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

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c.  S I M I L A R I T Y SOLUTION OF A FOKKER-PLANCK EQUATION WITH A MOVING, ABSORBING BOUNDARY by RICHARD TSAN MING LEE B.Sc,  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 MASTER  FOR THE DEGREE OF OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES The I n s t i t u t e  of A p p l i e d  M a t h e m a t i c s 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 c o n f o r m i n g to the required  standard  THE UNIVERSITY OF B R I T I S H September  (c)  COLUMBIA  1980  R i c h a r d T s a n M i n g L e e , 1980  In p r e s e n t i n g t h i s t h e s i s an a d v a n c e d d e g r e e a t the L i b r a r y I further for  agree  the U n i v e r s i t y  make  it  freely  this  written  thesis for  It  Mathematics  The  of  British  Columbia  2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  September 30> 1980,  requirements I agree  r e f e r e n c e and copying of  this  that  not  for  that  study. thesis  b y t h e Head o f my D e p a r t m e n t  financial gain shall  permission.  the  B r i t i s h Columbia,  i s understood  Department of  Date  of  that permission for extensive  representatives.  University  fulfilment of  available for  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  by h i s of  shall  in p a r t i a l  or  copying or p u b l i c a t i o n  be a l l o w e d w i t h o u t  my  ii  ABSTRACT  A one derive  parameter L i e group of  a c l o s e d form u  = u  t  similarity + (xu)  x x  x  transformations  s o l u t i o n of the  ,-«><x<r{t)  with  a(-o»,t) = u ( r ( t ) , t ) = 0 ,  where  r (t)  0  the  problems. A of  A new  time  for  application  passage time  (Kolraogorov f o r w a r d )  we  asymptotic expansion i n o r d e r t o be  of  find  of  distribution  used t o  four  find  equivalent  of the i n t e g r a l  that  i t  Taylor expansions, transforming  equivalent  and  i t i s not  regular  one  a p p l y i n g some e x p l i c i t  criteria  are derived  admits  arbitray  r (t) .  for  the  an  the  the f e a s i b i l i t y  is  of  Stability  solution  p i e c e d t o g e t h e r t o form  method and  (*)  i s found.  powerful  an  \  e q u a t i o n i n an  i s subsequently  a more g e n e r a l r (t) . E x a m i n i n g  method,  ~  '  distributions  distribution  i s calculated  solution this  passage  s m a l l time  {*)  t > 0 ,  t  domain i s d e r i v e d , which  first  to  equation:  = Y e* + Y e * .  the g e n e r a l Fokker-Planck  irregular  used  u ( x , 0 ) = <j>(x) ,  A simple expression f o r the f i r s t of  #  is  irregular  to  a  simple  b e t t e r than domain  the  to  schemes. C o n v e r g e n c e explicit  method  of  a and  which  iii  TABLE  OF  CONTENTS  ABSTRACT  i i  TABLE OF CONTENTS  i i i  L I S T OF TABLES  ••••••  •  v  L I S T OF FIGURES  vi  ACKNOWLEDGEWfWT'.  viii  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  1.4  RELATION TO THE HEAT EQUATION  CHAPTER  II  10 ..  FIRST PASSAGE TIME DISTRIBUTION  15 22  II. 0  INTRODUCTION  22  II. 1  EQUIVALENCE TO u/^-«J = 0 WITH FIXED BOUNDARY  23  II. 1  FIRST PASSAGE TIME PROBLEM  31  t  CHAPTER I I I  ASYMPTOTIC AND NUMERICAL  III. O  INTRODUCTION  111.1  ASYMPTOTIC EXPANSION  111.2  A  FINITE  FOR SMALL TIME  BIBLIOGRAPHY  45  DIFFERENCE METHOD FOR ARBITRARY MOVING  COMPUTATIONS  CHAPTER IV  42 42  BOUNDARY III. 3  PROCEDURES  DISCUSSION AND CONCLUSIONS  55 58 75 82  APPENDIX  V  LIST  Table  2. 1  With  OF TABLES  C o n n e c t i o n s Between Moving  Boundary  The  r U ) = Y. * e  Fokker-Planck +  Equation  Yi £ * And The D i f f u s i o n  Equation Table  3.1  Ay=0.1: T a b l e 3.1  .26  a)  Case  Of  Approximate b)  Case  Y„=-1,  Y = x  2,  r.-x,, = 1,  &t=1/200,  Solution  Of  Y» =-1,  ...69 Yi =  2,  At=1/200,  r.-x#=1,  Ay=0. 1: E x a c t S o l u t i o n Table  3.2  Case Of Y « = - 1 , Y = 2, A t = 1 / 2 0 0 , r - x = 1, A y = 0 . 2 :  3.4  b)  Y = t  2, A t = 1 / 2 0 0 , r „ - X o = 1 , A y = 0 . 5 :  Solution Case  Of  a y = 0 . 1: A p p r o x i m a t e Table  ....71  0  a)  e  Solution  Case Of Y =-1,  Approximate T a b l e 3.4  e  x  Approximate T a b l e 3.3  ...70  Case  71 Y, =-9  r  YA=10,  At=1/200,  r , - x . = 1,  Solution  Of  I . =-9,  .72 Yi=10,  6t=1/200, r o - x  0  = 1,  Ay=0.1: E x a c t S o l u t i o n T a b l e 3.4  c)  Case  Ay=0. 1: E r r o r  Of  Y„ =-9,  73 Yi=10  r  At=1/200,  •  r -Xo = 1 0  r  74  vi  LIST OF  Figure  0,1  Various  Fiyure  2.1  a)  Yi =1  Forms Of V a r y i n g  3 1  C  ( s e e T a b l e 2.1)  2.2  a)  Similarity  3.1  Solution  Variable  Of  In  t  A  Variable  30  X = nt> - Y,  (see C h a p t e r  Graphs  Graphs  For r ( t ) = e "  ( s e e T a b l e 2.1)  Graphs  b) Similarity  29  Green's  T r a n s f o r m e d Domain  Figure  Thresholds  G r e e n ' s S o l u t i o n F o r r (t) = Y e* + Y, e" . Y»=0,  b)  Figure  FIGURES  Of  + Yi £ * ,  H  Is  The  I)  37  3 = fl«) ~ *Y.$ + Y»  ,  ^  Is  The  (see C h a p t e r I)  Of f(ot,z)  Showing  38 Where  The  Major  C o n t r i b u t i o n s Are Located Figure  3.2  Illustrating  a)  Ay=0.1,  b)  Ay=0.1,  Apprarent  47  Instability  Due To fit/ ( A x) > 1/2: 2  6t= 1/200: S t a b l e At=1/195:  From  The  60  Unstable,  Numerical  But  Result  In  It  Is  This  Not Time  Interval c) Figure  61  Ay=0.1, 3.3  At=1/190: Unstable  Illustrating  The  62  Propagation  Of  Error:  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  Figure 4* Figure  3.4  S o l u t i o n With  Approximate  Boundary  Condition  =0 3.5  6  S o l u t i o n With E x a c t  Boundary C o n d i t i o n s  7  68  Figure 4. 1  Graphs  Of g (t) t  As  Y  0  /  A  Parameter In The  T h r e s h o l d , Changes F i g u r e 4.2  Graphs  ••••  Of q  i  (t)  As r  4  - x» ,  The  Between The I n i t i a l Membrane P o t e n t i a l And The  6  Distance Threshold 77  At t = 0, changes F i g u r e 4.3  7  Graphs Of g (?) As Y 3  0  Changes  .78  viii  ACKNOWLEDGEMENT  I  am  grateful  t o my  supervisor,  i n t r o d u c i n g me t o t h i s t o p i c a n d g u i d i n g the  Dr. George  Bluman, f o r  me d u r i n g t h e c o u r s e o f  research. I would l i k e  t o thank Drs. U r i Ascher and John  s u g g e s t i o n s which l e d t o the improvement Thanks attention  a r e a l s o due t o D r . H e n r y  t o some u s e f u l  financial  of  Applied  T u c k w e l l f o r d i r e c t i n g my  Department  o f Mathematics  and t h e  Mathematics  and S t a t i s t i c s  for  s u p p o r t s , and t o a l l  the completion o f t h i s  thesis.  references.  I am t h a n k f u l t o t h e Institute  of t h i s  Petkau f o r  work  t h o s e whose  possible.  encouragements  the  make  1  CHAPTER 0 INTRODUCTION This  t h e s i s i s concerned  numerical computation Planck  of the s o l u t i o n t o  a  e g u a t i o n with a moving, a b s o r b i n g  The  Fokker-Planck  physical  evolution  from  a diffusion  of  a probability  approximation  Wang and U h l e n b e c k have  [4],  contributed  very  general,  some  often  in  a  i n the  i t governs  f u n c t i o n which continuous  £ 2 ] , Uhlenbeck [5],  Chandrasekhar  the arises  Markovian  and O r n s t e i n [6]  Doob  and  t o t h e d e r i v a t i o n and development o f 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 particles  Fokker-  boundary.  density  of  s o l u t i o n s and t h e  particular  arises  E i n s t e i n £ 1 ] , Smoluchowski  process..  others  eguation  and b i o l o g i c a l s c i e n c e s . I n  time  [3],  with s i m i l a r i t y  force  field.  motion  of  The e g u a t i o n t a k e s t h e f o l l o w i n g  form:  Bluman  [ 7 ] , Bluman  and  Cole  [ 8 ] obtained  solutions  of  Orthogonal  p o l y n o m i a l s o l u t i o n s c a n be d e r i v e d f o r some f o r m s o f  f{x)  t h i s e g u a t i o n f o r some f i x e d b o u n d a r y  similarity conditions.  £9]..  A Fokker-Planck appears  in  a  model  being s i m p l i f i e d , VP  _  eguation  with  f o r the f i r i n g  moving  boundary  o f neurons  £ 1 0 , 11].  After  t h e e g u a t i o n has t h e form: V_p  ^(*P>  -oo < x < rct> , t >o  w i t h boundary c o n d i t i o n s : P(-oo,t) = P ( r ( t ) , t ) and  (b=b(t))  i n i t i a l condition:  P(x,0)  =^(x),  t  = 0  2  where P ( x , t ) = t r a n s i t i o n  probability  membrane p o t e n t i a l ,  density f o r the  which  satisfies  t h e Chapman-Kolmogorov e q u a t i o n ; t = time; x = a measure o f t h e membrane r (t)  = a measure o f t h e t h r e s h o l d ( s e e F i g u r e 0 . 1 ) ;  t(x)  = an a r b i t r a r y if  The time  potential;  d e n s i t y f u n c t i o n , f o r example,  x = x, a t t = 0, f (x)  = £(x-x ). 0  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  problem  will  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  & number o f f o r m s f o r r ( t ) h a v e been s u g g e s t e d 81,  the  passage  thesis.. [12,  pp76-  corresponding references are also given here.]. In the  following  list,  the subscript s i n  l a b e l S o f t h e graph  1r It)  i s  s  referred  to  the  i n F i g u r e 0.1.. OL  (i)  Hagiwara  (1954),  f It) = h  C  T  ,* > »  [ a l s o i n 11, e q u a t i o n 4 ] ; (ii)  Buller,  N i c h o l l s and S t r o m  (iii)  C a l v i n and S t e v e n s  (iv)  Geisler  (1965),  and G o l d b e r g  (1953),  r (t> =  V It) ~  ;  t  ( 1 9 6 6 ) , 1-j Li.) = / oo V&  (v) In parameter  Weiss Chapter  (1964),  ;  fc  e  T  i  i f o^t<«., i f t >* •  r^(t) * <x! €  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  f a m i l y o f r (t)  which  i n c l u d e s (v) a s a  special  case.  It)  r  •  . o  —j  i i  i i i i  0  F i g . 0.1  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  1  2  Various Forms of Varying  3  Thresholds  i i  4  i i i  [  i i  i i  5  i i i i > | l i i i i  6  i i  i i | i  7  i i i i  4  Then  we  t r a n s f o r m t h e e q u a t i o n t o t h e s i m p l e h e a t e q u a t i o n and  consider  the  several to  transformed  t h e two In  parameter  Chapter  II  Ornstein-Uhlenbeck  solutions.  we  the  f u n c t i o n s t o each o t h e r .  The  [13,  i s formulated generally  and  i n t e r p r e t a t i o n g i v e n . This f o r m u l a t i o n i s then  used  The  models  c o n n e c t i o n between  Green's  the f i r s t  problems.  the  the  time problem  find  by  and  passage  to  method  process  first  physical  this  passage  the  14]  Wiener  time d i s t r i b u t i o n  process,  r e s u l t s of t h i s chapter demonstrate are  actually  that  e q u i v a l e n t when v i e w e d  distribution  i n the o r i g i n a l  other  the  parameter  case,  f a m i l y o f r (t) , i s w e l l  Chapter computing  special  variables  I I I i s devoted  to  together  integral  s o l u t i o n s f o r t h e two four  the  distinct  small  the  [10]  is  the  setting  up o f  an  explicit  transformation  to  the  a  one  16].. feasibility  moving b o u n d a r i e s  of by  time asymptotic expansions of the  most  Finally  time  On  parameter and  problem  powerful computation  moving  boundaries.  we  We  show t h a t t h e method o f to one.  a  fixed  We  demonstrate  scheme  based  boundary  on  method and d e r i v e t h e a s s o c i a t e d c o n v e r g e n c e  criteria.  of  proper  passage new.  15,  examining  t r a n s f o r m i n g t h e moving b o u n d a r y  stability  known [11,  algorithms  problem  is  some  i n the  corresponds  the s o l u t i o n u(x,t) f o r a r b i t r a r y  piecing  define  (x, t)  which  and  of four e q u i v a l e n t  c o o r d i n a t e s . To o u r k n o w l e d g e t h e f o r m o f t h e f i r s t  hand  used  r(t).  establish  relate  its  Consequently,  i n [ 8 ] c a n be  We i l l u s t r a t e  f a m i l y of  subsequently  the  conditions.  c l a s s e s o f moving b o u n d a r i e s d e r i v e d  construct similarity  using  boundary  g i v e some n u m e r i c a l  examples  the the and to  5  i l l u s t r a t e the  stability  problem.  6  CHAPTER I S I M I L A R I T Y SOLUTIONS  Ls.9.  INTRODUCTION In  section  we f i n d  (1)  sufficient  we d e f i n e t h e p r o b l e m .  c o n d i t i o n s which t h e  I n s e c t i o n (2)  infinitesimals  of  the  one p a r a m e t e r L i e g r o u p o f t r a n s f o r m a t i o n s must  satisfy  in  order  without  to  consideration section that  leave of  i n v a r i a n t the eguation  initial  and  and  boundary  under t h e t r a n s f o r m a t i o n s . similarity  solution  one  heat  group of t h e s i m p l e heat similarity  Subsequently  section  f o r the  In  i n order  conditions are also invariant  which admits  moving b o u n d a r i e s . . I n  THE  conditions..  (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  initial  into  boundary  itself  there  emerges  a  a two p a r a m e t e r f a m i l y o f  (4) we t r a n s f o r m t h e  problem  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 e q u a t i o n c a n be u s e d t o  s o l u t i o n s of our o r i g i n a l  construct  problem.  PROBLEM Consider  eguation  following  with one-sided  - u where  the  t  + cx u)  x  - «*> < x <• r i t )  subject  moving  linear  partial  differential  boundary,  =o  (1 • 1)  , t > o ,  t o t h e boundary c o n d i t i o n s f o r t >• o ,  u (-«>-+.> « u cirtt), t ) = o ui*,o) : U x We s h a l l  find  the  ,  (1.1b)  Xo<rc>).  forms  of  r (t)  (1- l a )  admitted  by  the  7  similarity  I_2  method. .  INVARIANCE OF THE EQUATION We  shall follow  t h e general set-up  2.0 a n d 2 . 1 ] , C o n s i d e r  [8,  part I I section  Lie  group o f t r a n s f o r m a t i o n s : u* = U * ( x , t . u  ;  o f Bluman a n d C o l e  £) = u + £•? ( * , t ,  t  a one p a r a m e t e r  0(£ )  ,  l  X* = X * U , t , R ; t ) = X + £ * Cx,t,U>  + Oil*)  i*  + 0(t ) .  = T * U , i , u ; Z) = t •» i T C x , t ,  u = 9 (*, -t )  If  I v  (  U  2  )  J  i s a n i n v a r i a n t s o l u t i o n o f (1.1), i t  satisfies  Expanding  <*x.*x* , U^* and U * i n  one h a s f r o m (1.3)  t = o,  * U + 11**' » t+  +  fl  *  0*K  - et + (x©)  where t h e f i r s t  and  t h e second  x  After  =o  series  about  lfi + C M + <*+ *>( * T) + 0 ( £ ) c  8  +i>  l  (1.4)  ,  extensions:  extension  *2.x*u e x 6* -  (1.4)  Taylor  x  fix*  - ?uu e x l e t  - 2.ext  -3  implies  the coefficient  8 «fix-  making t h e s u b s t i t u t i o n  fix*  &«x e t - 2 Tu e*t 0x . of 2  must be z e r o .  9„ r g - (x©)* r we h a v e t h e y  t  8  following  equation.:  t <M*u -  < *U«  +  *  + S )* 8  - * Sxu + a- * \»  +  )  l  " * !?«  " * * * > 6«  0±  T  °'  (1.5)  The  group  coefficients 0<0tr  • St +  *t  e  i s f o u n d by e q u a t i n g X  differential  eguations  1)  of the terras 0 , 8 fl*t  a n d  determining  (1.  of  6x •  The  eguations  in  equations. we  M*« -  It  ^  , f  +  X  , $ 6 x  X  linear  and  this  8  t r  resulting  infinitesimals  are l i s t e d  *  » 6 0* r S S  Solving  f i n d the  These e q u a t i o n s  t  to zero  T  set  the  , 0  t  partial  are c a l l e d of  the  determining  which s a t i s f y  (1.5).  below.  »  +  !  - •  ( 1  S„,o T„  =  (1.  *  +  -  THM  =  7  *  +  r  7 )  ° 0  = 0  10)  (1.11) (1.  12)  0  (1.  13)  o  (1.  14)  (1.  15)  (1.  16)  ? » «  f«H  ?x  "  n-9)  *U« - * ! H  ,  3 X  =  O  9  (1..16),  where  and  (1.9)  T=  give  (1.11)  ?<x,t>  (1.17)  5  r  f  r i <x,t> u + 3 <x,-fc)  $ (x,t )  ( 1 . 18)  ( 1 . 19)  f , 3 are a r b i t r a r y . (1.10),  (1.12),  and  (1.13)  are  (1 - 1 4 )  satisfied  consequently. and  (1.15)  Recalling  imply  (1.17)  that  u  t 0  ?-7<t).  («,t ) ,  (1.20)  from  (1.6)  and  (1.19)  we  have <7**  *  + *i<  Noticing  + (3x - 3 X  that  w= J U , * )  and  o  f»  -  K  *  (1.7)  +  t  =>  2f„  (1.8)  x  Solving  ' ^  I (x,t) =  where  t  x  * +  K  $  *  (1.1),  o +  t  +  $  * 0  (1.23)  „  ( 1 . 24)  we have  (1.23)  A') -  ( 1 . 25) *.  (t'+  1.") + 3 ( t )  5<t) a r e a r b i t r a r y f o r t h e t i m e  substitution  of  we  (1.22)  Alt),  - —(A +  A(t) and The  \£  z  and  (1.24)  i Ix,*) =  . |„  of  (1.21)  conditions:  a f«  *  • <*3>x ) * 0  i sa solution  have t h e f o l l o w i n g s u f f i c i e n t 9 *  t  (1.24)  and  (1.26)  ( 1 . 26)  being. into  (1.22)  gives X \ V " + 4 . T ' >  +  Hence  1  I A"-  2.  8  the  following  A  )  -  ((3't  l  4  sufficient  "  - 2)*o  .  J.  conditions  are  obtained: T " ' - 4 ?  =  0  (1.27)  10  (1. 29) Notice in  that there  a r e a l l together  t h e s o l u t i o n o f (1.27),  the  infinitesimals  7 , _j  s i x free  (1.28) and ( 1 . 2 9 ) , and \  .  We  parameters  and h e n c e i n  shall  show  the  following: (i)  I f a l l t h e boundary  similarity two  s o l u t i o n of (1.1),  parameter  (ii)  I f only  solution The  (1.1a) and  o f (1.1) and (1.1a) a d m i t s a f i v e  ( i i )i n  eguation  admits  a  (1.1) a n d (1.1a) a r e i n v a r i a n t , t h e s i m i l a r i t y  c o n d i t i o n a t t = 0 c a n be s a t i s f i e d  case  (1.1b)  rit). .  We s h a l l show c a s e  I_3  conditions are also invariant, the  section  i n t o t h e heat  (i) (4)  r(t)  .  by s u p e r p o s i t i o n . .  i n section by  parameter  (3) , and i n v e s t i g a t e  mapping t h e F o k k e r - P l a n c k  eguation.  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 b o u n d a r y  c o n d i t i o n s a r e i n v a r i a n t under t h e group t r a n s f o r m a t i o n s have a s i m i l a r i t y (a)  Invariance  o f (1.1a)  a two parameter  i s equivalent  w ( r i t * ) , t*) r o  u*im>,t)e  Mr(ti,t)  solution admitting  to  the  . Expanding about  we  rtt)  .  condition l ? 0 gives  + € 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  Thus by (1. 25)  \ ( r<0, t ) + H  t  (r(t),t) %-rim  ou ). l  11  T<t)r'(t^ = $ < m i , t ) (b)  Invariance x * *  of  T  + A(t>.  (1.30)  (1.1b) means i n v a r i a n c e o f t h e s o u r c e a t  f o r t = o „ Thus  0  X*=  X  t* = and It  £  Yli  =  a  t  t i [ ( X , , 0 ) r X, ,  o  +  £ T(-o) ^  u*(x,0= follows that  and by e x p a n d i n g  W(x*,o)r  S ( X*- X, ) .  (1.31)  T (o) r a  (1.32)  ]f O C o ) r f /  (1.31) a b o u t 6 t o , we  f ( X , o) ? -  have  ( x„. 0 ) ,  ;  (1.25) and  0  (1.32) g i v e  (1.33)  X» V c o )  +  A ( 0 >  -  0  I  r (1.30) a n d  (1.32) g i v e  r  ° ^  ( a )  + A to) = o .  ft*  Since  x» ^ r , i t f o l l o w s t h a t  have  the  8  Moreover  (c)  The  initial (1.26) and  T ' I O r ACO = o .  condition (1.33)  for  with i n i t i a l  (1.27) and  ordinary  (1.28).  eguations  conditions  TCo> = T'<») = A(») ^ 3(o) - |? ( A'fo)+ is  we  give  s o l u t i o n o f the system of l i n e a r  (1.27-29)  Thus  I J ^ ) r o  f o u n d t o be T It) r  Z0.  A (t) r b & (t) =  where  o., b  satisfy  S^t  t  -| ( C*»Ut - J ^ A 2 t ) -r £  are a r b i t r a r y  (1.30),  constants.  ( a x * + |>x  e  Since  -a)  tit)  we h a v e t h e f o l l o w i n g s o l u t i o n :  must  12  ret) = -c Q**k t + i s^k t , where  C = —  2. a.  = -r to) = - f  0  ,  £  i s a r b i t r a r y . Hence r(t)  c o n t a i n s two f r e e parameters. The s i m i l a r i t y solving  the  2.1]); and  | Q  invariant + T 6  K  solution  t  K.- 9 ( x , t )  ~  obtained  by  s u r f a c e c o n d i t i o n (see [ 8 , s e c t i o n  = t  t h e r e f o r e the c h a r a c t e r i s t i c  jJ(M)  is  T(t>>  eguations  -f(^u '  (1.34)  In t h i s problem T  It)  =  i».  5"J^K*t  + § C C o * 2t - S ^ k 2 t ) +  The f i r s t  eguation  of  (1.34)  £  ( Oi x, + t> x 0 - a. )  gives  the  t  similarity  varible  ZCS^t where  c  i s an a r b i t r a r y  The second eguation  (1.35)  constant,  assuming tx% o .  of (1.34) g i v e s the f o l l o w i n g :  (1. 36) where  A:  ^•(Xo+c)^,  convenience l e t  >  F(?) is  a  f u n c t i o n of £  f = c ( ^ - ^ ) . Then we have  o n l y . For  13  — ex  ucx,t) r i  (1.37)  Substituting differential it  i s found  e  e  0.37)  (1.1)  into  we  have  an o r d i n a r y  3 ( Y ) • A f t e r some a l g e b r a  eguation governing that  3 " _ A-i 3 = o •  The c o n d i t i o n  (1.38) s o  ulnt>,-t)  gives  3 ( Y.) = 0  (1.39)  y<> " £ ( * - ~ > .  where  c  The s o l u t i o n o f (1.38)  3(Y)= A * U J ? ( Y - Y 0 ) ) Thus  3(Y)= To  A*  u.(x,o) = S i * - * © )  as t  « (x,t)  , where  1/T -c - x . >o. r  X„tc) c Y - YoO  A* SUk  find  a n d (1.39) i s  such  that  i s satisfied  the  then  o •  3 = 3, t 3 i * e r e w i  3, =  u : R + P, where  ,  ±  / V l  It Yr  hSS.  i s easy  e  r x + c > < Y-Yo)  *  0  ~ e 1  i<«-e )Y lt  -£c*kt  p,. M i € e P> =  initial  e  *e  e  t o show t h e f o l l o w i n g :  *!£. - £  +  0  ( t ),  condition  we c o n d i d e r t h e b e h a v i o u r o f  A  Let  (1.40)  v  14  J-  Q  i  -  fl*  ^  t  O  \C f X y t Q  , YQ(X 4C) 0  •  •e ~ ~ ^  T  h  U  /x H)Y  S  =-AJ? e  *  1 .  i-rtx+e^OCoto) -0i+O  e  e  and +  t-»oT  ii I K o\ - £ f v- X-S u(x,o) = S(x-x0>  =>  A = — (> A = 7= e /TT  Hence «( *,t) =  (x, X , t) 0  47 Suit e -  e  e  where i -  e  ^  V r  15  solves  (1.1),  (1.2a) ana (1.2b),.  As a c h e c k f o r t h e a n s w e r ,  putting  Y = Yo  w  e  have  u ( r(t>, t ) = o  Ii. . 4  EIMIION  TO THE HEAT  EQUATION  Doob [ 6 ] u s e d a t r a n s f o r m a t i o n _ = x e*  ,  s = •£ e  .  l t  eguivalent t o  tr = u s '  to reduce  (1. 1) t o t h e h e a t e g u a t i o n  In  case the corresponding  this  transformed (i)  (1.42)  4  - v = o. s  b o u n d a r y c o n d i t i o n s c a n be  as f o l l o w s ;  Fort = o ,  u ( x, o)  s  J( x-x ) 6  F o r S r £ , v( j . t ^ s J T J i y - U ^ where «j„= x„ . 0  (ii)  For x : r | t ) , 4=* F o r  u( nti.t) = o  3 = & ( S ) r t ( i ^ ) /TT  where  n  s  -oo<|j<ft(S);  S  >  > V(rttt>,s)»o  ,  i s t h e domain.  Thus t h e 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. . •  where -«o < j < subject  < - > 1  S > i. ,  t o t h e boundary  V(»M,S) s V(A<<),S)  conditions  = o  for £> _ ,  OstfSCM,). The g r o u p a s w e l l a s g e n e r a l the  heat e g u a t i o n The g r o u p i s  43  (1.43a) (1.43b)  similarity  c a n be f o u n d i n [ 8 ] . .  solutions  of  16  ) where and  ^ , ? and f V  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 o f ,S M  respectively.  Let  ^  be  the  distinct  cases [ 8 ] :  easel.  |J ^*r.  similarity  variable, there are four  x  w  h  e  r  JT*O  A.litlie.  e  f  B s  V  -  JfLJsf  .  3-  ( A ?  t 6)  The moving b o u n d a r y h a s t h e f o r m :  s j ^ +ifs  + rj + As + s ^ 1  , T* ©  ^ = ( W J +  J<JlIf  where t h e p a r a m e t e r 6" i s t h e assumed  value  C a s e II.  similarity Case I I I .  variable  |3 r«<.r , v  where similarity  the  f = |1 = o , ' a * o  —L_  t = H-  p a r a m e t e r <T i s t h e assumed  = o  }  of the  )> •  variable  C a s e I V . u = (I = f  +  K ST  value o f t h e  p  ,  S~ ^ o  *-*  ^ = S ,  w h i c h means t h a t t h e r e i s no moving b o u n d a r y . In the f o l l o w i n g  e x a m p l e we s h a l l c o n s i d e r t h e  b o u n d a r y c o r r e s p o n d i n g t o a two p a r a m e t e r  l*<t).  moving  17  _______  __1  ret)  then  ft<s)  r  -  -c  + -fe s^k t =  e* +  i*s r (J3«777) s tfi { l£±J-/T + ^S=M  e  s  = *r,s 4 Y,  where y _ -c + , v _ >c»ft. ft _— '« " z  Thus i t i s i n the form of Case I I i f  I n v a r i a n c e of S = £ I n v a r i a n c e of (J =  =>  7  .'.  o(+  f •£ )  (I -t  of  + ^  it  -  + **"(:»>.  0 .  at s t  •,'. k + i Invariance  =  and  o  =  i) = <5?(s)=»  5  C  ,  tl  >  | (&(*>, 5 )  =  Thus « = £ , (i r - t > ^ = 1 r "=  a -.  K =  .  "1 -  The group ( 1 . 4 4 ) i s reduced to  j , . (|\ Solving  the  4 ,  variable  £J  •  x .  characteristic v  similarity  -  M+C  ~—- .  equation  we  have  the  18  The s i m i l a r i t y v  and  form f o r v  s-t  i s ( c f [ 8 , p217])  '  F(fc) s a t i s f i e s  which has s o l u t i o n  —  — 3.  < \  f (\) = B (\1 Si~.k(M  ,  a s e i g e n v a l u e s we would  find  < PO  Considering  X  AfX)  such  that  V(<j,S) =  j  A ( X) Sink (  (  4-  W ))•____;  <U  (1.45)  z s a t i f i e s the i n i t i a l Changing then  condition  .  v a r i a b l e : A = |^pi r A  - AlMiA,  (1.45) i s e g u i v a l e n t t o  -2—  = {* f i A tA) e  A l ^ l e e s"t  - e e *-*  ...  g  ( A+ lli:)V  -Awe i ~ = £ = = 7 ^ \  iA  e  19  ^ince It  -c s A( _) >5  follows  For  ,  H+f < o .  that  v(«j,i> =  ft 5(1-1*)  .  Thus  Substituting  ^(J>>  into  (1.46)  we  have  - A :  f(s--i)« *  The the  solution i s eguivalent  transformation  ) For yijj.t) w e  T  h  h  u  s  a  v  e  (1.41) a f t e r  (1.42) and r e c a l l i n g  -  V C j , i ) = $<!>> = 3 ?  A<p)  to  -JTe^ff+f)  that  ^ A(^±)e  * ••  making  20  (1.46)  v a r i a b l e s a g a i n w i t h A + |- ~ % we  By c h a n g i n g general  The  -e~ $(A+|)sa(MV-«))e  => V(D,S)  preceding  example  shows t h a t g e n e r a l  eguation  solutions  mapping e x i s t s  between them. Bluman  and  partial  sufficient  one d i m e n s i o n a l h e a t  and  invariant  under  transformations. clearly  falls  A c l a s s of eguation  another  eguation  into this  moving  used  to  equation  [17]  to  shows  for  similarity construct provided a that  a second  be  six-parameter  Fokker-Planck  Lie  group  e g u a t i o n we  to  of  considered  class.  boundaries  of  the  Fokker-Planck  presented  in  the  beginning  Hence we have  •  order  equivalent  section.  C a s e !•  the  c a n be o b t a i n e d by c o n s i d e r i n g t h e t r a n s f o r m a t i o n  (1.42) a n d t h e f o u r c a s e s this  the  eguation are that i t i s parabolic  a  The  be  conditions  differential  the  here  for  can  corresponding  linear  have  result  s o l u t i o n s o f t h e heat  necessary  Ah.  ret) = {^  2(j.x 4rctt r lt  +  e  lt  +Aie  l  t  + -t  where p % * j f ,  0 ,  s}e  of  21  The  Case  s i m i l a r i t y  i i " . _  v a r i a b l e  ^  ^  >  Case  where  f'=<*t' ,  As  case  II  4e %  i n  r(t) Case  case  +  +  +t  III  us  ^  the  parameter  i n  r(±>  .  --t  e  + * e  4*  o .  it  ,  *  j £ ( - t ) = G'  g i v e s  us  the  parameter  i n  .  I V  .  No  moving  Remark:  g i v e s  $-f=o,  where  t2^e  « ._  Jt  r  1  As  e  -2(t)=<T  III .  r(t> s  k-S|3  t  I  z  i n  i s  boundary  A l t e r n a t i v e l y  boundaries  by  F o k k e r - P l a n c k c o n d i t i o n s o l u t i o n  *l-{Lifzo,  one  can  c o n s i d e r i n g e g u a t i o n  to i s  for  be  a l s o  e i g e n s o l u t i o n s  d i r e c t l y  i n v a r i a n t .  s i m i l a r  to  ^ o .  o b t a i n  without  o b t a i n e d  I  these the  r e q u i r i n g In  by  that  case  groups  of  the  the  i n i t i a l  the  complete  s u p e r p o s i t i o n  (1.45).  moving  of  the  22  CHAPTER I I FIRST PASSAGE TIME 11-. Q.  DISTRIBUTION  INTRODUCTION In to  this  the  diffusion  associated general of  a  connect the Fokker-Planck  (heat)  first  eguation  passage  time  e x p r e s s i o n f o r the f i r s t diffusion  density  process  function  (Kolmogorov  passage  thereby  R e c a l l the f o l l o w i n g * - u  + (x u )  t  where  u (-<*> , t )  x  is  l  time  derive  a  new  distribution  transition general  probability Fokker-Planck  moving boundary r ( t ) .  used  to  find  eguivalent  the  first  problems.  A  result  i n Chapter  I:  = o  n-t), ±>  t  We  the  appears.  e* t Y, e  t o the boundary * u  a  of four  - oo < x < r(t)Y.  subject  passage  eguation  investigate  problem.  the  formula  time d i s t r i b u t i o n s  x  and  e g u a t i o n f o r any  this  distribution  with  satisfying  forward)  Subsequently,  new  c h a p t e r we  -  t  >o  conditions for  o  - (e -i)/Y lt  +  t > o  *•-^M  1.  where  and  (2, 1)  23  let  a = x e* , s = i £  v  t  v=uS*.  .  The f o l l o w i n g i s e g u i v a l e n t  V\n - V  r  where  - <*> < J <  s  subject  (s) =• 2Y S + Yj , S > ± » 0  t o t h e boundary c o n d i t i o n s  $ («f) *  <f><3> . -<T(_Vj_)  -ir.-y  5 = •£ Y e  .  X * X ,  EQUIVALENCE TO u) - UJ = 0 WITH F I X E D  BOUNDA_Y  where  ^ = - JkJL± ,  }J  Let  ?  3 = ft<s) - y = 2 Y s + Y, - 9 , 0  L  then  -  2!  2  v  0  T * .s - _ ,  + V? i  V  The b o u n d a r y c o n d i t i o n s a r e t r a n s f o r m e d We have  one.  o  V (3 , ±> =  II__  t o the preceding  V(*>,t)  *  VCo,T)  * o  V (* o> = y* ^ ( r 0 t V , - | > £ f <*> f  , _1  accordinglyforr >«,  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 a s f o l l o w s :  ?  24  Notice  s. .  that  *  - ft -1  "  -  *Y  6  2  - .  Yo t Y, - 3  _  t  "  _  S-  -  t(fr -zY,*) T  ±  Ye  i  o The  Green's f u n c t i o n  The density  factor  is  in brackets  functions,  which  i s a difference  suggests  the  two  Gaussian  following:  Define  v  JUM..o.^(e First  of  we  write  -  •* the  e  -  ).  s o l u t i o n i n the  following  form:  •'ft  Hence  The  V  (}  ,t)  relevant  e"  Y e V +  ^ = jf  <f  transformation  «)e ^  +  V  i s observed  ' ° Xtt<») < U • C2.2) to  be  - YoV + Y.'T  The  eguation  boundary  conditions  thus are  becomes  eguivalent  lO^-iO^ro, to  the  whereas  following:  the  25  The  , %>° •  <v>  i  Green's f u n c t i o n f o r t h i s i n i t i a l  p r o b l e m i s o b t a i n e d from  x  Thus  we  have t r a n s f o r m e d  -f (ol) "K (fr,oL, T) J e t the  original  problem t o  the  b o u n d a r y v a l u e p r o b l e m of t h e d i f f u s i o n e g u a t i o n  a semi-infinte interval, summarize the c o n n e c t i o n s Bluman to find  -miff,  solution i s  lA>($,"0* f  initial  value  (2.2):  -(±iio)  The  boundary  [17]  to  [18]..  We  2.1,.  problems i n Table  diffusion  process  which  can  be  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 diffusion  of these  solved  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 h e most g e n e r a l  transformed  which i s r e a d i l y  in  which can  be t r a n s f o r m e d  to  the  pure  e g u a t i o n , t h e r e f o r e t h e t r a n s f o r m a t i o n f r o m Form  1 t o Form 2 o f T a b l e  2.1  i s a s p e c i a l case  i n that  general  setting. Plots Figures  2.1a  of and  t h e G r e e n ' s f u n c t i o n Qj and 2.1b  respectively.  Gf+ a r e shown i n  26  Table Connections  2±\  between t h e F o k k e r - P l a n c k  Boundary  f (t) - Y „  Equation  + Y i £ * and t h e D i f f u s i o n  w i t h Moving. Eguation  +  Diffusion  Diffusion  Diffusion  Eguation  Eguation  Eguation  with  Drift  *Y,  P.D. E.  Underlying  Ornstein-  Stochastic  Ohlenbeck  Wiener  Wiener  Wiener  Process  + 1st  +-  Inde-  pendent  3 = same a s (*Y© T +Y<,*  Y('})  = *Y s+Y,-!l 8  Variable (11)  J rt zt  and  ( c o n t i n u e d on n e x t  page ....).  preceding  27  Mappings  2nd I n d e -  S -  t = £ JUxS  pendent  + T  2  T =• same a s  =  C  5-i  preceding  Variable (12) and Mappings +— Dependent  w cx,t >  Variable  VIfr.T)  and  u)^,T)e  Mappings  e = same as preceding  Y.S-Y.V  e  -t  -Y»e\y eVY,e-^ e  Y : ( C ^ I L )  C  e Y.**Y.T  Domain o f  X  €  ( - o o ,  3 6 (-«>, fll*>>  f ( t ) )  0.  00 1  11: D ( I 1 )  Moving  Hti«Y,€*+Y,e"  Boundary  r0=r(o) = Y. + Y,  t  ^ ( $ ) = i Y 0 s + Yi  +Domain o f  t *  ( o.  OO)  T * ( o , OO)  12: D ( I 2 )  ( c o n t i n u e d on n e x t page ...)  *T  6 tO, 0*)  28  V(-*>,s>  Boundary Conditions  u(rct),t) = o  s V (0^(S>,5) = O  VI26D(I2) +  Init ial  u (x,o) =  4) ( x ) Y.J  Conditions V I 1 € D ( I 1)  Green's Function  Qr, ( x , X  0  , t >  t+Y e <nt)-x) -e 0  - e  (e  -e Q*(e (r(«-K>, 4  V  Solution  U( x , t 1 =  -00  Fig.  2.1  a)  Green's S o l u t i o n f o r Alt) = Y„ e*+ Yi=i ( Table 2 . 1 )  YoE0'  s e e  Y.  0.0 Fig. 2.1  1.0  b)  2.0  3.D  4.0  Green's S o l u t i o n f o r nrt) r e in a Transformed Domain (see Table 2 . 1 ) - t  5.0  6.0  31  ____  FIRST PASSAGE TIME PROBLEM For passage 2.1.  future  reference  time d i s t r i b u t i o n  The  distributions  we  shall  compute  of the f o u r cases l i s t e d obtained  are  transformation,  to the i n v e r s e Gaussian  was  Schrodinger  given  by  i n 1915  distribution  (1945,  [ 2 0 a , b, c ]. Wald  this of  1957)  distribution  sample s i z e  Folks  and  with  (1978)  application  first  fixed  passage  boundaries  o b t a i n e d can  moving  b o u n d a r i e s . . We  extension treating  t = t . The 0  ft,, (t)  { X(°) « x , e  first  passage  [13,  is 23,  give  T  RxL-t)  derived  test.  review  24]  of  the  a  considered although  into  of  the  t h a t f o r some  straightforward directly.  process s t a r t i n g X It) t o  (&, Ct) < ftUt), V t > t„)  X(t) <•  Tweedie  distribution  usually  transformed  shall  time  which  ratio  a  to  distribution.  14,  a diffusion  (t)  R.lt) <  [21]  g e n e r a l moving b o u n d a r i e s  be  r  gave  problem  t h e n be  X (t]  and fl  of t h i s time  results  Let  (1947)  probability  [22]  up  statistical  were e s t a b l i s h e d by  seguential  Chhikara  statistical The  a  i n Table  distribution  as a l i m i t i n g f o r m f o r t h e  in  first  equivalent,  [ 1 9 ] , Some  p r o p e r t i e s of t h i s 1956,  the  the can  , Vt.<t<T,  at x  0  when  boundaries be d e f i n e d by  X<T)ert, <T)  or  X (T) * To  avoid  {T>,t} = { f t , ( t ' K Suppose P ( x t , . x, t ) 0 )  trivialities  x  <t'>  <  the of  fait')  assume ,  process  0  Vt.<t'<t>  transition the  <ft,(t )<X < ft* ( t » ) .Then 0  .  probability satisfies  the  density Kolmogorov  32  e q u a t i o n , namely t h e f o r w a r d e q u a t i o n  (the g e n e r a l  Fokker-  P l a n c k equation) , i.  H  (c4<x,t>P> -  = - J l  (MM»P)  X  and t h e backward e q u a t i o n  where and  i s the i n f i n i t e s i m a l mean  ot(x,t)  (J(x,t) i s t h e i n f i n i t e s i m a l v a r i a n c e of t h e p r o c e s s .  Then we have  P{T£t) tL.it) = Probability  t h a t a b s o r b i n g does not occur  prior  to time t . Let  3 11 I t , X. , ft.Ct), ftilt) )  be t h e p r o b a b i l i t y  0  density  f u n c t i o n o f T , then  r  By changing  "ft J  P  variable,  ( X  ».t.;  K , t ) Jx ,  _ 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 ('  Pt«o t ;<(,ltH|(^(t).R,(t»), t ) (  (  = ~ j' [ M M ' !  x =ft,<t>f j- (<IMt>  (d^lt)-  71 U x l O - O ^ . C t ) ) + P  Ct')) <*g-  <R,(t))«J^  (X ,t ;X,t)] 4  0  (2.3)  33  9 = - f It  [P„(x ,t .x,t> e  0  i-(^U)-<R,(t)) +  p219].  t>t„.  t  0  /t  i s r e a d i l y shown t h a t t h e a p p r o p r i a t e b o u n d a r y c o n d i t i o n  for absorbing barriers i s that [23,  ^ > » ] Jx .  P ix ,t  P(x  Hence  0 (  t  0}  P ( X « , t , j x , t ) must  <R, (t), ± )  =  vanish  P (x , t j ^ C t ) , - t ) e  0  there = 0 for  Conseguently,  3 r - [i_ ( < R  t t ) [ pfXo.t.j^lt^t) -P(X ,t  v  0  P+ i x  Using  t o .. x ,  6  t h e boundary  t  a J  «.(t),t ) 1  W x.  condition  and  the  Fokker-Planck  e g u a t i o n , we have  3(t |-t.,x  Hence  0|  <R,<t>(<Mtn = - J o t ( x , t ) P ( x . , i . x, t ) | x  the f i r s t  passage  time  density i s related  slopes of the t r a n s i t i o n probability density barriers,  and t h e a p p r o p r i a t e  (2-4)  0  t o the  a t the absorbing  proportional  factor  i s  the  i n f i n i t e s i m a l mean.. From an e l e m e n t a r y  concept  of transport  processes,  one  observes  t h a t 3<"t) i s t h e n e t f l u x o f t h e t r a n s i t i o n p r o b a b i l i t y  density  transferring  The  proportionality  transport  process,  in heat t r a n s f e r ,  o u t o f t h e domain t h r o u g h  the boundaries.  c o n s t a n t -J-oiCX/t^ i s t h e d i f f u s i v i t y and i t i s a n a l o g o u s  momentum d i f f u s i v i t y  t o t h e heat  of the  diffusivity  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  t r a n s p a r e n t i f we w r i t e  n^.  where similar walk  the  (2.4) a s  unit  interpretation  vector  the  normal  f o r the f i r s t  t o a fixed absorbing barrier Once  is  is  e t c . . The a n a l o g y becomes more  t o t h e boundary * . A  passage  appears  time  compute t h e f i r s t  passage  the f o u r cases l i s t e d  to  use  (2.4)  time density.  i n Table  a  random  i n[5].  t r a n s i t i o n p r o b a b i l i t y density  r e l a t i v e l y much e a s i e r  of  i s determined, i t  instead  of  We s h a l l a p p l y  (2.3)  to  (2.4) f o r  2.1.  We h a v e  7TT*  0  35  = e - ( r . - k+ i Y . ^ - t ) )  4  <o  d  )  J,(t  \',*o;  -0°,  -€rtx  '  = -e e  -  1 ,  t )  ^  -^,„(ic x..t)|  r  /  )  |  B  ret)  t-Y.N£^)-Y.fr .x.>,  i  it  0  -I»*(c -O-Y,(»;-x ) lt  e  e  0  x  3^(t(€%t-»>|o)r,-xoso,oo)  e - (*-,-x ) e  v  it  = e »r  3j(T|0/l; o,eo) i s distribution, S i ( * I °, *o  Wald's ',  i  s  3, ( t I o,X, j -c», m > ) J t Moreover ,  e  x  known  4  as  distribution  the  [22].,  inverse  Gaussian  I t i s obvious  o f 3j (t 1 e, Jr ; o, oo) s i n c e  a transformation -  ( , e^.p  0  3i f ^ I 0 . o - y ; o, oo) r  o  that  36  34-  can  be  reduced  to  a  gamma d i s t r i b u t i o n  by c h a n g i n g  variables. It  s h o u l d be n o t e d  that  J ( 1 1 t* . X© . A,(-t),tf <t>)Jt  L  may n o t  %  to be  egual  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  X may n e v e r expands.  h i t the boundaries,  that the state  particularly,  when  variable  the  domain  We h a v e t h e p r o b a b i l i t y o f e v e r r e a c h i n g t h e b o u n d a r i e s  V  -  p ^ o l o  jL  ,99  le  " ^ X -  I?"  €.  Hence Yo o i s t h e c r i t i c a l s  Once  value  we have t h e f i r s t  o  if  Yo S  i f  Yo > o .  (see F i g u r e 2 . 2 ) .  passage time d i s t r i b u t i o n , i n order  to  study  i t s p r o p e r t i e s , f o r e x a m p l e , 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}  is readily  found  "> -it  r J  o  The  (see [ 2 3 , p221j) -Yok-oSCToV*)*  where Y £ o .  "* Laplace  t o be  0  transform  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 V a r i a b l e (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 V a r i a b l e (see Chapter I)  PAGE 39 DOES NOT EXIST  PAGE 40 DOES NOT EXIST  41  h a n d l e . We  where  have  V-x„  , \ >, 0  o .  However, f o r a s p e c i f i e d given,  one  integration.  can  distribution  easily  obtain  i . e . where Y„ and the  moments  by  J-  0  are  numerical  42  CHAPTER I I I ASYMPTOTIC AND NUMERICAL _____  PROCEDURES  INTRODUCTION Our Planck  a i m i s t o compute eguation  we a l r e a d y form  with a r b i t r a r y  have t h e s o l u t i o n  r e t ) = Y<>e* + Yi C ,  a  +  seguence •Let  the  Algorithm  {t;.} be a s e g u e n c e  3.1:  the  approach  each  i s to  points  We  interval  (t i ,  We  for  r  A t l  r't)  compute  K(*,T) by  )  we  algorithm.  approximates (ii)  t*+i  i.e.  algorithms:  A global look  the  f i ta  i n t°;Tj ,  t  (i)  of  Y H) .  form t o of  Fokker-  r(t) . since  f o r a moving b o u n d a r y  of b o u n d a r i e s o f t h i s  the following  of  moving boundary  naive  For consider  solution  (t) = Y»>'  i n this  an  the  £  which  interval.  approximate  using  e +• Yi,i*i  -t  solution  following  Utf(x,T) t o recurrence  relation. . r (o) 4  (3.1)  where  t - _ , < t * t,- , t-  •e- Y , , ( f t P ) - * i ) a  *<t-tf.,>  43  (see T a b l e  Algorithm  2. 1) .  3.2: An a s y m p t o t i c  (i)  Same as A l g o r i t h m  (ii)  We the  use  the  algorithm. 3. 1 ( i ) .  s m a l l time  asymptotic  i n t e g r a l i n (3.1) i n s t e a d o f  integration  over  a  expansion  performing  semi-infinite  of the  domain..(see  s e c t i o n I I I . 1 f o rt h e d e t a i l s of the expansion.) Algorithm  3.3: A T a y l o r e x p a n s i o n  (i)  Same as A l g o r i t h m  (ii)  We u s e Form 4 i n T a b l e following  3. 1 ( i ) . 2.1, i . e . . we  apply  the  transformations:  If z e * c r ( t ) - x )  (iii)  algorithm.  ,  t - ± ( e ^ - o  We c o m p u t e t h e T a y l o r  expansion  t  o f cd(},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  in 7  directly. .  Thus  + where  *(*)(*, 7;)  0 C( T A)  (  j  jfl  - 7^)  7)  )  (3.2)  44  The  d e r i v a t i v e s i n J- c a n be a p p r o x i m a t e d  appropriate  Algorithm  3.4:  (i)  We the  finite  An e x a c t  d i f f e r e n c e [25,  boundary  the  p17].  algorithm.  use t h e t r a n s f o r m a t i o n moving b o u n d a r y  by  JJ = f ( t j - x  problem  [10] to  into a fixed  map  boundary  problem.. Hence  u  where  t  u.^  s  o < <j < oo  t ((|J t F(i>) u ) , ,  (3.3)  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 (ii)  Same  as  partial  We  Algorithm  algorithm'  3. 3 ( i i i )  d i f f e r e n t i a l eguation  observe  that  initial  condition.  except  that  used i s (3.3)..  Algorithm  3.1  i n the sense t h a t f o r each  is x  a  'global  i n t i m e tj- , / +  we h a v e 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  ij  a  3.2,  semi-infinite  and  3.4  are  domain,  'local  derivatives  of  the  approximated  locally.  From since  the  Algorithm  whereas A l g o r i t h m s  algorithms'  since  I n t h i s sense  over  the  dependent v a r i a b l e i n tj  not b e t t e r than the other  the  Algorithm  can 3.1  3.3 X be is  algorithms.  p o i n t of view of p r a c t i c a l 3.3(i) i n v o l v e s  an  computation  approximation  of  45  the  moving  boundary  f(t)  does n o t , a n d A l g o r i t h m 3 . 4 { i i ) , i ti s obvious Now t h e r e m a i n i n g a  new  method?  How  whereas A l g o r i t h m  3.3 ( i i i )  i s eguivalent  asymptotic  expansion  guestions  are: I s Algorithm  different  this  Algorithm  Algorithm  to  t h a t 3.4 i s b e t t e r t h a n 3.3. 3.2  i s i t from t h e o t h e r  Algorithms. I n s e c t i o n 1 of t h i s chapter the  3.4 ( i )  we show  that  3.2 i s t h e same a s t h e T a y l o r  3.3. T h i s i s o u r m a j o r r e s u l t i n  chapter. In  s e c t i o n 2 we u s e A l g o r i t h m 3.4 t o s e t up an  explicit  finite  difference  scheme.  We  find  the  c o n d i t i o n s f o r c o n v e r g e n c e o f t h e scheme. In  s e c t i o n 3 we a p p l y t h e  section  2  to  do  procedure  some n u m e r i c a l  s e t up i n  computations.  s o l u t i o n s f o r t h e t w o p a r a m e t e r moving b o u n d a r i e s evaluated  i n order  to  make  a comparision  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  Exact are  with the  conditions f o r  t h e scheme a r e t e s t e d .  III.1  ASYMPTOTIC EXPANSION FOR SMALL TIME We w r i t e t h e s o l u t i o n i— u(x t) = J2£ e  t + Y  '  ;  where  >  2,  (x.-t) r  -Y oi  e  a  3-  .  (1.41) i n t h e f o l l o w i n g f o r m : Yo  %  [lilx t>(  -xCob-S) "  $<r.-.l>e  Ulx,t>]  1  de*  ,°° ~ *AU> , . =J 4> «> e <K  46  r*u,t>* ( e 3= e C Y . c t t t  *iw)e  Observe t h a t J moving  * *  r(t) .  each g i v e n i n i t i a l of  .  e  ^ T 7  o i s a measure o f  boundary  expansion  ?c*> r ' Y o 0 t <fr(>w> ,  Y, € * * ) ,  Moo = - l*-V*  .  jot = J 4>w>e d o t ,  I  As t-»  0  condition  •  the distance  from  the  , X"*°°, a n d t h e r e f o r e f o r  4(x)  utx,o> =  and I j ,  (  }  ,  an  asymptotic  a n d hence o f 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 o f  II^MsfJuie^" ^'^ 1  where  -j ( o l . %) = - c « - f ) \  ^0  - *o <  L  Clearly, behaviour  of  v e r y much  on \  and I  Z  a r e i n t h e form  the asymptotic  expansion  ( s e e F i g u r e 3 . 1 ) . . We  I<  fro .  of I  of I  . The depends  distinguish  the  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,  expansion  _  where  V?-<*> .  We s e p a r a t e i u  ;  I  A!  i n t o two p a r t s , i . e .  r (/; • j . ) < M * > e '  o*  T  '* j e t ,  4 8  The  second p a r t i s  J_ = j .  $  e  <U  it  P(j>  where and  i s t h e gamma f u n c t i o n ,  the f i r s t  part i s j ,  =  j  (ot;  (ol) e  0 0°  Hence  ,.,*«/ *t'  oo  Notice that t h e formal expansion even d e r i v a t i v e s o f ^ U )  o n l y depends  on t h e  a t «<•=$-.  |=o.  (ii)  Obviously,  we  only  have t h e second p a r t o f t h e  i n t e g r a l i n (i) , hence  In general the formal expansion for  |=o . S i n c e  we have d i f f e r e n t  of J  (  expansions  does n o t v a n i s h for  1 -o  and  |sS>o  and a s 3 " * ° t h e e x p r e s s i o n s do n o t c o n v e r g e t o e a c h  other,  a  uniform  asymptotic  expansion  a  maximum  global  expression  i s sought.  The  uniform  for this class of integrals containing near  PP167-170, 27]. B a s i c a l l y ,  an e n d p o i n t  has been s t u d i e d [ 2 6 ,  i t simultaneously  takes  into  4 9  account the  the contributions  endpoint  of  B l e i s t e i n [27], 1^)1)  i  a t t h e maximum o f  integration.  Applying  and a t  the  result  of  we have t h e f o l l o w i n g :  = J*$W>e*  f W t ; i r )  Jel.  where  -f<*;|> * " <*-f>  X  ft  Strictly  speaking,  oi ^ -^-j—J , s u c h t h a t that  $(«0  - L . —  t h e two s e r i e s a r e  D  u  "  t  should 1 1 1  be  written  Appendix  I  we  as show  equivalent.  The s y m b o l s i n t h e s e r i e s  mean t h e f o l l o w i n g :  ft  w,'u) y i i e  1  S  ^fc(^) -  l = * w^)  -  I  ,  .00  where  _ , »  Subseguent  The eguations:  terms  i n the s e r i e s are determined  functions  { Gf (l) } n  satisfy  the  recursively  following  50  More  precisely,  we  calculate  the f i r s t  few t e r m s i n t h e  following:  G(tf*>  -ill-  _  4>ifr)  n  Jf : Gr,(o)r  where  14  ( 4 ) ) ' (^ .  (By L ' H o s p i t a l ' s  i s  found  application of L'Hospital's  by  rule)  differentiation  and  rule:  rr'l -  Hence  ^  *»• Notice the  t h a t when t h e s a d d l e  point  endpoint of i n t e g r a t i o n (ot=o),  d e f i n e d . I n t h i s c a s e we have  the  now  expansion of  turn  fx A.  are  on  still  JT, = Jfx = o ,  2 We  (et =• £) c o a l e s c e s  '  5  our a t t e n t i o n t o the uniform I t i s e a s y t o show t h a t  asymptotic <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 r e a d i l y shown by i n d u c t i o n f o r n=0(1)N,  U ]  =  that  ,  fx* t ft] = - C - *1 . f o r n=0(1)N,  Q _i e.) - -G*C* -a) n  and  ;  ;  ^ [ 2 ftl = ;  $  1-2;  ,  Thus  I ( X; -j) =  where  ,  --  Notice  Hence  4» u> e  that  W  and  vv);<4) -  II  - 1(A ;  0  *  ( ^ t  W  0  (-*>  *Xi =  JMI  r /Ti ^  X  Now expansion:  we  have  *7*  £  the  / z  X»«  small  time  Jrt'i-TTZ: . r  A  A *  uniform  fa*  K l t t * .  J  asymptotic  7)  52  Using  the  first  UU,t> r  Comparing  2 terms of  Y , 5 e  *  (3.8)  Y , T  yip  with  (3.7)  we  get  { f x [ ? 0 > + T f * J > ] + 0 lT*>^  the  solution  in  (3.8)  T a b l e 2.1< >, l  we  o b s e r v e t h a t by l e t t i n g  + 0(T )  W (fr) + T the with  solution  3.2  u)(j.,?>  boundary  It  is  V  0  a  which  asymptotic  satisfies  expansion  w^j, i  of  ) - tO (},t) =• o T  conditions  should  be  clear  i s same as A l g o r i t h m  relate  uniform  Yi.'s  a l l the  t h a t we 3.3  up  to the  have shown t h a t  t o 0 (T*) .-We  initial  now  Algorithm proceed  distribution  to  w(J,o)  directly. Since r  f  *  xn  _  , and  s i n c e we  (*»'  4io) ~(i  can  write  U=-o.  0.  we  can  ( o)  relate  explicitly; guantity i s  however, r  %n  and  (3.6)  - 0.  Differentiating  shows t h a t t h e -  (3.5)  to  < * <*« ( 2 > ) ' t w i c e , we  | , _ 2  A  interesting  .  have f o r n=1(1)N  < > 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 motivates f o u r t h e g u i v a l e n t f o r m i n T a b l e 2.1. l  only  more  one  to  find  the  53  Differentiating repetitively,  again  twice  and  shifting  n t o n-1  we g e t  i*>  tSi  (•)  4* -ft.  2  Differentiating  Q  IZ)  =  n!  (3-4)  ( » " H H i ( 5 ,  1  (2n+2)  ^»  times  ?  e -  a.  •  gives  •+ <«**>(«$,(*))  Thus  Substituting  (3.9) i n t o  ( 3 . 6 ) we  have  the  explicit  series I ( X ; \) - ICX; -%)  R e c a l l i n g t h a t «i = - y i j we h a v e  f  = t and Q (.*) r  y  I(A;\) - I (X; -J)  = / f f f*>+ £ t t  • O C T  S  -  " ) }  and U ( X  From asymptotic  Table  2.1  expansion  and t h e p r e c e d i n g we h a v e t h e u n i f o r m f o r ^Ij-.T)  :  54  UM}.,!} = oOl^-.o) + Since  we  have  ^,<»)T^ + O C T  ).  (3.10)  gives  consistency.  expansion partial  (3. 10)  j f <*>  i s about  We  (3.10)  the  ( J-,0) and i t d e p e n d s on t h e e v e n  order  d e r i v a t i v e s of  observe  that  (0(^,-0 w . r . t .  in  .  Comparing  this  e x p a n s i o n w i t h t h e T a y l o r e x p a n s i o n f o r w (j.,z) a b o u t T= o , i.e.  we  see t h a t  (3.10) and  s a t i s f i e s the  (3.11) a r e e g u i v a l e n t s i n c e  diffusion  eguation  IOI^"*)  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 Algorithm  have 3. 3.  shown  ^ that  Algorithm  3.2  i s eguivalent  to  55  III_2  A F I N I T E DIFFERENCE METHOD FOR ARBITRARY MOVING Algorithm up  more  solution  3.4 i s a g e n e r a l s t a r t i n g  detailed  a t t " tj we c a n compute  noniteratively, idea  behind  we  Applying  ( Nt  i s a  j:U . ( U i , "tj ) •  schemes. S i n c e h a v i n g t h e  have a 'one s t e p e x p l i c i t  scheme*. The  3.4  the  i s  not  new  3.4,  with uniform  Using  .  3  K = ©. |j  such  be  the  ,  zk  0  -  Let  j e oco  ,  t h e boundary c o n d i t i o n s imply = U Uk) O  £2  for f  o  r  od> oo j :  o ( i i ^  be t h e o p e r a t o r  +  (I-  if) u?  domain  and •£= At = -j~  t h a t t - t j - j-k exact  <^<+ *W*L -  =  A  the  solution  and at  ^  + ocV) , and  the  differences f o r the derivatives, ^  ((CJfF(t))u)J.. 3  discretize  spacing  to  centered  we h a v e  we  integer) ,  denote  where  [ 2 5 , and  there].  positive  We  ;  setting  at t = tj+|  Algorithm  T > t >o  R  point f o r  solution  Algorithm  references given  04j<M  computational  BOUNDARY  ,  U_, ' 0 ,  _  such  that  lfc-. + F t t , » U J . ,  56  o  where  We  _  have t h e  We  denote  satisfies and  the  tT<» =  f o l l o w i n g approximation  approximate  \J±  boundary c o n d i t i o n s  We  solution  e x a c t l y the eguation  , Vl  0  the  - o  follow a similar  convergence  eguation:  by  \J£  (3.13)  0  t  stability,  Subtracting  (3.13) f r o m  oo ,  .  d e v e l o p m e n t of Ames [ 2 5 ]  and  which  «  M CAKi f o r X. - oin  =  j r o tl) N  for  f o r the  conditions  for  to  this  obtain  explicit  method.  difference  £g Assuming  -  satisfied  0  ,  S  by t h e  0  0>  (3.12),  we  obtain  the  finite  error  hJ . t  o < r < •/_  (3* 15)  and  } then  ,En'|  < ir+^u  U l  tF(t,-)J)  (3. 16)  |£iJ  57  = <it*)i|E*ll + AC i ' + i K " ] A  where :  Conseguently  where  £j  |(  „  j, ^ m i ) | | E ' | |  £  iCjt, - C l+ 1) ^ + 1  follows that  „ i*r„ £  ^ £ [ i  (3. 18)  Jf*. s " £ [ ( < + - f t . ) ] 1  (  +  ^ . , J  T  (3.19)- i m p l i e s t h a t solution  The  A[  *fc*]  (AX) ] 1  l l f ^ ' l l ^ t f as A t - * o ,  of the f i n i t e  converges t o t h e s o l u t i o n eguation  and  At£N*^],  +  $ Ae CA t * the  t  the f o l l o w i n g d i f f e r e n c e eguation:  The s o l u t i o n o f (3.18) i s It  t  = o we have  satisfy  {  .17)  I Ej|| -  U  and s i n c e  ( 3  on the values of u  i s a constant depending HE'II = * P  im»  t  difference  .  (3.19)  4x-»©„  eguation  of the p a r t i a l  Thus (3.13)  differential  (3.3) . condition  (3.15) i s t h e 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  >, which imposes a f u r t h e r  (3.16) i s e g u i v a l e n t t o  raj) + f'ctj)  ^-(JL+  H^  restriction  on  We  h.  ( 3  . O) 2  have the  58  following  theorem:  4 i  Theorem 3. 1 and  L e t u € C ' be t h e s o l u t i o n o f e g u a t i o n (3.3) a s s o c i a t e d b o u n d a r y c o n d i t i o n s , a n d l e t "U" be t h e  the  solution  o f ( 3 . 1 3 ) . . I f ° < r£ _ and Ax  i s small  enough  so  that  fi***  -&*|» I U J ~  then  III_3  * k -?(tj> +r'(tj) >, - l J K  [ £ Ae  T  +AX)  in  J l r [ 0 « ) < [ » J l  [ At + (AX) ] f o r 1  (  l^^,t^)6SL  COMPUTATIONS In  the  following  illustrate  mainly  impossible  to  infinite time, the  domain  we  Example Let  the  stability  problem.. Since  compute  numerical  solutions  with uniform  restrict  boundary  we g i v e some n u m e r i c a l e x a m p l e s t o  ourself  condition:  in  mesh s i z e i n f i n i t e to consider  Y,  -  1 ,  U(S,t)=o.  h =  a  semi-  computing  } J t C 0 ; 5 ] and i m p o s e  3. 1  Y o ^ ~  i t is  Aij r 0. |  otherwise  59  The r e s u l t s Since  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.  (3.20) h o l d s f o r h = °-l  the i n s t a b i l i t y i s  T > •£ .  due t o (3.15) where we have  Example  (•$9*S5")r  3.2 Y, = -1 , Y, - 2 , K * o.i , 0.2 , 0 . 5  Let  •<(  ,  I  0.1) =  ^* 0  otherwise  U, C J ) .  =  We a p p l y t h e 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 The  results  * = '/loo  are  shown  i n Table  at t =  instability  condition  in  (3. 20) ,  Table  3.3  where  we  +  k.  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 The  0. 1  is  due  three  to  cases.  violation of  1 +1):  ,  but  3.4 shows t h a t i n s t a b i l i t y  occurs  have  n  <k - Y (tj) + Ir f t p c a n be a s l a r g e a s 5 + 2e. .  ExamjDle Let  3^3 Y =- l 0  ,  Ul«J,0.l) s  Y. - 'o  r(t)  violated  ^  0.1 ,  Vzoo ,  U , ( t j )  The r e s u l t s i n T a b l e when  .  decreases  i n the i n t e r v a l  so r a p i d l y (J  .  t h a t c o n d i t i o n (3.20) i s  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 a) A t = - £ - i Stable  to  J±±  >J., 1  U (y . t) D . 75  H  At=1/195  D . 50  H  0.25  H  4y = 0 . 1  y o. 00  ' I r i r i i ! I I I I i i T ^ - ^ ^ ^ ^ r r " ? i T i i i ! ^i I  2 .5  Fig.  3.2  b) AS = o. 1, o-t=ri?« Unstable, but I t i s not Apparent Numerical. R e s u l t i n t h i s Time I n t e r v a l  3 .0  from  (  the  iF  1 7  I  I  T  p  3 .5  Error  F i g . 3.3  (xio-  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 = ^  in  r  .  3)  o o "  .  •  C\J  II  n  o o in ro  o o o o  o a in  o  a o m  OJ  a o a in  i  i  i  66  Y  L e t  The  » Z ,  0  results  Y, =  ,  in  h = o.l, ft.*, o.S  Figure  3.4  ,  - i - '/»«> , 2  demonstrates  that  an  A  increasing  r(t)  will  a p p r o x i m a t i o n of the accurate. Example  d e c a y s o l u t i o n and (0, 5)  the  is  not  i n Example  3.4,  i s o b s e r v e d f o r h=0.5.  3._5  that  = Gj(r(t)-5, that  a slowly  s e m i - i n f i n i t e domain by  Instability  Using except  give  the  same  we  use  1 , t)  instability  s e t o f p a r a m e t e r s as the  boundary  condition  f o r t > 0. 1 i n s t e a d .  occurs  b o u n d a r y c o n d i t i o n s . The  in  the  case  r e s u l t s are  This of  u(5,  t)  e x a m p l e shows  applying  shown i n F i g u r e  exact 3.5.  y Fig.  '}A  S o l u t i o n with Approximate Boundary C o n d i t i o n  Uj. =o 5  as  0  3  2  3  5  4  y F i g . 3.5  S o l u t i o n with Exact Boundary Conditions  69  Table  3.1  Case o f Y.=-1, Yt = 2, r a) A p p r o x i m a t e S o l u t i o n  I  t=0. 1  0  - X o  = 1, 4 t = 1 / 2 0 0 , Ay=0. 1:  +•  3  y 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  t=0. 2 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  t=0.6 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  t=1. 0 0.0 0.2693E- 02 0.4006E- 02 0.4418E- 02 0.4282E- 02 0.3845E- 02 0.3277E- 02 0.2684E- 02 0.2128E- 02 0.1642E- 02 0.1237E- 02 0.9115E- 03 0.6584E- 03 0.4667E- 03 0.3250E- 03 0.2226E- 03 0. 1499E- 03 0.9938E- 04 0.6487E- 04 0.4171E- 04 0.2643E- 04 0.1650E- 04 0. 1015E- 04 0.6158E- 05 0.3683E- 05 0.2172E- 05 0. 1263E- 05 0.7246E- 06 0.4098E- 06 0.2286E- 06 0.1258E- 06 0.6828E- 07 0. 3655E- 07 0. 1930E- 07 0. 1005E- 07 0.5162E- 08 0.2615E- 08 0. 1306E- 08 0.6435E- 09 0.3125E- 09 0.1495E- 09 0.7049E- 10 0.3270E- 10 0.1491E- 10 0. 6660E- 11 0.2901E- 11 0.1220E- 11 0.4853E- 12 0. 1737E- 12 0.4732E- 13 0.0  T a b l e 3.1 y  I  0. 0 i 0. 1 I  0.2  |  0. 3 I 0.4 | 0. 5 I 0. 6 I 0. 7 | 0.8 | 0. 9 I 1. 0 I 1. 1 I 1. 2 I 1. 3 I 1. 4 I 1. 5 I 1. 6 j 1. 7 I 1. 8 I 1. 9 I 2. 0 I 2. 1 i 2. 2 I 2. 3 | 2. 4 I 2. 5 I 2. 6 | 2. 7 j 2. 8 1 2. 9 1 3. 0 1 3. 1 I 3. 2 I 3. 3 I 3. 4 i 3. 5 I 3. 6 I 3. 7 | 3. 8 1 3. 9 1 4. 0 I 4. 1 I 4. 2 I 4. 3 J 4. 4 1 4. 5 1 4. 6 1 4. 7 | 4. 8 1 4. 9 I 5. 0 1  Case o f Y»=-1, Yi = 2, r „ - x = 1 , A t = 1 / 2 0 0 , A V = 0 . 1: b) E x a c t S o l u t i o n 0  t=0.1  J  0.0 | 0.2160E+00 J 0.4013E + 00 J 0.5668E+00 | 0.7122E+00 J 0.8293E+00 | 0.9069E+00 | 0. 9361E + 00 J 0.9134E+00 | 0.8432E+00 | 0. 7364E + 00 J 0. 6086E+00 | 0.4760E+00 | 0.3523E+00 | 0.2468E*00 | 0.1636E+00 | 0. 1026E + 00 J 0.6090E-01 | 0.3421E-01 | 0. 1818E-01 | 0.9147E-02 | 0. 4354E-02 | 0.1961E-02 | 0. 8362E-03 J 0.3373E-03 | 0. 1288E-03 | 0.4653E-04 J 0. 1591E-04 J 0.5147E-05 | 0. 1576E-05 J 0.4566E-06 | 0. 1252E-06 J 0.3248E-07 | 0. 7977E-08 1 0.1854E-08 | 0. 4076E-09 J 0.8482E-10 J 0. 1670E-10 | 0.3113E-11 | 0. 5489E-12 1 0.9161E-13 | 0. 1447E-13 J 0.2162E-14 | 0. 3058E-15 | 0.4093E-16 | 0.5185E-17 | 0.6214E-18 J 0. 7049E-19 J 0. 7566E-20 | 0.7686E-21 J 0.7388E-22 |  t=0.2  |  0.0 | 0.2337E+00 | 0. 4076E + 00 | 0. 5273E + 00 | 0.5993E+00 J 0.6301 E + 00 | 0.6265E+00 | 0.5958E+00 | 0.5452E+00 | 0.4815E+00 | 0.4114E+00 J 0.3403E+00 | 0.2729E+00 | 0.2121E+00 | 0. 1599E + 00 J 0.1169E+00 | 0.8288E-01 | 0.5701E-01 | 0. 3804E-01 | 0.2463E-01 j 0.1547E-01 | 0.9421E-02 | 0.5568E-02 | 0.3192E-02 | 0.1776E-02 | 0.9581E-03 J 0.5016E-03 | 0.2547E-03 j 0.1255E-03 | 0.5997E-04 J 0.2781E-04 | 0.1251E-04 J 0.5458E-05 | 0.2311E-05 | 0. 9489E-06 | 0.3780E-06 J 0.1461E-06 | 0.5479E-07 | 0.1993E-07 | 0.7033E-08 | 0.2408E-08 | 0.7997E-09 | 0. 2576E-09 ( 0.8053E-10 ] 0.2442E-10 | 0.7184E-11 J 0.2050E-11 | 0.5676E-12 | 0.1524E-12 | 0.3972E-13 | 0. 1004E-13 |  t=0.6  I  t=1. 0  0. 0 0.0 0.4265E- 0 1 | 0.2850E- 02 0.6980E- 01 | 0.4273E- 02 0.8462E- 01 | .0.4752E- 02 0.9C08E- 01 | 0.4646E- 02 0.8881E- 01 | 0.4212E- 02 0.8302E- 01 | 0.3626E- 02 0.7452E- 01 J 0.3002E- 02 0.6473E- 01 | 0.2408E- 02 0.5466E- 01 | 0.1880E- 02 0.1434E- 02 0.4502E- 01 j 0.3626E- 01 | 0.1071E- 02 0.2860E- 01 | 0.7851E- 03 0.2212E- 01 i 0.5650E- 03 0.1679E- 01 | 0.3998E- 03 0.1253E- 01 | 0.2784E- 03 0.9184E- 02 | 0. 1909E- 03 0.6622E' 02 | 0.1289E- 03 0.4698E- 02 | 0.8582E- 04 0.3280E- 02 | 0.5633E- 04 0.2254E- 02 | 0.3647E- 04 0.1525E- 02 | 0.2329E- 04 0.1017E- 02 | 0.1468E- 04 0.6673E- 03 J 0.9132E- 05 0.4315E- 03 | 0.5607E- 05 0.2749E- 03 | 0.3399E- 05 0.1726E- 03 | 0.2034E- 05 0. 1203E- 05 0.1067E- 03 ! 0.6506E- 04 | 0.7020E- 06 0.3908E- 04 | 0.4048E- 06 0.2313E- 04 | 0.2305E- 06 0.1350E- 04 | 0.1297E- 06 0. 7762E- 05 | 0.7209E- 07 0.3959E- 07 0.4399E- 05 J 0.2458E- 05 | 0.2148E- 07 0.1353E- 05 | 0. 1151E- 07 0.7345E- 06 | 0.6098E- 08 0.3192E- 08 0.3929E- 06 1 0.2072E- 06 | 0.1651E- 08 0. 1077E- 06 | 0.8437E- 09 0.5520E- 07 | 0.4261E- 09 0.2127E- 09 0.2788E- 07 J 0.1049E- 09 0.1388E- 07 j 0.51 15E- 10 0.6814E- 08 J 0.3297E- 08 | 0.2465E- 10 0.1572E- 08 j 0.1174E- 10 0.7393E- 09 | 0.5523E- 11 0.3427E- 09 j 0.2569E- 11 0.1566E- 09 | 0.1181E- 11 0.7051E- 10 ] 0.5367E- 12 0.3131E- 10 | 0.2410E- 12  71 T a b l e 3.2  y  0.0 0. 2 0.4 0. 6 0. 8 1. 0 1. 2 1. 4 1. 6 1. 8 2. 0 2. 2 2. 4 2. 6 2. 8 3. 0 3. 2 3. 4 3. 6 3. 8 4. 0 4. 2 4. 4 4. 6 4. 8 5. 0  I  I I  1 I I I I  I I I I  I I I I  I ] I I I  I I I I  ]  I  T a b l e 3.3 y  I  0.0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 3. 5 4. 0 4. 5 5. 0  | I I I I I i I I  I j  Case o f Y. = - 1 , Ji=2, r . - x < > = 1 , Approximate S o l u t i o n t = 0. 1  J  0. 0 0.4013E+00 0.7122E+00 0.9069E+00 0.9134E + 00 0.7364E+00 0.4760E+00 0.2468E+00 0. 1026E+00 0.3421E-01 0.9147E-02 0.1961E-02 0. 3373E-03 0.4653E-04 0.5147E-05 0. 4566E-06 0.3248E-07 0.1854E-08 0.8482E-10 0.3113E-11 0.9161E-13 0.2162E-14 0. 4093E-16 0.6214E-18 0. 7566E-20 0.0  | J | | | | | | | | j | j | J | j | j | | | I | J |  t = 0.2  |  0.0 | 0.4158E+00 j 0.6106E + 00 J 0.6377E+00 j 0. 5534E+00 J 0.4149E + 00 | 0.2720E+00 J 0.1565E+00 J 0.7903E-01 | 0.3503E-01 | C.1362E-01 J 0. 4645E-02 | 0.1387E-02 J 0.3625E-03 | 0.8283E-04 J 0.1654E-04 j 0.2880E-05 | 0. 4372E-06 j 0. 5773E-07 j 0.6620E-08 | 0.6578E-09 j 0. 5649E-10 | 0.4179E-11 J 0.2654E-12 | 0. 1422E-13 J 0.0 1  t=0.6  0.0 0.8293E+00 0. 7364E+00 0.1636E+00 0.9147E-02 0. 1288E-03 0. 4566E-06 0.4076E-09 0. 916 I E - 1 3 0.5184E-17 0.0  J  t=0.2  J  I  0. 0 0.7488E- 01 J 0.9457E- 01 | 0.8494E- 01 | 0.6421E- 0 1 | 0.4304E- 01 | 0.2615E- 01 | 0.1456E- 01 J 0.7479E- 02 | 0.3552E- 02 J 0.1562E- 02 | 0.6362E- 03 | 0.2399E- 03 | 0.8366E- 04 | 0..26 94E- 04 J 0.7990E- 05 | 0. 2178E-05 J 0. 5443E- 06 | 0.1242E- 06 | 0.2578E- 07 | 0.4844E- 08 J 0.8194E- 09 | 0.1237E- 09 | 0.1634E- 10 | 0.1700E- 11 | 0. 0 I  C a s e o f Y e = - 1 , Y i = 2, r . - x . =1, Approximate S o l u t i o n t=0.1  & t = 1 / 2 0 0 , A y = 0 . 2:  At=  t=1. 0 0. 0 0.4298E- 02 0.4346E- 02 0.3128E- 02 0.1895E- 02 0.1017E- 02 0.4943E- 03 0.2196E- 03 0.8970E- 04 0.3374E- 04 0. 1 169E- 04 0.3728E- 05 0.1091E- 05 0.2922E- 06 0.7127E- 07 0.1574E- 07 0.3123E- 08 0.5516E- 09 0.8562E- 10 0.1148E- 10 0.1297E- 11 0.1188E- 12 0.8220E- 14 0.3624E- 15 0.4304E- 17 0.0  1 / 2 0 0 , A y = 0 . 5:  t=0i6  | o.o | 0.0 0.6954E+00 J 0. 1486E+00 j | 0.4636E+00 | 0.3022E- 01 | 0". 9035E-01 | -0.3717E- 02 0.1557E-02 | - 0 . 2 6 2 4 E - 03 J | -0.2102E-03 | 0.2250E- 03 j 0. 1185E-04 | -0.6658E- 04 | -0.4714E-06 | 0.1465E- 04 0. 1206E-07 J - 0 . 2 6 5 3 E - 05 J | 0. 4367E-10 | 0.2838E- 06 0.0 J 0.0 }  I  J | | | | | | |  t=1. 0 0.0 0.5030E- 02 -0.3333E- 02 0.1068E- 02 - 0 . 1 7 4 9 E - 03 - 0 . 2 0 9 4 E - 04 0.2994E- 04 -0.1245E- 04 0.4824E- 05 0.1198E- 06 0.0  72  Table  3.4  + I  C a s e o f Y = - 9 , Y»=10, r - x , = 1 , A t = 1 / 2 0 0 , ay=0. 1: a) A p p r o x i m a t e S o l u t i o n  I  t=0. 1  y  0  J  0  . t=0. 2  J  t=0.6  j  +  +. J +  i 1 | | | | | | | | | | | | | | j | | | | | | j 1 I | j I 1 | | | j | | | 1 | J | i | 1 | | | 1 | |  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  1  5.0  +  t=1.0  | 0.0 J 0.0 | 0. 3790E-01 | 0. 8683E-10 j 0.2909E-01 | 0.1174E-10 | 0.1697E-01 | -0.7850E-11 | 0.8808E-02 | - 0 . 3 6 7 6 E - 1 1 | 0. 4236E-02 J - 0 . 5408E-12 | 0.1914E-02 | 0.2231E-12 | 0.8160E-03 | 0.1423E-12 | 0.3289E-03 J 0.2590E-13 | 0.1254E-03 | -0.6200E-14 J 0. 4524E-04 j - 0 . 4940E-14 | 0. 1545E-04 I - 0 . 1066E-14 l 0. 4990E-05 | 0.1510E-15 | 0.1526E-05 | 0.1610E-15 J 0. 4414E-06 j 0.3815E-16 | 0. 1208E-06 | - 0 . 3391E-17 | 0.J131E-07 j -0.4756E-17 | 0. 7677E-08 | -0.1201E-17 | 0.1781E-08 j 0.5919E-19 | 0.3911E-09 | 0. 1274E-18 | 0.8127E-10 | 0.3368E-19 | 0. 1598E-10 | - 0 . 7698E-21 | 0.2974E-11 | - 0 . 2 9 6 0 E - 2 0 | 0.5237E-12 | -0.8036E-21 | 0. 8727E-13 ] - 0 . 1498E-22 | 0.1376E-13 | 0.5512E-22 | 0.2054E-14 } 0.1823E-22 | 0.2900E-15 | 0.1499E-23 ] 0. 3876E-16 J - 0 . 9270E-24 | 0.4903E-17 | -0.3842E-24 J 0. 5868E-18 | - 0 . 4408E-25 | 0.6646E-19 | 0. 1302E-25 j 0.7123E-20 j 0.6281E-26 | 0. 7225E-21 j 0. 9722E-27 | 0.6935E-22 | -0. 8826E-28 J 0. 6299E-23 j -0.8150E-28 J 0.5414E-24 | - 0 . 1853E-28 J 0.4404E-25 j - 0 . 7 5 1 9 E - 3 0 | 0.3390E-26 | 0.7798E-30 j 0. 2470E-27 J 0. 2560E-30 | 0.1703E-28 | 0.3340E-31 | 0. 1111E-29 J - 0 . 2755E-32 I 0.6857E-31 | -0. 2346E-32 1 0. 4006E-32 ] - 0 . 5444E-33 | 0.2215E-33 J - 0 . 5047E-34 J 0.1159E-34 J 0.8936E-35 | 0.5737E-36 | 0.4516E-35 | 0. 2688E-37 | 0.9138E-36 | 0.1192E-38 | 0.9998E-37 | 0. 5001E-40 | - 0 . 1561E-37 I 0.0 | 0.0  J 0.0 J - 0 . 1828E-16 | 0.4149E-16 J 0.1648E-16 | -0.1853E-16 | -0. 9087E-17 | 0.5665E-17 | 0.4191E-17 J -0.1283E-17 | -0.1693E-17 J 0. 1532E-18 | 0.6114E-18 \ 0.4412E-19 | -0.2002E-18 } -0.4213E-19 | 0. 5998E-19 J 0.2051E-19 | - 0 . 1652E-19 | -0.7930E-20 | 0.4192E-20 j 0. 2671E-20 ( - 0 . 9821E-21 | -0.8121E-21 | 0.2132E-21 J 0.2269E-21 | -0.4353E-22 | -0.5883E-22 | 0.8693E-23 j 0. 1428E-22 | -0.1845E-23 J -0.3270E-23 | 0. 4567E-24 J 0.7153E-24 | - 0 . 1324E-24 J - 0 . 1522E-24 j 0.4105E-25 | 0. 3228E-25 | -0.1249E-25 | -0.7010E-26 J 0. 3575E-26 | 0.1579E-26 | -0.9454E-27 | - 0 . 3643E-27 | 0. 2288E-27 | 0. 8251E-28 j -0.4965E-28 | -0.1701E-28 J 0. 8978E-29 | 0.2645E-29 J - 0 . 9083E-30 I 0.0 -r  | 0.0 J | -0.4110E+03 | J -0.3597E+03 J | 0.5144E+03 | | 0.3175E+03 | | - 0 . 3553E+03 | ) -0.2154E+03 | j 0.2045E+03 | ] 0.1317E+03 | | -0.1071E+03 | | -0.7589E+02 | | 0.5272E+02 | | 0.4181E+02 J | -0.2479E+02 | J -0.2219E + 02 | | 0.1122E+02 | I 0.1138E+02 j J -0.4918E+01 | | -0.5651E+01 j | 0.2095E+01 | | 0. 2724E+01 J | - 0 . 8703E+00 | | -0.1276E+01 J ( 0.3542E+00 | | 0.5814E+00 \ | -0.1420E+00 | 3 -0.2582E+00 | | 0.5639E-01 J I 0.1119E+00 \ | -0.2235E-01 | \ -0. 4735E-01 | | 0. 8902E-02 | | 0. 1959E-01 | J -0. 3577E-02 | j -0.7927E-02 j | 0. 1450E-02 | i 0.3135E-02 J I -0.5896E-03 | | -0.1210E-02 | | 0.2378E-03 j | 0.4534E-03 | J -0. 9356E-04 j | - 0 . 1633E-03 | J 0.3503E-04 J | 0. 5533E-04 | | -0.1198E-04 j | -0.1674E-04 | | 0.3405E-05 J | 0.3817E-05 | | - 0 . 5299E-06 J | 0.0 |  73  Table + J  + —y  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  3.4 1  C a s e o f Yo=-9 Yi = 10, r - x b) E x a c t S o l u t i o n f  t=0. 1 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. 8 1 2 7 E - 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  e  t=0. 2 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 0.2001E-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  I  a  =1  r  a t = 1 / 2 0 0 , Ay=0.1  t=0.6 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. 6 8 0 8 E - 7 3 0.7558E-74 0.8272E-75 0. 8924E-76 0. 9491E-77 0.9949E-78 0.0  t=1. 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  •- + I  74 T a b l e 3.4 y  i  0.0 | 0. 1 1 0.2 | 0. 3 | 0.4 | 0.5 | 0.6 | 0. 7 | 0.8 | 0.9 j 1.0 | 1. 1 | 1.2 I 1. 3 | 1.4 | | 1.5J 1.6 | 1.7 I 1.8 | ] 1. 9 | I 2.0 | I 2. 1 | I 2.2 | 1 2. 3 | | 2. 4 | 1 2.5 | | 2.6 | | 2. 7 J | 2.8 | I 2.9 1 1 3.0 1 3 3. 1 1 1 3.2 | 3 3. 3 J 1 3.4 | | 3. 5 1 | 3.6 | 1 3.7 1 1 3.8 | 3 3. 9 j 1 4.0 | i 4. 1 | | 4.2 I 3 4. 3 J 1 4.4 1 1 4.5 | 1 4.6 | J 4. 7 J 3 4.8 | J 4.9 | 1 5.0 |  Case o f Yo=-9, Y = 10, r - x , = 1, A t = 1 / 2 0 0 , Ay=0.1: c) E r r o r t=0.1  J  0.0 | 0.0 3 0.0 | 0.0 J 0. 0 | 0.0 I 0.0 | 0.0 ] 0. 0 | 0.0 | 0.0 | 0.0 | 0. 0 | 0.0 | 0.0 3 0.0 | 0. 0 | 0.0 3 0.0 | 0. 0 } 0.0 | 0.0 | 0. 0 | 0.0 } 0.0 | 0.0 ) 0.0 J 0.0 J 0.0 3 0.0 3 0.0 | 0.0 J 0. 0 | 0. 0 J 0.0 | 0.0 1 0.0 | 0.0 J 0.0 | 0. 0 | 0. 0 | 0.0 3 0.0 | 0.0 3 0.0 | 0.0 1 0.0 3 0.0 1 0. 0 | 0. 0 J 0.1986E-41 |  t  0  t=C.2  J  t=0.6  I  .0.0 I 0.0 0.1828E- 16 J 0. 7496E-06 \ 0. 4920E-06 1 - 0 . 4 1 4 9 E - 16 3 0.2396E-06 J -0.1648E- 16 J 0.1025E-06 | 0.1853E- 16 | 0. 4056E-07 | 0.9087E- 17 J 0. 1518E-07 | -0.5665E- 17 | 0.5434E-08 | -0.4191E- 17 j 0.1283E- 17 | 0.1871E-08 J 0.1693E- 17 J 0.6221E-09 J 0.2001E-09 | -0.1532E- 18 | 0.6230E-10 J -0.61 14E-18 | 0.1880E-10 3 -0.4412E- 19 3 0. 5500E-11 | 0.2C02E- 18 j 0. 1560E-11 | 0.4213E- 19 3 0.4294E-12 | -0.5998E- 19 | 0.1146E-12 | -0.2051E- 19 J 0.2968E-13 | 0.1652E- 19 } 0.7453E-14 J 0.7930E- 20 | 0.1816E-14 J - 0 . 4 1 9 2 E - 20 J 0.4292E-15 3 -0.2671E- 20 | 0.9821E- 21 | 0.9842E-16 J 0.2190E-16 | 0.8121E- 21 | 0. 4726E-17 J -0.2132E- 21 ] 0. 9892E-18 | -0.2269E- 21 I 0.4353E- 22 | 0. 2008E-18 J 0.5883E- 22 | 0.3956E-19 J -0.8693E23 J 0.7564E-20 J 0. 1404E-20 | -0.1428E- 22 | 0.1845E- 23 } 0.2528E-21 J 0.3270E-•23 | 0. 4409E-22 3 0. 7444E-23 J -0.4567E- 24 | 0.1218E-23 | - 0 . 7.153E- 24 | 0.1942E-24 ] 0. 1324E-•24 J 0. 1522E--24 | 0.3025E-25 3 0.4605E-26 \ -0.4105E-•25 J 0.6766E-27 | -0.3228E-•25 | 0.1249E-•25 3 0.9363E-28 } 0.1194E-28 | 0.7010E-•26 | 0.1433E-29 3 -0.3575E-•26 j 0. 1843E-30 | - 0 . 1579E--26 | 0.9454E--27 | 0.2996E-31 J 0. 5646E-32 | 0. 3643E--27 | 0.9326E-33 3 - 0 . 2288E--27 J 0.9478E-34 | -0.8251E--28 | -0.4030E-35 ] 0. 4965E--28 | -0. 3989E-35 | 0. 1701E--28 | -0.8589E-36 J - 0 . 8978E--29 3 -0.9443E-37 J - 0 . 2645E--29 | 0.9083E--30 j 0.1616E-37 J 0.5180E-40 | 0.0  t=1.0  J  0.0 | 0.4110E+03 3 0.3597E+03 J -0.5144E*03 J -0.3175E + 03 J 0. 3553E+03 \ 0.2154E+03 | -0.2045E+03 | -0.1317E+03 | 0.1071E+03 J 0. 7589E*02 J -0.5272E+02 | -0.4181E+02 | 0.2479E+02 J 0.2219E+02 | -0.1122E+02 I -0.1138E+02 J C.4918E+01 | 0.5651E+01 | -0.2095E+01 J -0.2724E+01 | 0.8703E+00 | 0.1276E+01 | -0.3542E+00 j -0.5814E*00 3 0. 1 420E + 00 | 0.2582E+00 | -0.5639E-01 | -0.1119E+00 | 0.2235E-01 J 0. 4735E-01 3 -0.8902E-02 | -0.1959E-01 | 0.3577E-02 0. 7927E-02 -0.1450E-02 -0.3135E-02 0.5896E-03 0. 1210E-02 -0.2378E-03 -0.4534E-03 0. 9356E-04 0. 1633E-03 -0.3503E-04 - 0 . 5533E-04 0. 1198E-04 0. 1674E-04 -0.3405E-05 -0.3817E-05 0. 5299E-06 0.0  75  CHAPTER IV DISCUSSION AND  In  CONCLUSIONS  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  underlying problem  Lie  group  structure  calculated the  of t h e i n i t i a l boundary  ( 1 . 1 , 1.1a, 1.1b) f o r r (t) = Y e * + Y i e .  Knowing  t  c  the Fokker-Planck eguation  e g u a t i o n c a n be t r a n s f o r m e d  Chapter  I . The f i r s t  c a l c u l a t e d by  procedures  passage time d i s t r i b u t i o n  ( 2 . 4 ) , which  The a p p l i c a b i l i t y  by a p p l y i n g o f example 1  can  then  Although modellers  e x p r e s s i o n of g of  gj  g  of the f i r s t  h a s been In  the  the f i r s t  has  3  with Y  4  been  have  passage time  distribution to  or  16]  the  closed  expression  ignored.  case  of neural modelling a further step i s t o f i t  estimate  the  interval  histogram  directly  [29,  Pfeiffer  and K i a n g [ 3 2 ] d i s p l a y  g, a n d  15,  = 0, i t seems t h a t t h e f u l l  to  of  [11,  passage time d i s t r i b u t i o n  histogram  of the  known f o r a w h i l e and a number o f  mentioned 0  be  i s simple t o apply i n general..  m o d e l l i n g d e p e n d s , t o a c e r t a i n e x t e n t , on t h e s i m p l i c i t y  neural  heat  we c o u l d t h u s o b t a i n t h e L i e g r o u p s and t h e s i m i l a r i t y  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  form.  that  t o the simple  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 o f moving b o u n d a r i e s  in  value  g , (see  to the  moments 30,  interspike from  3 1 ] . . The  the  interval interspike  histograms  of  very s i m i l a r s t r u c t u r e s t o those  F i g u r e 4.1, 4.2, 4.3) . The f i t t i n g  of g  3  to  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„-x , the Distance between the I n i t i a l Membrance P o t e n t i a l and the Threshold a t t*o, Changes 0  79  the r a t c o c h l e a r nucleas found  that the modified  the other Least of  other  was  reasonably  parameters  (r  estimation  of  period,  A  as L e v i n e  the  Y  0  to  g ,  of the  of u(x,t) i n  eguation  satisfied  t  by  satisfy  differential of the  scheme  the  order  where  solutions.  stability  two  provide  an  refractory measure  i n t o the  may  of c e r t a i n  u(x,t)  types  obtained of  extend  eguations.  y partial  i t  hidden  that cannot [36].  of  a  by  time Taylor  transformed  t h i s idea to integrals  Algorithm  3.4  d e r i v a t i v e s g i v e s us a  find which  plus  the  complete  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  to have convergence of t h e a p p r o x i m a t e  exact  cannot  substitution  u ( x , t ) . We  expansion  discretization  solution  and  the asymptotic some  can  also  of t h e  of r e l a t i v e  i . e . we  one  [36]  have shown i n d a t a i l t h a t t h e s m a l l  expansion  expansion  hypernormal  fitting  w h i c h p r o v i d e s some i n s i g h t  mechanism i s c e r t a i n l y b e t t e r t h a n  asymptotic  He  superior to  Shefner's  however,  A  an o b s e r v a b l e ,  I I I we  was  [33]..  distribution  underlying behaviour  model  In Chapter  and  S m i t h ' s Gamma  and  0  which i s not  directly.  Nilsson  good f i t t o o t h e r d a t a . The  - x )  0  by  Maximum L i k e l i h o o d m e t h o d s . A number  such  d i s t r i b u t i o n [ 3 5 ] and C. give  out  Marguardt a l g o r i t h m [34]  S g u a r e s and  functions,  carried  This r e s u l t  problems are  also  to  the  i s s u m m a r i z e d i n Theorem 3 . 1 .  The  clearly  solutions  illustrated  by  explicit  computations. We  s u m m a r i z e t h e m a j o r 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  following : (1)  We  successfully  Planck  eguation  obtained  an a n a l y t i c  with  two  a  solution for a  parameter  family  of  Fokkermoving  80  b oun d a r i e s . (2)  We  derived  a new  general  time d i s t r i b u t i o n probability Planck  expression  of a d i f f u s i o n  f o r the  process  with a  density f u n c t i o n s a t i s f y i n g the  (Kolmogorov forward)  eguation  first  transition  general  f o r any  passage  Fokker-  smooth  moving  b oundaries. (3)  We  showed  problem and (4)  We  three  f o r t h e two  defined four moving  that  a  the  moving  Finally asymptotic  we  p r o b l e m . We  asymptotic  the  boundaries,  to  solve  c a l c u l a t e d the  algorithm  the small  s o l u t i o n only  is  to  eguivalent to a  algorithm.  numerical  general  aiming  e x p a n s i o n of the i n t e g r a l  c o n v e r g e n c e and  for  moving  of  distributions.  algorithms  boundary  Taylor expansion The  formulations  passage time  numerical  time asymptotic find  eguivalent  p a r a m e t e r f a m i l y of  their related f i r s t  general  (5)  other  stability scheme  conditions  applied  to  were this  established eguation  with  boundaries.  should  point  out  that  e x p a n s i o n o f an i n t e g r a l  the  u ( x , p)  idea  that  with r e s p e c t  the  to  the  p a r a m e t e r 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  with  1,  2,  the  s u b s t i t u t i o n of the by  partial derivatives ( — \  a differential ralation Up = g ( x  u(x  r  p)  ), k =  may  general  be  much  demonstrated beginning  r  p,  worth n o t i c i n g easier in  to  one  Ux  since  apply  Chapter I I I .  of t h i s c h a p t e r  u,  , u the  , ...)  X K  later  satisfied method  t h a n t h e f o r m e r one As  can  we  have  as  mentioned  is  by in  we  have  in  the  f i n d the s i m i l a r i t y s o l u t i o n s  81  for a l lthree moving with which  c l a s s e s o f moving b o u n d a r i e s , i n  boundaries  6 = 0 and may  of c l a s s I I ' , which i s d e f i n e d  & < 0 providing  contain  a  3  parameter  t h a n t h e two  t h i s case i s i n terms  of  Airy  parameter and  I,  boundary  closer to the  f a m i l y . The  Bessel  the  i n Chapter  moving  more s t r u c t u r e s s o t h a t i t f i t s  r e a l moving b o u n d a r y in  particular,  solution  functions  [8],  82  BIBLIOGRAPHY  [ 1 ] A.  Einstein,  Ann.  d. P h y s i k 17,  549  (1905).  [2]  M. .V. . S m o l u c h o w s k i ,  [3]  G . . E . . U h l e n h e c k and L . S . O r n s t e i n , On the theory of the B o r w n i a n m o t i o n . P h y s i c a l Review 36, 823 (1930).  [4]  M. .C. .Wang and G. E. U h l e n b e c k , Brownian motion I I , Reviews (1945) .  [5]  S..Chandrasekhar, Stochastic problems in a s t r o n o m y . Reviews o f Modern P h y s i c s 15,  [6]  J . L. Doob, The Brownian movement e g u a t i o n s . . A n n a l s o f M a t h e m a t i c s 43,  Phys.  Z e i t s . . 17, 557  ( 1916).  On the theory of the o f Modern P h y s i c s 17, 323 physics and 1 (1943)..  and stochastic 351 ( 1 9 4 2 ) . .  [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 p a p e r s on n o i s e and s t o c h a s t i c p r o c e s s e s . D o v e r , New York ( 1 9 5 4 ) .  [7]  G..W.  Bluman, S i m i l a r i t y s o l u t i o n s o f t h e one-dimensional F o k k e r - P l a n c k e g u a t o n . I n t . J . N o n - L i n e a r M e c h a n i c s 6, 143 (1971) .  [8]  G..W.  Bluman and J . D . . C o l e , Similarity methods for d i f f e r e n t i a l egutions. Applied Mathematical Sciences 13, S p r i n g e r - V e r l a g , pp258-274 (1974).,  [9]  R. . I . . C u k i e r , K. . L a k a t o s - L i n d e n b e r g and K..E. . S h u l e r , Orthogonal p o l y n o m i a l s o l u t i o n s of the F o k k e r - P l a n c k e g u a t i o n . . J . . o f S t a t i s t i c a l p h y s i c s 9, 137 ( 1 9 7 3 ) .  [ 1 0 ] J..D. Cowan, S t o c h a s t i c models o f n e u r o e l e c t r i c activity.. In S. A. R i c e , K. F. F r e e d and J . C. L i g h t : S t a t i s t i c a l M e c h a n i c s , New C o n c e p t s , New P r o b l e m s , New Application, University of C h i c a g o P r e s s , 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 a neuron with v a r y i n g t h r e s h o l d . J . Theor. B i o l . pp633-644 (1973) . [ 1 2 ] A.  V. H o l d e n , Models neurons. Lecture V e r l a g (1976).  of the stochastic notes, Biomathematics  of^ .39/  activity of 12, S p r i n g e r -  [ 1 3 ] A. . S i e g e r t , On t h e f i r s t p a s s a g e t i m e p r o b a b i l i t y P h y s i c a l Review 81, 617 ( 1 9 5 1 ) . .  problem.  83  £14]  Do ft. D a r l i n g and A. S i e g e r t , The first passage problem for a c o n t i n u o u s M a r k o v p r o c e s s . .Ann. .Math. S t a t i s t . . 24, 624 (1953) .  [ 1 5 ] P. . I . M. J o h a n n e s m a , D i f f u s i o n m o d e l s f o r t h e stochastic activity of neurons.. In E. E . . C a i a n e l l o : N e u r a l N e t w o r k s , S p r i n g e r - V e r l a g , pp116-144 ( 1 9 6 8 ) . [ 1 6 ] S.  S h u n s u k e , On t h e moments o f t h e f i r i n g i n t e r v a l o f the diffusion approximated model neuron. Mathematical B i o s c i e n c e s 39, 53 ( 1 9 7 8 ) .  [ 1 7 ] G.  W. B l u m a n , On t h e 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 t h e W i e n e r p r o c e s s . . (To a p p e a r i n SIAM J . A p p l i e d Mathematics).  [ 1 8 ] G.  F. D. D u f f and D. N a y l o r , D i f f e r e n t i a l Applied Mathematics..Wiley (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, [ 2 0 a ] M. [20b]  [20c]  [20d]  C. K. 155,  Tweedie, I n v e r s e 453 ( 1 9 4 5 ) . .  Gaussian (1956) . Gaussian (1957) .. Gaussian (1957) .  289  eguations  of  (1915)..  statistical  variates.  Nature  Some s t a t i s t i c a l p r o p e r t i e s o f distributions. Virginia J. Sci.  inverse 7, 160  Statistical properties of inverse d i s t r i b u t i o n s I . Ann. M a t h . S t a t i s t . . 2 8 , 362 Statistical properties of d i s t r i b u t i o n s I I . Ann. M a t h . S t a t i s t .  [ 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  inverse 28, 696  (1947)..  [ 2 2 ] J . L. F o l k s and E. S. C h h i k a r a , The inverse Gaussian distribution and its statistical application - a r e v i e w . , J . R. S t a t i s t . S o c . B 40, 263 ( 1 9 7 8 ) . . [ 2 3 ] D.  R. Cox and H. D. M i l l e r , The theory p r o c e s s e s . L o n d o n , Methuen ( 1 9 6 5 ) . _  of  [ 2 4 ] M.  T. Wasan, F i r s t p a s s a g e t i m e d i s t r i b u t i o n of motion with positive drift (inverse distribution). Queen's paper i n Pure and M a t h e m a t i c s 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 e g u a t i o n s . Academic P r e s s ( 1 9 7 7 ) .  stochastic Brownian Gaussian Applied  differential  84  [ 2 6 ] L.  S i r o v i c h , T e c h n i q u e s of a s y r a p t o t i c s analysis., applied 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 ( 1 9 7 1 ) .  [ 2 7 ] N.  Bleistein, 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 with stationary p o i n t near algerhric singularity. Communication on P u r e and A p p l i e d M a t h e m a t i c s 19, 353 (1966) . .  [ 2 8 ] F.  John, Partial differential eguations. Applied 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 ( 1 9 7 8 ) . .  [ 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 o f t h e i n t e r - s p i k e t i m e s o f neurons r e c e i v i n g randomly arriving post-synatik 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 ( 1 9 7 5 ) . [ 3 0 ] H..C.  Tuckwell, On the first-exit time problem for temporally homogeneous Markov processes. J. Appl. P r o b . 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 interspike time d i s t r i b u t i o n s and t h e e s t i m a t i o n o f 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. Biol. 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 , S p i k e d i s c h a r g e p a t t e r n s of s p o n t a n e o u s 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 nucleus 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 in a m o d e l . C o m p u t e r s and Biomedical Research (1977) .  [ 3 4 ] R.  diffusion 10, 191  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 nonlinear least sguares. H a r w e l l R e p o r t , AERE-R, 6799 (1971) .  [ 3 5 ] M. . W. . L e v i n e and J . M. . S h e f n e r , A model for the variability 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. Biophysical Journal 19, 241 (1977). A comment on a retinal [ 36 ] C. E. S m i t h , B i o p h y s i c a l J o u r n a l 25, 385 ( 1 9 7 9 ) . .  neural  model.  85  APPENDIX I  DEFINITIONS^  + r€[0,oo),NtfW,  where  G ( r - J ; Z) = H ( Z ) , HeiQ,  Z  r ) , H (Z) i s r e g u l a r ,  (1)  where H (Z) h a s a z e r o a t Z = 0 o f d e g r e e r , U (s) = I Z e  '  r  r-i  r  dZ,  (2)  (r) = 0,  (3)  (r) = G (r;0) ,  0  (4)  r, ( r ) = ( G ( r ; 0 ) - G ( r ; - a ) ) / a ,  (5)  r  (6)  (r) =  in  r  ilW  ,  (r+1)  (r) =  G (r;0) , n  (r+1)  (Gn (r;0)-G„ (r ; - a ) ) / a + Gj, ( r ; - a ) ,  G ( r ; Z ) = ft (r) + r,(r) Z+Z (Z+a) G, (r;Z) ,  (7) (8)  (r + 1) G ( r ; Z) +ZG* ( r ; Z ) n  =  r  an  ( r ) + r -,  ( r ) Z+Z  an  f o r n= 1  (1)  (Zta)Ghfi  (r;Z) ,  (9)  N. .  NOTATION G' (r;Z) = ^ G « ( r ; Z ) . n  LEMMA A l l fc„(0)  =  t l n - t O ) ,  £« i(0) +  f o r n=0 ( 1 ) N;  = A.  (l)-a^i(l),  G { 0 ; Z ) = Z G ( 1 ; Z ) + Jfl*.|{1) , n  n  for Proof:  n=1 (1) N.  (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) |  By  J s o  = 0 = fit.  ( 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) . (ii)  n=1: By ( 1 ) , (8) , ( 1 1 ) 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) G i (0;Z) . Divided G(1;Z) = By  by Z, we have fi(1)-a  fi(1)+(Z+a)Gj  (0;Z). .  (8) a g a i n  G(1;Z) = Eguating  lfi(1)Z + Z(Z + a ) G i ( 1 ; Z ) . ( 1 2 ) , (13)  gives  r ( 1 ) - a eT, (1) + (Z*-a)G, (0;Z) 0  =  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) By  = *i<1) .  (7) , (15) , (14)  and (9)  f, (0) = (Gi ( 0 ; 0 ) - G i (0;-a))/a+GH0;-a) =  ( f i ( 1 ) - ( ( - a ) G i (1;-a)+ • (Gj (1;-a)+ (-a)G'i (1;-a) )  = 2Gt (1;-a)-aG«i(1;-a) .  fi(1)))/a  87  r,(0)  =  r*(i)-ar3(i).  (14) , (15) , (16) i m p l y  (16) t h e lemma i s t r u e  Assume t h e lemma i s t r u e *U(0)  =  f o r n=1.  f o r n=k, i . e . (17)  ^ i - l d ) r  (18) G*(0,Z) = ZGftJ1;Z) + fik-l  (iii)  (19)  (1) ,  n=k+1: By (9) , (19) G4L(0;Z) +ZG ^(0 ;Z)  = fa (0) +  + » (0) Z + Z (Z+a) G t + 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  o f t h e e g u a t i o n by  Z, 2GA(1;Z)+ZG»4(1;Z)•= By  (9) a g a i n  Ifafi + i (0)+(Z + a)G^.H-i ( 0 ; Z ) .  f o r r=1  2G«.(1;Z) +ZG'A(1;Z) = r l ( 1 ) +  *U ,(1)Z  2  +  + Z(Z + a ) G f t Eguating  (20)  + 1  (1,Z)  (20) ,(21) and a p p l y i n g  (21)  ( 1 8 ) , we have  ^ ( 1 ) - a j T f t , (1)+ (Z*a)G#. + i ( 0 ; Z ) z  =  t  A * ( 1 ) + Jfifctid) Z+Z(Z+a)Gfcti(1;Z) Divided  G i t i (0;z) or  by (Z+a) , we have = ZGA+I (1 ;Z) +  G* ,(0;Z) +  By  (1)  = ZG * (1; Z) +tf£ jU.>-«. (1) . . +t  c  (22)  (6) , (22)  oWo(0) = G f c ( 0 ; 0 ) = j \ U . ) - l ( 1 ) . lt  +l  By  (7) , (22) , (23)  ft<4«-i>+ i =  ( 0 ) = ( G t f i ( 0 ; 0 ) - G i , (0;-a))/a+G'*+i(0;-a) +  ( *\cA+o-i (D-(-aG**, (1;-a) + r*ift+i>-i(D ) ) / a  (23)  88  +  ( -JUi  (1;-a)-aG^ (1;-a)  G  + l  = 2G*+i ( 1 ; - a ) - a G ' * + i n ; - a ) . By  (9) we have  Aci*«>+1  (0) =  (1) -a j T i t ^ + o t i (1) o .  fol*.*')  (24)  (22) , (23) and (24) i m p l y t h e 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  t o see from  (2)  t h a t U (s) = - 0 '  U (s) = J V 2 e e r f c (s/i"2) and U i ( s ) =  (s) ,  s0o<s)-1._  T  e  We h a v e S(0)  =  ^-[u.f/St)f  ( 1 )  =  f  s  C(Q-ftC„^  [ K , I J X 4 ) ^  (l)  N o t i c e t h a t U (s) ^ 0  , ±  +  —ft.Tin 4|  £ H) JT/1*"*  ^vtr £ * hence, s t r i c t l y  prove  speaking, the f i r s t  S(0) = S ( 1 ) ,  p a r t s o f S (0) a n d S ( 1 )  series.  however,  i t i s s u f f i c i e n t t o show  that (0)  ft* (0)  =  a s s -» +*  a s s -* - *>,  a r e p r o d u c t s o f two a s y m p t o t i c To  x £  X\« (1) - a it„ <1) a n d +l  = j f t n - i (1) , n=0 (1) N.  By Lemma A 1 , t h i s i s i n d e e d t h e c a s e .  

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