - 0 . Invariance of (J = at s t •,'. k + i = o and a - . t l > Invariance of i) = <5?(s)=» = | (&(*>, 5 ) . Thus « = £ , (i r - t > ^ = 1 r 5"= C , K = "1 -The group (1.44) i s reduced to j , . ( | \ 4 , - £ J • x . Solving the c h a r a c t e r i s t i c equation we have the v M + C s i m i l a r i t y variable ~—- . 18 The s i m i l a r i t y form f o r v i s ( c f [ 8 , p217]) vs-t ' and F(fc) s a t i s f i e s which has s o l u t i o n f (\) = B (\1 Si~.k(M , — — < \ < PO 3. C o n s i d e r i n g X as e i g e n v a l u e s we would f i n d A f X ) such t h a t V(<j,S) = j A ( X) Sink ( ( W ))•____; 4-<U (1.45) z s a t i f i e s t h e i n i t i a l c o n d i t i o n . Changing v a r i a b l e : A = |^pi r A - A l M i A , then (1.45) i s e g u i v a l e n t t o - 2 — A l ^ l e e s " t - e e * - * e = {* f i A tA) e g... ( A + l l i : ) V -Awe i ~ =£==7^ \iA 19 ^ i n c e -c s A( _) >5 , H+f < o . I t f o l l o w s that For v(«j,i> = ft 5(1-1*) . Thus S u b s t i t u t i n g ^(J>> i n t o (1.46) we have - A : f(s--i)« * The s o l u t i o n i s e g u i v a l e n t t o (1.41) a f t e r making the t r a n s f o r m a t i o n (1.42) and r e c a l l i n g that ) For yijj.t) - ^ w e h a v e V C j , i ) = $<!>> = 3? A ( ^ ± ) e * •• T h u s A<p) -JTe^ff+f) 20 -e~ $(A+|)sa(MV-«))e Ah. (1.46) => V(D,S) By changing v a r i a b l e s a g a i n w i t h A + |- ~ % we have the g e n e r a l r e s u l t The p r e c e d i n g example shows t h a t g e n e r a l s i m i l a r i t y s o l u t i o n s o f t h e heat e g u a t i o n can be used t o c o n s t r u c t c o r r e s p o n d i n g s o l u t i o n s f o r a n o t h e r e q u a t i o n p r o v i d e d a mapping e x i s t s between them. Bluman [ 1 7 ] shows t h a t the l i n e a r p a r t i a l d i f f e r e n t i a l e g u a t i o n t o be e q u i v a l e n t t o the one d i m e n s i o n a l heat e g u a t i o n a r e t h a t i t i s p a r a b o l i c and i n v a r i a n t under a s i x - p a r a m e t e r L i e group of t r a n s f o r m a t i o n s . The F o k k e r - P l a n c k e g u a t i o n we c o n s i d e r e d here c l e a r l y f a l l s i n t o t h i s c l a s s . A c l a s s of moving b o u n d a r i e s of the F o k k e r - P l a n c k e g u a t i o n can be o b t a i n e d by c o n s i d e r i n g the t r a n s f o r m a t i o n (1.42) and t h e f o u r c a s e s p r e s e n t e d i n the b e g i n n i n g o f t h i s s e c t i o n . Hence we have n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r a second o r d e r Case !• • ret) = {^ + 2(j.xelt4rcttltr + A i e l t + s } e - t where p % * j f , 0 , 2 1 T h e s i m i l a r i t y v a r i a b l e i s C a s e i i " . _ ^ ^ > t k - S | 3 z I e t2^e w h e r e f'=<*t' , « ._ A s i n c a s e I I - 2(t)=<T g i v e s u s t h e p a r a m e t e r i n r(±> . C a s e I I I 1 . r Jt + - - t r ( t > s 4 e + e + * e , w h e r e $-f=o, 4* o . % +t ^ i t * A s i n c a s e I I I j£(-t ) = G' g i v e s u s t h e p a r a m e t e r i n r(t) . C a s e I V . N o m o v i n g b o u n d a r y f o r *l-{Lifzo, I ^ o . R e m a r k : A l t e r n a t i v e l y o n e c a n o b t a i n t h e s e m o v i n g b o u n d a r i e s b y c o n s i d e r i n g d i r e c t l y t h e g r o u p s o f t h e F o k k e r - P l a n c k e g u a t i o n w i t h o u t r e q u i r i n g t h e i n i t i a l c o n d i t i o n t o b e i n v a r i a n t . I n t h a t c a s e t h e c o m p l e t e s o l u t i o n i s a l s o o b t a i n e d b y s u p e r p o s i t i o n o f t h e e i g e n s o l u t i o n s s i m i l a r t o ( 1 . 4 5 ) . 22 CHAPTER II FIRST PASSAGE TIME DISTRIBUTION 11-. Q. INTRODUCTION In t h i s chapter we connect the Fokker-Planck eguation to the d i f f u s i o n (heat) eguation and i n v e s t i g a t e the a s s o c i a t e d f i r s t passage time problem. We d e r i v e a new g e n e r a l e x p r e s s i o n f o r the f i r s t passage time d i s t r i b u t i o n of a d i f f u s i o n process with a t r a n s i t i o n p r o b a b i l i t y d e n s i t y f u n c t i o n s a t i s f y i n g the general Fokker-Planck (Kolmogorov forward) eguation f o r any moving boundary r ( t ) . Subsequently, t h i s formula i s used t o f i n d the f i r s t passage time d i s t r i b u t i o n s of four e g u i v a l e n t problems. A new d i s t r i b u t i o n thereby appears. R e c a l l the f o l l o w i n g r e s u l t i n Chapter I: x* - u t + (x u ) x = o where - oo < x < r(t) Y. e* t Y, e t > o s u b j e c t to the boundary c o n d i t i o n s u (-<*> , t ) * u l n-t), ±> - o f o r t > o t - (e l t - i)/Y + *•-^M1. where and (2, 1) 23 l e t a = x e* , s = i £ v t . v = u S * . The f o l l o w i n g i s e g u i v a l e n t t o t h e p r e c e d i n g one. V\n - Vs r o where - <*> < J < (s) =• 2Y 0 S + Yj , S > ± » s u b j e c t t o the boundary c o n d i t i o n s V (3 , ±> = $ («f) * <f><3> . -<T(_Vj_) - i r.-y v where ^ = - JkJL± , 5 = •£ Ye . X * X0 , I I _ _ EQUIVALENCE TO u) } J - UJ? = 0 WITH FIXED BOUNDA_Y L e t 3 = ft<s) - y = 2 Y0 s + Y, - 9 , T * .s - _ , t h e n L - 2! 2 + V ? i V , _1 The boundary c o n d i t i o n s a re t r a n s f o r m e d a c c o r d i n g l y -We have V(*>,t) * V C o , T ) * o f o r r > « , V (*f o> = y* ^ ( r 0 t V , - | > £ f <*> f o r j ? o . The Green»s f u n c t i o n s i m p l i f i e s as f o l l o w s : 24 N o t i c e t h a t s. . - ft -1 - . Yo t Y, - 3 _ t(fr -zY,*) * " S - _ t " T - * Y - ± i e o 6 2 Y The Green's f u n c t i o n i s The f a c t o r i n b r a c k e t s i s a d i f f e r e n c e of two Gaussian d e n s i t y f u n c t i o n s , which suggests the f o l l o w i n g : Define v J U M . . o . ^ ( e •* - e - ). F i r s t we write the s o l u t i o n i n the f o l l o w i n g form: •'ft Hence V (} , t ) e " Y e V + ^ = j f <f « ) e ^ + V ' ° Xtt<») < U • C2.2) The r e l e v a n t t r a n s f o r m a t i o n i s observed t o be - YoV + Y.'T The eguation thus becomes l O ^ - i O ^ r o , whereas the boundary c o n d i t i o n s are e g u i v a l e n t to the f o l l o w i n g : 2 5 - i <v> , % > ° • The Green's f u n c t i o n f o r t h i s i n i t i a l boundary v a l u e problem i s o b t a i n e d from ( 2 . 2 ) : - ( ± i i o ) x -miff, The s o l u t i o n i s lA>($,"0* f -f (ol) "K ( fr, oL, T) J e t Thus we have t r a n s f o r m e d the o r i g i n a l problem t o the i n i t i a l boundary v a l u e problem of the d i f f u s i o n e g u a t i o n i n a s e m i - i n f i n t e i n t e r v a l , which i s r e a d i l y s o l v e d [ 1 8 ] . . We summarize the c o n n e c t i o n s o f t h e s e problems i n Table 2 . 1 , . Bluman [ 1 7 ] r e c e n t l y used a g r o u p - t h e o r e t i c approach t o f i n d the most g e n e r a l d i f f u s i o n p r o c e s s which can be t r a n s f o r m e d t o t h e Wiener p r o c e s s e s , or e g u i v a l e n t l y , the Kolmogorov e g u a t i o n s which can be t r a n s f o r m e d t o t h e pure d i f f u s i o n e g u a t i o n , t h e r e f o r e the t r a n s f o r m a t i o n from Form 1 t o Form 2 of Table 2 . 1 i s a s p e c i a l case i n t h a t g e n e r a l s e t t i n g . P l o t s of the Green's f u n c t i o n Qj and Gf+ are shown i n F i g u r e s 2.1a and 2.1b r e s p e c t i v e l y . 26 T a b l e 2±\ C o n n e c t i o n s between t h e F o k k e r - P l a n c k E q u a t i o n w i t h Moving. Boundary f (t) - Y „ + Y i £ * and the D i f f u s i o n E g u a t i o n P.D. E. U n d e r l y i n g S t o c h a s t i c P r o c e s s 1 s t Inde-pendent V a r i a b l e ( 1 1 ) and O r n s t e i n -Ohlenbeck + (*Y©T+Y<,* Y ( ' } ) J rt zt +  D i f f u s i o n E g u a t i o n Wiener D i f f u s i o n E g u a t i o n w i t h D r i f t * Y , Wiener +-= * Y 8 s + Y , - ! l D i f f u s i o n E g u a t i o n Wiener 3 = same as p r e c e d i n g ( c o n t i n u e d on next page ....). 27 Mappings 2nd Inde-pendent V a r i a b l e (12) and Mappings +— Dependent V a r i a b l e and Mappings Domain of 11: D ( I 1 ) Moving Boundary Domain of 12: D(I2) t = £ JUxS w cx,t > Y.S-Y.V X € ( - o o , f ( t ) ) Ht i«Y,€*+Y ,e" t r0=r(o) = Y. + Y, t * ( o. OO) S - + T 2 C VIfr.T) u)^,T)e 3 6 (-«>, fll*>> ^ ( $ ) = i Y 0 s + Yi = 5 - i = same as p r e c e d i n g 0. 00 1 +-T * ( o , OO) T =• same as p r e c e d i n g - t e e-Y»e\yeeVY,e-^ C Y : ( C ^ I L ) Y . * * Y . T e *T 6 tO, 0*) ( c o n t i n u e d on next page ...) 28 Boundary C o n d i t i o n s V I 2 6 D ( I 2 ) I n i t i a l C o n d i t i o n s V I 1 € D ( I 1) Green's F u n c t i o n V S o l u t i o n u(rct),t) = o u (x,o) = 4) (x) Qr, ( x , X 0 , t > - e t+Y0e <nt)-x) Q*(e4(r(«-K>, V(-*>,s> s V (0^(S>,5) = O + - e Y.J (e - e U( x , t 1 = -00 F i g . 2 . 1 a) Green's Solution f o r Alt) = Y„ e*+ Y. Y o E 0 ' Y i = i ( s e e Table 2 . 1 ) 0.0 1.0 2.0 3.D 4.0 5.0 6.0 F i g . 2 . 1 b) Green's Solution f o r nrt) r e - t i n a Transformed Domain (see Table 2 . 1 ) 31 ____ FIRST PASSAGE TIME PROBLEM For f u t u r e r e f e r e n c e we s h a l l compute the f i r s t passage time d i s t r i b u t i o n of t h e f o u r cases l i s t e d i n Table 2.1. The d i s t r i b u t i o n s o b t a i n e d are e q u i v a l e n t , up t o t r a n s f o r m a t i o n , to t h e i n v e r s e G a u s s i a n d i s t r i b u t i o n which was g i v e n by S c h r o d i n g e r i n 1915 [ 1 9 ] , Some s t a t i s t i c a l p r o p e r t i e s o f t h i s d i s t r i b u t i o n were e s t a b l i s h e d by Tweedie (1945, 1956, 1957) [ 2 0 a , b, c ]. Wald (1947) [ 2 1 ] d e r i v e d t h i s d i s t r i b u t i o n as a l i m i t i n g form f o r the d i s t r i b u t i o n of sample s i z e i n a s e g u e n t i a l p r o b a b i l i t y r a t i o t e s t . F o l k s and C h h i k a r a (1978) [ 2 2 ] gave a review o f the s t a t i s t i c a l a p p l i c a t i o n of t h i s d i s t r i b u t i o n . The f i r s t passage t i m e problem i s u s u a l l y c o n s i d e r e d w i t h f i x e d b o u n d a r i e s [ 1 3 , 14, 23, 24] a l t h o u g h t h e r e s u l t s o b t a i n e d can then be t r a n s f o r m e d i n t o t h a t f o r some moving b o u n d a r i e s . . We s h a l l g i v e a s t r a i g h t f o r w a r d e x t e n s i o n t r e a t i n g g e n e r a l moving b o u n d a r i e s d i r e c t l y . L e t X (t] be a d i f f u s i o n p r o c e s s s t a r t i n g a t x 0 when t=t 0. The f i r s t passage time T of X It) t o t h e b o u n d a r i e s ft,, (t) and flr (t) (&, Ct) < ftUt), V t > t„) can be d e f i n e d by { X(°) « xe, R.lt) < X(t) <• RxL-t) , V t.<t<T, X<T)ert, <T) or X (T) * To a v o i d t r i v i a l i t i e s assume <ft,(t0)<X0< ft* ( t » ) .Then {T>,t} = { f t , ( t ' K x <t'> < fait') , V t . < t ' < t > . Suppose the t r a n s i t i o n p r o b a b i l i t y d e n s i t y P ( x 0 ) t , . x, t ) o f the p r o c e s s s a t i s f i e s the Kolmogorov 32 equation, namely the forward equation (the general Fokker-Planck equation) , i . H (c4<x,t>P> - X ( M M » P ) = - J l and the backward equation where ot(x,t) i s the i n f i n i t e s i m a l mean and (J(x,t) i s the i n f i n i t e s i m a l variance of the process. Then we have P { T £ t ) tL.it) = P r o b a b i l i t y that absorbing does not occur p r i o r to time t . Let 3 11 I t 0 , X. , ft.Ct), ftilt) ) be the p r o b a b i l i t y d ensity f u n c t i o n of T , then r " f t J P ( X » . t . ; K , t ) J x , (2 .3 ) By changing v a r i a b l e , _ j _ j - flC|(t) and d i f f e r e n t i a t i n g under the i n t e g r a l s i g n , we have 3 c "A (' P t « o ( t ( ; < ( , l t H | ( ^ ( t ) . R , ( t » ) , t ) (d^lt)- <R,(t))«J^ = ~ j ' [ M M ' ! 71 UxlO-O^.Ct)) + P ( X 4 , t 0 ; X , t ) ] x = ft,<t> f j- (<IMt> Ct')) <*g-33 9 = - f [ P „ ( x e , t 0 . x , t > i-(^U)-<R,(t)) + P t i x 0 , t ^> / t»] Jx . I t i s r e a d i l y shown t h a t the a p p r o p r i a t e boundary c o n d i t i o n f o r a b s o r b i n g b a r r i e r s i s t h a t P ( X « , t , j x , t ) must v a n i s h t h e r e [ 2 3 , p219]. Hence P ( x 0 ( t 0 } <R, (t), ± ) = P (x e, t 0 j ^ C t ) , - t ) = 0 f o r t > t „ . C o n s e g u e n t l y , 3 r - [ i _ ( < R v t t ) [ p f X o . t . j ^ l t ^ t ) - P ( X 0 , t a J «.(t),t ) 1 P+ i x 6 to .. x , t W x . U s i n g t he boundary c o n d i t i o n and the F o k k e r - P l a n c k e g u a t i o n , we have 3 ( t | - t . ,x 0 | <R,<t>(<Mtn = - J o t ( x , t ) P x ( x . , i 0 . x, t ) | ( 2 - 4 ) Hence the f i r s t passage time d e n s i t y i s r e l a t e d t o the s l o p e s of the t r a n s i t i o n p r o b a b i l i t y d e n s i t y a t the a b s o r b i n g b a r r i e r s , and the a p p r o p r i a t e p r o p o r t i o n a l f a c t o r i s t h e i n f i n i t e s i m a l mean.. From an elementary concept of t r a n s p o r t p r o c e s s e s , one obser v e s t h a t 3<"t) i s t h e net f l u x of the t r a n s i t i o n p r o b a b i l i t y d e n s i t y t r a n s f e r r i n g out of t h e domain through t h e b o u n d a r i e s . The p r o p o r t i o n a l i t y c o n s t a n t -J-oiCX / t ^ i s the d i f f u s i v i t y of the t r a n s p o r t p r o c e s s , and i t i s analogous t o t h e heat d i f f u s i v i t y i n heat t r a n s f e r , momentum d i f f u s i v i t y i n momentum t r a n s f e r and 34 mass d i f f u s i v i t y i n mass t r a n s f e r e t c . . The an a l o g y becomes more t r a n s p a r e n t i f we w r i t e (2.4) as where n .^ i s t h e u n i t v e c t o r normal t o the boundary * . A s i m i l a r i n t e r p r e t a t i o n f o r t h e f i r s t passage time of a random walk t o a f i x e d a b s o r b i n g b a r r i e r appears i n [ 5 ] . Once t h e t r a n s i t i o n p r o b a b i l i t y d e n s i t y i s de t e r m i n e d , i t i s r e l a t i v e l y much e a s i e r t o use (2.4) i n s t e a d of (2.3) to compute t h e f i r s t passage time d e n s i t y . We s h a l l a p p l y (2.4) f o r the f o u r c a s e s l i s t e d i n T a b l e 2 . 1 . We have 7TT* 0 35 = e - ( r . - k + i Y . ^ - t ) ) 1 , <o 4 - t ) d ) J , ( t \ ' , * o ; -0°, ' - € r t x ^ r - ^ , „ ( i c / x . . t ) | ) | B ret) = - e i t - Y.N£ ^ ) - Y . f r 0 . x . > , i t -I»*(c l t-O-Y,(»;-x0) - x e e 3 ^ ( t ( € % t - » > | o ) r , - x o s o , o o ) e e - ( * - , - x e ) v i t = e r » - x e 4 (, e^.p 3j(T|0/l; o,eo) i s known as t h e i n v e r s e G a u s s i a n d i s t r i b u t i o n , Wald's d i s t r i b u t i o n [ 2 2 ] . , I t i s ob v i o u s t h a t S i ( * I °, *o ', i s a t r a n s f o r m a t i o n o f 3j (t 1 e, Jr0 ; o, oo) s i n c e 3, ( t I o,X, j -c» , m> ) J t - 3i f ^  I 0 . r o-y o ; o, oo) Moreover , 36 34- can be reduced t o a gamma d i s t r i b u t i o n by c h a n g i n g v a r i a b l e s . I t s h o u l d be noted t h a t L J (11 t* . X© . A,(-t),tf%<t>)Jt may not to be e g u a l 1, s i n c e t h e r e i s a p o s s i b i l i t y t h a t the s t a t e v a r i a b l e X may never h i t t h e b o u n d a r i e s , p a r t i c u l a r l y , when the domain expands. We have the p r o b a b i l i t y of ever r e a c h i n g the b o u n d a r i e s V l e - jL p ^ o l o ,99 " ^ X - I?" i f Yo S o €. i f Yo > o . Hence Yos o i s the c r i t i c a l v a l u e (see F i g u r e 2.2). Once we have t h e f i r s t passage time d i s t r i b u t i o n , i n o r d e r to s t u d y i t s p r o p e r t i e s , f o r example, i t s moments, i t i s n a t u r a l to c a l c u l a t e i t s L a p l a c e t r a n s f o r m . The L a p l a c e t r a n s f o r m o f 3} i s r e a d i l y found (see [ 2 3 , p221j) t o be r"> -it -Yok-oSCToV*)* Jo "* where Y 0 £ o . The L a p l a c e t r a n s f o r m of Si**' / u n f o r t u n a t e l y , i s not easy t o F i g . 2.2 a) Graphs of x = M±)-Y.e*+ Y, £*, 2 i s the S i m i l a r i t y Variable (see Chapter I) F i g . 2.2 b) Graphs of 3 = <«<s> = 2 Y , s + Y,, 5 i s the S i m i l a r i t y Variable (see Chapter I) PAGE 39 DOES NOT EXIST PAGE 40 DOES NOT EXIST 41 h a n d l e . We have where V - x „ , \0>, o . However, f o r a s p e c i f i e d d i s t r i b u t i o n i . e . where Y„ and J-0 are g i v e n , one can e a s i l y o b t a i n the moments by n u m e r i c a l i n t e g r a t i o n . 4 2 CHAPTER I I I ASYMPTOTIC AND NUMERICAL PROCEDURES _____ INTRODUCTION Our aim i s t o compute t h e s o l u t i o n o f the F o k k e r -P l a n c k e g u a t i o n w i t h a r b i t r a r y moving boundary r ( t ) . s i n c e we a l r e a d y have the s o l u t i o n f o r a moving boundary of the form r e t ) = Y<>e* + Yi C + , a n a i v e approach i s t o f i t a c o n s i d e r the f o l l o w i n g a l g o r i t h m s : A l g o r i t h m 3 . 1 : A g l o b a l a l g o r i t h m . t - t ( i ) We l o o k f o r r A t l (t) = Y»>' e +• Yi,i*i £ which a p p r o x i m a t e s r ' t ) i n t h i s i n t e r v a l . ( i i ) We compute an approximate s o l u t i o n U t f(x,T) t o K(*,T) by u s i n g t h e f o l l o w i n g r e c u r r e n c e r e l a t i o n . . seguence of b o u n d a r i e s o f t h i s form t o Y H) . •Let {t;.} be a seguence of p o i n t s i n t°;Tj , Fo r each i n t e r v a l ( t i , t * + i ) i . e . we r 4 (o) (3 . 1 ) where t - _ , < t * t,- , t-•e - Y a , , ( f tP ) - * i ) *<t-tf.,> 43 (see T a b l e 2. 1) . A l g o r i t h m 3.2: An a s y m p t o t i c a l g o r i t h m . (i) Same as A l g o r i t h m 3. 1 ( i ) . ( i i ) We use the s m a l l time a s y m p t o t i c e x p a n s i o n of the i n t e g r a l i n (3.1) i n s t e a d of p e r f o r m i n g the i n t e g r a t i o n o v er a s e m i - i n f i n i t e domain..(see s e c t i o n I I I . 1 f o r t h e d e t a i l s of the expansion.) A l g o r i t h m 3.3: A T a y l o r e x p a n s i o n a l g o r i t h m . (i) Same as A l g o r i t h m 3. 1 (i) . ( i i ) We use Form 4 i n Table 2.1, i . e . . we a p p l y t h e f o l l o w i n g t r a n s f o r m a t i o n s : If z e * c r ( t ) - x ) , t - ± ( e ^ - o t ( i i i ) We compute the T a y l o r e x p a n s i o n of cd(},7) i n 7 by a p p l y i n g t h e d i f f u s i o n e g u a t i o n d i r e c t l y . . Thus + 0 C( T j f l - 7^) ) (3.2) where *(*)(*, 7;) A ) ( j 7 ) 44 The d e r i v a t i v e s i n J- can be approximated by the a p p r o p r i a t e f i n i t e d i f f e r e n c e [25, p17]. A l g o r i t h m 3.4: An e x a c t boundary a l g o r i t h m . ( i ) We use t h e t r a n s f o r m a t i o n JJ = f ( t j - x [ 1 0 ] t o map the moving boundary problem i n t o a f i x e d boundary problem.. Hence u t s u.^ t ((|J t F(i>) u ) , (3.3) where o < <j < oo , T > t > p(t) « - (ret) + r'(tl) , U ( o , t) - u (oo, t ) = 0 , t >,<> , = *oiy> = 4 ( f(•» - y), ^(*) i s t h e o r i g i n a l i n i t i a l c o n d i t i o n . ( i i ) Same as A l g o r i t h m 3. 3 ( i i i ) e x c e p t t h a t t h e p a r t i a l d i f f e r e n t i a l e g u a t i o n used i s (3.3).. We observe t h a t A l g o r i t h m 3.1 i s a ' g l o b a l a l g o r i t h m ' i n the sense t h a t f o r each x i n time tj- +, / we have t o do an i n t e g r a t i o n on a f u n c t i o n i n i j over a s e m i - i n f i n i t e domain, whereas A l g o r i t h m s 3.2, 3.3 and 3.4 are ' l o c a l a l g o r i t h m s ' s i n c e t h e X d e r i v a t i v e s o f t h e dependent v a r i a b l e i n tj can be a p p r o x i m a t e d l o c a l l y . I n t h i s sense A l g o r i t h m 3.1 i s not b e t t e r than the o t h e r a l g o r i t h m s . From the p o i n t of view of p r a c t i c a l c o m p u t a t i o n s i n c e A l g o r i t h m 3.3(i) i n v o l v e s an a p p r o x i m a t i o n o f 45 the moving boundary f(t) whereas A l g o r i t h m 3.4 ( i ) does n o t , and A l g o r i t h m 3.3 ( i i i ) i s e g u i v a l e n t t o 3 . 4 { i i ) , i t i s o b v i o u s t h a t 3.4 i s b e t t e r than 3.3. Now t h e r e m a i n i n g g u e s t i o n s a r e : I s A l g o r i t h m 3.2 a new method? How d i f f e r e n t i s i t from t h e o t h e r A l g o r i t h m s . I n s e c t i o n 1 of t h i s c h a p t e r we show t h a t the a s y m p t o t i c A l g o r i t h m 3.2 i s t h e same as the T a y l o r e x p a n s i o n A l g o r i t h m 3.3. T h i s i s our major r e s u l t i n t h i s c h a p t e r . In s e c t i o n 2 we use A l g o r i t h m 3.4 t o s e t up an e x p l i c i t f i n i t e d i f f e r e n c e scheme. We f i n d t he c o n d i t i o n s f o r convergence of the scheme. I n s e c t i o n 3 we a p p l y t h e procedure s e t up i n s e c t i o n 2 t o do some n u m e r i c a l c o m p u t a t i o n s . Exact s o l u t i o n s f o r t h e two parameter moving b o u n d a r i e s are e v a l u a t e d i n o r d e r t o make a c o m p a r i s i o n w i t h the f i n i t e d i f f e r e n c e scheme. The s t a b i l i t y c o n d i t i o n s f o r the scheme are t e s t e d . I I I . 1 ASYMPTOTIC EXPANSION FOR SMALL TIME We w r i t e t h e s o l u t i o n (1.41) i n the f o l l o w i n g form: i — t + Y 3 - Yo% u ( x ; t ) = J 2 £ e ' [ l i l x ( t > - U lx , t>] > -Yaoi . -xCob-S)1" ,°° ~ *AU> , . where 2, (x.-t) r e $<r.-.l>e de* =J 4> «> e <K 46 r*u,t>* ( e * i w ) e j o t = J 4>w>e d o t , 3= e t C Y . c t t Y, € * * ) , ?c*> r e ' Y o 0 t <fr(>w> , . Moo = - l*-V* . * * ^ T 7 } • Observe t h a t J o i s a measure of the d i s t a n c e from t h e moving boundary r ( t ) . As t-» 0 , X"*°°, and t h e r e f o r e f o r each g i v e n i n i t i a l c o n d i t i o n utx,o> = 4(x) , an a s y m p t o t i c e x p a n s i o n of I ( and I j , and hence of u ( * , t ) may be found..We s h a l l c o n c e n t r a t e on t h e a s y m p t o t i c e v a l u a t i o n of I I ^ M s f J u i e ^ " 1 ^ ' ^ where -j ( o l . %) = - c « - f ) \ ^0 - *o < I < fro . C l e a r l y , L and I Z a r e i n the form of I . The b e h a v i o u r of the a s y m p t o t i c expansion of I depends ve r y much on \ (see F i g u r e 3 . 1 ) . . We d i s t i n g u i s h t h e f o l l o w i n g two cases,, ( i ) I- > o . Assume f ( o O i s smooth, i . e . . I t has a Taylor, e x p a n s i o n where _ V?-<*> . A! We s e p a r a t e I i n t o two p a r t s , i . e . i u ; r (/; • j . ) < M * > e T '* j e t , ' o* 4 8 The second p a r t i s J_ = j . $ e <U it where P(j> i s t h e gamma f u n c t i o n , and t h e f i r s t p a r t i s j , = j ( o l ) e ( o t ; 0 0° , . , * « / * t ' Hence oo N o t i c e t h a t t h e f o r m a l e x p a n s i o n o n l y depends on the even d e r i v a t i v e s o f ^ U) a t «<•=$-. ( i i ) |=o. O b v i o u s l y , we o n l y have the second p a r t o f t h e i n t e g r a l i n ( i ) , hence I n g e n e r a l the f o r m a l e x p a n s i o n of J ( does not v a n i s h f o r |=o . S i n c e we have d i f f e r e n t e x p a n s i o n s f o r 1 - o and |sS>o and as 3 " * ° the e x p r e s s i o n s do n o t converge t o each o t h e r , a uniform e x p r e s s i o n i s sought. The u n i f o r m a s y m p t o t i c e x p a n s i o n f o r t h i s c l a s s o f i n t e g r a l s c o n t a i n i n g a g l o b a l maximum near an e n d p o i n t has been s t u d i e d [ 2 6 , PP167-170, 27]. B a s i c a l l y , i t s i m u l t a n e o u s l y t a k e s i n t o 4 9 a c c o u n t t h e c o n t r i b u t i o n s at the maximum of i and a t the e n d p o i n t o f i n t e g r a t i o n . A p p l y i n g the r e s u l t o f B l e i s t e i n [ 2 7 ] , we have the f o l l o w i n g : 1^)1) = J * $ W > e * f W t ; i r ) J e l . where -f<*;|> * " <*-f>X ft S t r i c t l y s p e a k i n g , $(«0 s h o u l d be w r i t t e n as oi ^ -^-j— J , such t h a t - L . — D u " t 1 1 1 Appendix I we show t h a t t h e two s e r i e s a re e q u i v a l e n t . The symbols i n the s e r i e s mean the f o l l o w i n g : ft w,'u) S y i i e 1 ^fc(^) - l = * w ^ ) - I , .00 where _ , » Subseguent terms in the series are determined recursively The functions { Gfn (l) } s a t i s f y the following eguations: 50 More p r e c i s e l y , we c a l c u l a t e t h e f i r s t few terms i n the f o l l o w i n g : G(t f*> _ 4>ifr) -ill- n Jf : Gr,(o)r (By L ' H o s p i t a l ' s r u l e ) where 14 (4))' (^  . i s found by d i f f e r e n t i a t i o n and a p p l i c a t i o n o f L ' H o s p i t a l ' s r u l e : rr'l -Hence *»• ^ N o t i c e t h a t when t h e s a d d l e p o i n t (et =• £) c o a l e s c e s on the e n d p o i n t of i n t e g r a t i o n (ot=o), t h e fx A. are s t i l l d e f i n e d . I n t h i s case we have JT, = Jfx = o , 2 ' 5 We now t u r n our a t t e n t i o n t o the uniform a s y m p t o t i c e x p a n s i o n of I t i s easy to show t h a t <p <°0 i s an 51 odd function of o< by using the r e s u l t i n Table 2. 1. Consequently, i t i s re a d i l y shown by induction that for n=0(1)N, U ] = , fx* t ft] = - C - *1 . for n=0(1)N, Q n_i ;e.) - -G*C* ;-a) and ^ [ 2 ; ftl = - $ 1-2; , Thus I ( X; -j) = 4» u> e *7* where - - , * *Xi Notice that W 0 ( ^ t W 0 ( - * > = J M I £ and vv);<4) - r /Ti ^  / z Hence II - 1 ( A ; X X » « A * A J Now we have the small time uniform asymptotic expansion: Jrt'i-TTZ: . r fa* K l t t * . 7) 52 U s i n g the f i r s t 2 terms of (3.7) we get UU,t> r e Y , 5* Y , Tyip { f x [ ?0>+T f*J>] + 0 lT*>^ (3.8) Comparing (3.8) w i t h the s o l u t i o n i n Table 2.1<l>, we o bserve t h a t by l e t t i n g W0(fr) + T + 0(TV) i s a u n i f o r m a s y m p t o t i c e x p a n s i o n of the s o l u t i o n u)(j.,?> which s a t i s f i e s w^ j, i ) - tOT(},t) =• o w i t h boundary c o n d i t i o n s I t s h o u l d be c l e a r t h a t we have shown t h a t A l g o r i t h m 3.2 i s same as A l g o r i t h m 3.3 up t o 0 (T*) .-We now proceed t o r e l a t e a l l the Yi.'s t o t h e i n i t i a l d i s t r i b u t i o n w(J,o) d i r e c t l y . S i n c e fxn * , and s i n c e we can w r i t e r _ 4io) ~(i (*»' U=-o. 0. we can r e l a t e ( o) and t o more e x p l i c i t l y ; however, (3.6) shows t h a t the o n l y i n t e r e s t i n g g u a n t i t y i s r%n - 0. - < * <*« ( 2 > ) ' | 2 , _ A . D i f f e r e n t i a t i n g (3.5) t w i c e , we have f o r n=1(1)N <l> A c t u a l l y , the a s y m p t o t i c r e s u l t m o t i v a t e s one t o f i n d the f o u r t h e g u i v a l e n t form i n Table 2.1. 53 D i f f e r e n t i a t i n g a g a i n t w i c e and s h i f t i n g n t o n-1 r e p e t i t i v e l y , we get tSi i*> (•) 4* -ft. 2 n ! 1 ? e - a . • D i f f e r e n t i a t i n g (3-4) (2n+2) t i m e s g i v e s Q IZ) = ( » " H H i ( 5 , ^ » •+ <«**>(«$,(*)) Thus S u b s t i t u t i n g (3.9) i n t o (3.6) we have the e x p l i c i t s e r i e s I(X; \) - ICX; -%) R e c a l l i n g t h a t «i = - y i j - f = t and Q (.*) r y we have I(A;\) - I (X; -J) = / f f ft*>+ £ t S • O C T - " ) } and U ( X From Table 2.1 and the p r e c e d i n g we have the u n i f o r m a s y m p t o t i c e x p a n s i o n f o r ^ Ij-.T) : 54 UM}.,!} = oOl^-.o) + j f <*> ^,<»)T^ + OCT ). (3.10) S i n c e (3. 10) g i v e s we have c o n s i s t e n c y . We o b serve t h a t i n (3.10) the e x p a n s i o n i s about ( J-,0) and i t depends on the even orde r p a r t i a l d e r i v a t i v e s of (0(^,-0 w.r.t. . Comparing t h i s e x p a n s i o n w i t h the T a y l o r e x p a n s i o n f o r w (j.,z) about T= o , i . e . we see t h a t (3.10) and (3.11) are e g u i v a l e n t s i n c e I O I ^ " * ) s a t i s f i e s the d i f f u s i o n e g u a t i o n W^t^-c) s oOjj,<£,T)f and hence The e r r o r i n t h e e x p a n s i o n i s t h e r e f o r e (Nti) 1. ^ We have shown t h a t A l g o r i t h m 3.2 i s e g u i v a l e n t t o A l g o r i t h m 3. 3. 55 I I I _ 2 A FINITE DIFFERENCE METHOD FOR ARBITRARY MOVING BOUNDARY A l g o r i t h m 3.4 i s a g e n e r a l s t a r t i n g p o i n t f o r s e t t i n g up more d e t a i l e d c o m p u t a t i o n a l schemes. S i n c e h a v i n g the s o l u t i o n a t t " tj we can compute t h e s o l u t i o n a t t = t j + | n o n i t e r a t i v e l y , we have a 'one s t e p e x p l i c i t scheme*. The i d e a b e h i n d A l g o r i t h m 3.4 i s not new [ 2 5 , and the r e f e r e n c e s g i v e n t h e r e ] . A p p l y i n g A l g o r i t h m 3.4, we d i s c r e t i z e t h e domain 0 4 j < M R T > t >o with uniform s p a c i n g K = ©. |j and •£= At = -j~ ( N t i s a p o s i t i v e i n t e g e r ) , such t h a t t - t j - j-k and j : U ; . We denote t o be the e x a c t s o l u t i o n a t ( U i , "tj ) • Using c e n t e r e d d i f f e r e n c e s f o r t h e d e r i v a t i v e s , we have . ^ ^ ( ( C J f F ( t ) ) u ) J . . = <^<+ *W*L - l f c - . + F t t , » U J . , 3 A3 zk + ocV) , where , j e oco , and t h e boundary c o n d i t i o n s i m p l y = U 0 U k ) f o r od> oo , U_, ' 0 , - O f o r j : o ( i i ^ _ L e t £2 be the o p e r a t o r such t h a t + ( I - if) u? 56 where o _ We have the f o l l o w i n g a p p r o x i m a t i o n f o r the e g u a t i o n : We denote the approximate s o l u t i o n by \J£ which s a t i s f i e s e x a c t l y t h e e g u a t i o n \J± « (3.13) and the boundary c o n d i t i o n s = M0CAKi f o r X. - oin oo , tT<» = 0 , Vl - o f o r j r o tl) N t . We f o l l o w a s i m i l a r development of Ames [ 2 5 ] t o o b t a i n convergence and s t a b i l i t y , c o n d i t i o n s f o r t h i s e x p l i c i t method. S u b t r a c t i n g (3.13) from (3.12), we o b t a i n the f i n i t e d i f f e r e n c e s a t i s f i e d by t h e e r r o r £g - 0 , S 0 0> hJt . Assuming o < r < •/_ (3* 15) and } (3. 16) then , E n ' | < i r + ^ u U l tF(t,-)J) | £ i J 57 = <it*)i|E*ll + AC i ' + i K " ] t im» ( 3.17) where A i s a constant depending on the values of u t t and : HE'II = * U P I Ej|| -Conseguently „ £ j, ^  m i ) | | E ' | | + A t £ N * ^ ] , and since |( = o we have where £j s a t i s f y the following difference eguation: iCjt, - C l+ 1) ^  + 1 { (3. 18) The s o l u t i o n of (3.18) i s Jf*. s " £ [ ( < + - f t . ) 1 ] I t f o l l o w s t h a t „ £i*r„ ^ £ [ i ( + ^ . , J A [ *fc*] $ A eT C A t * (AX)1] . (3.19) (3.19)- i m p l i e s t h a t l l f ^ ' l l ^ t f as A t - * o , 4x-»©„ Thus the s o l u t i o n of the f i n i t e d i f f e r e n c e e g u a t i o n (3.13) converges t o t h e s o l u t i o n of t h e p a r t i a l d i f f e r e n t i a l e g u a t i o n (3.3) . The c o n d i t i o n (3.15) i s the s t a b i l i t y c o n d i t i o n [ 2 8 , p 1 8 3 ] , and the c o n d i t i o n (3.16) i s e g u i v a l e n t t o >, raj) + f'ctj) ^ - ( J L + H^ ( 3. 2O) which imposes a further r e s t r i c t i o n on h. We have the 58 f o l l o w i n g theorem: Theorem 3. 1 4 i L e t u € C ' be the s o l u t i o n of e g u a t i o n (3.3) and the a s s o c i a t e d boundary c o n d i t i o n s , and l e t "U" be t h e s o l u t i o n o f (3. 1 3 ) . . I f ° < r£ _ and Ax i s s m a l l enough so t h a t t h e n -&*|» I U J ~ [ £ Ae T [ At + (AX)1] f o r l^^,t^)6SL I I I _ 3 COMPUTATIONS In the f o l l o w i n g we g i v e some n u m e r i c a l examples t o i l l u s t r a t e m ainly the s t a b i l i t y problem.. S i n c e i t i s i m p o s s i b l e t o compute n u m e r i c a l s o l u t i o n s i n a semi-i n f i n i t e domain w i t h uniform mesh s i z e i n f i n i t e computing t i m e , we r e s t r i c t o u r s e l f t o c o n s i d e r } J t C 0 ; 5 ] and impose the boundary c o n d i t i o n : U(S,t)=o. Example 3. 1 f i * * * * k -?(tj> +r'(tj) >, - l J K +AX) i n J l r [ 0 ( « ) < [ » J l L e t Y o ^ ~ Y, - 1 , h = Aij r 0. | o t h e r w i s e 59 The r e s u l t s of our e x p l i c i t method a r e shown i n F i g u r e 3.2. S i n c e (3.20) h o l d s f o r h = °-l (•$9*S5")r the i n s t a b i l i t y i s due t o (3.15) where we have T > •£ . Example 3.2 L e t Y, = -1 , Y, - 2 , K * o.i , 0.2 , 0 .5 , * = '/loo •<( 0.1) = I *^ 0 o t h e r w i s e = U, CJ ) . We a p p l y the f i n i t e d i f f e r e n c e s s t a r t i n g a t t = 0. 1 + k. The r e s u l t s a r e shown i n T a b l e 3.1, 3.2, 3.3 and F i g u r e 3.3. C o n d i t i o n (3.15) c l e a r l y h o l d s i n t h e s e t h r e e c a s e s . The i n s t a b i l i t y i n T a b l e 3.3 i s due t o v i o l a t i o n o f c o n d i t i o n (3. 20) , where we have 1 + 1 ) : , but n <k - Y (tj) + Ir ftp can be as l a r g e as 5 + 2e. . ExamjDle 3^3 L e t Y 0 = - l , Y. - 'o . ^ 0.1 , Vzoo , Ul«J,0.l) s U , ( t j ) The r e s u l t s i n T a b l e 3.4 shows t h a t i n s t a b i l i t y o c c u r s when r ( t ) d e c r e a s e s so r a p i d l y t h a t c o n d i t i o n (3.20) i s v i o l a t e d i n the i n t e r v a l (J . F i g . 3 . 2 I l l u s t r a t i n g I n s t a b i l i t y due to J±± > J . , a) A t = - £ - i Stable 1 D . 75 H D . 50 H 0 . 2 5 H o. 00 U (y . t) A t = 1 / 1 9 5 4 y = 0 . 1 y ' I r i r i i ! I I I I i i T ^ - ^ ^ ^ ^ r r " ? i T i i i ! i 2 . 5 ^ I ( i F 1 7 I I T p 3 .0 3 . 5 F i g . 3.2 b) AS = o. 1, o-t=ri?« Unstable, but I t i s not Apparent from the Numerical. Result i n this Time Interval E r r o r ( x i o - 3 ) F i g . 3.3 I l l u s t r a t i n g the Propagation of E r r o r i a) 63.o.J, * t = ^ r in . 3 ) • o o " . C\J II n o o o o a a o o a o o in o in a m ro o OJ in i i i 66 L e t Y0 » Z , Y, = , h = o.l, ft.*, o.S , - i - '/2»«> , The r e s u l t s i n F i g u r e 3.4 demonstrates t h a t an A i n c r e a s i n g r ( t ) w i l l g i v e a s l o w l y decay s o l u t i o n and the a p p r o x i m a t i o n of t h e s e m i - i n f i n i t e domain by (0, 5) i s not a c c u r a t e . I n s t a b i l i t y i s observed f o r h=0.5. Example 3._5 U s i n g the same s e t of parameters as i n Example 3.4, e x c e p t t h a t we use t h e boundary c o n d i t i o n u ( 5 , t) = G j ( r ( t ) - 5 , 1 , t) f o r t > 0. 1 i n s t e a d . T h i s example shows t h a t i n s t a b i l i t y o c c u r s i n the case of a p p l y i n g exact boundary c o n d i t i o n s . The r e s u l t s a r e shown i n F i g u r e 3.5. y F i g . '}A Solution with Approximate Boundary Condition Uj. 5=o as 0 3 2 3 4 5 y F i g . 3 .5 Solution with Exact Boundary Conditions 69 T a b l e 3.1 +• 3 y I Case of Y.=-1, Yt = 2, r 0 - X o = 1, 4t=1/200, Ay=0. 1: a) Approximate S o l u t i o n t=0. 1 t=0. 2 t=0.6 t=1. 0 0. 0 0. 1 0.2 0.3 0.4 0. 5 0.6 0.7 0. 8 0. 9 1.0 1.1 1.2 1.3 1.4 5 6 7 8 9 0 1 2. 2 2. 3 2. 4 2. 5 2.6 2. 7 2. 8 2.9 3.0 3. 1 3. 2 3. 3 3.4 3. 5 3. 6 3. 7 3.8 3.9 4. 0 4. 1 4. 2 4. 3 4.4 4. 5 4. 6 4. 7 4.8 4.9 5. 0 0. 0 0.2160E+00 0. 4013E+00 0.5668E+00 0.7122E+00 0.8293E+00 0.9069E+00 0. 9361E + 00 0. 9134E+00 0.8432E+00 0.7364E+00 0.6086E+00 0.4760E+00 0. 3523E + 00 0. 2468E+00 0.1636E+00 0. 1026E+00 0.6090E-0 1 0.3421E-01 0. 1818E-0 1 0. 9147E-02 0. 4354E-02 0. 1961E-02 0.8362E-03 0.3373E-03 0. 1288E-03 0. 4653E-04 0. 1591E-04 0. 5147E-05 0. 1576E-05 0.4566E-06 0.1252E-06 0.3248E-07 0.7977E-08 0. 1854E-08 0. 4076E-09 0. 8482E-10 0.1670E-10 0.3113E-11 0. 5489E- 12 0.9161E-13 0. 1447E-13 0.2162E-14 0.3058E-15 0.4093E-16 0.5185E-17 0.6214E-18 0.7049E-19 0.7566E-20 0. 7686E-21 0.0 0.0 0.2359E+00 0.4109E+00 0.5310E+00 0.6025E+00 0.6324E+00 0.6279E+00 0.5961E+00 0. 5446E+-00 0. 4803E+-00 0.4098E+00 0.3386E+00 0.2712E+00 0.2106E+00 0. 1586E+-00 0.1158E+00 0. 8203E-01 0. 5635E-01 0. 3754E-01 0. 2424E-01 0. 1518E-01 0. 9213E-02 0.5419E-02 0.3089E-02 0. 1705E-02 0.9120E-03 0.4722E-03 0. 2367E-03 0. 1148E-03 0. 5385E-04 0.2444E-04 0. 1072E-04 0. 4544E-05 0. 1860E-05 0. 7356E-06 0. 2807E-06 0. 1034E-06 0.3670E-07 0. 1256E-07 0. 4140E-08 0. 1314E-08 0.4014E-09 0. 1 179E-09 0.3330E-10 0.9034E-11 0.2353E-11 0.5881E-12 0. 1409E- 12 0. 3210E-13 0.6462E-14 0.0 0. 0 0.4263E-01 0.6960E^01 0.8417E-0 1 0.8935E-0 1 0.8782E-01 0.8182E-0 1 0.7319E-0 1 0.6332E-01 0. 5325E-0 1 0.4366E-0 1 0.3500E-01 0. 2746E-0 1 0.2112E-01 0. 1594E-0 1 0.1181E-01 0.8603E-02 0. 6158E-02 0.4334E-02 0.3001E-02 0.2044E-02 0.1370E-02 0. 9040E-03 0.5871E-03 0. 3753E-03 0.2362E-03 0. 1464E-03 0.8930E-04 0.5364E-04 0.3172E-04 0. 1847E-04 0.1059E-04 0. 5979E-05 0. 3323E-05 0. 1818E-05 0.9796E-06 0. 5195E-06 0. 2712E-06 0. 1394E-06 0. 7049E-07 0. 3508E-07 0. 1718E-07 0. 8277E-08 0.3919E-08 0. 1821E-08 0. 8277E-09 0. 3654E-09 0. 1539E-09 0. 5910E-10 0,. 1757E-10 0.0 0.0 0.2693E-0.4006E-0.4418E-0.4282E-0.3845E-0.3277E-0.2684E-0.2128E-0.1642E-0.1237E-0.9115E-0.6584E-0.4667E-0.3250E-0.2226E-0. 1499E-0.9938E-0.6487E-0.4171E-0.2643E-0.1650E-0. 1015E-0.6158E-0.3683E-0.2172E-0. 1263E-0.7246E-0.4098E-0.2286E-0.1258E-0.6828E-0. 3655E-0. 1930E-0. 1005E-0.5162E-0.2615E-0. 1306E-0.6435E-0.3125E-0.1495E-0.7049E-0.3270E-0.1491E-0. 6660E-0.2901E-0.1220E-0.4853E-0. 1737E-0.4732E-0.0 02 02 02 02 02 02 02 02 02 02 03 03 03 03 03 03 04 04 04 04 04 04 05 05 05 05 06 06 06 06 07 07 07 07 08 08 08 09 09 09 10 10 10 11 11 11 12 12 13 T a b l e 3.1 Case of Y»=-1, Yi = 2, r„-x 0=1, At=1/200, A V=0. 1: b) Exact S o l u t i o n y I t=0.1 J t=0.2 | t=0.6 I t=1. 0 0. 0 i 0.0 | 0.0 | 0. 0 0.0 0. 1 I 0.2160E+00 J 0.2337E+00 | 0.4265E- 0 1 | 0.2850E- 02 0 . 2 | 0.4013E + 00 J 0. 4076E + 00 | 0.6980E- 01 | 0.4273E- 02 0. 3 I 0.5668E+00 | 0. 5273E + 00 | 0.8462E- 01 | .0.4752E- 02 0.4 | 0.7122E+00 J 0.5993E+00 J 0.9C08E- 01 | 0.4646E- 02 0. 5 I 0.8293E+00 | 0.6301 E + 00 | 0.8881E- 01 | 0.4212E- 02 0. 6 I 0.9069E+00 | 0.6265E+00 | 0.8302E- 01 | 0.3626E- 02 0. 7 | 0. 9361E + 00 J 0.5958E+00 | 0.7452E- 01 J 0.3002E- 02 0.8 | 0.9134E+00 | 0.5452E+00 | 0.6473E- 01 | 0.2408E- 02 0. 9 I 0.8432E+00 | 0.4815E+00 | 0.5466E- 01 | 0.1880E- 02 1. 0 I 0. 7364E + 00 J 0.4114E+00 J 0.4502E- 01 j 0.1434E- 02 1. 1 I 0. 6086E+00 | 0.3403E+00 | 0.3626E- 01 | 0.1071E- 02 1. 2 I 0.4760E+00 | 0.2729E+00 | 0.2860E- 01 | 0.7851E- 03 1. 3 I 0.3523E+00 | 0.2121E+00 | 0.2212E- 01 i 0.5650E- 03 1. 4 I 0.2468E*00 | 0. 1599E + 00 J 0.1679E- 01 | 0.3998E- 03 1. 5 I 0.1636E+00 | 0.1169E+00 | 0.1253E- 01 | 0.2784E- 03 1. 6 j 0. 1026E + 00 J 0.8288E-01 | 0.9184E- 02 | 0. 1909E-03 1. 7 I 0.6090E-01 | 0.5701E-01 | 0.6622E' 02 | 0.1289E- 03 1. 8 I 0.3421E-01 | 0. 3804E-01 | 0.4698E- 02 | 0.8582E- 04 1. 9 I 0. 1818E-01 | 0.2463E-01 j 0.3280E- 02 | 0.5633E- 04 2. 0 I 0.9147E-02 | 0.1547E-01 | 0.2254E- 02 | 0.3647E- 04 2. 1 i 0. 4354E-02 | 0.9421E-02 | 0.1525E- 02 | 0.2329E- 04 2. 2 I 0.1961E-02 | 0.5568E-02 | 0.1017E- 02 | 0.1468E- 04 2. 3 | 0. 8362E-03 J 0.3192E-02 | 0.6673E- 03 J 0.9132E- 05 2. 4 I 0.3373E-03 | 0.1776E-02 | 0.4315E- 03 | 0.5607E- 05 2. 5 I 0. 1288E-03 | 0.9581E-03 J 0.2749E- 03 | 0.3399E- 05 2. 6 | 0.4653E-04 J 0.5016E-03 | 0.1726E- 03 | 0.2034E- 05 2. 7 j 0. 1591E-04 J 0.2547E-03 j 0.1067E- 03 ! 0. 1203E-05 2. 8 1 0.5147E-05 | 0.1255E-03 | 0.6506E- 04 | 0.7020E- 06 2. 9 1 0. 1576E-05 J 0.5997E-04 J 0.3908E- 04 | 0.4048E- 06 3. 0 1 0.4566E-06 | 0.2781E-04 | 0.2313E- 04 | 0.2305E- 06 3. 1 I 0. 1252E-06 J 0.1251E-04 J 0.1350E- 04 | 0.1297E- 06 3. 2 I 0.3248E-07 | 0.5458E-05 | 0. 7762E-05 | 0.7209E- 07 3. 3 I 0. 7977E-08 1 0.2311E-05 | 0.4399E- 05 J 0.3959E- 07 3. 4 i 0.1854E-08 | 0. 9489E-06 | 0.2458E- 05 | 0.2148E- 07 3. 5 I 0. 4076E-09 J 0.3780E-06 J 0.1353E- 05 | 0. 1151E- 07 3. 6 I 0.8482E-10 J 0.1461E-06 | 0.7345E- 06 | 0.6098E- 08 3. 7 | 0. 1670E-10 | 0.5479E-07 | 0.3929E- 06 1 0.3192E- 08 3. 8 1 0.3113E-11 | 0.1993E-07 | 0.2072E- 06 | 0.1651E- 08 3. 9 1 0. 5489E-12 1 0.7033E-08 | 0. 1077E- 06 | 0.8437E- 09 4. 0 I 0.9161E-13 | 0.2408E-08 | 0.5520E- 07 | 0.4261E- 09 4. 1 I 0. 1447E-13 J 0.7997E-09 | 0.2788E- 07 J 0.2127E- 09 4. 2 I 0.2162E-14 | 0. 2576E-09 ( 0.1388E- 07 j 0.1049E- 09 4. 3 J 0. 3058E-15 | 0.8053E-10 ] 0.6814E- 08 J 0.51 15E- 10 4. 4 1 0.4093E-16 | 0.2442E-10 | 0.3297E- 08 | 0.2465E- 10 4. 5 1 0.5185E-17 | 0.7184E-11 J 0.1572E- 08 j 0.1174E- 10 4. 6 1 0.6214E-18 J 0.2050E-11 | 0.7393E- 09 | 0.5523E- 11 4. 7 | 0. 7049E-19 J 0.5676E-12 | 0.3427E- 09 j 0.2569E- 11 4. 8 1 0. 7566E-20 | 0.1524E-12 | 0.1566E- 09 | 0.1181E- 11 4. 9 I 0.7686E-21 J 0.3972E-13 | 0.7051E- 10 ] 0.5367E- 12 5. 0 1 0.7388E-22 | 0. 1004E-13 | 0.3131E- 10 | 0.2410E- 12 71 T a b l e 3.2 Case of Y. =-1 , Ji=- 2, r. -x< > = 1 , &t = 1/200, A y=0. 2: Approximate S o l u t i o n y I t = 0. 1 J t = 0.2 | t=0.6 I t=1. 0 0.0 I 0. 0 | 0.0 | 0. 0 0. 0 0. 2 I 0.4013E+00 J 0.4158E+00 j 0.7488E- 01 J 0.4298E- 02 0.4 1 0.7122E+00 | 0.6106E + 00 J 0.9457E- 01 | 0.4346E- 02 0. 6 I 0.9069E+00 | 0.6377E+00 j 0.8494E- 01 | 0.3128E- 02 0. 8 I 0.9134E + 00 | 0. 5534E+00 J 0.6421E- 0 1 | 0.1895E- 02 1 . 0 I 0.7364E+00 | 0.4149E + 00 | 0.4304E- 01 | 0.1017E- 02 1 . 2 I 0.4760E+00 | 0.2720E+00 J 0.2615E- 01 | 0.4943E- 03 1 . 4 I 0.2468E+00 | 0.1565E+00 J 0.1456E- 01 J 0.2196E- 03 1 . 6 I 0. 1026E+00 | 0.7903E-01 | 0.7479E- 02 | 0.8970E- 04 1 . 8 I 0.3421E-01 | 0.3503E-01 | 0.3552E- 02 J 0.3374E- 04 2. 0 I 0.9147E-02 j C.1362E-01 J 0.1562E- 02 | 0. 1 169E-04 2. 2 I 0.1961E-02 | 0. 4645E-02 | 0.6362E- 03 | 0.3728E- 05 2. 4 I 0. 3373E-03 j 0.1387E-02 J 0.2399E- 03 | 0.1091E- 05 2. 6 I 0.4653E-04 | 0.3625E-03 | 0.8366E- 04 | 0.2922E- 06 2. 8 I 0.5147E-05 J 0.8283E-04 J 0..26 94E-04 J 0.7127E- 07 3. 0 I 0. 4566E-06 | 0.1654E-04 j 0.7990E- 05 | 0.1574E- 07 3. 2 ] 0.3248E-07 j 0.2880E-05 | 0. 2178E-05 J 0.3123E- 08 3. 4 I 0.1854E-08 | 0. 4372E-06 j 0. 5443E-06 | 0.5516E- 09 3. 6 I 0.8482E-10 j 0. 5773E-07 j 0.1242E- 06 | 0.8562E- 10 3. 8 I 0.3113E-11 | 0.6620E-08 | 0.2578E- 07 | 0.1148E- 10 4. 0 I 0.9161E-13 | 0.6578E-09 j 0.4844E- 08 J 0.1297E- 11 4. 2 I 0.2162E-14 | 0. 5649E-10 | 0.8194E- 09 | 0.1188E- 12 4. 4 I 0. 4093E-16 I 0.4179E-11 J 0.1237E- 09 | 0.8220E- 14 4. 6 I 0.6214E-18 | 0.2654E-12 | 0.1634E- 10 | 0.3624E- 15 4. 8 ] 0. 7566E-20 J 0. 1422E-13 J 0.1700E- 11 | 0.4304E- 17 5. 0 I 0.0 | 0.0 1 0. 0 I 0.0 T a b l e 3.3 Case of Y e = - 1 , Y i = 2, r . - x . =1, A t = 1/200, A y=0. 5: Approximate S o l u t i o n y I t=0.1 J t=0.2 J t=0i6 I t=1. 0 0.0 | 0.0 | 0.0 | o.o 0.0 0. 5 I 0.8293E+00 j 0.6954E+00 J 0. 1486E+00 J 0.5030E- 02 1 . 0 I 0. 7364E+00 | 0.4636E+00 | 0.3022E- 01 | -0.3333E- 02 1 . 5 I 0.1636E+00 | 0". 9035E-01 | -0.3717E- 02 | 0.1068E- 02 2. 0 I 0.9147E-02 J 0.1557E-02 | -0.2624E- 03 | -0.1749E- 03 2. 5 I 0. 1288E-03 | -0.2102E-03 | 0.2250E- 03 | -0.2094E- 04 3. 0 i 0. 4566E-06 j 0. 1185E-04 | -0.6658E- 04 | 0.2994E- 04 3. 5 I 0.4076E-09 | -0.4714E-06 | 0.1465E- 04 | -0.1245E- 04 4. 0 I 0. 916 IE-13 J 0. 1206E-07 J -0.2653E- 05 | 0.4824E- 05 4. 5 I 0.5184E-17 | 0. 4367E-10 | 0.2838E- 06 0.1198E- 06 5. 0 j 0.0 J 0.0 } 0.0 0.0 72 Table 3.4 Case of Y 0=-9, Y»=10, r 0 - x , = 1 , At=1/200, ay=0. 1: a) Approximate S o l u t i o n + +. I y I t=0. 1 J . t=0. 2 J t=0.6 j t=1.0 J + + i 0.0 | 0.0 J 0.0 J 0.0 | 0.0 J 1 0.1 | 0. 3790E-01 | 0. 8683E-10 J -0. 1828E-16 | -0.4110E+03 | | 0.2 j 0.2909E-01 | 0.1174E-10 | 0.4149E-16 J -0.3597E+03 J | 0.3 | 0.1697E-01 | -0.7850E-11 J 0.1648E-16 | 0.5144E+03 | | 0.4 | 0.8808E-02 | -0.3676E-11 | -0.1853E-16 | 0.3175E+03 | | 0.5 | 0. 4236E-02 J -0. 5408E-12 | -0. 9087E-17 | -0. 3553E+03 | | 0.6 | 0.1914E-02 | 0.2231E-12 | 0.5665E-17 ) -0.2154E+03 | | 0.7 | 0.8160E-03 | 0.1423E-12 | 0.4191E-17 j 0.2045E+03 | | 0.8 | 0.3289E-03 J 0.2590E-13 J -0.1283E-17 ] 0.1317E+03 | | 0.9 | 0.1254E-03 | -0.6200E-14 | -0.1693E-17 | -0.1071E+03 | | 1.0 J 0. 4524E-04 j -0. 4940E-14 J 0. 1532E-18 | -0.7589E+02 | | 1.1 | 0. 1545E-04 I -0. 1066E-14 | 0.6114E-18 | 0.5272E+02 | | 1.2 l 0. 4990E-05 | 0.1510E-15 \ 0.4412E-19 | 0.4181E+02 J | 1.3 | 0.1526E-05 | 0.1610E-15 | -0.2002E-18 | -0.2479E+02 | | 1.4 J 0. 4414E-06 j 0.3815E-16 } -0.4213E-19 J -0.2219E + 02 | | 1.5 | 0. 1208E-06 | -0. 3391E-17 | 0. 5998E-19 | 0.1122E+02 | j 1.6 | 0.J131E-07 j -0.4756E-17 J 0.2051E-19 I 0.1138E+02 j | 1.7 | 0. 7677E-08 | -0.1201E-17 | -0. 1652E-19 J -0.4918E+01 | | 1.8 | 0.1781E-08 j 0.5919E-19 | -0.7930E-20 | -0.5651E+01 j | 1.9 | 0.3911E-09 | 0. 1274E-18 | 0.4192E-20 | 0.2095E+01 | | 2.0 | 0.8127E-10 | 0.3368E-19 j 0. 2671E-20 | 0. 2724E+01 J | 2. 1 | 0. 1598E-10 | -0. 7698E-21 ( -0. 9821E-21 | -0. 8703E+00 | | 2.2 | 0.2974E-11 | -0.2960E-20 | -0.8121E-21 | -0.1276E+01 J j 2.3 | 0.5237E-12 | -0.8036E-21 | 0.2132E-21 ( 0.3542E+00 | 1 2.4 | 0. 8727E-13 ] -0. 1498E-22 J 0.2269E-21 | 0.5814E+00 \ I 2.5 | 0.1376E-13 | 0.5512E-22 | -0.4353E-22 | -0.1420E+00 | | 2.6 | 0.2054E-14 } 0.1823E-22 | -0.5883E-22 3 -0.2582E+00 | j 2.7 | 0.2900E-15 | 0.1499E-23 | 0.8693E-23 | 0.5639E-01 J I 2.8 ] 0. 3876E-16 J -0. 9270E-24 j 0. 1428E-22 I 0.1119E+00 \ 1 2.9 | 0.4903E-17 | -0.3842E-24 | -0.1845E-23 | -0.2235E-01 | | 3.0 J 0. 5868E-18 | -0. 4408E-25 J -0.3270E-23 \ -0. 4735E-01 | | 3. 1 | 0.6646E-19 | 0. 1302E-25 | 0. 4567E-24 | 0. 8902E-02 | | 3.2 j 0.7123E-20 j 0.6281E-26 J 0.7153E-24 | 0. 1959E-01 | j 3.3 | 0. 7225E-21 j 0. 9722E-27 | -0. 1324E-24 J -0. 3577E-02 | | 3.4 | 0.6935E-22 | -0. 8826E-28 J -0. 1522E-24 j -0.7927E-02 j | 3. 5 J 0. 6299E-23 j -0.8150E-28 j 0.4105E-25 | 0. 1450E-02 | | 3.6 J 0.5414E-24 | -0. 1853E-28 | 0. 3228E-25 i 0.3135E-02 J 1 3.7 J 0.4404E-25 j -0.7519E-30 | -0.1249E-25 I -0.5896E-03 | | 3.8 | 0.3390E-26 | 0.7798E-30 | -0.7010E-26 | -0.1210E-02 | J 3.9 j 0. 2470E-27 J 0. 2560E-30 J 0. 3575E-26 | 0.2378E-03 j | 4.0 | 0.1703E-28 | 0.3340E-31 | 0.1579E-26 | 0.4534E-03 | i 4. 1 | 0. 1111E-29 J -0. 2755E-32 | -0.9454E-27 J -0. 9356E-04 j | 4.2 I 0.6857E-31 | -0. 2346E-32 | -0. 3643E-27 | -0. 1633E-03 | 1 4. 3 1 0. 4006E-32 ] -0. 5444E-33 | 0. 2288E-27 J 0.3503E-04 J | 4.4 | 0.2215E-33 J -0. 5047E-34 | 0. 8251E-28 | 0. 5533E-04 | | 4.5 J 0.1159E-34 J 0.8936E-35 j -0.4965E-28 | -0.1198E-04 j | 4.6 | 0.5737E-36 | 0.4516E-35 | -0.1701E-28 | -0.1674E-04 | 1 4.7 | 0. 2688E-37 | 0.9138E-36 J 0. 8978E-29 | 0.3405E-05 J | 4.8 | 0.1192E-38 | 0.9998E-37 | 0.2645E-29 | 0.3817E-05 | | 4.9 | 0. 5001E-40 | -0. 1561E-37 J -0. 9083E-30 | -0. 5299E-06 J 1 5.0 I 0.0 | 0.0 I 0.0 | 0.0 | + -r 73 T a b l e 3.4 + J y 1 + — Case of Yo=-9 f Yi = 10, r e - x a = 1 r at=1/200, Ay=0.1 b) Exact S o l u t i o n t=0. 1 t=0. 2 I t=0.6 t=1. 0 • - + I 0.0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0.8 0. 9 1.0 1. 1 1.2 1.3 1.4 1. 5 1. 6 1.7 1.8 1.9 2.0 2. 1 2. 2 2. 3 2. 4 2. 5 2.6 2. 7 2.8 2. 9 3. 0 3. 1 3. 2 3. 3 3.4 3. 5 3. 6 3. 7 3.8 3. 9 4. 0 4. 1 4.2 4. 3 4.4 4. 5 4. 6 4. 7 4. 8 4.9 5. 0 0 . 0 0. 3790E-0 1 0.2909E-0 1 0.1697E-01 0.8808E-02 0.4236E-02 0.1914E-02 0.8160E-03 0. 3289E-03 0.1254E-03 0. 4524E-04 0. 1545E-04 0. 4990E-05 0. 1526E-05 0.4414E-06 0. 1208E-06 0. 3131E-07 0. 7677E-08 0. 1781E-08 0. 3911E-09 0. 8127E- 10 0. 1598E- 10 0.2974E-1 1 0. 5237E- 12 0.8727E-13 0. 1376E- 13 0. 2054E-14 0. 2900E-1 5 0.3876E-16 0. 4903E- 17 0.5868E-18 0. 6646E-19 0. 7123E-20 0. 7225E-21 0.6935E-22 0. 6299E-23 0. 5414E-24 0. 4404E-25 0. 3390E-26 0. 2470E-27 0. 1703E-28 0. 1111E-29 0.6857E-3 1 0. 4006E-32 0. 2215E-33 0. 1159E-34 0.5737E-36 0. 2688E-37 0. 1192E-38 0. 5001E-40 0.1986E-41 0.0 0.7437E-06 0.4921E-06 0.2396E-06 0. 1025E-06 0. 4056E-07 0.1518E-07 0.5434E-08 0.1871E-08 0.6221E-09 0.2 0 0 1 E-09 0.6230E- 10 0. 1880E- 10 0.5500E- 11 0.1561E-11 0.4294E- 12 0.1146E-12 0.2967E-13 0.7453E-14 0.1816E- 14 0.4292E-15 0. 9842E- 16 0.2189E-16 0.4725E- 17 0.9891E-18 0. 2009E- 18 0. 3958E- 19 0". 7566E-20 0. 1 403E-20 0.2524E-21 0. 4405E-22 0.7457E-23 0. 1225E-23 0.1952E-24 0.3017E-25 0. 4524E-26 0.6581E-27 0. 9288E-28 0. 1272E-28 0. 1689E-29 0.2177E-30 0. 2721E-31 0. 3300E-32 0. 3882E-33 0. 4431E-34 0. 4906E-35 0. 5270E-36 0. 5491E-37 0. 5551E-38 0. 5444E-39 0. 5180E-40 0. 0 0.1468E-38 0.5592E-39 0.1578E-39 0.3911E-40 0. 8974E-4 1 0.1953E-41 0.4080E-42 0.8249E-43 0.1621E-43 0.3109E-44 0.5828E-45 0.1070E-45 0.1926E-46 0.3405E-47 0.5911E-48 0.1009E-48 0. 1693E-49 0.2796E-50 0. 4544E-5 1 0. 7269E-52 0.1145E-52 0.1776E-53 0.2714E-54 0.4086E-55 0.6059E-56 0.8853E-57 0.1275E-57 0.1808E-58 0.2528E-59 0.3484E-60 0.4732E-61 0.6334E-62 0.8356E-63 0.1087E-63 0.1393E-64 0.1760E-65 0.2191E-66 0.2690E-67 0.3255E-68 0.3882E-69 0.4565E-70 0.5290E-71 0.6045E-72 0. 6808E-73 0.7558E-74 0.8272E-75 0. 8924E-76 0. 9491E-77 0.9949E-78 0.0 0.0 0. 0 0. 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0.0 0.0 0.0 74 Ta b l e 3.4 Case of Yo=-9, Yt = 10, r 0 - x , = 1, At=1/200, Ay=0.1: c) E r r o r y i t=0.1 J t=C.2 J t=0.6 I t=1.0 J 0.0 | 0.0 | .0.0 I 0.0 0.0 | 0. 1 1 0.0 3 0. 7496E-06 \ 0.1828E- 16 J 0.4110E+03 3 0.2 | 0.0 | 0. 4920E-06 1 -0.4149E- 16 3 0.3597E+03 J 0. 3 | 0.0 J 0.2396E-06 J -0.1648E- 16 J -0.5144E*03 J 0.4 | 0. 0 | 0.1025E-06 | 0.1853E- 16 | -0.3175E + 03 J 0.5 | 0.0 I 0. 4056E-07 | 0.9087E- 17 J 0. 3553E+03 \ 0.6 | 0.0 | 0. 1518E-07 | -0.5665E- 17 | 0.2154E+03 | 0. 7 | 0.0 ] 0.5434E-08 | -0.4191E- 17 j -0.2045E+03 | 0.8 | 0. 0 | 0.1871E-08 J 0.1283E- 17 | -0.1317E+03 | 0.9 j 0.0 | 0.6221E-09 J 0.1693E- 17 J 0.1071E+03 J 1.0 | 0.0 | 0.2001E-09 | -0.1532E- 18 | 0. 7589E*02 J 1. 1 | 0.0 | 0.6230E-10 J -0.61 14E-18 | -0.5272E+02 | 1.2 I 0. 0 | 0.1880E-10 3 -0.4412E- 19 3 -0.4181E+02 | 1. 3 | 0.0 | 0. 5500E-11 | 0.2C02E- 18 j 0.2479E+02 J 1.4 | 0.0 3 0. 1560E-11 | 0.4213E- 19 3 0.2219E+02 | | 1 . 5 J 0.0 | 0.4294E-12 | -0.5998E- 19 | -0.1122E+02 I 1.6 | 0. 0 | 0.1146E-12 | -0.2051E- 19 J -0.1138E+02 J 1.7 I 0.0 3 0.2968E-13 | 0.1652E- 19 } C.4918E+01 | 1.8 | 0.0 | 0.7453E-14 J 0.7930E- 20 | 0.5651E+01 | ] 1. 9 | 0. 0 } 0.1816E-14 J -0.4192E- 20 J -0.2095E+01 J I 2.0 | 0.0 | 0.4292E-15 3 -0.2671E- 20 | -0.2724E+01 | I 2. 1 | 0.0 | 0.9842E-16 J 0.9821E- 21 | 0.8703E+00 | I 2.2 | 0. 0 | 0.2190E-16 | 0.8121E- 21 | 0.1276E+01 | 1 2. 3 | 0.0 } 0. 4726E-17 J -0.2132E- 21 ] -0.3542E+00 j | 2. 4 | 0.0 | 0. 9892E-18 | -0.2269E- 21 I -0.5814E*00 3 1 2.5 | 0.0 ) 0. 2008E-18 J 0.4353E- 22 | 0. 1 420E + 00 | | 2.6 | 0.0 J 0.3956E-19 J 0.5883E- 22 | 0.2582E+00 | | 2. 7 J 0.0 J 0.7564E-20 J -0.8693E- 23 J -0.5639E-01 | | 2.8 | 0.0 3 0. 1404E-20 | -0.1428E- 22 | -0.1119E+00 | I 2.9 1 0.0 3 0.2528E-21 J 0.1845E- 23 } 0.2235E-01 J 1 3.0 1 0.0 | 0. 4409E-22 3 0.3270E-•23 | 0. 4735E-01 3 3 3. 1 1 0.0 J 0. 7444E-23 J -0.4567E- 24 | -0.8902E-02 | 1 3.2 | 0. 0 | 0.1218E-23 | -0. 7.153E- 24 | -0.1959E-01 | 3 3. 3 J 0. 0 J 0.1942E-24 ] 0. 1324E-•24 J 0.3577E-02 1 3.4 | 0.0 | 0.3025E-25 3 0. 1522E--24 | 0. 7927E-02 | 3. 5 1 0.0 1 0.4605E-26 \ -0.4105E-•25 J -0.1450E-02 | 3.6 | 0.0 | 0.6766E-27 | -0.3228E-•25 | -0.3135E-02 1 3.7 1 0.0 J 0.9363E-28 } 0.1249E-•25 3 0.5896E-03 1 3.8 | 0.0 | 0.1194E-28 | 0.7010E-•26 | 0. 1210E-02 3 3. 9 j 0. 0 | 0.1433E-29 3 -0.3575E-•26 j -0.2378E-03 1 4.0 | 0. 0 | 0. 1843E-30 | -0. 1579E--26 | -0.4534E-03 i 4. 1 | 0.0 3 0.2996E-31 J 0.9454E--27 | 0. 9356E-04 | 4.2 I 0.0 | 0. 5646E-32 | 0. 3643E--27 | 0. 1633E-03 3 4. 3 J 0.0 3 0.9326E-33 3 -0. 2288E--27 J -0.3503E-04 1 4.4 1 0.0 | 0.9478E-34 | -0.8251E--28 | -0. 5533E-04 1 4.5 | 0.0 1 -0.4030E-35 ] 0. 4965E--28 | 0. 1198E-04 1 4.6 | 0.0 3 -0. 3989E-35 | 0. 1701E--28 | 0. 1674E-04 J 4. 7 J 0.0 1 -0.8589E-36 J -0. 8978E--29 3 -0.3405E-05 3 4.8 | 0. 0 | -0.9443E-37 J -0. 2645E--29 | -0.3817E-05 J 4.9 | 0. 0 J 0.1616E-37 J 0.9083E--30 j 0. 5299E-06 1 5.0 | 0.1986E-41 | 0.5180E-40 | 0.0 0.0 75 CHAPTER IV DISCUSSION AND CONCLUSIONS In t h e p r e c e d i n g c h a p t e r s we have e x p l i c i t l y c a l c u l a t e d the u n d e r l y i n g L i e group s t r u c t u r e of the i n i t i a l boundary value problem (1.1, 1.1a, 1.1b) f o r r (t) = Y c e * + Y i e t . Knowing t h a t the F o k k e r - P l a n c k e g u a t i o n can be t r a n s f o r m e d t o the s i m p l e h e a t e g u a t i o n we c o u l d t h u s o b t a i n the L i e groups and the s i m i l a r i t y s o l u t i o n s f o r t h e t h r e e c l a s s e s of moving b o u n d a r i e s by a p p l y i n g r e s u l t s i n [ 8 ] and f o l l o w i n g t h e s i m i l a r procedures o f example 1 i n C h apter I . The f i r s t passage time d i s t r i b u t i o n can th e n be c a l c u l a t e d by ( 2 . 4 ) , which i s s i m p l e t o ap p l y i n g e n e r a l . . The a p p l i c a b i l i t y of the f i r s t passage time d i s t r i b u t i o n t o m o d e l l i n g depends, t o a c e r t a i n e x t e n t , on the s i m p l i c i t y o f the form. A l t h o u g h g 3 has been known f o r a w h i l e and a number o f n e u r a l m o d e l l e r s have mentioned [ 1 1 , 15, 16] t h e c l o s e d e x p r e s s i o n of g 4 w i t h Y 0 = 0, i t seems t h a t t h e f u l l e x p r e s s i o n of gj has been i g n o r e d . In the case o f n e u r a l m o d e l l i n g a f u r t h e r s t e p i s t o f i t t h e f i r s t passage time d i s t r i b u t i o n t o the i n t e r s p i k e i n t e r v a l h i s t o g r a m o r t o e s t i m a t e t h e moments from the i n t e r s p i k e i n t e r v a l h i s t o g r a m d i r e c t l y [ 2 9 , 30, 3 1 ] . . The h i s t o g r a m s of P f e i f f e r and Kiang [ 3 2 ] d i s p l a y v ery s i m i l a r s t r u c t u r e s t o t h o s e of g, and g, (see F i g u r e 4.1, 4.2, 4.3) . The f i t t i n g of g 3 t o e x p e r i m e n t a l data f o r the d i s c h a r g e p a t t e r n of s i n g l e neurons i n F i g . *J-.l Graphs of 9,(t) as Y.» a Parameter i n the Threshold, Changes es F i g . 4.2 Graphs of 3t<*> as /t„-x0, the Distance between the I n i t i a l Membrance Potential and the Threshold at t*o, Changes 79 t h e r a t c o c h l e a r n u c l e a s was c a r r i e d out by N i l s s o n [ 3 3 ] . . He found t h a t t h e m o d i f i e d Marguardt a l g o r i t h m [ 3 4 ] was s u p e r i o r t o the o t h e r L e a s t Sguares and Maximum L i k e l i h o o d methods. A number of o t h e r f u n c t i o n s , such as L e v i n e and S h e f n e r ' s hypernormal d i s t r i b u t i o n [ 3 5 ] and C. Smith's Gamma d i s t r i b u t i o n [ 3 6 ] a l s o g i v e r e a s o n a b l y good f i t t o o t h e r d a t a . The f i t t i n g of t h e two parameters ( r 0 - x 0 ) and Y 0 t o g A , however, can p r o v i d e an e s t i m a t i o n of the u n d e r l y i n g b e h a v i o u r of r e l a t i v e r e f r a c t o r y p e r i o d , which i s not an o b s e r v a b l e , i . e . we cannot measure i t d i r e c t l y . A model which p r o v i d e s some i n s i g h t i n t o the h i d d e n mechanism i s c e r t a i n l y b e t t e r than one t h a t cannot [ 3 6 ] . I n Chapter I I I we have shown i n d a t a i l t h a t the s m a l l time a s y m p t o t i c e x p a n s i o n of the s o l u t i o n u ( x , t ) o b t a i n e d by T a y l o r e x p a n s i o n of u ( x , t ) i n t and s u b s t i t u t i o n of a t r a n s f o r m e d e g u a t i o n s a t i s f i e d by u ( x , t ) . We may extend t h i s i d e a t o f i n d t h e a s y m p t o t i c e x p a n s i o n of c e r t a i n t y p e s of i n t e g r a l s which s a t i s f y some d i f f e r e n t i a l e g u a t i o n s . A l g o r i t h m 3.4 p l u s the d i s c r e t i z a t i o n of the y p a r t i a l d e r i v a t i v e s g i v e s us a complete scheme where the mesh s i z e s must s a t i s f y c e r t a i n c o n d i t i o n s i n o r d e r to have convergence of t h e approximate s o l u t i o n s t o t h e e x a c t s o l u t i o n s . T h i s r e s u l t i s summarized i n Theorem 3 . 1 . The s t a b i l i t y problems a r e a l s o c l e a r l y i l l u s t r a t e d by e x p l i c i t c o m p u t a t i o n s . We summarize the major c o n t r i b u t i o n s of t h i s t h e s i s i n t h e f o l l o w i n g : (1) We s u c c e s s f u l l y o b t a i n e d an a n a l y t i c s o l u t i o n f o r a F o k k e r -P l a n c k e g u a t i o n w i t h a two parameter f a m i l y of moving 80 b oun d a r i e s . (2) We d e r i v e d a new g e n e r a l e x p r e s s i o n f o r t h e f i r s t passage t i m e d i s t r i b u t i o n of a d i f f u s i o n p r o c e s s w i t h a t r a n s i t i o n p r o b a b i l i t y d e n s i t y f u n c t i o n s a t i s f y i n g t h e g e n e r a l F o k ker-P l a n c k (Kolmogorov forward) e g u a t i o n f o r any smooth moving b o u n d a r i e s . (3) We showed t h r e e o t h e r e g u i v a l e n t f o r m u l a t i o n s of the problem f o r the two parameter f a m i l y of moving b o u n d a r i e s , and t h e i r r e l a t e d f i r s t passage time d i s t r i b u t i o n s . (4) We d e f i n e d f o u r n u m e r i c a l a l g o r i t h m s aiming t o s o l v e t h e g e n e r a l moving boundary problem. We c a l c u l a t e d the s m a l l t i m e a s y m p t o t i c e x p a n s i o n of the i n t e g r a l s o l u t i o n o n l y t o f i n d t h a t t h e a s y m p t o t i c a l g o r i t h m i s e g u i v a l e n t t o a T a y l o r e x p a n s i o n a l g o r i t h m . (5) The convergence and s t a b i l i t y c o n d i t i o n s were e s t a b l i s h e d f o r a n u m e r i c a l scheme a p p l i e d t o t h i s e g u a t i o n w i t h g e n e r a l moving b o u n d a r i e s . F i n a l l y we s h o u l d p o i n t out t h a t t h e i d e a t h a t t h e a s y m p t o t i c e x p a n s i o n of an i n t e g r a l u (x, p) with r e s p e c t t o the parameter p i s e g u i v a l e n t t o the T a y l o r e x p a n s i o n of u i n p w i t h the s u b s t i t u t i o n of the p a r t i a l d e r i v a t i v e s ( — \ ) , k = 1, 2, by a d i f f e r e n t i a l r a l a t i o n Up = g ( x r p, u, Ux , u X K , ...) s a t i s f i e d by u ( x r p) may be worth n o t i c i n g s i n c e t h e l a t e r method i s i n g e n e r a l much e a s i e r t o a p p l y than the former one as we have demonstrated i n Chapter I I I . As we have mentioned i n t h e b e g i n n i n g of t h i s c h a p t e r one can f i n d t h e s i m i l a r i t y s o l u t i o n s 81 f o r a l l t h r e e c l a s s e s o f moving b o u n d a r i e s , i n p a r t i c u l a r , t h e moving b o u n d a r i e s of c l a s s I I ' , which i s d e f i n e d i n Chapter I , wit h 6 = 0 and & < 0 p r o v i d i n g a 3 parameter moving boundary which may c o n t a i n more s t r u c t u r e s so t h a t i t f i t s c l o s e r t o t h e r e a l moving boundary than the two parameter f a m i l y . The s o l u t i o n i n t h i s c a s e i s i n terms of A i r y and B e s s e l f u n c t i o n s [ 8 ] , 82 BIBLIOGRAPHY [ 1 ] A. E i n s t e i n , Ann. d. Physik 17, 549 (1905). [ 2 ] M. .V. . Smoluchowski, Phys. Z e i t s . . 17, 557 ( 1916). [3] G..E..Uhlenheck and L . S . O r n s t e i n , On the theory of the Borwnian motion. P h y s i c a l Review 36, 823 (1930). [ 4 ] M. .C. .Wang and G. E. Uhlenbeck, On the theory of the Brownian motion I I , Reviews of Modern P h y s i c s 17, 323 (1945) . [5] S..Chandrasekhar, S t o c h a s t i c problems i n p h y s i c s and astronomy. Reviews of Modern Physics 15, 1 (1943).. [6] J . L. Doob, The Brownian movement and s t o c h a s t i c eguations..Annals of Mathematics 43, 351 (1942).. [3, 4, 5, 6] r e p r i n t e d i n N. Wax: S e l e c t e d papers on n o i s e and s t o c h a s t i c processes. Dover, New York (1954). [ 7 ] G..W. Bluman, S i m i l a r i t y s o l u t i o n s of the one-dimensional Fokker-Planck eguaton. I n t . J . Non-Linear Mechanics 6, 143 (1971) . [ 8 ] G..W. Bluman and J. D..Cole, S i m i l a r i t y methods f o r d i f f e r e n t i a l e g u t i o n s . A p p l i e d Mathematical S c i e n c e s 13, S p r i n g e r - V e r l a g , pp258-274 (1974)., [ 9 ] R. . I . .Cukier, K. .Lakatos-Lindenberg and K..E. . Shuler, Orthogonal p olynomial s o l u t i o n s of the Fokker-Planck eguation. . J. . of S t a t i s t i c a l physics 9, 137 (1973). [10] J..D. Cowan, S t o c h a s t i c models of n e u r o e l e c t r i c a c t i v i t y . . In S. A. R i c e , K. F. Freed and J. C. L i g h t : S t a t i s t i c a l Mechanics, New Concepts, New Problems, New A p p l i c a t i o n , U n i v e r s i t y of Chicago Press, pp104-130 (1972) . [ 1 1 ] J . .R. .Clay and N. S . G o e l , D i f f u s i o n models f o r f i r i n g of^ a neuron with varying t h r e s h o l d . J . Theor. B i o l . .39/ pp633-644 (1973) . [12] A. V. Holden, Models of the s t o c h a s t i c a c t i v i t y of neurons. L e c t u r e notes, Biomathematics 12, S p r i n g e r -Verlag (1976). [ 1 3 ] A. . S i e g e r t , On the f i r s t passage time p r o b a b i l i t y problem. P h y s i c a l Review 81, 617 (1951).. 83 £14] Do ft. D a r l i n g and A. S i e g e r t , The f i r s t passage problem f o r a c o n t i n u o u s Markov p r o c e s s . .Ann. .Math. S t a t i s t . . 24, 624 (1953) . [ 1 5 ] P. . I . M. Johannesma, D i f f u s i o n models f o r the s t o c h a s t i c a c t i v i t y of neurons.. In E. E . . C a i a n e l l o : N e u r a l Networks, S p r i n g e r - V e r l a g , pp116-144 (1968). [ 1 6 ] S. Shunsuke, On the moments of t h e f i r i n g i n t e r v a l of the d i f f u s i o n approximated model neuron. M a t h e m a t i c a l B i o s c i e n c e s 39, 53 (1978). [ 1 7 ] G. W. Bluman, On the t r a n s f o r m a t i o n o f d i f f u s i o n p r o c e s s e s i n t o the Wiener p r o c e s s . . (To appear i n SIAM J . A p p l i e d M a t h e m a t i c s ) . [ 1 8 ] G. F. D. Duff and D. N a y l o r , D i f f e r e n t i a l e g u a t i o n s of A p p l i e d M a t h e m a t i c s . . W i l e y (1966).. [ 1 9 ] E . S c h r o d i n g e r , P h y s . . Z e i t s 16, 289 (1915).. [ 2 0 a ] M. C. K. Tweedie, I n v e r s e s t a t i s t i c a l v a r i a t e s . Nature 155, 453 (1945).. [ 2 0 b ] Some s t a t i s t i c a l p r o p e r t i e s of i n v e r s e G a u s s i a n d i s t r i b u t i o n s . V i r g i n i a J . S c i . 7, 160 (1956) . [ 2 0 c ] S t a t i s t i c a l p r o p e r t i e s of i n v e r s e G a u s s i a n d i s t r i b u t i o n s I . Ann. Math. S t a t i s t . . 2 8 , 362 (1957) .. [ 2 0 d ] S t a t i s t i c a l p r o p e r t i e s of i n v e r s e G a u s s i a n d i s t r i b u t i o n s I I . Ann. Math. S t a t i s t . 28, 696 (1957) . [ 2 1 ] A..Wald, S e g u e n t i a l a n a l y s i s . . N e w York, Wiley (1947).. [ 2 2 ] J . L. F o l k s and E. S. C h h i k a r a , The i n v e r s e G a u s s i a n d i s t r i b u t i o n and i t s s t a t i s t i c a l a p p l i c a t i o n - a r e v i e w . ,J. R. S t a t i s t . Soc. B 40, 263 (1978).. [ 2 3 ] D. R. Cox and H. D. M i l l e r , The t h e o r y of s t o c h a s t i c p r o c e s s e s . London, Methuen (1965)._ [ 2 4 ] M. T. Wasan, F i r s t passage t i m e d i s t r i b u t i o n of Brownian motion w i t h p o s i t i v e d r i f t ( i n v e r s e G a u s s i a n d i s t r i b u t i o n ) . Queen's paper i n Pure and A p p l i e d Mathematics 19, Queen's U n i v e r s i t y ( 1 9 6 9 ) . . [ 2 5 ] W. .F. Ames, N u m e r i c a l methods f o r p a r t i a l d i f f e r e n t i a l e g u a t i o n s . Academic P r e s s (1977). 84 [ 2 6 ] L. S i r o v i c h , Techniques of a s y r a p t o t i c s a n a l y s i s . , a p p l i e d M a t h e m a t i c a l S c i e n c e s 2, S p r i n g e r - V e r l a g (1971). [ 2 7 ] N. B l e i s t e i n , Uniform a s y m p t o t i c e x p a n s i o n s of i n t e g r a l s w i t h s t a t i o n a r y p o i n t near a l g e r h r i c s i n g u l a r i t y . Communication on Pure and A p p l i e d Mathematics 19, 353 (1966) . . [ 2 8 ] F. John, P a r t i a l d i f f e r e n t i a l e g u a t i o n s . A p p l i e d M a t h e m a t i c a l S c i e n c e s 1, S p r i n g e r - V e r l a g (1978).. [ 2 9 ] H. .C. . T u c k w e l l , D e t e r m i n a t i o n of t h e i n t e r - s p i k e t i m e s of neurons r e c e i v i n g randomly a r r i v i n g p o s t - s y n a t i k p o t e n t i a l s . B i o l . C y b e r n e t i c s 18, 225 (1975). [ 3 0 ] H..C. T u c k w e l l , On t h e f i r s t - e x i t time problem f o r t e m p o r a l l y homogeneous Markov p r o c e s s e s . J . Appl. Prob. 13, 39 (1976) . . [ 3 1 ] H. . C. . T u c k w e l l and W. R i c h t e r , Neuronal i n t e r s p i k e t i m e d i s t r i b u t i o n s and the e s t i m a t i o n of n e u r o p h y s i o l o g i c a l and n e u r o a n a t o m i c a l p a r a m e t e r s . . J . Theor. B i o l . 71, 167 (1 978) . [ 3 2 ] R..R. P f e i f f e r and N. Y-S. K i a n g , Spike d i s c h a r g e p a t t e r n s of spontaneous and c o n t i n o u s l y s t i m u l a t e d a c t i v i t y i n the c o c h l e a r n u c l e u s of a n e s t h e t i z e d c a t s . B i o p h y s i c a l J o u r n a l 5, 301 (1965) . [ 3 3 ] H..G. N i l s s o n , E s t i m a t i o n of parameters i n a d i f f u s i o n model. Computers and B i o m e d i c a l Research 10, 191 (1977) . [ 3 4 ] R. F l e t c h e r , A m o d i f i e d Marguardt s u b r o u t i n e f o r non-l i n e a r l e a s t s g u a r e s . H a r w e l l R e p o r t , AERE-R, 6799 (1971) . [ 3 5 ] M. . W. .Levine and J . M. .Shefner, A model f o r t h e v a r i a b i l i t y of i n t e r s p i k e i n t e r v a l s d u r i n g s u s t a i n e d f i r i n g of a r e t i n a l neuron. B i o p h y s i c a l J o u r n a l 19, 241 (1977). [ 36 ] C. E. Smith, A comment on a r e t i n a l n e u r a l model. B i o p h y s i c a l J o u r n a l 25, 385 (1979).. 85 APPENDIX I DEFINITIONS^ + where r € [ 0 , o o ) , N t f W , Z G ( r - J ; Z) = H ( Z ) , HeiQ, r) , H (Z) i s r e g u l a r , (1) NOTATION G' n(r;Z) = ^G«(r;Z). LEMMA A l l fc„(0) = t l n - t O ) , £ « + i ( 0 ) = A . ( l ) - a ^ i ( l ) , f o r n=0 ( 1 ) N; (2) where H (Z) has a z e r o at Z = 0 of degree r , U r (s) = I Z e ' dZ, r-i (r) = 0, (3) r0 (r) = G (r;0) , (4) r , (r) = ( G ( r ; 0 ) - G ( r ; - a ) ) / a , (5) r i n (r) = (r + 1 ) G n (r;0) , (6) r i l W, (r) = (r + 1 ) (Gn (r;0)-G„ (r ; - a ) ) / a + Gj, (r;-a) , (7) G(r;Z) = ft (r) + r,(r) Z+Z (Z+a) G, (r;Z) , (8) (r + 1) G n ( r ; Z) +ZG* (r;Z) = ran (r) + ran-, (r) Z+Z ( Z t a ) G h f i (r;Z) , (9) f o r n= 1 ( 1 ) N. . G n{0;Z) = ZG n(1;Z)+ Jfl*.|{1) , f o r n=1 (1) N. P r o o f : (By i n d u c t i o n ) ( i ) n=0: By (4) , (1) , (3) To(0) = G (0;0) = ZG (1;Z) | J s o = 0 = fit. By ( 5 ) , (10) , (1) ,<8) l*i(0) = (G(0;0) -G (0;-a) ) /a = -G (0 ; - a ) / a = ZG(1;Z)/(-a) = -aG(1;-a) )/(-a) . /. o*i (0) = iTo(1)-a rj<1) . ( i i ) n=1: By ( 1 ) , (8) ,(11) G(0;Z) = ZG(1;Z) = t o (0) + ifi (0) Z +Z (Z + a) Gi (0; Z) = 0 + ( r«(1) -a £ (1) ) Z+Z (Z + a) Gi (0;Z) . D i v i d e d by Z, we have G(1;Z) = f i ( 1 ) - a fi(1)+(Z+a)Gj (0;Z). . By (8) a g a i n G(1;Z) = lfi(1)Z + Z(Z + a ) G i (1;Z). E g u a t i n g ( 1 2 ) , (13) g i v e s r 0(1)-a eT, (1) + (Z*-a)G, (0;Z) = o*o(1)+ Z+Z(Z>a)Gi (1;Z) G, (0;Z) = ZGj (1; Z) + r»{1) . By (6) , (14) 8^(0) = G i (0 ; 0) = *i<1) . By (7) , (15) , (14) and (9) f, (0) = (Gi (0;0)-Gi (0;-a))/a+GH0;-a) = ( f i ( 1 ) - ( ( - a ) G i (1;-a)+ fi(1)))/a • (Gj (1;-a)+ (-a)G'i (1;-a) ) = 2Gt (1;-a)-aG«i(1;-a) . 87 r,(0) = r * ( i ) - a r 3 ( i ) . (16) (14) , (15) , (16) imply the lemma i s t r u e f o r n=1. Assume the lemma i s t r u e f o r n=k, i . e . * U(0) = ^ i - l d ) r G*(0,Z) = ZGftJ1;Z) + fik-l (1) , (17) (18) (19) ( i i i ) n=k+1: By (9) , (19) G4L(0;Z) +ZG ^(0 ;Z) = fa (0) + + » (0) Z + Z (Z+a) Gt+i (0 ; Z) = ZG*(1;Z) + r»ft-i (1) +Z(G*.(1;Z)+ZG%(1;Z)) . By (17) and d i v i d i n g both s i d e s of the eguation by Z, 2GA(1;Z)+ZG»4(1;Z)•= Ifafi + i (0)+(Z + a)G^.H-i (0;Z). (20) By (9) again f o r r=1 2G«.(1;Z) +ZG'A(1;Z) = r 2 l ( 1 ) + *U +,(1)Z Eguating (20) ,(21) and a p p l y i n g (18), we have ^ ( 1 ) - a j T z f t t , (1)+ (Z*a)G#. + i(0;Z) = A*(1)+ Jfifctid) Z+Z(Z+a)Gfcti(1;Z) Divi d e d by (Z+a) , we have G i t i (0;z) = ZGA+I (1 ;Z) + (1) o r G* +,(0;Z) = ZG * +t (1; Z) + tf£cjU.>- «. (1) . . (22) By (6) , (22) oWo(0) = Gfc + l(0;0) = j \ l t U . ) - l (1). (23) By (7) , (22) , (23) ft<4«-i>+ i (0)=(Gtfi ( 0 ; 0 ) - G i + , (0;-a))/a+G'*+i(0;-a) = ( *\cA+o-i (D-(-aG**, (1;-a) + r*ift+i>-i(D ))/a + Z(Z + a)Gft + 1 (1,Z) (21) 88 + ( G-JUi ( 1 ; - a ) - a G ^ + l ( 1 ; - a ) = 2G*+i ( 1 ; - a ) - a G ' * + i n ; - a ) . By (9) we have A c i * « > + 1 (0) = fol*.*') (1) -a j T i t ^ + o t i (1) o . (24) (22) , (23) and (24) i m p l y the Lemma i s t r u e f o r n=k+1. 2ID THEOREM i l l S{0) = S ( 1 ) . P r o o f : I t i s easy to see from (2) t h a t U (s) = -0 ' (s) , U e (s) = J V 2 e T e r f c (s/i"2) and U i ( s ) = s 0 o < s ) - 1 . _ We have S(0) = ^ - [ u . f / S t ) f C ( Q - f t C „ ^ x £ s ( 1 ) = f [ K , I J X 4 ) ^ (l) — ft. Tin 4| N o t i c e t h a t U 0 (s) ^  , ± + £ H) JT/ 1 *"* as s -» +* ^vtr £ * as s -* - *>, hence, s t r i c t l y s p e a k i n g , the f i r s t p a r t s o f S (0) and S(1) are p r o d u c t s of two a s y m p t o t i c s e r i e s . To prove S(0) = S ( 1 ) , however, i t i s s u f f i c i e n t t o show t h a t (0) = X\« (1) -a it„ + l<1) and ft* (0) = j f t n - i (1) , n=0 (1) N. By Lemma A1, t h i s i s i n d e e d t h e case. 

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