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Numerical algorithms for the solution of a single phase one-dimensional Stefan problem Milinazzo, Fausto 1974

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NUMERICAL ALGORITHMS FOR THE SOLUTION OF A SINGLE PHASE  ONE-DIMENSIONAL  STEFAN PROBLEM by Fausto B.Sc,  University  A THESIS THE  Milinazzo  of  British  SUBMITTED IN  Columbia,  1970  PARTIAL FULFILMENT OF  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in  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 rtrvd S t a t i s t i c s  We a c c e p t required  THE  this  thesis  as c o n f o r m i n g  to  standard  UNIVERSITY  OF B R I T I S H COLUMBIA  April,  1974  the  In p r e s e n t i n g t h i s  thesis  in p a r t i a l  fulfilment  o f the  requirements  an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I a g r e e the L i b r a r y s h a l l make i t f r e e l y I further  agree  available  for  that  f o r r e f e r e n c e a n d study  t h a t p e r m i s s i o n for e x t e n s i v e copying o f t h i s  thesis  for s c h o l a r l y purposes may be granted by the Head o f my Department n r by h i s of  this  representatives.  It i s understood that copying o r p u b l i c a t i o n  thesis f o r financial  gain s h a l l not  be allowed without my  written permission.  Department  of  Aymlipr)  I" a -h>iPTr);T+. -j  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  June I > . ~i 97/  n«  rmrl  fit.gtistins  ABSTRACT .  A one-dimensional, single considered.  This problem  is  phase S t e f a n  shown to h a v e  Problem  a unique  is  solution  w h i c h d e p e n d s c o n t i n u o u s l y on t h e  boundary d a t a .  In  two  its  numerical  algorithms  are  solution. '  formulated  algorithm  based on S i m i l a r i t y ,  c o n v e r g e n c e between examples  In  is  s u g g e s t e d by t h e  the  numerical  comparison  is  difference  scheme.  of  differential  to  reduce  equations.  ordinary  differential  examples  indicate  the  achieving very  the It  accurate  this  is  is  shown to  and a  a set  observed that is  stiff.  a  approximations.  It  which  improve  brief  well-known  of  this  results  ordinary set  Moreover,  a high order  are  algorithm  (a C o l l o c a t i o n Scheme) problem to  of  algorithm  the  which  numerical  Furthermore,  algorithm  equations  that  this  convergence are  The second a l g o r i t h m an a t t e m p t  Moreover,  aspects of  significantly.  made b e t w e e n  Algorithm),  converge with order  m o d i f i c a t i o n s to  proof  results  Similarity  and o n e .  various  particular,  are  (the  shown t o  one h a l f  illustrating  presented.  from  approximate  . The f i r s t  is  for  addition  of numerical  scheme c a p a b l e is  of  observed that  th  (ii)  apparent renders  stiffness this  second  of  the  system o f  algorithm  ordinary  relatively  differential  inefficient.  equations  TABLE- OF CONTENTS  ABSTRACT  (i)  TABLE OF CONTENTS  (iii)  L I S T OF T A B L E S  (vi)  L I S T OF FIGURES  (vii)  ACKNOWLEDGMENTS < .  (viii)  INTRODUCTION  1 The  Problem  Existence  and U n i q u e n e s s o f  The N u m e r i c a l  CHAPTER  I  CHAPTER I I  2  Stefan  the  Solution  Problem  4 5  Non-dimensionalizing  7  EXISTENCE AND UNIQUENESS  11  The  11  Integral  Equations  Existence for Small  and U n i q u e n e s s o f Time  the  Existence  and U n i q u e n e s s o f  the  Solution 18 Solution  for A l l t 6 OiTl  29  THE S I M I L A R I T Y METHOD  32  Lie  33  Group o f  Transformations  35  Invariance The M o s t G e n e r a l of  m-Parameter  Transformations  Reducing  the  The,Classical  Leaving  Number o f Group o f  (iii)  Lie  Group  Invariant  (2.1)  Variables the  Heat  37 42  Equation  44  (iv)  CHAPTER I I I  A USEFUL SIMILARITY SOLUTION  OF THE HEAT  EQUATION F O R . T H E STEFAN PROBLEM  46  CHAPTER IV  THE S I M I L A R I T Y ALGORITHM  60  CHAPTER V  THE CONVERGENCE OF THE S I M I L A R I T Y ALGORITHM  65  Continuous  Dependence  for  Small  65  Continuous  Dependence  for  All  Convergence Order CHAPTER V I  of  of  the  Similarity  Time  Time  80  Algorithm  82  Convergence  83  THE SIMILARITY ALGORITHM -  NUMERICAL  RESULTS :  86  Numerical  Examples  Optimization Order -  of  of  the  Convergence  Small  Time  the  the  of  Similarity  the  the  with  Difference  Lotkin's  Large  Similarity  Scheme  Equations for  Conduction  Time Solution  102  Algorithm  A COLLOCATION SCHEME  112 115  Heat 115  A Galerkin  Scheme  A Collocation Numerical  95  Similarity  Versus  Comparison of  The L a g r a n g i a n  BIBLIOGRAPHY  of  Algorithm  98  Representation  CHAPTER V I I I  Similarity  Algorithm The  CHAPTER V I I  86  Scheme  Results  CONCLUSIONS  127 135 142 150 152  (v)  APPENDIX A  156  . APPENDIX B  162  APPENDIX C  167  APPENDIX D  171  APPENDIX E  174  APPENDIX F  179  LIST OF TABLES  TABLE 6.0  TABLE 6.1  E r r o r s Versus Time Increment Data (6.0)) E r r o r Versus Time Increment M o d i f i c a t i o n s I and I I  (Boundary  88 Ai.  Using 97  TABLE 6.2  Observed Order of Convergence  101  TABLE 6.3  Approximate O p e r a t i o n Count  109  TABLE 6.4  S i m i l a r i t y A l g o r i t h m Versus D i f f e r e n c e Scheme  TABLE 7.0  TABLE 7.1  Lotkin's 1.13  E r r o r s i n UU,t),W^' t\M (« t)and (Exact S o l u t i o n (7.25)) 1  lw  E r r o r s i n v*w,t>, u. (Exact S o l u t i o n (7.26))  (vi)  1  JCtV 146  and J t t ) 147  LIST  OF FIGURES  FIGURE  0.1  A Melting  Slab  FIGURE  1.0  The (weak) Maximum P r i n c i p l e r e c t a n g u l a r Domain  FIGURE  4.0  The S i m i l a r i t y  FIGURE  6.0  Approximate T =.40  FIGURE!  6.1  for  Data 6.2  FIGURE  6.3  Data FIGURE  6.4  Temperature the  62 Distributions  Boundary Data to  the  >5C-i)up t o  90  P o s i t i o n of  T=.4  for  at  (6.0)  the  the Boundary  H(10  and  Vi("t) f o r  the  91  Boundary  (6.0)  92  The A p p r o x i m a t e for  13  (6.0)  Comparing Data  on a N o n -  Algorithm  Approximations Boundary  FIGURE  3  "t  Between  Temperature 0 and  .1  for  Distribution the  Boundary  (6.1)  94  Comparing the Generated  G i v e n Heat  by t h e  Flux with  Similarity  Using M o d i f i c a t i o n  II  (vii)  that  Algorithm 97  ACKNOWLEDGMENTS  I for  having  guidance  during  for  thank  also  carefully  British  I  have  the  Dr.  for  George  his. patient,  Bluman,  encouraging  work.  James V a r a h  for  many h o u r s  of  Research C o u n c i l of  Columbia Mathematics  Canada  and  Department  for  assistance. e x p r e s s my a p p r e c i a t i o n  I  A p p l i e d Mathematics  whatever  to  and  Dr.  discussions.  Finally,  Doedel;  topic  course of  typing  Eusebius  to my s u p e r v i s o r ,  the N a t i o n a l  of  financial I  the  am g r a t e f u l  University  their  the  and u s e f u l I  the  grateful  suggested  I helpful  am v e r y  for  the  to my w i f e ,  Beverly,  manuscript.  t h a n k my f r i e n d s and S t a t i s t i c s , e a c h one has  achieved.  (viii)  in  the  Institute  particularly  contributed  Hart  of Katz  significantly  and to  INTRODUCTION  This through  Thesis  is  a medium w h i c h  concerned with  is  experiencing  Characteristically s u r f a c e made up o f  points  If  the  the  the  p o s i t i o n of  problem,  essentially  known as to one o f  if  the  Inverse Stefan  the  the  solving a parabolic  "moving" Problem  "moving" to  on an  of  linear  position  of  this  time,  reduces  irregular  is  another.  then  equation  domain. so  is  the  Problem. when  problem,  the  referred  distribution  surface. is  a  differential  system  - B o u n d a r y P r o b l e m , , becomes one o f temperature  involve  Problem,  heat  phase.  as a f u n c t i o n  Stefan  conditions  of  phase changes  given  Inverse  differential  However, a priori,  which.one is  diffusion  a change o f  such problems  surface  w i t h a s s o c i a t e d boundary Evidently  at  the  of  the  to  as  surface  a Direct  finding  is  Stefan  or  simultaneously  medium and  As c a n be s e e n r e a d i l y  the the  position Direct  not  given  Free the of  the  Stefan  non-linear. A l t h o u g h F r e e Boundary Problems  « G . Lame a n d B . P .  Clapeyron published  - 1 -  in  date back  1831  and t o  to  a work  several  of  - 2 -  papers  of  nineteen During  J . Stefan which appeared thirties  the  published Stefan  Problems. for  In  their  until  begin  in  analytic  addition, numerical  properties  a number o f  of  the earnest.  has  been  one-dimensional  schemes h a v e  been  solution.  Problem  Stefan  Problem.  melting  of  between  ^= 9  and is  distribution, slab  for  change.  p r e c i s e l y , we w i s h  >  3© ^ )  .  a  W  heat  heat  H (f)  flux  0  the melt  our a t t e n t i o n  in such a  to  the  space  temperature interior  slab  to  the  be  upon  to  the  insulated  "moving" boundary  isothermal  immediately  govern the  the  temperature  p o s i t i o n of  s o l i d phase  phase  o c c u p i e s the  equation  c a u s e s an  removed  equations slab.  the  we assume t h e > the  single  describe  initial  assume t h a t  e  Furthermore,  The f o l l o w i n g distribution  •  to  initially  , and whose  Q  at  By h a v i n g  we r e s t r i c t  £  > obeys the  t  T>o  2"  ^ *  ^\y Z)  o , while time  More  one-dimensional  a homogeneous s l a b , w h i c h  distribution  at  not  a c o n s i d e r a b l e amount  We c o n s i d e r a p a r t i c u l a r  at  1889,  on s u c h p r o b l e m s  twenty years  documenting the  developed  The  past  d i d work  in  phase  formation,  only. temperature  - 3 (1) (0.1)  J-Crf Cr;, r) J 7-^ -O  JyiOyZ)  r « Co.Tp),  (0.1a)  recoup),  (0.1b)  r t  to,T ),  (0.1c)  F  (O.ld)  (O.le)  W»(r> Fig.  0.1  A Melting  (  derivatives  1  )  of  Here  X  T  JT(y;r)  Slab  i V . r ) , J>, ( y , * ( y , * ) a  n  d  denote  J ' t i d_ ['jS'tJ:)]  partial  - 4-  Here, specific .\»  -  to  heat,  the  and a r e the  p  latent  taken  -  that  the  heat  of  case  the  heat  flux,  .J'Gs'Or), ~)  i.e.  boundary  is  never  temperature.  f u s i o n are  characteristic  problem  s  the  to  for  of  n  e  the  and material  our  fall  will  (0.1,a,b,c,d,e)  attention  c a n be  a n a l y s i s we w i l l  of  J*  to  the  below  arbitrary  a priori  t  conductivity,  we r e s t r i c t  sufficient  temperature  e  "  H„(T)fc6.  throughout  allowed  Whether  c a n be g u a r a n t e e d  ,  general  n  the  Moreover,  6  t  -  c  H  V - i ( r ) » i»  temperature,  density,  * 3-**.  X  with numerically, the  melting  to be c o n s t a n t .  Although dealt  the  maintain  slab  vTwv  at  melting,  the  ~ the and  0  assume  M  melting  melting this  6  be d i s c u s s e d b r i e f l y  at  a  later  time. Under the of  Stefan  or  available  the  above a s s u m p t i o n , c o n d i t i o n  F r e e B o u n d a r y c o n d i t i o n and d e t e r m i n e s energy  to  The c o m p l e t e pair  of  functions  Existence  the  diffusion  s o l u t i o n of  ( T ^ Z ) ,  and U n i q u e n e s s o f  and t h e  are  solution  is  allocation  processes.  is  then  the  »SHr>) ,  the  Solution  e s t a b l i s h e d by one o f expressed in  becomes  the  melting  (0.1,a,b,c,d,e)  The e x i s t e n c e and u n i q u e n e s s o f Problems  (O.le)  terms  of  solutions  s e v e r a l methods. a set  of  coupled  to  Stefan  Usually Volterra  the  - 5 -  •.  I n t e g r a l E q u a t i o n s , then the proof proceeds  by e i t h e r u s i n g the  Maximum P r i n c i p l e or a f i x e d p o i n t argument. Cannon and Denson H i l l  £  7 ]J use the Strong Maximum  P r i n c i p l e t o g e t h e r w i t h a r e t a r d e d argument approach the e x i s t e n c e and uniqueness consider.  o f the s o l u t i o n o f the problem  they  Priedman £ 17 3» r e f i n i n g the work of Rubinstein  |~ 29 ]],' t r e a t s the same problem '  to e s t a b l i s h  u s i n g a f i x e d p o i n t argument.  Using methods as o u t l i n e d  the e x i s t e n c e , uniqueness  (Chapter I) and continuous  (Chapter V) on the boundary data s o l u t i o n to the system  i n Friedman f 17 1 we e s t a b l i s h  o f equations  y  H» ( r ^ J" (*)| o f the 0  (0.1,a,b,c,d,e).  of the S i m i l a r i t y A l g o r i t h m (see Chapter  The Convergence  IV) then f o l l o w s from  the continuous dependence of (0 .1 ,a,b ,c ,d,e)  The Numerical  dependence  on  Vl l^» 0  S t e f a n Problem  U s u a l l y n u m e r i c a l schemes d e a l i n g w i t h Free Boundary Problems a r e p a r t i c u l a r to the boundary c o n d i t i o n s b e i n g c o n s i d e r e d . For i n s t a n c e , the f i n i t e d i f f e r e n c e scheme developed and G a l l i e £ 11 ] uses  by Douglas  two boundary c o n d i t i o n s to e s t a b l i s h an  i t e r a t i o n which a t each step i n time l o c a t e s the p o s i t i o n o f the "moving" boundary.  S i m i l a r l y the continuous methods o f Mason and  Farkas £ 24 J r e l y on the appearance o f  ^{X)  twice i n the system  of equations so t h a t a g a i n an i t e r a t i o n to the s o l u t i o n can be  - 6 -  established. S e v e r a l a u t h o r s , f o l l o w i n g the l e a d of Landau [ 22 ] , X  make the t r a n s f o r m a t i o n  s  ¥/jS'ltr)  , then c o n s t r u c t  schemes f o r the r e s u l t i n g system on the f i x e d space  £o,lJ.  For example L o t k i n £  space  i n t e r v a l [o,lJ  has been to reduce  "J(X <r) #  The  referred  This  was  temperature  system (0.1, a ,b,,c ,d , e)  yields a set of ordinary d i f f e r e n t i a l  We  equations.  as an a p p r o p r i a t e F o u r i e r S e r i e s w i t h  time dependent c o e f f i c i e n t s .  C o e f f i c i e n t s and  fixed  the S t e f a n Problem to  25 J by e x p r e s s i n g the  a c h i e v e d by Melamed £  distribution,  f o r (0.1 ,a ,b ,c ,d , e) .  a l t e r n a t i v e to d i f f e r e n c e schemes on the  a countable s e t o f o r d i n a r y d i f f e r e n t i a l first  interval  23 ~\ uses t h i s t r a n s f o r m a t i o n  to o b t a i n a f i n i t e d i f f e r e n c e approximation One  approximating  equations  then  f o r the F o u r i e r  the p o s i t i o n of the boundary.  propose s e v e r a l schemes.  The  to as the S i m i l a r i t y A l g o r i t h m  first  (Chapter  scheme, IV) i s based  on an exact s o l u t i o n of the Inverse S t e f a n Problem o b t a i n e d through  the S i m i l a r i t y Method  (Chapter I I ) .  That  i s , solutions  of the I n v e r s e S t e f a n Problem are p i e c e d together i n such a as to given an approximate In Chapter  VI we  the S i m i l a r i t y A l g o r i t h m .  way  s o l u t i o n of the D i r e c t S t e f a n Problem. g i v e numerical examples ~  illustrating  -  The s e c o n d related, Problem  and a r i s e  extensions taking as  and t h i r d  equations.  approach.  That  combination  Hence they  which  functions)  (with support  are global  Stefan  ordinary  instead of on £ o , l ] ( s u c h  a finite  vT( ' ^ v  in a subinterval  c a n be o b t a i n e d  However,  T  (finite  element  by a f i n i t e  coefficients)  equations  are closely  the D i r e c t  system o f  we a d o p t  time dependent  VII)  c a n be l o o k e d u p o n as  i s , we a p p r o x i m a t e  A system o f d i f f e r e n t i a l rt'{Z)  to a c o u n t a b l e  functions  the Trigonometric  (Chapter  to reduce  o f t h e method o f M e l a m e d .  as a b a s i s ,  which have  schemes  from attempts  (0.1,a,b,c,d,e)  differential  7 -  of  linear  functions  element)  of  £o,l].  f o r t h e c o e f f i c i e n t s and  i n s e v e r a l ways by u s i n g  equations  (0.1,e). We w i l l is  derive  two s y s t e m s o f e q u a t i o n s .  known a s a c o n t i n u o u s G a l e r k i n s y s t e m w h i l e  referred  t o as a C o l l o c a t i o n s y s t e m .  The f i r s t  the second  Some n u m e r i c a l  is  results are  given.  Non-dimensionalizing Before  proceeding further  system o f e q u a t i o n s  (0.1,a,b,c,d,e)  we n o n - d i m e n s i o n a l i z e t h e by i n t r o d u c i n g  the  following  -  8 -  variables: (2) , where " " is a characteristic length  X= >Vet T  a  pea* > T  ucx.tvi CTo?r)- T v A i / J v ^  ,  (0.2)  •T -  K  T  Substituting the variables (0.2) into (0.1,a,b,c,d,e) ve obtain The c h a r a c t e r i s t i c length "a" can be taken to be the i n i t i a l length of the slab v  - 9  O < X < 4 Ct) < JC«Js b  <*,*V= M f c U , * )  (0.3)  I « C o,T^  J  i<t>* o  U  U U I D ^ J - O  t6(o,T),  (0.3b)  t t  (0.3c)  M t«>*)-© R  .'  Utx.o)-.  that  (0.3a)  ( o j l j  (OjT),  (0.3d)  X t C«,  it!) s « * | M « U U ^ - 0 -  We n o t e  J ( T ) - b„>o  (0.3e)  with  j(o);  h(t\J  b  t«Co,TK  (0.3e)  can be w r i t t e n  equivalently  as  (0.3f)  We now s e e k of  e C'Co.b]  with  \-»Ct) c o n t i n u o u s  .discontinuities of  solution  equations ( 0 . 3 , a , b , c , d , e ) .  V.U') and  the  but  on [o,l]J.  (0.3,a,b,c,d,e)  for  More  ( -u. (.*, i), j ct>) precisely,  v.lOi© a finite  satisfying  the  for  of  system  A (»)= U ( 0  0  b)r  o  jump  a solution  conditions:  the  take  on [ 0 , b ] ,  number  Then we l o o k  we  to  (*u.  j ct)).  (a)  M t * , t ) , M lw,t) 6 C t>,.scu), t <s < O , T ) ;  (b)  v.(.x,t^ <S.  (c)  x x  cro^ct)!  U „ C*,t> €  We b e g i n  CCo.iti^  in Chapter  equations(0.3,a,b,c,d,e)  ,  I  t C  Co.T);  t e Co,T);  by showing t h a t  has a unique  solution.  the  system  CHAPTER I  EXISTENCE AND  UNIQUENESS  In t h i s c h a p t e r we show t h a t the system o f equations (0.3 ,a ,b ,c , d ,e) has a unique s o l u t i o n (14,5) f o r a l l  ,  <£) s j U,t) i o< x < 4<tl , o < t < T  To  accomplish  constructing  an e q u i v a l e n t  Equations and showing that "K< 9-  the l e a d of Friedman [ 1 7 J , by  t h i s we f o l l o w  system o f Coupled V o l t e r r a there  exists a  C>© such t h a t  for a l l  , these i n t e g r a l equations have a unique s o l u t i o n .  then show t h a t  this  procedure can be repeated to y i e l d  and  uniqueness of the s o l u t i o n o f the system  the  i n t e r v a l o f time  The  I n t e g r a l Equations ..  s e v e r a l u s e f u l lemmas.  existence  (0.3,a,b,c,d,e) f o r  >  the i n t e g r a l e q u a t i o n s , we s t a t e  The f i r s t ,  working lemma used e x t e n s i v e l y integral  We  (o T).  Before constructing  -SCt.^  Integral  due to Friedman [ 17 ] , i s a  throughout the c o n s t r u c t i o n  o f the  e q u a t i o n s ; w h i l e the other two e s t a b l i s h p r o p e r t i e s o f  ( t h e p o s i t i o n o f the f r e e boundary) and  -  11  -  vt> (..St-U^-fc^ x  (the  -  amount''of. h e a t a l l o c a t e d to Defining of  the  heat  12  the  -  diffusion  K ix,-t;^ t) t  equation,  that  to  be  Lemma  the  following  usual  source  solution  Cx-g)*  lemma.  1.1. Let . I n  for  the  respectively.  is, -  we h a v e  process)  <fit^i  -set)  addition,  some c o n s t a n t  sctw©  M  .  be c o n t i n u o u s let  jet)  Then f o r  functions  satisfy  all  the  Lipschitz  "k £ i , "! 0  on t h e  0  we  interval condition  have  Ox ^  Proof. The p r o o f ,  given  i n Appendix A ,  c o n s i s t s of  showing  -  r  13  -  1  In o r d e r to e s t a b l i s h following auxiliary  the next two lemmas we use the  p r o p o s i t i o n s whose p r o o f s are g i v e n i n  Appendix B.  P r o p o s i t i o n 1.1  (The (weak) Maximum P r i n c i p l e ) .  Suppose "UAx^)  with  <St-t)  a positive  satisfies  continuous f u n c t i o n and  'M„U,t>, vi^ <.«,*> £ C (•& V <8 ) X  Fig.  1.0  The  where .£)  -uCx,-t> e C  i s the c l o s u r e o f ^  (weak) Maximum P r i n c i p l e on a N o n - r e c t a n g u l a r Domain  ) , and  -  then  v.U,t)  assumes i t s maximum and minimum v a l u e s  boundary  1.2  If  (The n e c e s s a r y c o n d i t i o n  (-u,s)  for  i s a s o l u t i o n o f the system  Vj.uui.t^ j o  for a l l  We a r e now r e a d y  Lemma  SC*.)  (0.3,a,b,c,d,e)  all  to e s t a b l i s h  ( 0 . 3 , a , b , c ,d ,e)  the f o l l o w i n g  is a solution  then  .SCiV  o f the system of  satisfies  the L i p s c h i t z  t., , t j . €.  O . T ]  .  Proof. Consider  condition  (0.3f)  «  t,  melting).  t e CO,T3 .  (1)  at  data  properties.  1.2. If  for  on t h e  - <Sy •  Proposition  then  14 -  and  i  o Then  i  Notation:  lt*il,-  equations condition  where  o Since  A'  implies  that  o  - - H, (Jlt.+j),!,^)  /Wo) = o  and  ^ ^ ^ 5 > <3*°*  , Proposition  1.2  T n u s  t, and h e n c e  ••Uct,..>-s(i,)| > «<*«mi » t - t , l . T  (  Lemma 1 . 3 . If  ( 0 . 3 , a , b , c , d ,e)  for  all  Proof:  ' M * C*,tV  then  (x,t} C c5b  Proposition  is a solution  ("U,5)  , provided  satisfies  vt^t^-O  satisfies  the hypotheses  of  trivial. established  construct  the integral  establish  existence  equations  these  equations  preliminary  which w i l l  results  ultimately  and u n i q u e n e s s o f t h e s o l u t i o n  we c a n allow  us t o  to the system  (0 »3 , a , b , c , d , e ) .  To t h i s half  equations  1.1.  Having  of  of the system of  end. we i n t r o d u c e  the Green's  plane 6*Ci,t;5,T)t K ( * , t ^ t ) + i  KU,i;-5,:r),  functions  f o r the  -  and n o t e  that  satisfies  any  solution  Green's  Identity  §LfG £M +  - ^  the  v.C^,t)  3 6 l  -  +  of  the  &L f ( 5 * v l  heat  equation  =6  (1.0)  J  domain  •  where  -  5 f l - art  Sk in  16  £  O8 = JC?,^£  >  o<f<T<t-f^  >o. Integrating  (0.3b,c)  o<^,<s(rv  and  (1.0)  over  6 ^ t * , i ; o,2*V~ O  and u s i n g we  conditions  obtain  r° O-  ^ vt(^,£) G C K t ; ^ r ) c | F +  l  <  <  .sen + \ vUE;,t-0 G (*,t;^, W ^ g ; +  In  Appendix  C we show t h a t  as  f->o  (1.1)  (1.1)  becomes  (1.2)  -  Differentiating Lemma 1 . 1  as  17 -  (1.2)  x-»J<-k)-o  with respect  we f i n d  to  x  and a p p l y i n g  -o l - O 2 -u c 5 t t i  that  n  satisfies  •--vm* *  .}* <3>G£Utti,t;e,o)d£  ^ vcr> G* W ( t > , t i 3 ( * ^ r ) d r .  t  Since after the  (1.3)  0  <S^.(*,i>? o)*- <»*<*•,t;£,o) , G~(x, t • o , o ) s o  and  t  making  first  the a p p r o p r i a t e  integral  substitution,  expression of  °  (1.3)  to  -u t»=o, 0  we i n t e g r a t e b y  parts  obtain  (1.4)  it • ©  Moreover, (0.3e)  as does F r i e d m a n £  and f i n d  17 } we i n t e g r a t e  the  condition  that  .sm-  b  \ ( v ( r l - h ( r ) ) dt.  +  (1.5)  ©  We h a v e by  (1.5),  that  W-t)  h e n c e we r e f e r  Furthermore, solution, of  we h a v e (1.4),  satisfies  to v ( t )  where  as t h e s o l u t i o n o f  the f o l l o w i n g  (1.5))  (1.4)  and ( v , s )  equivalence  s(t)  (1.4),  between  (the s o l u t i o n  is  of  given (1.5).  v(t)(the (0.3 , a , b , c ,d ,e))  - 18 -  Lemma 1 . 4  (The e q u i v a l e n c e o f the d i f f e r e n t i a l  and i n t e g r a l  sys terns). If  v ( t ) i s a s o l u t i o n o f ( 1 . 4 ) , where s(t>  (1.5), then ( v , s ) ( v < ( * , t )  d e f i n e d by (1.2) and j t i )  (1.5)) forms a s o l u t i o n o f (0.3,a,b,c,d,e). i s a s o l u t i o n o f (0.3,a,b,c,d,e),  then  i s g i v e n by  d e f i n e d by  Conversely i f  \> (t> * H » U t t ) , t )  (M s) t  is a  s o l u t i o n of (1.4). The proof i s standard and i s g i v e n i n Appendix D.  E x i s t e n c e and Uniqueness  of the S o l u t i o n f o r Small Time  From the e q u i v a l e n c e of the system of d i f f e r e n t i a l equations  (0.3,a,b,c ,d,e) and thesystem o f coupled  equations  ((1.4),  ( 1 . 5 ) ) , we see that showing that the former has a  unique s o l u t i o n reduces unique  integral  to demonstrating  that the l a t t e r has a  solution. To e s t a b l i s h t h a t  we make the f o l l o w i n g  ( ( 1 . 4 ) , (1.5)) has a unique  definitions.  D e f i n i t i o n 1.1:  the s e t o f bounded continuous f u n c t i o n s on |0,tf"  D e f i n i t i o n 1.2:  the c l o s e d M-sphere i n C  r  .  solution  -  Definition  -  1.3. Define  (1.5),  19  that  • H"*  to  be t h e  transformation  is,  + 2 \vt*v  G;csto,t;5«*i,t;H-t  where  It  is  e a s y ,to s e e  Moreover,  Theorem  we h a v e  Hf  1  Proof.  the  following  theorem.  1.1. There  where  that  exists  : C ^  M  ->  a  <r>o  CO-.M  such  that  g i v e n , by  (1.4),  - 20 Suppose  Let  V«  CO-,M  Uvll^ < M  then  x  and hence  s a t i s f y the f o l l o w i n g i n e q u a l i t y  (1.6)  0" 5 "<^ ~ v  where  L  N*\\WI| . T  N  Then f o r a l l  which i n t u r n i m p l i e s  1 C- [ o ^ ]  that  < *ft) t 1 b  for all  v  **  < r  -  (1.7)  Since  from ( 1 . 4 )  we conclude that  \\T^1|  Writing  where  *  a n i A  +  M ^ ^ ^ ^ M ^  •  .  (1.8)  - 21 -  G, = \\K  &i  we e s t i m a t e  G  and  continuous we f i n d  i n turn.  a  N o t i n g t h a t J(t) i s L i p s c h i t z  that  \ G,l i -?L? , ( ^ ^ N ) -t"« .  To e s t i m a t e  (1.9)  S, we use (1.7) to o b t a i n .  A  (2)  16,1 «: a «*&e(-^. ) «  (1.10)  i  Now a p p l y i n g  the i n e q u a l i t y  etftcfO,* — • JL  to (1.10) we  have  IGJ* J. / t » Combining  .  (1.11)  (1.9) and (1.11) we see t h a t  HTvll  t  f  2»* rt' 0  b  +  M  (1.8) becomes  (o('(M4W)+  J.) t  ty'/j  Hence the c o n c l u s i o n also s a t i s f y  a  .  b  o f the theorem f o l l o w s  i f we i n s i s t  that  o-  the i n e q u a l i t y C- J X  !  .  (2) ' We use the n o t a t i o n v  V  (1.12)  „  ,  t  erucvM- —  \  Vc  -**  At ,  -  The restrict on has  the size  ^C,M  a  a unique  Theorem  following  n  d  of  22 -  theorem  <r  so t h a t  hence allows  solution  shows t h a t  in  T  1  we c a n  further  is a contraction  us to c o n c l u d e t h a t C  f o r a small  C>o  such that  mapping  (1.4),  (1.5)  time.  1.2. There  mapping o n  exists  Cg- ^  a  for a l l  | (  t  is a  contraction  6 C o , «rl,.  Proof. Initially satisfies satisfy  (1.6)  (1.5)  Since  be s u c h  c  and ( 1 . 1 2 ) .  with  -o'Ct)  t-fe) ^ v'tt^  <c  f«  From ( 1 . 5 )  let  we h a v e  <r i «r«  0%  ^ i t ) , v> Ct> C- C ^  If  and  that  1  Vtf  -JCt)  respectively  where let  c  e  V <fc>,5 Ct>  and d e f i n e  we have  for a l l  the f o l l o w i n g  | sit) - j'(t\V * e «•*  • .  inequalities: (1.13) (1.14) (1.15)  - 23 and  as b e f o r e V  $ 5lkV, S»<*>-.$ 3 b .  Now  f o r a l l r,t  « Z*,<*1 .  (1.16)  consider  TvTv'=  v, - V,  where  We can w r i t e  V, - V,' + V,"  where  2 ^o(%)[wCS(^^o)-W(5'(t)^;^ o)]a^  V>  J  %  o  v;. s-2^v.^)[KUrt),t f,o)-K(s a) t -^o)] d | . ,  ,  r  J  i  o  Applying  t h e Mean Value Theorem and the i n e q u a l i t y  (1.16) to V,'  we o b t a i n IVI  To estimate  *  V,'  £ nv.il.  we assume that  the p o s s i b l e c a s e s :  (1.17)  .s'ct) > .S ("fc)  and c o n s i d e r  24 C a s e I:  o< b * A U > < V<t > i  Case  o < h *  II:  Case I I I :  o < |  *  |  x<  b  >  * |  b  ,  * 5It) < « ' < t ) .< b .  Considering  v; ^^C5>[K_<SW,t f o>-KCS <i) t;f,o)]e»5 s  ,  3  ©  and a p p l y i n g times,  at  the  I:  IV.'W  Case  II:  W\\ J  Case  III:  I v'. I * ^  (1.18)  W,l  with  S  i n Case  I,  in  Case  II,  in  Case  III.  Theorem a n a p p r o p r i a t e  number o f  estimates:  Case  Combining  >  Jt-fc)  t h e Mean V a l u e  we a r r i v e  I  i.  (1.17)  tlfM^A  II  • «t't "*  nv.il^ t"* 1  •<*  ^  (1.18)  t"'  we s e e t h a t  *'V  (1.19)  25  To e s t i m a t e  we w r i t e  where  V  *  =  ^[\  v l  * > ^ l ± i + i £ * l > K ( s < t ) , t - - s «->,*> d r  o  Since  Jit)  *Ct-r)  i s L i p s c h i t z c o n t i n u o u s we s e e t h a t  (1.20)  17 "* Applying  t h e Mean V a l u e Theorem t o  I (w> we  obtain  (i"*>  J  i  -  lw*J $  t"*  26 -  .  (1.21)  ft"*-  Writing  W j  as  . _1_ \ -u'trvU'ttr- &'c^)  w  we s e e t h a t  the l a s t  term  _ (sttwwf  «<fc-*>  e  c a n be e s t i m a t e d  as  fi  follows  \ (S'CO- -S'ttV)*- <SU>- 3  <  Hence t a k i n g  2.'(M+A/)  a-  S U'til  to f u r t h e r  3 M (ot^ C M +//)  and u s i n g t h e i n e q u a l i t i e s we f i n d  -  satisfy  o- J  i  (1.22)  I I ~ « ~ & | $ 3 l ^ l . ( l ^ l j i)  and ( 1 . 1 5 )  that Wv,» i 3 M  To c o m p l e t e  (+*  +  the estimate  t ' 3  of  2  .  Vg , we w r i t e  (1.23)  as  -  27 -  where L  = 9 \ (•-Qtt)-i>'<*»(.-SC-fc.)-«-3(tn e  The e s t i m a t i o n (1.7),  of  (1.10)  of  involves  a straightforward  t  application  of  and y i e l d s  3£.t>, Tr"a b  *  To o b t a i n  U,  d  an e s t i m a t e  two v a r i a b l e s  (a-any non-zero  (1.24)  f o r L , t h e Mean V a l u e 2  must b e a p p l i e d constant).  Theorem f o r a  C*-*^)e  to the f u n c t i o n  A simple c a l c u l a t i o n  function  then  leads  to t h e  estimate VLJ  Using  (1.25)  <  (1.20),  (1.25)  and ( 1 . 2 4 )  |V'| < £  Hence  »*«  we s e e t h a t  (lVM*'+3) t ' '  (1.21),  (1.23)  (1.26)  1  and ( 1 . 2 6 )  imply  that  satisfies  28 -  IV  Now  a  l < J. T H M * *  +1 +• 3 M ( K  combining the estimates  f  (M4«))' +  -^W+iv)] t  *.  (1.27)  (1.19) and (1.27) we see that (1.28)  where  >A  i s a constant  dependent o n l y on the data (1.29)  O"  Taking  to f u r t h e r  A  <  o-"»  the c o n c l u s i o n  \  1.2)  (1.30)  ,  o f tire Theorem  Theorems Theorem  satisfy  1.1 and 1.2  (1.4),  Co-,*"1  where  the d a t a  (1.29) .  follows.  M -  a«"u,t|^ 1 . +  Note that  o~  depends o n l y  (1.5) we must show that any s o l u t i o n of (1.4),  " <r"  of Theorem  C <j-!^ #  say  -o  on  o f uniqueness of the s o l u t i o n o f  i r r e s p e c t i v e of whether i t belongs t o - C,.  in  o->o(given i n  (1.5) has a unique s o l u t i o n f o r a l l t < <r i n  To complete the proof (1.4),  imply that f o r  M  (where  <r  (1.5), i s the  1.2), must c o i n c i d e w i t h the f i x e d p o i n t o f , i n t h e i r common i n t e r v a l of  existence.  H"  1  -  If  C°,?l  interval where when  v(tV i s  we h a v e  Theorems that  general  v  it  is  both  from  M  is  fixed  of  \V  v(U,tf(l)  to  of  }  was  that  it  vtt^  satisfy  origin  c-, < 5-  of  the  time  where  in  interval  O J  O  (1.5)  that  boundary  are  at  i n Theorems  there  exists only  we c o n c l u d e their  t  "t <f, r  v< (5,0-,) s ^-C^.o-^  then of  n-Gw« fA^  <r,) , s < <M]  t  S i n c e the  L"«>»3  on  C<J.',M'  (1.4),  that  -w\«-y ^  on t h e  , ^ u  IT,  S  Co, 0-, + 4) ,  on  vt^ivU^  Note  -0<*l=O(i) on  and p o s i t i o n s  the  in  1  the  Co,5-]  M*=  "v<^=v(t>  respectively  Shifting  T  equations  15 <rj  on  existence.  we~'can a g a i n c o n c l u d e t h a t  such t h a t  (  if  (1.5)  r e p l a c e d by  points  integral  distribution  to  c  the  (1.4),  v < - k ) « v < i * on  5) i s s u c h t h a t  Hence  >= si*-,}.  •u<^ s  show t h a t  common i n t e r v a l  or, (<r, <  clear  corresponding  on  of  a * < ? . H e n c e we c o n c l u d e t h a t  temperature  and 1 . 2  -  solution  and 1 . 2 :  are  v,-C  <«•,)! ••*>< «-,) ,•  the  the  )  1.1  Now i f then  another  t h e n we must  J : •MW|»,»| in  29  an  t  1.1 f>o  restriction  that  common i n t e r v a l  of  existence.  Existence  and U n i q u e n e s s of Let  .then the  there origin  o-  exists of  time  the  satisfy  W )  a unique to  i  s  ^  Solution (1.6),  for  (1.12),  s o l u t i o n of we c a n f i n d  (1.4), a  all  1 6 Q. 1 T  (1.22) (1.5)  and for  such  (1.30); Moving  t<  that  the  -  solution  of  Continuing  (1.4),  (1.5)  inductively  30 -  exists  and i s u n i q u e  for a l l  we s e e t h a t we c a n g e n e r a t e  t i o* ^ 1  a sequence  _ **  ^ o^*^._ for  such t h a t  all  1 j. i  such f o r  (1.4),  .  *"  If  has a u n i q u e  we c a n show t h a t  solution  there  exists  a  0"  0)  >  0-°  (1.31)  t h e n we c a n c o n c l u d e t h a t  and h e n c e  (1.4),  (1.5)  However,  this  for  f o r some  (1.6),  A/  has a u n i q u e  solution  i s immediate F o r then  inequalities  (1.12),  (1.22)  if  o-°  we c a n f i n d determined  and ( 1 . 3 0 )  t C- <°,T).  for a l l  with  global  upper  by t h e M  replaced  by  and  replaced Since  1.3  by  "MKU,!)  is applicable  Therefore  V  each  o-  bounds  (1.5)  we have  T*  satisfies  is continuous  (1.31). on ob  we s e e t h a t  Lemma  and hence  that  (1.4),  (1.5)  and h e n c e  (0.3,a,b,c,d,e)  -  has a u n i q u e bounded.on  £O,T]  In which w i l l III)  solution for  be u s e d to  upon which  is  all  is  0  II  we w i l l derive  based the  -  "t-* ( 0 , T V -  v <x)  and  Chapter  31  the  provided  uniformly  outline  the  Similarity  Similarity  is  b o u n d e d on  Similarity Solution  Algorithm  £o,bJ. Method  (Chapter  (Chapter  IV).  CHAPTER  II  THE SIMILARITY METHOD  The a l g o r i t h m on p a r t i c u l a r  solutions of  S i m i l a r i t y Method. basis of  to be  as w e l l  the  which the  give  recipe  for  in order  i s more  results  original  the  IV  is  based  found by  the  theoretical  c o n s t r u c t i n g such  solutions  equations.  change v a r i a b l e s  whose s o l u t i o n  Chapter  equation  provides  procedure for  A common method o f to  in  diffusion  The f o l l o w i n g  as t h e  differential  introduced  finding  to  transform  the  easily obtainable.  are  system.  solving differential  often  such  equation  The  one  a symmetry  Method p r o v i d e s a  transformations  to  is  transformations  those which e x p l o i t  The S i m i l a r i t y  equations  using Lie  of  systematic  (continuous)  Groups. Sophus L i e differential  equation under  transformations of  an o r d i n a r y  "Lie"  showed t h a t  Group o f  leads  a one p a r a m e t e r  directly  differential  i n v a r i a n c e of  to  ordinary  c o n t i n u o u s group  a r e d u c t i o n b y one i n  equation.  transformations  an  leaving  - 32 -  He showed how t o invariant  an  the find  ordinary  of order the  - 33 differential  equation^^  and  found  a subgroup of the f u l l  group of  (2) the heat  equation.  recent y e a r s to  reduce  However, i t remained f o r authors of more  to show how  to use continuous  groups of t r a n s f o r m a t i o n s  the number of v a r i a b l e s , and hence f i n d  particular  (3) s o l u t i o n s , of p a r t i a l  differential  contributions  regard have come from Ovsjannikov £ 28  Matschat  i n this  and M u l l e r  £33»L"^3  Bluman  £ ^]» 2  a  nd  equations.  Bluman [  2  has a p p l i e d the S i m i l a r i t y  J.  The  major J,  More r e c e n t l y ,  Method to boundary  v a l u e problems.  (4) Lie  Group o f  Transformations  C e n t r a l to the theory i s the concept  of a L i e Group of  Transformations. D e f i n i t i o n 2.1: A one  -(-a L i e Group of T r a n s f o r m a t i o n s ) . parameter f a m i l y of t r a n s f o r m a t i o n s  For a treatment of the S i m i l a r i t y Method a p p l i e d to o r d i n a r y d i f f e r e n t i a l equations see Bluman and Cole P a r t I. (2)  L i e d i d not see how to use i n v a r i a n c e to c o n s t r u c t p a r t i c u l a r s o l u t i o n s to p a r t i a l d i f f e r e n t i a l e q u a t i o n s .  (3) For a thorough treatment of the S i m i l a r i t y Method as a p p l i c a b l e to p a r t i a l d i f f e r e n t i a l equations see Bluman and Cole [ 6 3 Part I I . :  (4)  differential v a r i a b l e s we  Since we are i n t e r e s t e d only i n a p a r t i a l equation i n v o l v i n g one dependent and two independent r e s t r i c t our a t t e n t i o n to t h i s case.  - 34 -  where  and  forms  a Lie (a)  Group o f  T r a n s f o r m a t i o n s w i t h parameter  (Associative Property)  there  exists  a  £  if:  function  with  for  all  a.bjCCQ  (b)  for  all  (Identity  for  Element)  there  x*x**jXeS  exists  an  satisfying  €„£Q  such  that  exists  an  x €S ;  (c) such  such t h a t ,  (Inverse  Element)  for  every  £ £Q  there  € Q  that  We n o t e a group o f •J'CXJO,  that  conditions  transformations, make  it  while  (a), the  (b),  (c)  continuity  a L i e Group o f  make  the  family  conditions  transformations.  on We  - 35 -  remark that by a s u i t a b l e rep-arameterization, the i d e n t i t y element  can be assumed  to be z e r o .  To apply the S i m i l a r i t y partial  differential  Method  to a second order  equation we c o n s i d e r the f o l l o w i n g L i e  Group o f t r a n s f o r m a t i o n s :  X  where  *=  X<U,X,t;£)  (2.0)  >  -u. i s the dependent v a r i a b l e and X, "t a r e the independent  variables.  Invariance. A partial  differential  equation  G('U^ M ,M ,v<k,v ,-w< x -fc)= o >  together with  x t  t t  < t  >  l  2  <  1  )  the boundary c o n d i t i o n s  By(-u«,v4 u,x,-b)so  (2.1a)  tl  on the boundary  curves  W tx,t>= v  is  (  o  i n v a r i a n t under ( 2 * 0 ) provided  Y=  '.•••,p  (2.1b)  -  36  -  (2.2)  and  (2.2a)  on the  boundary  curves  (2.2b)  h o l d whenever  (2.1,a,b)  equation,  boundary  these  the  curves  take  the  hold.  That  c u r v e s and same f o r m  is,  the  the  governing  differential  boundary c o n d i t i o n s  in both  transformed  and  on  original  variables. Since rich  enough to  a partial leave  we s e e k a g r o u p equation  invariant  leaving  (2.1).  differential  What  []  Chapter  numerical  II  only  data the  boundary c o n d i t i o n s  be s a t i s f i e d  J Part  boundary  invariant  can f r e q u e n t l y 6  equation  In  addition  s o l u t i o n s by " a l m o s t " s a t i s f y i n g  conditions'^ be o b t a i n e d  Further,  useful  by f o r m u l a t i n g  (5) by  the  invariant  (0.3,a,b,c),  and by " a l m o s t "  Similarity  (cf.  we c a n certain  (0.3e).  is  be  differential left  in  invariant  B l u m a n and  boundary  terms  (2.1) of  may the  (0.3,a,b,c,d,e) obtained  (0.3d)  by  Cole  construct  solutions of  s o l u t i o n of  Algorithm  by s a t i s f y i n g  satisfying  cannot  boundary c o n d i t i o n s  The a p p r o x i m a t e  generated  particular  group  such a s . ( 2 . 1 , a , b ) ,  governing  by s u p e r p o s i t i o n  11 ) .  seldom has a  by  leaving  superposition  invariants of the group leaving invariant  (2.1).  The Most General m-Parameter Lie Group of Transformations Leaving Invariant ( 2 . 1 ) F i r s t we note that the transformations ( 2 . 0 ) on the variables induce transformations on the derivatives, which together with ( 2 . 0 ) constitute what are referred to as the Extended Transformations.  These also form a Lie Group of  transformations. Before proceeding further i t i s necessary to reformulate invariance i n a more useful way.  To this end, we introduce the  i n f i n i t e s i m a l transformations. Noting that we expand about  f= o  i n f i n i t e s i m a l form,  where  T J ' C H ^ t - O , X(-*,w \,i) t  , T (M,x,t <r) 6 C ° ° ( / / ? , ) 3  ;  (the identity) to obtain ( 2 . 0 ) i n  38  »i^v,^«l>T(u »,tiOl )  The t r a n s f o r m a t i o n s derivatives, i . e . ,  and  (2.3) induce t r a n s f o r m a t i o n s on the  where  Similarly  we  obtain  where  To o b t a i n  we w r i t e  the  second  extensions  - 40 -  i.e.  7x*  £5*  dv.  \S*4v<  8K'/  M<  C>*»  *  3R  Now (2.2) can be expanded about  Ve>~  ,3*0.*/  1  <JW  *  V  fro  to  yield  *  - 41 -  G IK-*-. K- e» *Vt»»  + f  • **V»**>**) = °  GC^x «,UMt v» - v«,M ,w ic -t) . l  J  iB  tl  t  y  l  +• ot£»)  where we  have i n t r o d u c e d the f i r s t  order d i f f e r e n t i a l  operator  I t can be seen that i n v a r i a n c e of (2.1) under (2.0) i s e q u i v a l e n t to  XG =0 G  whenever  =  (2.4)  O,  With t h i s f o r m u l a t i o n of i n v a r i a n c e we are prepared to  find  the most g e n e r a l m-parameter Substituting  and u s i n g the r e l a t i o n equations of  for  f,?>r, G~0,  *?  group l e a v i n g  , 9?  t  xt  ,r?  it  (2.1) i n v a r i a n t into  (2.4)  we o b t a i n the determining  by s e t t i n g equal to zero the c o e f f i c i e n t s  the independent d e r i v a t i v e terms  (^KI  l e f t with a set of 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  )• We are equations f o r  - 42 -  Reducing the Number of V a r i a b l e s To every L i e Group of t r a n s f o r m a t i o n s  there  corresponds  a s e t of C a n o n i c a l C o o r d i n a t e s , i n which the group i s a of one  of the v a r i a b l e s .  Using  translation  these Canonical Coordinates i t  can be shown that i f the t r a n s l a t e d v a r i a b l e i s an  independent  v a r i a b l e then i n v a r i a n c e o f a p a r t i a l  equation under  a one  differential  parameter L i e Group of t r a n s f o r m a t i o n s leads to a r e d u c t i o n  by one  i n the number of independent v a r i a b l e s provided  s o l u t i o n i s unique ( c f . Bluman and Cole [^6j I t should be noted  that i n t h i s  number of independent v a r i a b l e s by one differential  the  Part I I Chapter  instance reducing  3). the  leaves us with an o r d i n a r y  equation.  Suppose (2.1,a,b) i s i n v a r i a n t under (2.0), whose infinitesimal If  t r a n s f o r m a t i o n s are g i v e n by w*  v«s &ix*,t') Now  assuming  V, w. find  and (2.2)  about  v  toe*,*)  i s a s o l u t i o n o f (2.1)  vv."*U(©.**'tiO has  £zo  (2.3). then  both  are s o l u t i o n s of  a unique s o l u t i o n then and gathering-terms  v =^*' v  .  (2.2). Expanding  i n powers of  <f  we  that  (2.5)  (The  I n v a r i a n t S u r f a c e Condition) must be s a t i s f i e d  if viv*  and  - 43 -  conversely. The g e n e r a l the c h a r a c t e r i s t i c c\x  s o l u t i o n o f (2.5)  equations . -  s  • g<G.«,±Y  can be found by s o l v i n g  d  ®  (2.6)  >  •  (6) If % / t (2.6)  i s independent of  takes  the form  ©  ,  then the general  © = €(*,±> \,  (the S i m i l a r i t y V a r i a b l e ) and generated b y - s o l v i n g u s i n g the r e l a t i o n equation  for  reduced by  ^C^)  (2.6).  where are the two  Substituting ©  $ V*,*) we  s  constants  i n t o (2.1)  o b t a i n an o r d i n a r y  and  differential  Hence the number of v a r i a b l e s has  complete s o l u t i o n o f (2.1)  the o r d i n a r y d i f f e r e n t i a l  equation  been  i n v a r i a n t and  can be  found by s o l v i n g  f o r £ t 0 . However, i f a  parameter L i e Group of t r a n s f o r m a t i o n s  leaves  two  (2.1,a,b)  the i n v a r i a n t s (the s i m i l a r i t y v a r i a b l e s )  a s s o c i a t e d with  (2.6)  f $(*|t)  one. The  of  s o l u t i o n of  the two  parameters are f u n c t i o n a l l y independent  (8)  I f 2*/T depends on © then the general s o l u t i o n i s of the form O = «&<€>.*, t;i,V CO) and $-^(G>,.*,tK  The boundary c o n d i t i o n s c o n d i t i o n s to be s a t i s f i e d by ?^V>-  (2.1  a,b)  become boundary,  (Q\  Two i n v a r i a n t s are f u n c t i o n a l l y independent provided t h e i r r e s p e c t i v e i n f i n i t e s i m a l operators are l i n e a r l y independent over the f i e l d of complex f u n c t i o n s ( c f . Bluman and Cole [ 6 ^ Part I I Chapter 8 ) .  - 44 then the  s o l u t i o n of  (2.1)  can  i n v a r i a n t s without recourse Part I I Chapter  be  found d i r e c t l y u s i n g  to (2.1)  ( c f . Bluman and  the  Cole [ 6  leaves  i f an m-parameter L i e Group of invariant a partial  differential  equation  with accompanying boundary c o n d i t i o n s , i t i s n e c e s s a r y that associated  the  i n v a r i a n t s ( s i m i l a r i t y v a r i a b l e s ) be f u n c t i o n a l l y  independent b e f o r e we can be  ]  8).  In g e n e r a l , transformations  -  reduced by  are assured that the number of v a r i a b l e s  m.  (9) The  Classical  Group of the Heat  Equation  Considering  the  invariance condition  (2.4)  implies  whose s o l u t i o n y i e l d s the s i x parameter group:  7  c w u e [ . * \ j \ | ^ | x * i ]  (9)  See  Bluman and  Cole \j>~\-  • Ct> j A  (2.7)  Here is  «<, v^'V, 6, K, \  an a r b i t r a r y  are  the  s o l u t i o n of  The g r o u p  (2.7)  parameters  the  in  heat  the  of  th.e g r o u p w h i l e  •  equation.  (x,t)  plane  is  a subgroup  of  (10) the  eight  parameter  represent  "O  find we  translations  represents  while  ^ the  projective  is  In construct  the  or  associated with  the  The r e s u l t i n g  the  a stretching  form of  solve,  in  the  set  of  global  i.. and  The parameters  Galilean  chapter  are  respectively;  transformation;  associated with  given  V  by  central  to  (2.7) the  will  be u s e d  Similarity  Algorithm.  See Bluman and C o l e [^6]  To  equations  a subgroup of  solution  <*, K  transformation.  transformation  characteristic  similarity  directions  x  similitudinuous the  transformations  the next  group.  Part  I  Chapter  7.  to  CHAPTER I I I A USEFUL SIMILARITY SOLUTION OF THE HEAT EQUATION FOR THE STEFAN PROBLEM In t h i s chapter we w i l l use the ..Similarity Method, as does Bluman [ 4 ], to derive the solution to an Inverse Stefan Problem corresponding to the boundary melting at a constant v e l o c i t y . Given  We proceed as follows.  set) , the system (0.3,a,b,c,d) reduces  to the Inverse Stefan Problem: (3.0)  u(ito,t)-o  M <»,±)^ K  *A(*,O\-  o (1)  Vi„( > V  i^Co.T)-,  (3.0a)  't«Co/r>,  (3.0b)  tCCo.r),  (3.0c)  xeC«iCl.  (3.0d)  ^ The methods o f t h i s c h a p t e r may be u s e d t o d e a l w i t h t h e b o u n d a r y c o n d i t i o n s vWo,t>- PCt) or u (o,t>-RC-O • K  -  46  -  - 47 -  We  will  show u s i n g the S i m i l a r i t y  Method t h a t f o r  a member o f . a two  parameter f a m i l y of c u r v e s ,  (3.0,a,b,c,d) has  a c l o s e d form a n a l y t i c  solution.  For convenience the group (2.5) extension  the system  together with  a  first  Is g i v e n below,  «t + 2M t  + Vt  %<*>±\-- K+• S t f » x  l  + Yxt  (3-1)  where FCX,+,)=-Y[|\ J ] - |*  dx  +- A  V  ^x/ * -ax'  a* We  c o n s i d e r (3.1) with  the boundaries  l " ^ ' ^ °, 6  The  1  condition  x-o X J O implies  x=o,  C3.0,a,b,c)  if''^"»o  that  .  is l e f t  note that i f  i n v a r i a n t by  is satisfied  i s i n v a r i a n t under (3.1) , i . e . , S*K=o  and  are i n v a r i a n t under (3.1)  x.-sit)  then  ^cx,t)5o  The  satisfy  i f and  only i f  provided x*"=  o  invariance ofx=^Ct)  the d i f f e r e n t i a l  equation  and (3-1). &=o  ;  whenever under  (3.1)  -  48  -  ^"(.SCt),*) «. i r t ) 3T(k) .  •When c o m b i n e d w i t h  Hence invariant  given  the  corresponding  boundary  parameter  that  subgroup o f  (3-1)  leaving  whenever  the  the  (cf.  consider of  x ( v f  1  ) ^  (3-3)  Vt)  Similarity  to  be c o n s t r u c t e d  Solution  implies  by:  (3.4)  we o n l y  three  (3.2)  ( c ' u v t + y t  ?(<,t)t  Using  .S<<rt = c  (3.0,a,b,c)  Jt-kJ«-  is  the  (3.2)  most Bluman  Solution general [  4 ]  a subgroup of system  (3.4)  f  of  the  system  ( 3. 0 , a , b , c , d)  "moving" boundary ).  (3.3)  (3.0,a,b,c,d)  However, to  obtain  for the  (3.3) our  can  purposes  Similarity  c o r r e s p o n d i n g to  the  "moving"  - 49 -  Letting  v--/3c > V - ^ i *  +  +  (3.4)  >  ^ |  reduces t o  ^  (3-5)  where  The i n f i n i t e s i m a l s  From t h e f i r s t  (3.5)  yield  e q u a l i t y of (3.6)  Variable V  =  x  / ( c y 3 i )  »  where K=o  <=>  $ = O,  the s e t of c h a r a c t e r i s t i c  we o b t a i n the S i m i l a r i t y  equations  I n t e g r a t i n g the second e q u a l i t y o f (3.6) curves  ^ = constant  along  the s i m i l a r i t y  we o b t a i n the s o l u t i o n s u r f a c e  (3.7)  of  (3.6).  Here  differential  must s a t i s f y a c e r t a i n o r d i n a r y  ^((ijjx)  equation  together  with  - ^ - - • j x ) - - X^Oj/*)  if  -  conditions  o for a l l ^  (3.8)  v-CXjij^*) i s to be a s o l u t i o n o f (3.0,a,b,c) with  To d e r i v e the d i f f e r e n t i a l write  vet*,!)  equation  r e s u l t i n g expression Introducing  (3.7)  satisfied  eyai',  by  we  (the s o l u t i o n o f (3.0,a,b ,c ,d) with  as a s u p e r p o s i t i o n o f f u n c t i o n s ,  into  the boundary  into  £i  iCt)-c-/3±  j ^ O , and s u b s t i t u t e the  (3.0).  the v a r i a b l e s  we o b t a i n  v ^ t j ^ - . v U . t ^ ) - . ^ J F ^ p e * /U>c/(*,P> < *. • tr+  The  )  form o f (3.9)  suggests we take  P  (3.9)  -  51 -  (3.10)  Substituting boundary  (3.10)  conditions  into  (3.0)  (3.8),  and t a k i n g  ve see that  into  account  c^ (^ j ) f  >  ;  the must  satisfy  -As*  ^  1  (3.11)  The s o l u t i o n o f  (3.11)  is  (3.12)  where we h a v e  introduced  $C^)-  Substituting  Sc e ^  (3.12)  into  v . <«^>  (3.10)  and e v a l u a t i n g  the L a p l a c e  - 52 Transform by c l o s i n g  the contour  i n the l e f t  h a l f plane we o b t a i n  /3-aL <*=> - u £ t U(*,0»J^erteytiJ^T.e « 7 y i U co«(w .a_) H  .  (2) (/S)  (3.13)  where  ©  If we e v a l u a t e for large  p  the L a p l a c e  we o b t a i n the small  Transform by expanding  time r e p r e s e n t a t i o n of ( 3 . 1 3 )  UCx,-t>-- ^ ^ ( S p G C x / t ^ ) d S - , where  n  r x*  G(M-^)-- =Ue-  '  T  f  (3.14).  a  J  E(-») V  + <?  (2)  _ The e x p r e s s i o n %*. C*, ) of ( 3 . 7 ) can be s u b s t i t u t e d d i r e c t l y into ( 3 . 0 ) . The r e s u l t when combined with the boundary c o n d i t i o n s ( 3 . 8 ) i s a Regular S t u r m - L i o u v i l l e System f o r the eigenfunctions ^-?((il**M}. > '•N* t * - i [ h r . The s o l u t i o n ( 3 . 1 3 ) i s then obtained as a l i n e a r combination o f the e i g e n f u n c t i o n s s  -  We n o t i c e usual  fixed  Fourier  derivatives  p r o b l e m and ( 3 . 1 3 )  the s e r i e s  (3.13)  that  convergent  t h e sum i n ( 3 . 1 5 )  ~°  on  O.cl  then at  "fc>0.  we w i l l  -t - c  show  , and  vv C O . e  jc»s(i«3„)f){  satisfy  a  function  the  C«J(W^,1^ ,  is uniformly  convergent  (3.15)  we a l s o  have  ( W K ) ^ V c ^ i c o s K ^ «J.»j .  (3.16)  A  following.  3.1. Let  sums i n  tCx)  f o r any f i x e d  e  its  we c a n w r i t e  V'OO  the  well-known  t^Cy^t]  P r o b l e m on C©. »"V > f o r any  T"(x)= ZZZzos^x)} fct*0  Lemma  to'  the f u n c t i o n s  Sturm-Liouville  We c l a i m  to t h e  with a l l  c  converges  ^<JO C C* C°I»"}  on  c  is uniformly  Since  If  reduces  to t h e  together  and "u ( O =, i u l )  v „ U , i : o )  regular  (3.13)  converges u n i f o r m l y  "U^txi € c ' C o j C l  that  ( 3. 0 , a , b , c , d ) r e d u c e s  t  Solution.  Clearly  If  t h a t when y3-o  boundary  Series  53 -  ^Cx) 6 C* Co,."]  (3.15)  and V ( x )  and (3.1-6) respectively.  with  £cn=V'<oV-o  converge uniformly  on  then C°,i l  to  - 54 -  Proof. Let  J (x)  be t h e f o l l o w i n g  (l) 5co*  to C - ? ) ! -  2  of  * s C°-.»l,  (2)  §^  (3)  i ( « + i ) s - i <x-i)  ^ ( 0  Clearly  *M  extension  x  = $(->o  is piecewise  eC-a,*7,  x e C*l,.ll. '  on  C*  we  claim  A  € C*C-2, i l , to  ensure  this  Only the p o i n t s  x-o, 1  need be c h e c k e d  result.  X=o:  £(0-)- cfv^. L  <-> o  I ©-C  j  « ? ( o + ) =. - +"'(0+) = O ; c->o  L  c ->•© L  Since have  $"00 that  is piecewise  C*  «r  CfM)-l  J  J  as w e l l as  C  we  -55  -  «l  0 0  4  where the sura i s u n i f o r m l y convergent on and H i l b e r t  [8]  C©,il  ( c f . Courant  Chapter 2) .  That the sum i n (3.15) converges u n i f o r m l y on C°> i l follows t r i v i a l l y Hilbert  ^8]  since  § <*> G C* C-*,*J  ( f . Courant and c  Chapter 2 ) , and hence the lemma i s proven.  A p p l y i n g Lemma 3.1 to  - *€  V c  vt (.e*> 6  ZT t ^ K ^ ) ). « '  we see that  cci(w,«l^  (3.17)  and  e - 2 ^  S ^ K f )  \ e  where both sums converge u n i f o r m l y on  % , . ( ^  a  (3.18)  C°itl.  It should be noted that the r e s u l t s are e s s e n t i a l  -,Cw^U  (3.17) and (3.18)  s i n c e the F o u r i e r S e r i e s Expansions  for  v*,U,t>  - 56 and  U „ Cx*i(t),-t)  Algorithm. satisfied initial  are  Further at  step  note  (see  hypotheses  of  (0.3b,c).  If  affected,  that  is,  [S,c  any  the  that  to melt)  "H Cx) e  the  sum i n  0  (3.18)  However  if  as  fixed  \*.(c t) =o  (the  )  given i n Appendix  at  any  time  the  determine  if  satisfy is  not  converges u n i f o r m l y .  the  (the  slab  algorithm  (3.0,a,c,d)  boundary s o l u t i o n )  standard derivation  there  heat  flux  is  of  is  on  hot  and u s e  and  with  \$(i}=o  this solution  some d i f f i c u l t y  is  insufficient  S i n c e we p r o v i d e no m e c h a n i s m f o r step,  the  is  E).  We s e e t h a t if  the  satisfy  Algorithm  v , t t ^ o  (we  until  still  t h e n we must m o d i f y  the  are  only converges u n i f o r m l y  s o l u t i o n of  this  Lemma 3.1  need not  <x) may n o t  (3.17)  Series  to  Similarity  Algorithm after  Similarity  the u s u a l F o u r i e r  refer  the  hypotheses of  is, n  sum i n  .  b>o  in  IV).  condition  the  i-o  Similarity  V„(x5o)^o  while  prepared  the  the  Chapter  Lemma 3 . 1 ,  on [ o,C ] , for  that  each s t e p o f  The i n i t i a l  ]  used near  the  heat  flux  freezing, is  with  this  to m a i n t a i n we m u s t ,  sufficient  at  algorithm melting. each  to m a i n t a i n  time  melting,  - 57  i.e.,  -  determine the s i g n of  S(t;),t;) - W\ (!«•)),  If  A>o  heat f l u x  we u t i l i z e  p r o c e e d i n g as above we  (3.0,a,b,c,d) simplicity,  IV)  the  assume the heat f l u x to always  to m a i n t a i n m e l t i n g .  Returning to the group aid  solution until  Chapter  to m a i n t a i n m e l t i n g .  what f o l l o w s we w i l l  be s u f f i c i e n t  6  the f i x e d boundary  i s again s u f f i c i e n t In  *; ^~.(cf.  (3.1)  we note that s e t t i n g  Y  5  °  generate the p a r t i c u l a r s o l u t i o n s o f  g i v e n by Sanders £3l]  .  Because of t h e i r  relative  the Trigonometric f u n c t i o n s lend themselves much more  e a s i l y to n u m e r i c a l c a l c u l a t i o n than "do the Confluent Hypergeometric f u n c t i o n s which are the b a s i s o f Sanders' s o l u t i o n s . from the i n h e r e n t d i f f i c u l t i e s Hypergeometric  Apart  i n c a l c u l a t i n g with the C o n f l u e n t  f u n c t i o n s , there are a l s o convergence  would p l a c e i n doubt the u t i l i t y  of such a scheme.  q u e s t i o n s which For much the  same r e a s o n s , an a l g o r i t h m , s i m i l a r to that g i v e n i n Chapter based on the most g e n e r a l "moving  would encounter d i f f i c u l t i e s  boundary"  from the o n s e t , as here too the  s o l u t i o n o f (3.0,a,b ,c,d) i s expressed as a sum o f C o n f l u e n t Hypergeometric  functions.  IV,  - 58 /  We essentially  remark that i n 1939  O  \  Huber [ 2 0 ^  proposed  the a l g o r i t h m of Chapter IV to approximate  s o l u t i o n s of c e r t a i n one-dimensional two-phase S t e f a n Problems. Huber's s o l u t i o n , however, i s not based on a s i m i l a r i t y s o l u t i o n . He  e l i m i n a t e s the i n t i a l  source  c o n d i t i o n by  s o l u t i o n of the heat equation  i n t r o d u c i n g the  (J*C (*,i:,i;,r.) of .Ch.ap.ter ..I.).,  then uses a set of A p p e l l Transformations X-JC-t)=. c.-y3t invariant sum  to the  f i x e d boundary  the heat e q u a t i o n .  of a source  The  usual  to  transform  y- X , w h i l e  leaving  s o l u t i o n i s then g i v e n as a  term plus a complicated  convolution  integral.  Huber's s o l u t i o n is. too unwieldy f o r numerical Recently  R u b i n s t e i n [ 3 0 j has  be s i g n i f i c a n t l y  simplified  By  that Huber's method  so doing  transformations  can  by u s i n g a Green's f u n c t i o n on  x e ( 0 , cys-fc)^ t <c (o.,i)j f i r s t  the domain [33].  suggested  purposes.  d e r i v e d by  the need f o r the complicated  i s e l i m i n a t e d and  Soloviev  Appell  the s o l u t i o n can be  given  directly  (3) Recently A. Fasano and M. P r i m c e r i o [ l 4 ] have demonstrated convergence of Huber's Method f o r a one-dimensional s i n g l e phase S t e f a n Problem. (4) The r e p r e s e n t a t i o n suggested by R u b i n s t e i n i s *u<*,tU % C$) /rrx,t-,5,.w •*•<*,t',§>}«l$ where .5'(*,f,^ ) • i s the Green's f u n c t i o n of S o l o v i e v , and v , , ^ ) i s the i n i t i a l c o n d i t i o n . We remark that t h i s i s b a s i c a l l y the r e p r e s e n t a t i o n given by (3.14). However, the Green's f u n c t i o n GCx.-t-,^) i n (3.14) i s g i v e n i n a more compact form than i s the Green's f u n c t i o n , K(x,t' 5,o) + ^ ( x , . r  c  >  - 59 -  The c a l c u l a t i o n s i n v o l v e d i n u s i n g Ruber's Method, Rubinstein's  s i m p l i f i c a t i o n are very  i n the S i m i l a r i t y A l g o r i t h m representation  similar  with  to those  i f we were to use the small  (3.14) o f the S i m i l a r i t y s o l u t i o n .  We  argue i n Chapter VI t h a t the l a r g e time r e p r e s e n t a t i o n i s more p r a c t i c a l and  than i t s small time c o u n t e r p a r t ,  hence that the S i m i l a r i t y A l g o r i t h m  Huber's  involved  Algorithm.  !  time  will (3.13)  (3.14),  i s more u s e f u l than  CHAPTER  THE SIMILARITY  Having derived ready  to o u t l i n e  IV  ALGORITHM  the S i m i l a r i t y  the S i m i l a r i t y  the system  approximation  the p a i r  (0.3,a,b,c,d,e)  n e c e s s a r i l y a uniform  we a r e now  of functions  and we w i s h  to f i n d  C"VA,S) an  I[O,T].  ( £ , $ ) to ( v i s i o n  We p r o c e e d by p a r t i t i o n i n g  (not  (3.13)  Algorithm.  S u p p o s e t h a t we a r e g i v e n satisfying  Solution  the time  partition)  interval  and e s t i m a t e  >5("t)  on  £to,\,  by  Sit)'- C - /3„ ( t - t o ) ,  t  e  0  < t < t ,  where  As we h a v e Methods  seen  yield  (cf.  Chapter  a closed  III),  for  form s o l u t i o n  -  60 -  of  5 C t ) so d e f i n e d (0.3,a,b,c,d).  Similarity We d e n o t e  -  this the  s o l u t i o n by  61  V ' t K , i't,)  -  and r e m a r k  that  it  is valid  on  domain  To e x t e n d  our estimate  <t-f >,  |<t>*  (^»;^al  to  t, < t 5 - t  w  e  define  4  where  and to  /3,  is  c a l c u l a t e d by s u b s t i t u t i n g  y.*(e, -ai,) i n t o ;  (0.3e)  obtain  /3,« *'< Wet,) Now c o n s i d e r i n g Vt°Cx, ^ i | ) generate  U„*t,)) .  as t h e o r i g i n o f  as t h e  initial  a solution  condition  M.' (* •i.-t,) )  of  time i n U tO  w  (0 . 3 , a , b , c , d ) e  e  c a n , as  (0.3 , a , b , c ,d)  valid  before on  domain  Continuing s o l u t i o n on t h e  where  i n d u c t i v e l y we o b t a i n  interval  ( t^ii+j^  by  the  defining  and  approximate  the  - 62 -  We o b t a i n  "U* Ix, i - t ^ )  (0.3,a,b,c , d )  by t a k i n g  and the i n i t i a l  to be the o r i g i n o f time i n  •-  The S i m i l a r i t y  Lx)  c  to be  w**'( x>  •  "u^x, t - t ) on  Again the S i m i l a r i t y Method y i e l d s  F i g . 4.0  U  condition  {  Algorithm  Time - t  The approximate  We  solution  C  t  S)  i s then taken to be  prove i n Chapter V that as we r e f i n e  i n such a manner  the p a r t i t i o n  that  -w\ <x^ J4,t^ —> O  the  approximation  C"M.j5 )  converges  i n the supremum norm to  (\* s) t  - 63 That  i s , given  w\ew»  £>o  there  < &  exists  a  S>e  such  that  implies  t  and  .  In c a s e where t-i  of  (0.3e)  the f l u x  condition  the s u b i n t e r v a l s  advantage of  p a s s i n g we remark  anywhere  that  is satisfied  . I*  C "t«^  fact,  1  i n the s u b i n t e r v a l .  c o n s i d e r e d the at  the  endpoints  we c o u l d h a v e  satisfied  However we s e e no  particular  become a p p a r e n t ,  the choice  as w i l l  i n s i d e the s u b i n t e r v a l  ?  h e r e we h a v e  (0.3e)  in doing s o , while,  a point  (1)  A  complicates  the a c t u a l  numerical  procedure. A.  Note approximation  that  a heat  {y^ t) t  produces  the m e l t i n g  for  "kj € 7T  all  i.e.,  V* C t ) i s  d e s c r i b e d by  of  Notation:  the  be shown i n C h a p t e r  VI,  flux flux  which satisfies  c a n be  errors  given  4~Cx,t > > <^t* t) ,  (  ,  4  the heat  T h i s heat  «»i-.«-jt--.«iU M:f « i\ * *'* -'S W  i n d u c e d by t h e  one c a n c a l c u l a t e  , and as w i l l  u s e d as a i n d i c a t o r  ^  flux  k  i  ,  , K A ,  ' Jl  and  e!tt)^0  then  - 64 -  and  ( SCt), 3 (tl)  i.e.,  the  quantity  \  U U > - K C T > | 4*  (4.0)  c a n be c o m p u t e d and u s e d t o refined it  further.  requires  Although  a good d e a l  determine  whether  in principle  of  labour,  (4.0)  s h o u l d be  77*  c a n be  and i n s t e a d we  computed,  define  ECi) = «<* \ (van- wunar c  and  calculate  C  /  1  0 (a  relatively  an i n d i c a t i o n Moreover,  if  ^  inexpensive of £(t)  calculation).  how c l o s e l y is  small  to  the  that  s o l u t i o n < v\,s)  (0.3,a,b,c,d)  exactly  will  property  of  use t h i s  the  Similarity  °f  (0.3f)  for  t o be a g o o d a p p r o x i m a t i o n We remark  o  to  tu,s)  is  all  We s e e t h a t satisfied  it  provides  "almost"  in Chapter  Algorithm.  by  J £Ct)  gives  CviJ )  we e x p e c t  (vlji)  . an a n a l y t i c  ( 0 . 3 , a ,b , c , d , e ) .  while  1  o  Co,T]  Cvt.s)  C i >  V to  That  approximation  is j  satisfies demonstrate  satisfies  (0.3e). the  We  convergence  CHAPTER V  THE  CONVERGENCE OF  THE  SIMILARITY ALGORITHM  S i n c e the p a i r of f u n c t i o n s  generated by  the S i m i l a r i t y A l g o r i t h m i s an exact s o l u t i o n of (0 .3 ,a,b ,c ,d, e) A  with  r e p l a c e d by the i n t e r p o l a t e , V\(t)  V^tt)  that the convergence  , i t follows  of the S i m i l a r i t y A l g o r i t h m i s e q u i v a l e n t  to the continuous dependence of the s o l u t i o n of (0.3,a,b,c,d,e) (1) A on the boundary data V\,(i) , i f we can show that Vti) tends uniformly  to V\(t) on £ ; T * 3 as we  r e f i n e the  v^»^  1 At;|->o,  0  ( c f . Chapter  IV) such that  Continuous Dependence f o r Small Time We  partition  r  proceed by demonstrating that the system of equations  (0.3,a,b,c,d ,e) i s c o n t i n u o u s l y dependent on the boundary j b , Vl l.x), M t l j c  and  data  f o r a small time o - > o . More p r e c i s e l y , g i v e n C M , S )  satisfying  (0.3 , a,b , c ,d, e) w i t h  H(t)  Co, b,]  K , U ) ! THx"* on  j  In f a c t we w i l l show that <V,A> on the boundary data | b, "uu>, W t-uj . c  - 65 -  (5.0)  depends c o n t i n u o u s l y  and  W(-M- R ( t ) (5.1) V.OO*  (2) respectively, where  on  Co,b l a  j (T> - b,  W, > bj and  -r(T),  we  have the f o l l o w i n g Theorem.  Theorem 5.1. If (5.1)  U,s),  satisfy  r e s p e c t i v e l y , then  following  inequalities  there e x i s t s  a  cl  a  c->o  (5.0),  such t h a t the  hold:  r ^ - s i i ^ j : d>)v>;-fc i +  where  (0 .3 ,a ,b ,c ,d ,e) w i t h  -r^l^. ^ T l i ) , s t t ^  o-Vuv- 1 ^  A  3  c-|lH-R«  t  (5.2)  and the constants  depend o n l y on data  £ T  ;  H  < « , R ' t ^ u ^ ^ ^ o o ,  That i s , the system of equations (2) sustain  The f u n c t i o n s melting.  b ^ b ^ d f o ) , ^ • (5.4)  (0.3,a,b,c,d,e) i s c o n t i n u o u s l y  H(t^, R ( t ^  are taken  l a r g e enough to  -  dependent words,  on the boundary d a t a  the,system  variations  67 -  |b M Cx^ h (Uj .  ( 0 . 3 , a , b , c , d ,e)  )  In  0  i s ,s.t a b l e . w i t h  i n the boundary d a t a ,  other  respect  to  .  Proof.  With the  (1.4),  (1.5))  we n o t e  that  definitions  the f o l l o w i n g  equations  hold  (cf.  equations  (5.5)  X (5.5a)  and  o (5.6)  c  .t o  (5.6a)  -  In  addition,  subtracting  (5.6a)  « lb,-b J 4  To o b t a i n from  the  (5.5).and write  68  +  -  from  v->i¥--t  inequality  the  (5.5a)  resulting  (5.2)  +  we f i n d  that  K'-rw-'RU^t  we f i r s t  .  subtract  (5.7)  (5.6)  e x p r e s s i o n as  where  and  Since the  G^-Ot,t  ;  <5*<*,t; £  r\--  expression for  where  r * ©  I  and  c a n be r e w r i t t e n  NKb^- 6 " < * , t ; o ^ j 0  in  the  form  O,  69  o  It  i s easy to see t h a t  w,\ *  To estimate  "y-^'ib,  Vj  V/f  satisfies  (5.8)  .  we w r i t e  where  o  Proceeding, as i n Theorem 1.2 |v;i < v  A  «i\  w i t h v^' we o b t a i n  ,.  while a p p l y i n g the Mean Value Theorem to the r e s u l t i n g  (5.9)  v^"  and e v a l u a t i n g  i n t e g r a l we can show that  t V " |< JL 4  N  liil^  Hs-*H, .  ( 5  .  1 0 )  - 70 From (5.9) and (5.10) we conclude  I V.I $ 1  Finally  lla-  that  satisfies  .  we estimate  (5.11)  by w r i t i n g  where  V^, -z)  *>(?>[ <ii!!if>  v* (Sl*)-%)/zt *  Substituting  into  v  inequality  Iwle  $1  >] AS,.  we  \VJl *  Vj  and u s i n g  the  obtain  ftJPflv,,  (5.12)  T*» t. S i m i l a r l y we f i n d  that  Tt"t The estimates  V"  satisfies  t  (5.12) and (5.13) imply that  satisfies  - 71 -  7T  Combining  (5.8),  (5.11) and (5.14) we  \1\<  We  '* t  ±.  now  find  that  W" H*->H. L  estimate  7"  by w r i t i n g  where w  t  *  2  J ) | vet)  isa^sm  KU<t), t -jStr > ti t  3(t-t)  J  72  The term  W, =  c a n be e x p r e s s e d  WJ  VA/,'  + Wj" +  W  as t h e sum  M I (  where  2 V -tSiS (»(t)»»(«> J ,< (.set,, t • jew, r)  Since  .5 <t.V  is Lipschitz  iw/i < x  Then n o t i n g  and  that  continuous  we s e e  »HJ! II^.^II t " » . T  that  (5.i6)  - 73 -  we f i n d that \ Vct1-5<f)X _ .(t(t>-r(t)H < o t i t f t y - ^ H . + itH-nirl  and hence t h a t  w,"  satisfies  Iw,"!* Q('u^*»i;  Finally  [ l \ v - x n  VA/,'"  to estimate  t  +  l > H - ^ n ] t"*.  V-CtV  and use the f a c t t h a t  we f i r s t  apply the Mean Value  +U) and  X  to K (*, t ; ^f, t )  i s L i p s c h i t z continuous.  To the  expression  —  (where  (5.18)  T  Theorem f o r a f u n c t i o n o f two v a r i a b l e s (.x,^)  resulting  (5 17)  *  i s between  -cctN  J  Ct-*>  U-ti  and 4<t^ and Cj  and H z ) ) we apply the i n e q u a l i t i e s  I  i s between |v|e  * I  (5.17) to o b t a i n  lw,'"| 5  ll/^i^nftll^. ^Ilv-,*.!^  4  |IH-RI |t. T  (5.19)  -  Together  (5.16),  (5.18)  74  -  and ( 5 . 1 9 )  imply  that  Ti"t (  Now-we w r i t e  (5.20)  VA/^ as  where  - K(f<t\,t;-*<*),7U dr  Since  b, > * ( t ^  s(Vt  > b*  we o b t a i n  (5.21)  and  - 75 -  |W;w  To estimate w^'" two variables  ajMI U3-**.  \  4  1  ± * >A  we apply the Mean Value Theorem f o r a function  (*,£) to K(*,t;-f, <r) and obtain  A  .A.  (where sm  if and  (5.22)  7  5 ft*  i s between •+(*) ) .  and  Since  r(t>  and  * b« $ 2+  a t>,  tw^'l*  CV^'**  «^ w  have  e  d  i s between  r  .  (5.23)  o  Combining (5.21), (5.22) and (5.23) we see that  **** 1 Making the substituting that  (t-t>*4  W  satisfies  A  J  v = W/(t-*) * v  it  i s  e a s  Y  t o  s  e  e  - 76 and h e n c e u s i n g ™ 3 <2 J ~  w  the i n e q u a l i t i e s  e  «t^e  _L J.  obtain  r  ( 5  fit — <''4><V)! Thus  -X  and  *  2 4 )  satisfies  (5.25)  Combined w i t h  (5.15),  (5.25)  implies  (5.26) k  ill  -  where  C,  77 -  i s a c o n s t a n t d e p e n d e n t o n l y on t h e d a t a Now u s i n g ( 5 . 2 6 )  n v s  and ( 5 . 7 )  < C ^ 1 I «,,. ^ , z  ^  +  (5.4).  we o b t a i n  II V- v «  b j  (5.27)  + t'SlVviAi^  where  C »- *s , /  and  «j?c(<v)  I-y .  For  hence u s i n g (5.7)  fc*-*^  That  v  T  i s a c o n s t a n t d e p e n d e n t o n l y on t h e d a t a Now t a k e  J  + t * IIH-RH ^  i s , (5.2) To  satisfies  j  and l e t i«(o,»)  t  (5.4).  c-> o  satisfy  we s e e f r o m  (5.27)  that  we h a v e  A , lV>,-b l 2  +^,* ' ,  4 |  ^- f l|, v  |  1 1  A  j»'  l , H  '  f i l l  T.  holds. obtain  the heat  j C'f.tV:  (5.3)  we n o t e  equation  that  eu,U">  w(«,i)  i n the region  o<-t<-r| .  Hence b y t h e Maximum P r i n c i p l e  ( P r o p o s i t i o n 1.1)  we s e e t h a t  the  - 78 maximum and minimum values o f x  -  o  or  x  -  9Cx,i.)  )e(*,i)|  attains  II6C -,a-> II  Case I I :  where  <  l e ( x , i>| "fc  (say f o r f i x e d  o ,  d(t) .  There a r e three p o s s i b l e  Case I :  occur at  ^ € Cx, + t t ^  cases.  i t s maximum value at ||  1=0,  here  Vii.,  (5.29)  takes on i t maximum v a l u e at e U t ) - •*•(+.) ) ,  *  d Ci)  here  and hence  xe(©,5(t.) t€<o,T)  Using Lemma 1.3 we o b t a i n  net  Case I I I :  .  ' * " d ( 0 ™**{ !V, * k,| l ' ^ V • ,  >tl  5  BH  rt  U  takes on i t s maximum v a l u e at  here we use the i n t e g r a l equation  where  M  (1.2) to o b t a i n  (5.30)  -X=o ,  - 79 -  o  It  is  easy  and h e n c e  to  that  see  that  - 80 -  Using see  (5.29),  (5.30),  (5.28)  and ( 5 . 2 )  it  is  easy  to  that  Continuous  Dependence If  ftVllu  by  ft^rl^  by  by  H VII^  t h e n we c a n f i n d CT  f o r A l l Time  i n T h e o r e m 5.1  <*>H.  say  (5.31),  a new  c  (5.2)  T  replace  rtv(-,tH! , 3(t  u ( . , t)H . ... W  ft-u,  by  such t h a t  i , C C°. T - <r ]  we  ^ £°> 3  (-.t^,,.,  « ^ . ( - j t i H ^ ,  t \ say and ( 5 . 3 )  with  S>(^)= w ( ^ , ) ,  and h e n c e hold  at  a new  any t i m e  C"  y  -  That  81 -  is  |j(t)-v<t)| (5.32)  V o-„ H H - f t l l  +  d(i  <  for  new c o n s t a n t s  Now  (5.32)  dependent  to  cr.)  +• T ^ l l V i C - . t . l - v A / t e . l l l j ^  t  (5.33)  4  imply  on the boundary  and (5.33)  }  ^ A , A^ , A j , Q, , B, j 8 ^ .  and ( 5 . 3 3 )  s i n c e we c a n t a k e (5.32)  G , ISCV>-V<i,)\  1 +  T  A>  that  data  such  ( 0 . 3 , a , b , c , d , e)  is  continuously  | b, "M <*I V>(t)| f o r a l l 6  that  s u c c e s s i v e l y at  (  '//j < ^ the times  time  *t € C o i T J ^  and a p p l y  < y^ ^ N  ^ i * o , <r , a <r .. •, Vi(.*/-/)J 0  0l  obtain  M-+»I  T  \  / Ai'- A, <r/» A o ; \ 3  \  / li>,-ij  IIH- Rll,  -  Hence  ( 0 . 3 , a , b , c , d ,e)  data  \b, M o o , M t t | f o r  of  all  that  the c o n s t a n t s  importance. of  i s c o n t i n u o u s l y dependent  0  We r e m a r k  t h e . method  they  t  i* •» *j 3 i s  }  appear  of proof  (vi,4)  (0.3,a,b,c,d,e) (the  of  (cf.  flux)  IV)  J  T ^  we f e e l  by t h e S i m i l a r i t y  as  Similarity  >  and (vi.,s)  i s the exact  fts-slb  and T  >  the a c t u a l  system.  of  V\tt>  solution of  \UV-l-^U-  u  (  E»'.*rJ (  satisfies  Hence we must  #  (cf.  e  Algorithm Suppose  will  failing  •  ,  1  •*wu^l.j 4>i J  J(.T)-b^ )  depend  A  1\*A»H\\ —  vv^ft^t <^-t^ _  size  Algorithm with  <v o n l y on  is a  i s the s o l u t i o n  • t-«  T = - \ " T  the  Algorithm  Chapter  ( 0 . 3 , a , b , c , d , e ) we s e e t h a t (where  initial  considerable  and n o t c h a r a c t e r i s t i c . o f  generated  induced heat  considerations  t o be l a r g e ,  Convergence o f the S i m i l a r i t y If  on t h e  i» <c O ,T3 .  f o r numerical  Q  That  82 -  Chapter  IV)  show t h a t in order  V\  to prove  tends  to  that  the  converges.  "tj  i s a point  of the p a r t i t i o n  7T  t h e n we  show  To a c c o m p l i s h t h i s Similarity  we r e t u r n  Solution.  we d i f f e r e n t i a t e  to the e x p r e s s i o n  S i n c e we want  (3.14)  and e v a l u a t e  (3.14)  f o r the  *u.* ( S ( i ^ t ^ i ) f o r s m a l l K  "M. * ( s ( - v -i), t )  "t  - 83 asymptotically  for  in Appendix F ) ,  small  "t  To f i r s t  (the  order  we  detailed  calculation  is  given  obtain  "II 4  If  K tx)  is  continuous  the  partition  from we  7t  for  4 o(i)  i <f ( ©,  V\(t}  tends  and h e n c e we h a v e in  the  Order  sense of  of  points  of  as  t->o  +  to  on £ O ^ T ]  shown t h a t  Chapter  the  Similarity  as max ^1Algorithm  —> o coverges  IV.  shown t h a t  t u r n our a t t e n t i o n of  all  Convergence Having  order  at  ,)  ©<* ( K l i i + t ) * h  Thus  right  have  hCti+t>*htt^  and h e n c e  the  to  convergence.  the  the  Similarity  r a t e at  which  it  Algorithm converges,  converges i.e.  its  we  - 84 (u,s), ( u , S )  If  Cltx, tW> and  p  :  a r e as a b o v e ,  j ttl ->  and respectively  t h e n we s a y t h a t  with order  Sit)  of convergence  provided  and  respectively  as  (<vt ) —> d . t  To e s t a b l i s h  the order  A l g o r i t h m we assume t h a t  W (t)  of convergence of the  Similarity  satisfies  W U i + t ) * K(ti)-+O.C.t.) for  a l l points  "fc.^  «W-Ul  Since the  V.Cx.'t)  expression  T  and h e n c e  that  of the p a r t i t i o n  < 1  ^^^M^Cc  satisfies  the heat  tL P vi. ( s t i X t l l - o  ett  (5.34)  J  I f  ,  t h e n we h a v e  that  < t ) - - / J . < ' ( c ^ t . U ( ^ t ; )' i + 0 ( * t /  ; j  i  i  <  equation implies  at  that  x=.S(i)  + (  - 85 Thus we  have  Hence  if  V\ Ct) s a t i s f i e s  order  of  convergence of  Since  in  practise  order  of  convergence of  one h a l f  and  one.  the  Similarity is  the  (5.34),  small  Similarity  we have  Algorithm we a s s e r t Algorithm  is that is  that one the  the half. effective  between  CHAPTER VI  THE SIMILARITY ALGORITHM NUMERICAL  RESULTS  In t h i s chapter the r e s u l t s of our numerical with the S i m i l a r i t y A l g o r i t h m are g i v e n . examples i l l u s t r a t i n g including  experiments  We present s e v e r a l  the p r o p e r t i e s of the a l g o r i t h m ,  i t s order o f convergence,  increasing i t s accuracy.  and suggest  two ways o f  In a d d i t i o n , we attempt  to j u s t i f y  the use o f the l a r g e time r e p r e s e n t a t i o n (3.1.3) r a t h e r than the small time r e p r e s e n t a t i o n (3.14) i n the S i m i l a r i t y We conclude  Algorithm.  the chapter by comparing the S i m i l a r i t y  A l g o r i t h m w i t h L o t k i n ' s D i f f e r e n c e Scheme.  Numerical  Examples  B y p r e s e n t i n g the f o l l o w i n g n u m e r i c a l attempt  t o b r i n g to l i g h t  the advantages as w e l l as the d i s -  advantages o f u s i n g the S i m i l a r i t y We f i r s t  examples, we  Algorithm.  c o n s i d e r (0.3,a,b,c,d,e) w i t h  - 86 -  - 87 -  3  For  the d a t a (6.0) Sanders  m« t)' (  J t  J  /  (6.0)  [ 31 ] has g i v e n the exact  - " ( i - a )  on  o < * •<  solution  .sc-n * 0 - * t) *. v  Comparisons o f the exact s o l u t i o n w i t h the approximating s o l u t i o n s  are- summarized by Table 6.0  F i g u r e s 6.0,  Here s i x terms  Solution are used.  6.1,  6.2.  (3.13) and  equal time increments  of the S i m i l a r i t y ( AI^ - at  In each case the approximation i s used  t o t a l m e l t i n g time, i . e .  and  for a l l i  to 80% of the  T=.4.  In what f o l l o w s we use the n o t a t i o n  e^(Ty =  IU-J« , T  I f more than two or three terms of the s e r i e s (3.13) a r e used, we have found F i l o n ' s Rule f o r i n t e g r a t i n g S* * coi eU ( K a r e a l number) ( c f . F i l o n £15 ] , Davis and Rabinowitz [] 9 ] page 62) to be the most e f f i c i e n t method of g e n e r a t i n g the c o e f f i c i e n t s ^n(/3) ( c f . Chapter I I I ) .  )  - 88 -  where  ^ u.  > £ C*,*), J C-O, .r Ci)^ £ Ct) j  a r e d e f i n e d as i n  Chapter IV.  Table 6.0  E r r o r s Versus Time Increment (Boundary Data (6.0)) % Error  •<wt  g ( .4)  .200  .87(-l)  .05  w  (.4)  % Error  e (.4) w  11  .89(-l)  20  .96(^1)  .25(-l)  3  .27(-l)  6  .30(-l)  .01  .8 (-2)  1  .8 (-2)  2  .9 (-2)  .001  .6 (-2)  .75  .3 (-2)  .6  .4 (-2)  ( 2 )  From Table 6.0 and our 'numerical examples i t seems that an accuracy o f one to f i v e percent .is e a s i l y obtained.. However, h i g h e r accuracy we see t h a t w i t h  is difficult  to a c h i e v e .  For i n s t a n c e ,  -6.t =.01 the a l g o r i t h m r e q u i r e s 40 time  and leads to e r r o r s  ^(.4),  1% and 27„ r e s p e c t i v e l y .  steps  6 (.4) which.are s m a l l e r than S  However, f o r accuracy b e t t e r than  ? ^ ( . 4 ) = .006 (,757o) and  e (.4> = .003 (.6%) more than 400 s  time steps a r e n e c e s s a r y . The r e l i a b i l i t y of the l a s t column of Table 6.0, ©^CT)  t  difficult  as an i n d i c a t o r o f the e r r o r s to assess. (2)  a €  CT) and 6 CT)  However, the s o l u t i o n generated  Here we i n t r o d u c e the n o t a t i o n ( 0 , 1 ) , vi an i n t e g e r ,  is  a  by the v\  ek.C>M= a. x \o  - 89 -  S i m i l a r i t y Algorithm s a t i s f i e s  (0.3,a,b,c ,d) e x a c t l y . (,TV-> O  f o r the S i m i l a r i t y A l g o r i t h m sufficient  Hence,  i s a necessary  and  c o n d i t i o n f o r the convergence of the a l g o r i t h m .  In  a d d i t i o n , a rough c a l c u l a t i o n shows t h a t i f i  '  '  >  i  '  then vv% »-y /i I- —> <j 0  ' A  where  i s the  V\ CIO  f l u x generated  by  g i v e n heat  the a l g o r i t h m .  f l u x and Hence we  to be a "rough" i n d i c a t o r of the e r r o r s Moreover, v e  take  \-»(t)  i s the  heat ^j)  consider  S^CT)  the o r d e r of convergence of  a  e  «d  (.T)  s  Q. (y)->  o  * to be an e s t i m a t e of the orders o f convergence of and  C  solution of  remark t h a t f o r any  However, i n these cases  i n g e n e r a l be  corresponding  scheme.  schemes.  i s o n l y an  &^  (T)  ( T) -> u  e q u i v a l e n t to convergence of  d i f f e r e n c e schemes and w  ©^CTi-^o  scheme l e a d i n g to an  ( 0 . 3 , a , b , c , d , e ) , the q u a n t i t y  calculated.  C? K T )  A  -> O . We  not  •  approximate can  be  would  the  For i n s t a n c e , i n the cases o f  finite  the C o l l o c a t i o n scheme ( c f . Chapter  i n d i c a t o r of the t r u n c a t i o n e r r o r s o f  VII)  the  - 90 -  R  U(X.T)  VS X  TIME=G.40  -  Fig.  6.1  Approximations Boundary  91  to  Stt) up  -  the to  Boundary Data  Position  of  T=.4  the  for  (6.0)  the  - 92 -  Fig.  6.2  Comparing ^> (V> and Mi.) f o r the Boundary Data (6.0)  HEAT FLUX VS TIME  -i— a.is  -rII  - 93 -  Figures  6.0.,  .6.1  and  6.2'  show that the  accuracy of the S i m i l a r i t y A l g o r i t h m depends on how the generated flux,  V% tt} .  heat  flux,  W("t)  This i l l u s t r a t e s  closely  , approximates the g i v e n  heat  the proof of convergence.  For the S i m i l a r i t y A l g o r i t h m to be p r a c t i c a l  we  should have to use at most s i x .to eight" terms ,o,f~ the ser.i.es (3) (3.13) d u r i n g most of the c a l c u l a t i o n . smooth i n i t i a l  Experimentally, f o ^  temperature d i s t r i b u t i o n s , such.as the  given i n (6.0), we  find  that s i x to e i g h t terms i s more than  adequate f o r r e s u l t s s i m i l a r to those given i n Table However, more terms- of the s e r i e s the i n i t i a l  closely  is rich  i n the  higher  The number of r e q u i r e d terms i s governed by  (3.13) evaluated at " t Although  may  6.0.  i n (3.13) are necessary when  temperature d i s t r i b u t i o n  frequencies.  one  initially  5 0  reproduces  how  the i n i t i a l c o n d i t i o n .  a r e l a t i v e l y l a r g e number of terms  be r e q u i r e d , the f o l l o w i n g example (see F i g . 6.3)  that d u r i n g a r e l a t i v e l y short i n i t i a l  p e r i o d of time  shows (short  compared to the t o t a l m e l t i n g time) the h i g h e r f r e q u e n c i e s are largely attenuated. c h a r a c t e r of the.heat  T h i s i s a consequence of the equation.  dissipative  Hence o n l y the f i r s t  few  terms  (3) in  A bound on the e r r o r made i n t r u n c a t i n g the (3.13) w i l l be g i v e n l a t e r i n t h i s c h a p t e r .  series  - 94 of  (3.13)  n e e d be r e t a i n e d  f o r most o f  As an e x a m p l e , we c o n s i d e r  ~  u  0  u  \  the c a l c u l a t i o n .  (0.3,a,b,c,d,e)  with  - u~>) e (6.1)  (t) - '°/3  J> =  4  and u s e t h e S i m i l a r i t y  Algorithm with  •oA  terms  =.001  and f i f t e e n  of  equal  the s e r i e s  time  i n (3.13)  an a p p r o x i m a t e s o l u t i o n .  Fig.  6.3  increments  The Approximate Temperature Distr.ibu'tion f o r t. B'etw.een 0 .and .1 f o r the Boundary Data (6.1)  to  obtain  - 95 Figure at  t =0,  .001,  seen that time)  at  distribution than  .01  t. =.01  the high  fewer  6 . 3 shows t h e a p p r o x i m a t e  (about  initial  initially long to  five  percent  terms  (3.13)  distribution  the c a l c u l a t i o n .  term b e h a v i o u r  be i n d e p e n d e n t  a r e needed  our numerical  temperature  during  of  is of  melting temperature time  i n the c a l c u l a t i o n . indicate  importance  To a l a r g e  c a n be  Hence by t h i s  experiments  of the s o l u t i o n of  of  It  of the t o t a l  damped.  distribution  (6.1).  components o f t h e i n i t i a l  have been s i g n i f i c a n t l y  Furthermore, the  f o r the boundary data  frequency  fifteen  temperature  extent  only the  (0.3,a,b,c,d,e)  the shape o f the i n i t i a l  that  seems  temperature  distribution.  Optimization  of the S i m i l a r i t y  Algorithm  So f a r we h a v e made no a t t e m p t accuracy  of  choosing  an o p t i m a l  partition  each time  ~t±  at  the a l g o r i t h m .  It these  choices  is clear  ta<i}  least  j Kir),!? o  all  i.  .  that  closely  the  T h i s c a n be a c c o m p l i s h e d b y 7T  IT  a n d , o r an o p t i m a l  any s t r a t e g y  (the proof  approximate  B e l o w we i n t r o d u c e  value  . aimed a t  s h o u l d be g u i d e d by a d e s i r e  approximate that  of  to o p t i m i z e  optimizing  to have  V\ ( t 1  of convergence)  o r at  ) MtWr  for  o  two m o d i f i c a t i o n s  well to t h e  Similarity  - 96 -  Algorithm  as a step towards o p t i m i z a t i o n . We f i r s t note that the S i m i l a r i t y A l g o r i t h m  an exact  provides  s o l u t i o n when the boundary moves a t a constant  speed.  Hence when the boundary moves at a slowly v a r y i n g speed, i . e . |i*Ct)|  s m a l l , the s t r a i g h t  boundary should is large.  l i n e approximation  to the  l e a d to b e t t e r r e s u l t s than when  \^Ctl\  Thus we concentrate  d u r i n g p e r i o d s o f time when the most p a r t , corresponds changing most r a p i d l y . points  4.^  o f 77*  I:  p o i n t s o f the p a r t i t i o n \ J C--k\\  i s l a r g e , which, f o r  to p e r i o d s o f time when  V»(t) ±  As an example, f o r the data  can be taken  II  s  (6.0), the  to s a t i s f y  j U(r).lT= C  • ti  for  an a p p r o p r i a t e constant  C>0 .  The next m o d i f i c a t i o n i s motivated  by the proof o f  convergence and i s aimed at o p t i m i z i n g the c h o i c e o f the time i n t e r v a l The  £t^ "^<+\) v  f°  given p a r t i t i o n  ra  s t r a t e g y i s to add i n the time i n t e r v a l  p o r t i o n o f the heat which was m i s s i n g interval o  n  E^N'-I, "tj )  t * i , tj-M^  t  o  •  on 7T  r ^ o ^ < + i)  i n the previous  a  time  More p r e c i s e l y , we choose /3 (  satisfy  II: / 3 . - M « | M V ) +*-(W(i O-£<t;-))- < " f C ; , * t ; ) j i  - 97 >,«? (see F i g . 6 . 4 ) .  f o r some  F i g . 6.4  Comparing the Given Heat Flux with that Generated by the S i m i l a r i t y A l g o r i t h m Using M o d i f i c a t i o n I I  Time - t We remark that both m o d i f i c a t i o n s I and I I can be implemented at very l i t t l e computational 'expense. To i l l u s t r a t e the u t i l i t y o f the above m o d i f i c a t i o n s , we c o n s i d e r (0.3,a,b ,c,d,e) with the data C= )K(Mc!r following  and V, - \  fit  for a l l i  With  we o b t a i n the  results.  Table 6.1  (a)  (6.0).  E r r o r Versus Time Increment -<=>*t Using M o d i f i c a t i o n s I and I I  ( .4) • - E r r o r i n  Unaltered Algorithm .05  . -27(-l)  .01  •75(-2)  1.77.  .005  •58(-2)  1.3%  6%  JCt) Modifications i T22(-l)  II.  I and II  57.  .13C-1)  •65(-2)  1.5%  •40(-2)  .9%  •36(-2)  .8%  .44<-2)  1.07. •  .31(-2)  .7%  .29(-2)  .77.  37.  .99(-2)  27.  -  (b)  e..(.4) x  - Error i n  98  -  v. l*,t) Modifications  Unaltered Algorithm  I and II  II 2%  .ll(-l)  1%  .05  -25(-l)  31  .22(-l)  31  .13(-1)  .01  .84(-l)  IS  .80(-2)  1%  .68(-2)  .9%  .66(-2)  .8%  .005  .72(-l)  .64(-2)  .87.  .64(-2)  .8%  (c)  g (.4) h  A!  .9%  - Error i n  .70(-2)  .9%  HCA) Modifications  Unaltered Algorithm  I  I and II  II  .05  .30(-l)  •30(-l)  •15(-1)  .14(-1)  .01  .86(-2)  -76(-2)  .48(-2)  .44(-2)  .005  .58(-2)  .53 (-2)  .39(-2)  •36(-2)  As i s to be expected the m o d i f i c a t i o n s are most e f f e c t i v e f o r the l a r g e r time increments.  However, even f o r the  s h o r t e r time steps the improvement i n accuracy i s s i g n i f i c a n t . M o d i f i c a t i o n I I proves to be very u s e f u l , ^-u.  («4)>  Q_5(.4) and  e^(.4)  from ten to f i f t y  reducing percent.  We remark that M o d i f i c a t i o n I i n c r e a s e s the accuracy of the approximations although varying f o r t e  WCti  o f (6.0)  [o.,.4] ( h(0)=5.33,  Order of Convergence of the S i m i l a r i t y  i s actually  slowly  h(.4)=7.36). Algorithm  In Chapter V we showed that the order of convergence  - 99 -  of the S i m i l a r i t y A l g o r i t h m i s one h a l f . that f o r a l l T  That  l e s s than the t o t a l m e l t i n g  as ©  CT) " OU^**  4  ^ j , : ^:  where  .  i s , we showed  time  w i e-y -a "t • —>  i  Pt  This i s important  i n that i t . e x p l a i n s  our o b s e r v a t i o n that the S i m i l a r i t y A l g o r i t h m should be used only to o b t a i n coarse In Chapter multiplying of  accuracy. V we observed  t h a t the c o e f f i c i e n t  the o r d e r one h a l f term of the e r r o r  ? ("T)  and  U  values of  &  S  { T )  and  expansions  i s u s u a l l y s m a l l , hence, the e f f e c t i v e  are l a r g e r than one h a l f .  Hence the  S i m i l a r i t y A l g o r i t h m should be s i g n i f i c a n t l y b e t t e r than an order one h a l f scheme. which support  Here we g i v e some numerical examples  t h a t -claim.  F o r exact s o l u t i o n s we use those g i v e n by Sanders £ 31  In p a r t i c u l a r , f o r the data  (6,  fact  - 100 -  Sanders gives the solutions  J  on  o < A < i t t r - V \-«A t) *  Confluent Hypergeometric  .  Here  0-**t)  J  M(n;  Functions, and  related by the condition that: \,  t;^)  arid A  are the are  i s the smalle.s„t positive  a  root of the equation  2  M (-X ^ - i ; ^ ) - o 0  The data (6.0) corresponds  to A  =.5,  A =l. 0  We also consider the data  (6.3)  i with various i n i t i a l  •U, Ur-  temperature  x'-l  data  (6.3a)  )  U 0O = (x-Oe e  .  (6.3b)  V «,XN-- x- I •.  (6.3c)  0  To the data (6.2) and  (6.3) we apply the S i m i l a r i t y  Algorithm with equal time steps, A "t  , varying from f i v e to  t h i r t y percent of the t o t a l melting time. For the data (6.2) we are able to estimate the order of  convergence of  - 101 -  CT)  -50,  However, f o r the data  (6.3) we must s e t t l e f o r the order o f  convergence o f  s i n c e the c o r r e s p o n d i n g  exact s o l u t i o n s are not a v a i l a b l e .  In each case the a l g o r i t h m i s used fifty  percent o f the t o t a l m e l t i n g time.  to approximately  Table 6.2 p r o v i d e s  a summary o f the r e s u l t s .  Table 6.2  D  a  t  a  Observed Order o f Convergence,  g,(T)->o  <=>CT)->o V4  gy, (T) ->o  (6.2)  A=.5  .8  .8  .8  (6.2)  A = .85403  .8  .7  .7  (6.2)  A=l.  .7  .7  .6  (6.3,a)  -  -  .8  (6.3,b)  -  -  .8  (6.3,c)  -  -  .9  -  Table convergence of one.  6.2  supports our claim  the S i m i l a r i t y  Moreover,  it  provides  convergence of orders  The S m a l l  Time V e r s u s  Similarity  Solution  iteration  while  40 + 50 m series  in  performed  and  _>  to twenty  time)  t ^ ,  respectively.  and s q u a r e r o o t s and f i v e  + 3 n + 4 m n  involves  operations,  are retained  hand,  Algorithm using the small  t o be  where  A l g o r i t h m u s i n g the approximately n  terms  of the  and t h e n e c e s s a r y q u a d r a t u r e s  b y means' o f a 2 m - p o i n t On t h e o t h e r  operations,  respectively.  o f the S i m i l a r i t y (3.13)  (large  Multiplications,  are taken  operations  0  f o r one  a r e c l a s s i f i e d as e q u i v a l e n t  representation  (3.13)  to  counts  (3.13)  t-i  of  of the  A l g o r i t h m u s i n g the r e p r e s e n t a t i o n s (small  i.e.  operation  and  to the  of the  exponentiations  time  is closely related f>jCT)->o  step,  One i t e r a t i o n large  the o r d e r  t h e L a r g e Time R e p r e s e n t a t i o n  and a d d i t i o n s  equivalent  that  of  one h a l f  )  and (3.14)  divisions  the o r d e r  i s between  some e v i d e n c e  s e c t i o n we g i v e  (one time  Similarity time)  this  that  Algorithm  ;T> - > o  o f convergence of  In  102 -  Filon Integration  one i t e r a t i o n  time  of the  representation  are  Rule.  Similarity  (3.14)  requires  2 30 + 70 J + 90 J K + 40 J terms  of  the s e r i e s  for  + 80 J K  operations,  GCv,t-^)(see  (6 . 5 ) )  where  2 + K  are retained  and  -  the  integral  in  (6.5)  103  i s evaluated  using a  j  node  quadrature  rule. To i n i t i a t e (3.13))  and ( 3 . 1 4 )  (  t h e c o m p a r i s o n we w r i t e u*  4  <—>  (3.14))  (3.13)  ( "u.'^  as  (6.4)  s&n(P<y- ^i$ ' & C  S  ,«t;) c o s ( w ^ « l .  and  (6.5)  i  where  T^S  X/C,^,  Chapters  III  and I V h a s been  }  Ac C; (  to c a l c u l a t e  of  used.  Our a i m i s t o compare necessary  and t h e n o t a t i o n  4 L  u* (C^  t h e number o f  4 (  ^, -al±  + l  )  operations  to a g i v e n  accuracy  104  using  (6.4)  and  ? instead,  t h e number o f A  In error  ***  e  (6.4)  ^  t  made by t r u n c a t i n g the  We f i r s t  and  However,  (6.5)  t h e -term  motivates  o p e r a t i o n s n e c e s s a r y to  e a c h c a s e the  in evaluating  by  respectively.  i n both  ^•»+>\> "^t-n ^ c  (6.5)  o  us to c o n s i d e r  calculate s 0' ^  p r e s c r i b e d a c c u r a c y , say  a  error  the  enters  f r o m two  sources -  s e r i e s and q u a d r a t u r e , e r r o r  .  the  made  integrals. c o n s i d e r the  large  time r e p r e s e n t a t i o n  (6.4)  writing  Integrating  the  expression for  "U,(ft>l5  it  is  e a s y to  see  c 0  /  C  ,  ^  that 0  where  by p a r t s  (c" J. ^V (c,  -L-l ^ ' ^ ^  Hence  -^>^(j^-^ t w i c e  0  ,  ,  3 i  we  obtain  a ))ce,(^)cl^ i  105 E v a l u a t i n g the above i n t e g r a l by p a r t s we have  \  R  ^  -TT*(2n-l>H  S  t  (6.6)  Moreover, i f , asymptotic  TT,  (2^,-0 ^ *  is. lftfge, enp.ugh we,, o b t a i n the.  estimate  le^(oK< ^ - \ H L The  4  e_  .  e x p r e s s i o n s (6.6,a) give us a "rough"  number of terms,  n  , o f the s e r i e s  (6.6a)  estimate o f the  (6.4) necessary  to a c h i e v e  a pres'crrbed -accuracy f o r a g i v e n We now focus our a t t e n t i o n on the quadrature a r i s i n g from point F i l o n  the c a l c u l a t i o n o f  ^ ^ ^ ^ j^' !'"]* 1  1  by a  error 2 m -  I n t e g r a t i o n Rule.  Suppose  -Sv-(p')  i s the F i l o n approximation  to  y^^.(^-),  then we can w r i t e  where o<t^<i take  w  ( c f . Davis and Rabinowitz  l a r g e enough so that  [ 9  v-j^ / % ^ .$ "V^  ] p. 6 4 ) . t  I f we  then the t o t a l  - 106 -  e r r o r due to the quadratures, c a l l  it  , can be seen to  satisfy  where  E v a l u a t i n g the above trigonometric sum we a r r i v e at the estimate  Since  j^ixjjx  f o r x >, o  we can w r i t e  (6  Hence we have  Now representation  t u r n i n g our a t t e n t i o n to the small (6.5), we w r i t e  time  107  * <3>  s e  3  '^  C < , 4 , S  U*(C;  4 | j  ;*t  i + 1  )  11  • ^ 4-.  +e  S i n c e we h a v e a s e r i e s o f p o s i t i v e m o n o t o n i c a l l y d e c r e a s i n g terms we o b t a i n  iestoi.«  JL'rir  A« e-  WR,V  *«-  (6.8)  where K. = 0 < H <• »  To  investigate  the quadrature  error  involved  in  - 108 -  evaluating  to  integral  G ( C - ^, t ;  term o f is  the  A  4 l  say,  since  all  are well-behaved  for  determines  the  other  is  terms  °* 3 >  J  large  the  of  the  always  of  * '  that  the  t  n  e  enough to  ,\  nodes,  That  ^i^)  source term  quadrature  .  Q ' ^ & ' & ^ I  G ( C^, ^, >  dominant  largely J  evaluate  J  to  be  number o f n o d e s n e c e s s a r y to  evaluate  same a c c u r a c y . To c o m p a r e t h e  \TT js. 1  .  (6.8)  L  are  a  o p e r a t i o n c o u n t s we n o t e  H e n c e we s e t  l^C^l)  so t h a t of  j c-^)  a p r e s c r i b e d a c c u r a c y and assume t h i s  representative  to  + 1  we n o t e  r e q u i r e d number o f  H e n c e we c h o o s e  to  (6.5),  in  each l e s s  and  K,  j— =  of (6.6), than  i  )c^|  10* ' e  for  and f i n d  that n,  o f ( 6 . 7 ) and given values of  m  and K  Iej  |  . d  5  -  109 -  we take .001,  For values o f S; t y p i c a l range over which calculation). magnitude of  Since  . 0 1 , . 1 , 1.  (a  v a r i e s d u r i n g the course of a  K, has very l i t t l e  K ( X =0,1,2) we s e t  i n f l u e n c e on the  V<j=l.  Furthermore, to  assess the e f f e c t of the magnitudes o f L. and  on the  (  o p e r a t i o n count, for*, t-hse* larger time., representation-, we vary 3  both  between 1 and  and  summary of the  Table 6.3  (a)  results.  Approximate O p e r a t i o n Count  *d -=4  of Operations L a r g e Time Solution  # o f Operations  ,(4) .001  .01 .1 1.  100. Table 6.3 p r o v i d e s a  0  25+  1  25+  0  15  Small  Time  Solution  n  m  9  9  27,000 75,000  1,000  14  38  7,000  700  5  19  1,700  15  11,000  5  28,000  4  300  2  10  700  5  4,000  2  200  1  10  700  6,000  1  1 1 2  # of Operations Large Time . Solution L, : I , : 100  (4) Here we have used a Gaussian quadrature scheme ( c f . Isaacson and K e l l e r [ 2 l J p. 327) to e v a l u a t e the i n t e g r a l i n (6.5).  - 110  (b)  -  cl =6  t  of Operations Small Time Solu t i o n  # o f Operations Large Time Solution  r  K  J  .001  l  25+  75,000  14  36  .01  l  15  28,000  5  .1  l  10  13,000  2  5  6,000  We  n  # o f Operations Large Time Solution n  m  5,000  17  114  21,000  18  2,000  6  52  5,000  3  18  1,500  3  52  4,000  1.  9  700  2  29  2,000  remark that the number of nodes g i v e n i n Table  f o r F i l o n ' s I n t e g r a t i o n Rule i s l a r g e r than was our n u m e r i c a l experiments  (usually  m  was  used  6.3  f o r any of  taken between 10 and  6 20).  Moreover, we note that  10  i s n o r m a l l y the l i m i t  accuracy which one would want, s i n c e we time i f we  are wasting  t r y to make the t r u n c a t i o n and  significantly  of  computing  quadrature  errors  s m a l l e r than the i n h e r e n t e r r o r i n the a c t u a l  approximation to (0.3,a,b,c,d,e) generated by the  Similarity  Algorithm. Although the S i m i l a r i t y impractical  (3.14) i s the s m a l l time r e p r e s e n t a t i o n of  S o l u t i o n , Table 6.3  to use i f the i n t e g r a l  shows that i t i s n u m e r i c a l l y i s c a l c u l a t e d by a  - Ul c o n v e n t i o n a l quadrature r u l e . term o f  ^U,  G  C;^)  y  delta function in  That i s , because f o r small  about ^  the dominant  behaves l i k e a  , a c o n v e n t i o n a l quadrature scheme  r e q u i r e s a r e l a t i v e l y l a r g e number of nodes, c o v e r i n g the [0,lj,  whole of the i n t e r v a l Moreover, u  we  to a c h i e v e the n e c e s s a r y a c c u r a c y .  remark- that i n e s t i m a t i n g  v C,-^, <&X;)<2 *  a  J  we  to be a c o n s t a n t .  o p e r a t i o n s g i v e n i n Table 6 . 3  assumed Hence the number of  f o r the s m a l l time  representation  could be an u n d e r e s t i m a t e . Furthermore, s i n c e <it-  and  ^.  ^  (^>,")j  , the cosine, terms which  are independent  enter F i l o n ' s  Rule need  be c a l c u l a t e d o n l y once d u r i n g the e n t i r e c a l c u l a t i o n .  At  step these c o n s t i t u t e the weights of the F i l o n Quadrature This i s i n sharp c o n t r a s t with the c a l c u l a t i o n of the appearing i n the s m a l l time r e p r e s e n t a t i o n time step " i ;  }  G ( C;  +)  ^ t ^ , ' C;vj\  quadrature p o i n t s i n ^  and  in ^  These r e s u l t s ation  (3.13)  each Rule.  integral  ( 3 . 1 4 ) where at each  must be c a l c u l a t e d a t a l l .  That i s , i f  .  the nodes of the quadrature scheme then at each time GCCu,^,^.-/^^)  of  must be  a  r  e  "t^'  calculated.  i n d i c a t e t h a t the l a r g e time r e p r e s e n t -  i s b e t t e r than the s m a l l time r e p r e s e n t a t i o n  f o r the n u m e r i c a l s o l u t i o n of (0.3,a,b,c,d,e) u s i n g the  (3.14)  - 112 -  S i m i l a r i t y Algorithm. Table 7.3 a l s o p r o v i d e s us w i t h an estimate f o r the number o f r e q u i r e d terms f o r -the l a r g e time r e p r e s e n t a t i o n (3.13).  We can see t h a t u n l e s s  e x c e p t i o n a l l y misbehaved  -u.*"' (C; ^ > o,*t; )  (reflected  at most twenty terms of the s e r i e s  is  i n the v a l u e s o f  L,  and L  i  )  i n (3.1,3) need b.e us.ed.  For reasonably behaved f u n c t i o n s three to t e n terms are adequate.  Our n u m e r i c a l experiments  support  these  statements.  Comparison of the S i m i l a r i t y A l g o r i t h m w i t h L o t k i n ' s D i f f e r e n c e Scheme We conclude  t h i s chapter by comparing the S i m i l a r i t y  A l g o r i t h m with L o t k i n ' s D i f f e r e n c e Scheme. L o t k i n [ 23 ] transforms the t r a n s f o r m a t i o n  ^ = X/.SCi)  to a f i x e d boundary by making  i n (0.3 ,a ,b ,c , d , e) .  employs centered d i f f e r e n c e approximations  Then he  ( c f . Isaacson and  K e l l e r [ 21 J p . 445) f o r the s p a t i a l d e r i v a t i v e s appearing i n the r e s u l t i n g d i f f u s i o n e q u a t i o n , together w i t h backward d i f f e r e n c e approximations f o r both (0.3e).  -u..C»,"t) n d a  i(fc)  ( c f . Isaacson and K e l l e r [ 21 ] P. 445). i n the transformed  version of  The r e s u l t i n g non-linear system of d i f f e r e n c e  is solved i t e r a t i v e l y .  I f a uniform mesh i s taken  the above scheme i s second  >  order a c c u r a t e i n space  equations  in ^  , then  and f i r s t  order  -  accurate  113 -  i n time. In comparing the schemes, we use the data ( 6 . 2 )  with  A = . 5 , . 8 5 4 0 3 , and 1 .  are employed to approximately melting series 6 v i . CT)  time.  In each case the approximations n i n e t y percent  I n the S i m i l a r i t y A l g o r i t h m  o f the t o t a l  t h r e e terms o f t h e  i n ( 3 . 1 3 ) a r e used and the e r r o r s given a r e »?j(T) and •  For L o t k i n ' s Scheme n i n e  i n t e r i o r mesh p o i n t s a r e  used and the e r r o r s given are the maximum a b s o l u t e e r r o r s a t The r e s u l t s a r e summarized by Table 6 . 4 .  these mesh p o i n t s .  Table 6 . 4  S i m i l a r i t y A l g o r i t h m Versus L o t k i n ' s D i f f e r e n c e Scheme  Similarity  Algorithm  Lotkin's  Computer-(5), 'Error  T=.45 A=.5  T=.26 A-.85403  T=.22 A=l:  in  D i f f e r e n c e Scheme Error i n  Computer  g» CT)  Time(Sec)  uU.-t)  JCt)  Time(Sec)  .01  .62(-2)  .69(-2)  .14  .70(-2)  .97(-2)  .08  .005  ,57(-2)  .45(-2)  .24  •37(-2)  .55(-2)  .12  .24(-l)  .10(-1)  .07  .45(-l)  •16(-1)  .06  .005  ,24(-l)  .79(-2)  .16  •24(-l)  •96(-2)  .09  .01  .32C-1)  .12C-1)  .06  .89(-l)  •22(-l)  .05  .005  .32C-1)  •91(-2)  .12  •58(-l)  •13(-1)  .09  .01  (5)  All  c a l c u l a t i o n s were done on the IBM 3 7 0 / 1 6 8 .  -  It  c a n be  Scheme and t h e for that  if  t h e more  greater  seen f o r  Similarity  approximately  the  -  these  examples,  Algorithm  same amount  accuracy is  efficient  114  give  of  required,  algorithm.  that  Lotkin's  comparable  computing t i m e . then  Lotkin's  accuracy We  remark  Scheme  is  CHAPTER VII  A COLLOCATION SCHEME  t h i s chapter we c o n s i d e r ( 0 v 3 , a ,b;,c ,d , e) from a  In  ?  v a r i a t i o n a l p o i n t o f view approximating finite  element  i n order to develop a l g o r i t h m s f o r  its solution.  Our u l t i m a t e aim i s to a c h i e v e a  f o r m u l a t i o n o f ( 0 . 3,a,b,c ,d , e) .  The L a g r a n g i a n Equations f o r Heat.Conduction To i n i t i a t e a v a r i a t i o n a l - f o r m u T a t i o n o f ( 0 . 3 , a , b , c , d,e) we f o l l o w the l e a d o f B i o t £ referred  .1 ] by d e f i n i n g  §f-C*it),  to as the heat displacement f i e l d , to be the time i n -  t e g r a l o f the r a t e o f heat flow a c r o s s a u n i t c r o s s s e c t i o n a l area o f a g i v e n s l a b .  With t h i s d e f i n i t i o n the e q u a t i o n o f  heat c o n d u c t i o n can be w r i t t e n as  |M)*5  In  L[§C* t>}v (  ~  U  y  U,t).  a d d i t i o n , the law o f c o n s e r v a t i o n o f energy  (7.0)  i s expressed by  the r e l a t i o n  I x M :  - «*HU,t).  (7.1)  -  To o b t a i n  the  field  be t h e  temperature  [  1  arbitrary  which are  relation  (7.1)  ]  and t h e  let  and h e a t  of  the heat  "uC*,"i\  and  displacement T h e n we  displacement  conservation of  boundary c o n d i t i o n s  Su(«,u--  heat  ( 0 . 3 , a , b , c ,d ,e) .  variations  c o n s i s t e n t with the  for  first  we  distribution  r e s p e c t i v e l y associated with  consider field  -  Lagrangian equations  c o n d u c t i o n as d e r i v e d by B i o t  §(x,t)  116  energy  (0.3,c,e),  i.e.  § <«,t) K  and  F o r -any  (7.2)  interval  (a,.b)  along the  s l a b , (7.0)  implies  that  ^b . (1) ° - ) [ ^ ^ x ^ . t H § U,t>] S f U . t W U .  ( .3) 7  a Upon i n t e g r a t i n g constraining  by p a r t s  relations  the  (7.2)  first we  term o f  (7.3)  and u s i n g  the  obtain  4) x-a. where  Since  (7.3)  limits  of  integration  time.  In  fact,  (0.3,a,b,c,d,e)).  must be s a t i s f i e d  a and b c a n be t a k e n  we w i l l  take  e :o b--i(t) s  )  for to  all  time,  be f u n c t i o n s  ((/u,s)  the of  satisfying  -  The v a r i a t i o n a l equations for h e a t  referred  to  conduction,  as a g i v e n independent  -  principle  by B i o t  if  function  117  ^  1  (7.4) ]  as  we assume t h a t  of  x  parameters  and  %.  (generalized  leads the  to  Lagrangian  $(*,i>  and a t  a set  of  equations  can be  expressed  most a c o u n t a b l e  coordinates)  set  ^ ?«^^|-  of  »  i.e.  $(«,tu $<•$,,... 'hen f o r \ V^\-  arbitrary  it*  variation  In  and  variations  consistent with of  addition  the  heat  we h a v e  (7.2),  displacement  the  *  ^ ^ ^ ' ^ ^ j . &$(x,i), field  is  n  t  ^  i e  P  a r a m  eters  the  given  by  relation  hence  *J . Moreover, parameters  since ^  (7.6)  V(«i,b^"fcV Ct)|  %  we  is have  also a given  function  of  the  -  118  -  (7.7)  Introducing principle  (7.5), ( 7 . 6 ) (7.4)  we  and  (7.7)  we h a v e  Since  the  implies  introduced  parameters  the  solution of v  set  t , c  the  |  dissipation  ^  °i  (7  the  square  use  system of  be a s e t  of  integrable  \>; C o \ ~ -v-(I) = ° assume t h a t  to  5(«|t)  for  for ( 7.9)  ^ heat  independently,  (7.8)  basis  we t a k e  all  be  (0.3,a,b,c,d,e)  on  C°» l l  }  ,)  with  the and i"]  the  Furthermore  t  c a n be w r i t t e n as  L lT«,  for  i*v l ... , t  (  to  (w,i)  functions  functions  «  conduction.  equations  8)  function  c a n be v a r i e d  * l .  order  M.  of  ^ r?  +  Ctlj,  Lagrangian Equations In  i ;  variational  that  **« -  the  obtain  f °<M i?* F where  into  the  sum  let (the property we  -  119  ^;(5t,).  The  equations *  O  (7.9) T  h  a  t  <°> -s<*»;t)  i  t h e n become t h e S  '  W  into  S2?te,t)=o  to  (7.10)  e  t  a  k  S  C  (7.9)  k  :  o  i '  determining 3  :  S  and n o t e  (  t  ;  l  equations  substitute  that  for  Vfo.itt^t^  5 J  t)  = O,  obtain  S*<t) C,^(*j • ( A,- i(.tu(t)  Q,)  ^<t> = o  (7.11)  where  o .1  «  o  f (0=..(,|,C«,..; ^(«,...) . T  ;  Moreover,  if  (-u.s) i s  an a p p r o p r i a t e to  ^o")  where  is  satisfy  (0.3d)  condition.  If  then  |^<t) must  5 ( * < ° ) = °<* $0**)  satisfy  t h e n we  satisfy  ^  that  initial  to  [ f o ^ ' I ^ ^ ^ C ^ l ^ C t l ^ x ^  for a l l  y.l  J ? J  ... ;  take  - 120 -  < > 2  and  r Lf-V Roughly speaking, ^-(Vjj)  •  1  } $ U b ) ^-txUx.0  ^i(o)  Finally  i s the p r o j e c t i o n of  5 <x) 8  onto  the S t e f a n C o n d i t i o n , (0.3e), becomes (3)  (7  where iC©)=b  Even i f the i n i t i a l (7.13) has a s o l u t i o n  v a l u e problem  i t would be d i f f i c u l t  o f the c o u p l i n g o f the d e r i v a t i v e terms (7.11) and (7.13)* and  (7.11),  (7.12),  to o b t a i n because  -i(t),  »0  in  Hence we seek ways of r e f o r m u l a t i n g (7.11)  (7.13) i n order to a v o i d t h i s  difficulty.  (2) For a g i v e n temperature d i s t r i b u t i o n , Mt.Xj"fc) the heat displacement f i e l d , <f(x,t) , i s not u n i q u e l y determined.. S i n c e we are i n t e r e s t e d i n *u. O,-t,) and M^(v -t) o n l y we take § U,-tl- - 5 ^ ^ . " t ' ^ • ;  1c  As noted by B i o t £ 1 ] the d e t e r m i n a t i o n o f J C t ) i s not p a r t o f the v a r i a t i o n a l procedure but merely another o r d i n a r y d i f f e r e n t i a l e q u a t i o n added to. the L a g r a n g i a n equations  -  If  (7.13)  and  -  i n the Stefan C o n d i t i o n (0.3e)  ©('v^Utt), t>  for  121  instead of  we u s e  -§Ucu,-t)  <s<*\ ^  ~  J  then  becomes  the i n i t i a l  value  p r o b l e m t o b e s o l v e d becomes  - 0  (7 .  where  JCo)-b  A n o t h e r way by w h i c h we c a n e l i m i n a t e of  (7.11)  Instead  and ( 7 . 1 3 )  i s s u g g e s t e d by t h e work  of expressing  5(.*»t)  as t h e l i n e a r  we w r i t e  E p i t t e d - */ ). 4Ct> is, *w> J(t)  define  o  and  use (7.1)  to  obtain  the  difficulties  of Biot  [  combination  1  ].  (7.10),  122  jf <Pi<ky^  5  P r o c e e d i n g a s b e f o r e we o b t a i n  the i n i t i a l  *  n  S*(t> C j i p l O  -SU>= -  + ( A - iit>.sit> Q ) J i t ) * O  $ -f  C pto)= J . 4  .S<.o)-W where  ^  value  0  a  A  £ ft.(t)v?(o) + hCt)7  problem  -  It  is  interesting  V»v\ (x^* - ( v%-^)TT i uw-fC'n" j)TT xj ,  123  to  obtained Melamed  system o f  by V . G . Melamed  oo  equations  ^  an a p p r o x i m a t e  solution  is  coefficients  taking  becomes t h e  to  a  Moreover,  obtained  system  of  R u b i n s t e i n £ 3 0 ] Chapter  (cf.  by a s s u m i n g  (4) .  A/  by  (0.3,a,b,c,d,e)  reduces  differential  that  (7.15)  4  • U U A ) * X A^lOeoifVA-xj-Tr if  note  i  equations  -  8)  denumerable  that he h a s s h o w n that-  by c o n s i d e r i n g the  first  ,V  ]  Ct) (  then  as  /V->co  the  approximation converges. In first  A/  other  functions  approximation the  appropriately  approximate finite  with  of  these  an a p p r o x i m a t e  ( 0 . 3 , a , b ,c.,d ,e)  v e r s i o n of  of  will  by  the  an  solving  (7.15). i.e.  global  (7.14)  and  basis, c o n s i s t i n g of  basis  basis  and o b t a i n s  p r o c e e d i n g as a b o v e ,  formulation  s y s t e m s we o b t a i n and  to  f u n c t i o n s which are  an a p p r o x i m a t e  (7.14)  for  | cos j[(vv-l)'TrxJ |  truncated  basis  element  convenience  takes  solution  Instead  a  w o r d s , he  functions w i l l be a p p r o p r i a t e l y  for  on £ o , l ] ,  (7.15).  finite be  using,  we p r o p o s e  We w i l l  elements.  labelled  truncated  an  start  For  so t h a t versions  the of  (7.15). In  particular  we w i s h  to  express  to  write  (4) This motivated form g i v e n by ( 7 . 1 0 ) .  uCK,t)  as a  linear  T-  us  g(.x,"t)  in  the  124  combination of we d e f i n e  and  -  piecewise cubic Hermite  a partition  77"  of  the  polynomials.  interval  To t h i s  [o,l]:  let  where  9^-tlx)  is  the  cubic Hermite  x  with  VCx) =  and we h a v e  i n t r o d u c e d the  notation  polynomial defined  % Jf,  s  by  end  - 125  * vs > (cf.schultz  [ 32  ]  Chapter 3).  It. i s . w e l l £o,l],  known.that  which has s u f f i c i e n t l y  be a p p r o x i m a t e d  arbitrarily  combination of functions functions  fp„r f i x e d ; . "fc v  well  well  in mean^^  i n mean f o r f i x e d  fixed the  for  t  ^ *>'^  as  "t , t h e d e r i v a t i v e s  derivatives  systems  provided  for a set of  smooth  it  ( *, t)  G^U,t}  t  ;  ^ t / x . - t ' i , Q*« (*,i) , ^  w  (v,t)  i s known  that,  G ? IK, * >  for  tend  respectively  to  i n mean  taking  ^ . . W j •• •.  |  <^W*  !  (7.14) and (7.15) become 7  "* said  linear  V\ —> o .  jv.ort.VjU)  is  can  9^<*) ~ > g<*,*^  sufficiently  t  Hence  the  i.e.  by a  on  >  G 'U,i>*r E  Again,  behaved d e r i v a t i v e s ,  ^/^(Tf**),  in  a», fu.nct.ion•  The s e q u e n c e o f f u n c t i o n s  to converge S  V,*. ( O G L*C°,»]  i n mean t o a f u n c t i o n  - V W}'<1* -> o  as  v\ - > «o •  •^"(,0 6 L* C«*, il  -r\z\,3 , t  -  126 -  s'cof] Q ( i ) + (tf,-i<tmty3,) QCt) = o  5  (7.16)  and  (7.17)  respectively. Here  |  truncated versions  > /^*"\ > ' " ' *  are the  1  of  |  C ^ , A« ,  | , i -> 2. 1  appropriately  r e s p e c t i v e l y and  -  We s e e t h a t are  Gram M a t r i c e s  (7.16)  and  of l i n e a r l y  f"^  are nonsingular  independent  functions.  and (.7...17) c a n b e s o l v e d as i n i t i a l  question (7.17)  127 -  of  value  c o n v e r g e n c e o f t h e schemes o u t l i n e d  since  Thus  both  problems.  The  they  b y ( 7 . 1 6 ) and  is unresolved.  A Galerkin  Scheme We n o t e  that,  e a c h have  support'  are  tridiagonal  block  matrices.  However,  because the b a s i s  CX . , , x K  K+  ,]  (Xo-o,  matrices,  while  the estimate  for  X  functions  W 4?  = i )  ^ f\  }  H*\,..., A/+ / y  t  °i f^*-1  a  v  j P , , a/, ^ | ;  e  t}  •u ci(t»,t) x  I ^Pw^**|  used  in  (  -  (7.17)  is better  than  f o r m e r we u s e t h e  while  128 -  the estimate  used i n ( 7 . 1 6 ) ,  s i n c e in the  approximation  i n the l a t t e r  we u s e  tke estimate  v. .W.tt\ x  We w o u l d l i k e the b e t t e r  to a c h i e v e a f o r m u l a t i o n  approximation of  computational  advantage  -u (s(0,-r^  equations  of  i n (7.17)  x  of the sparse matrices  How t o p r o c e e d becomes a p p a r e n t the systems (7.14)  which combines  if  of  with  (7.16).  we c o n s i d e r t h e  and (7 .15) , ..r.espect i v e l y ,  and M r  1  "  1  I n t e g r a t i n g by parts appropriate  the middle  e x p r e s s ions f o r  term,then  the  substituting  5 ("i"^ > 5XK C><i't\, we o b t a i n  the  first  i.e..  129  and  respectively. than  force  That  t h e heat  the  heat  equation  the  introduction  v<Cx,-fc)  where  displacement  .  of  S§C*i"t}  That  i s we w r i t e  now  is a basis  0  ^  In o t h e r  ( S<  words,  This  V  to  suggests  for  L*C«> il  t h a t we f o r e g o with  with  (  and t h e p a r a m e t e r s ,  by the G a l e r k i n  conditions  1 V-(X) dJt ) cl; <o) r ^  V  we f o r c e  vt(.x,-t)  o  (  V  >  0  v  'to satisfy  the heat  i n a weak s e n s e . The s y s t e m  satisfy  a n d i n s t e a d work d i r e c t l y  I  0  do no more  j>(x,"fc)  ia-i, a, • • .  t o be d e t e r m i n e d  equations  field  i n a weak s e n s e .  x>J(o) =• v . O V - o are  i s , the Lagrangian  (0.3,a,b,c,d,e)  t h e n becomes  equation  -  130  -  *  (7  where  i U t ) C efct) 4 (A--i<k)i«t)Q)d(i)so  <2> :  .«=.»,-a, Now t o e x p r e s s  o  o  this  i n terms  of the f i n i t e  element  a p p r o x i m a t i o n we t a k e  M ; > = W«> j<*>^> v  Hence  (7.18)  becomes  ict)5U)-  « * (,&A/*»,»<t)  TOCo):' O b where  c  - JU)K(il)  ^  (7  -  131 -  0  ((SC,  C^il,  »" >' r  -  W*  fckls*  U)  132  V °  c  -  s  U  *  ^  a,...,*/  o,  D ( i ) ~ (0,,u>, 0 ,ct),..., D^c-t), D„ ,, (t»  T  A  = (  and we h a v e  4  ., cl (t))  T  3A/  introduced  a  the  notation  x  i  - 133 -  TJU,^).  The approximation  t  o  V-(x,i)  c a  n be written as  o„ct>  UC*,t>  Hence the v a r i a t i o n a l p r i n c i p l e (7.5) has led us to a semi-discrete or continuous Galerkin formulation for the Stefan Problem  {0o3-,a-,b,c ,d,e). That i s , the s p a t i a l variable of the  system of equations•(0.3,a,b,c,d,e) has been discretized while the time variable remains  continuous.  We remark -that the system  (7.19) combines t.h.e sparse  matrices of (7.16) with the better approximation to of  (7.17).  M„  <t},-t)  Moreover, unlike (7.16) and (7.17), the solution of  (7.19) yields d i r e c t l y and hence most accurately approximations to  the quantities of interest  ^ u.Cx,-t^, U* U ,-k>^ .  It should be noted that i f  "u OO <j C ' O i ^ l j e  then  an i n i t i a l value of "OCtV can-he, obtained by interpolating the i n i t i a l condition TT"  , i.e.,  "U^Cx^ at the points  of the p a r t i t i o n  O^, (oV- "U/b*^ , O j , loi= b -U (bXj)  instead of projecting  0  "U(>o 0  accuracy of the approximation Since  \ X. {  into U"lx,t)  j^^ilr")  ^=',*  . The order of  to \*. (x|t)  i s a Gram Matrix of S. A/  i s not affected. linearly  independent f u n c t i o n s the  •  1 3 4  over [ u , l ] i t i s i n v e r t i b l e and  system (7.19) i s s o l v a b l e  locally  has a unique s o l u t i o n f o r a l l i . show t h i s , we  i n time.  such that  hence  In f a c t  (7.19) To  •S(\t>c,  define  and note t h a t  which upon i n t e g r a t i o n by parts  '  becomes  °  o  that i s ,  Hence m u l t i p l y i n g  the f i r s t  equation of (7.19) by  D (t)  we  T  obtain  i[6 (t)TD(t>l 4<tW cU(t). f D ( t ) ^ 6(t)l - T  T  ' dt  cli  1  J  1  i(t)  0(t)  or cL n s < t ) - " 5 < t > ^ D ( i > l - D Ct)6( 5<t). cli JCi) T  z  T  J  Since  ,S(M>©  and  «<  i s p o s i t i v e d e f i n i t e , we  have  -  biSJVo,, ,  3<t) 0 l t > Y * D l t > $ T  We s e e t h a t D c t ) —> o equation  .  V\ (A) >, o,  i.  (.*)-><*? leads  -S(i)  that  remains  increases  <3(t)$o'  as t  then  s i n c e the second increases  since  bounded.  t h e n we must h a v e  to conclude that  such that  as t  to a c o n t r a d i c t i o n  implies  D It) -> °o  H e n c e we a r e a b l e all  then  Hence If  for  if  This  o f (7.19)  135 -  (7.19)  that  J c t ) —> o .  has a unique  solution  jtt)>o.  A C o l l o c a t i o n Scheme Because o f the p r o p e r t i e s is  a convenient  (Collocation) Galerkin  system to a n a l y z e .  formulation  system  (7.19)  of  \  However  >  ,  there  c l o s e l y associated with  which  i s more c o n v e n i e n t  is  (7.19) another  the continuous  f o r numerical  computation. To Galerkin  introduce  Conditions  and i n t e g r a t e  t h e C o l l o c a t i o n scheme we w r i t e t h e  ( 7 . 1 9 ) as  the middle  term by p a r t s  to  obtain  -  136  -  - A / 4 / , rf= 2 ,  Using  a  two  point  over  • C*v\, Xw+il  A/  O  where  Now  if  Gaussian we  2  'S, *t  scheme  to  evaluate  the  integral  obtain  h  Hi) w  t ' «• <  ?  t ,  -  w  >"  f*»Si*>-  ?«*«>*<t>  w,(v£t>? ^ W . j j v  -  then the f i r s t It using  has.been  cubic Hermite  approximations (cf.  Douglas  evaluating have  equation  of  137 -  (7.19) - i s s a t i s f i e d  shown t h a t  polynomials,  to s o l u t i o n s  and Dupont  [  11 ]  Collocation  ittViCtV=  wi^,o) * u cb^)  Galerkin  parabolic  greater  is clear  that  (7.20)  - J-lt)W itv)  Thus we  y  ^ i , . . . ^  c a n be w r i t t e n  in  (0.3,a,b,c,d,e)  *S<©)= b  It  systems  accuracy  i s - i n * a? s/en-s-e; " w a s t e d " . scheme f o r  scheme,  O ( W¥)  at best  boundary  ) hence  ( wyOjti  e  produces  of fixed  t h e above- i n t e g r a l -  the f o l l o w i n g  a continuous  OC^M  to  as t h e  system  (7.  -  •sVt)^J C K ^ +  138  -  + i<usct> B)  0  J  'f%  e,  Bet)  1  3(o^ - b  where  \  A-  ft,  (6)  For  a collocation  b a s e d on C u b i c S p l i n e s  see Doedel  formulation [lO  ].  139  /V-  A:  »K  a; KM,  6"  -  140 -  A/- ;  AK  (51  >?, <,07l) K  e: To show t h a t solved  locally  The f o l l o w i n g  the system o f equations  i n t i m e we must argument,  In [ 12 of a large  differential  was d e m o n s t r a t e d  provide  class  equations.  show t h a t  (7.21) is  c a n be  invertible.  due t o D o u g l a s and D u p o n t £ 12  J it  schemes s u c h as ( 7 . 2 1 ) solutions  \  that  collocation  O t V\*) a p p r o x i m a t i o n s  of second order  non-linear  to parabol  -  demonstrates  such  -  this.  The p r o o f exists  141  a nontrivial  p r o c e e d s by c o n t r a d i c t i o n .  Suppose  there  vector  that  b -o.  Upon c o n s t r u c t i n g  the  (7.22)  piecewise "  (7.22) . i m p l i e s  .we-see ..that  points  on  that  >  ?(x)*0  I :  the  together la^o  piecewise with  this  we s e e t h a t Hence  o  n  vanishes  at  three  either  CX^I]  cubic  , or  2(^)€C C°/l] > i t ,  (7.23) i m p l i e s t h a t is  £ (x)  <o.  II:'  Since  (7.23)  , ^t^...^/.  3J(l)*o  L^A/ YA/-*»1 •  polynomial  i  2<>?J>-0  Since  cubic  i m p o s s i b l e and h e n c e  j?(x) = 0 II  is  clear on  holds.  that  [o,l].  I Since  -  Z(X*) Z'(*~)<o  From we h a v e  that  the piecewise  l"')  a  n  is false We r e m a r k  generalizing single  (7.21)  phase S t e f a n  ease of  (7.-21)  The d i f f i c u l t y of the  quadratic '  inductively  equations methods  and  h  e  n  c  that to  there  is l i t t l e  However  from the extreme  o f Douglas  f  o  l  l  o  w  that  s  t  h  a  -O  in t  *!(O£Yo)<0  Hence  the  difficulty  i n c l u d e more g e n e r a l  Moreover,  C1?'')  original  nonsingular.  simplest  i s a d i f f i c u I t -one. stiffness  t h e non-1 i n e a r i t y  and Dupont £ 12  in  one d i m e n s i o n a l  even f o r the  the convergence q u e s t i o n  (7.21).  i f c  .  problems.  arises  e  we o b t a i n  is  £  must v a n i s h  i s -a., con t r a d i t Ion s i n c e  assumption  ? (J?,"*'^  and  ^ f ' j  d  Continuing which  142 -  J are not  o f the system i s such  that  easily  applicable. We c o n c l u d e t h i s piecewise No d o u b t  Cubic Hermite  section with  functions  were  a w e a l t h o f systems s i m i l a r  by c h o o s i n g b a s e s c o n s t r u c t e d  the remark  that  the  chosen f o r c o n v e n i e n c e .  to  from o t h e r  (7.21) finite  c a n be  obtained  element  funct ions.  Numerical  Results H e r e we g i v e  numerical  results  f o r the C o l l o c a t i o n  scheme  (7.21) o n l y .  We  do  this since for parabolic  on f i x e d s p a t i a l domains, C o l l o c a t i o n and based on order points  and  schemes,  P i e c e w i s e Cubic Hermite P o l y n o m i a l s , have the same  o f convergence, provided '  Galerkin  Galerkin  equations  Si.  we  n c e  scheme pro videos  hence  we  c o l l o c a t e at the  have no  Gaussian  reason to b e l i e v e that "Uk^A-^'h  a., bj&tker. est,ima>t;e> fo.r ; 4  i t t ) , than does the C o l l o c a t i o n scheme, we  the l a t t e r on  the b a s i s of computational  the ,  adopt  ease.  C o m p u t a t i o n a l l y the C o l l o c a t i o n scheme (7.21) i s much easier  to  implement than the G a l e r k i n  former r e q u i r e s quadratures.  *  increases  s  t  solve  system  w h i l e that of the G a l e r k i n  r  Hence the  the  initial  its stiffness.  "o^"'^ as  N  u  than the system  .  (7.19) and  system  Matrices  (7.21) i s  computationally  (7.19). value  problem  (7.21), we  must  That i s , the c o n d i t i o n number of  , a r i s i n g i n the G a l e r k i n system (7.19), , hence we  i n the s o l u t i o n of for large  o  is s i x .  more e f f i c i e n t  the m a t r i x  l a t t e r requ-ires  J  ^V^^C^y^!  contend w i t h  where the  the  Moreover, the "bandwidth o f the C o l l o c a t i o n M a t r i c e s  ^&><&J  To  function evaluations  scheme (7.19), s i n c e  see  that  the time constants  present  (7.19) have r a d i c a l l y d i f f e r e n t magnitudes  The the  intimate  r e l a t i o n s h i p between the  C o l l o c a t i o n system (7.21) leads  Galerkin us  to  -  suspect  -  that the system (7.21) i s a l s o This  experiment. The  144  first  i s s u b s t a n t i a t e d by  We  employ two  stiff.  the f o l l o w i n g  numerical  numerical  procedures to s o l v e  (7.21).  i s an Adams-Basford-Moulton M u l t i s t e p P r e d i c t o r -  C o r r e c t o r Method  ( c f . Isaacson and  K e l l e r [ 21  ] p. 388)  while  the second i s a M u l t i s t e p P r e d i c t o r - C o r r e c t o r Method due Gear £ 18  J constructed  cases t e s t e d , we maintain  specifically  for s t i f f  to  systems.  In a l l  have found that the time step r e q u i r e d  a given accuracy  i n the s o l u t i o n of  l a r g e r f o r Gear's A l g o r i t h m .  (7.21) was  to much  Consequently Gear's A l g o r i t h m  was  •found to execute f i v e to ten times f a s t e r than the "A'dams'-B'asTord Moulton A l g o r i t h m . is  indeed  We  conclude from t h i s that the system  stiff. In what f o l l o w s we  where  {M>0  (7.21)  adopt the  i s the s o l u t i o n of  notation:  (0.3 ,a ,b ,c , d , e) , ( U > ? )  i s the  -  solution  of  (7.21)  v. (.*,t) =  M To  accurate solve  and i t  W ) t  is understood  (x,"D ' o  illustrate  approximations  that  together  with  to the s o l u t i o n o f  "f"l*,i.)>  ( 0 . 3 , a , b , c , d , e) , we  equation  the boundary c o n d i t i o n s our f i r s t  .  t h e C o l l o c a t i o n Scheme p r o v i d e s  o< x  cx/O + * " l x » t )  We o b t a i n and p i c k i n g  that  J< > *<.i,\  for  t h e inhomogeneous h e a t  ••"M^  145 -  example  W (i),  and  < 4tt>  \ >o  y  (7.24)  ( 0 . 3 , a ,b , c , d , e) . by s e t t i n g w (*) 0  b=) j  so t h a t  the s o l u t i o n  becomes  (  We do t h i s spacing  for  A =10,  and u s e  20 and 5 0 .  the approximation  (7.25)  In e a c h c a s e we t a k e to T = . 4 .  Table  uniform  7.0  provides  a summary o f t h e r e s u l t s .  To d e a l obvious  with  c h a n g e s a r e made t o  the inhomogeneity, (7.21).  V***"!) >  t  n  e  -  Table  7.0  Errors  i n l U x . i ) , "Uy  (Exact  A=10  Observed of  Observed  JCt)  (7.25)) e,(.4)  3  •30(-l)  .30 (-1)  5.2(0)  •58(-3)  5  .20(-2)  .95(-2)  3.5(0)  .80(-5)  7  • 55(-3)  . 4 4 (-2.)  2.3(0)  .60(-5)  9  .13(-3)  .14(-2)  1.4(0)  •12(-4)  Order  4.9  e ^ 4 l  1.2  2.7  3  .25(0)  .30(0)  6.5(0)  .40(-l)  5  •ll(-l)  .22 ( - 1 )  9.5(0)  .19(-4)  7  .24(-2)  .14(-1)  7.1(0)  .20(-5)  9  .80(-3)  .73(-2)  4.8(0)  • 25(-4)  Order 5.2  A=50  .90(0)  1.10(0)  18.3(0)  .15(0)  5  .25(0)  .38(0)  18.2(0)  .41(-1)  7  .16C-1)  •30(-l)  24.2(0)  .62(-3)  8  .94(-2)  •32(-l)  22.6(0)  •36(-3)  7 . 0 one c a n s e e t h a t  the large  the temperature  --  6.5  6.9  From T a b l e inspite  1.2  3.3  4  Observed Order of Convergence  of  Solution  and  e,(.4)  Convergence  obtained  (,x,-t), TAX«  N  Convergence  A=20  of  146, -  values  distribution  good a c c u r a c y  o f the s p a t i a l  near  x - o .  is  derivatives  - 147 -  Next we c o n s i d e r •^tx,*.); W i t )  t  u , lx>  chosen so t h a t  VUx/k)*  b - l » a^* l  (7.24) w i t h  cosx (  =  and  the s o l u t i o n becomes  x*-  (7.26)  Approximations to the, above solutio.n- are obtained- fo,-r. S^. =100, 500 and 10,000. We are i n t e r e s t e d  i n the accuracy o f the C o l l o c a t i o n  a p p r o x i m a t i o n when  \j(t)|  approximation u n t i l  a p p r o x i m a t e l y 20% o f the s l a b remains.  each case the ' p a r t i t i o n results  a r e summarized  Table 7.1  Errors  B=100  Observed Order of Convergence  i s l a r g e s t , hence we employ the  Tf"  i s taken to *be •"uniform.  In  The  by Table 7.1.  i n U C x . - t l ^ . t x . - t ) , u«,(x,t) and si-t) (Exact S o l u t i o n (7.26)) N  e (T)  3  .13(-3)  .80(-3)~  .11(0)  .96(-4)  5  .19(-4)  .ll(-3)  .43(-l)  .16(-4)  9  .40(-5)  .24(-4)  .14(-1)  .60(-5)  B  ^  e ( ) t  3 2  T  e (T) ?  1.9  e.,(T)  -  B=500  3  . 1 4 ( - 3)  • 1 4 ( - 2)  .12(0)  .97(-4)  4  . 3 9 ( - 4)  . 4 8 ( - 3)  •67(-l)  •23(-4)  5  . 1 3 ( - 4)  . 1 6 ( - 3)  •43(-l)  .80(-5)  6  . 1 8 ( - 4)  . 9 0 ( - 4)  •30(-l)  .80(-5)  Observed Order o f Convergence  3  . 1 7 ( - 3)  . 8 4 ( - 2)  .17(0)  • 80(-4)  4  , 4 6 ( - 4)  . 2 5 ( - 2)  •67(-l)  .22(-4)  5  . 2 5 ( - 4)  . 1 1 ( - 2)  .47(-l)  .10(-4)  6  . 1 6 ( - •4)  .45 (- 3)  .31(-1)  ..60.(-.5)  Observed Order o f Convergence  and  \*<.*,*)  during  the p e r i o d s Tables  of  time under  order,  surprising  however,  s i n c e the order  the asymptotic  estimates obtained  behaviour  f o r the order when f i n e  that  the  the r e s u l t s  large  Collocation c o n c e r n i n g the  and do n o t g i v e  of convergence.  of convergence o f the e r r o r .  This  is a Hence  77^  of [o,i] are  a good i s not  characterization accurate  of convergence can u s u a l l y  partitions  J(i.)  consideration.  indicate  order  f o r both  is relatively  of convergence are s c a t t e r e d  e s t i m a t e f o r the a c t u a l  of  good a p p r o x i m a t i o n s  although  7 . 0 and 7.1  Scheme i s o f h i g h order  shows t h a t  are obtained  2.4  4.2  3.4  7.1  2.0  4.1  4.6  B=10,000  Table  148 -  o n l y be  taken.  - 149 -  The  above r e s u l t s i n d i c a t e that the C o l l o c a t i o n  Scheme (7.21) can be used to o b t a i n a c c u r a t e the s o l u t i o n of (0.3,a,b,c,d ,e) .  approximations to  However, i t should be  emphasized that the s t i f f n e s s o f (7.21) makes i t a n u m e r i c a l l y inefficient  scheme.  (7.21) e f f i c i e n t l y ,  Hence u n t i l  a method i s devised  to s o l v e  the u t i l i t y o f t h i s scheme, i s i n dqu,bt.  CHAPTER  VIII  CONCLUSIONS  T h i s - t h e s - i s has; pr-esent.ed two- a l g o r i t h m s - f o r numerical  s o l u t i o n of We h a v e  the  Stefan Problem  seen that  the  us w i t h a r e a s o n a b l y e f f i c i e n t approximations for  the  to  the  practical  distribution promises  situation  Furthermore, can  be u s e d b o t h  error  of  the  time  is  increment  the  the  required  the  "rough" Moreover,  a smooth i n i t i a l  flux,  telnp'ter'a'tu're  the. S i m i l a r i t y  flux  generated  by  a c c u r a c y and to  We o b s e r v e t h a t  the  to  provides  Algorithm  algorithm.  Similarity  The l a r g e  required  obtaining  ( 0 . 3 , a ,b , c , d , e ) .  both  heat  improve  Algorithm  algorithm  estimate  if  a very  Algorithm  number o f  achieve this  the  terms  and  is  the  accurate not  the  accuracy results  an  small in  times.  The applying  to  algorithm.  computation  of  approximation.  approximation efficient  method o f  and a c o n s t a n t h e a t  t o be an e f f i c i e n t  ( 0 . 3 , a ,b , c , d , e) .  Similarity  s o l u t i o n of  the  Similarity Similarity  Algorithm is Method  to  the  direct  a system of  -'150-  result  of  differential  long  -  equations.  151  The above procedure  Method can be used  effectively  s o l u t i o n s of n o n - l i n e a r While  -  i l l u s t r a t e s how  the  Similarity  to o b t a i n approximate  numerical  problems.  the S i m i l a r i t y A l g o r i t h m g i v e s "rough"  accuracy,  the C o l l o c a t i o n scheme i s capable of a c h i e v i n g high a c c u r a c y . Although both, the. S.imilar.ity,. A l gpr;i£"hm> and .Lp.tltin-.'s . d i f f.er.ence scheme execute found  f a s t e r than the C o l l o c a t i o n scheme, we  t h a t f o r r e l a t i v e l y coarse p a r t i t i o n s ,  Tr"  have  , the C o l l o c a t i o n  scheme a c h i e v e s a c c u r a c i e s which the other schemes cannot " a t t a i n . We  have seen that  the apparent  of o r d i n a r y d i f f e r e n t i a l equations inefficient  performance  s t i f f n e s s o f the  (7.21") i s 'the caus'e'of th'e  of "the C o l l o c a t i o n scheme.  that the simple form of the n o n - l i n e a r i t y appearing will  We conjecture, i n (7.21)  a l l o w us to c o n s t r u c t a scheme which deals with the  ness of the equations We be used  system  i n an e f f e c t i v e  stiff-  way.  a l s o c o n j e c t u r e that t h i s C o l l o c a t i o n Method  can  to deal with S t e f a n Problems i n v o l v i n g both more g e n e r a l  boundary c o n d i t i o n s and more g e n e r a l governing differential  equations.  parabolic  BIBLIOGRAPHY  M.A.  B i o t , V a r i a t i o n a l P r i n c i p l e s In Heat T r a n s f e r , Oxford U n i v e r s i t y P r e s s , E l y House,London W.l, 1970.  G.W.  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Friedman, Free Boundary Problems f o r P a r a b o l i c Equations I . M e l t i n g o f S o l i d s , J . Math. Mech., V o l . 8, No. 4, pp. 499-517, 1959. W. Gear, Numerical I n i t i a l Value Problems i n O r d i n a r y D i f f e r e n t i a l Equations, P r e n t i c e - H a l l , Inc., Englewood C l i f f s , New J e r s e y , 1971. H e l l w i g , P a r t i a l D i f f e r e n t i a l Equations An I n t r o d u c t i o n , B l a i d e l l P u b l i s h i n g Co., New York, 1964. Huber, Hauptaufsatze liber das F o r t s c h r e i t e n der Schmelzgrenze i n einem L i n e a r e n L e i t e r , ZAMM, V o l . 19, pp. 1-21, 1939. Isaacson & H.B. K e l l e r , A n a l y s i s o f Numerical John Wiley & Sons, Inc., New York, 1966.  Methods,  - 154  H.G*  M.  -  Latidau, Heat Conduction i n a M e l t i n g S o l i d , A p p l . Math., V o l . 8, pp. 81-94, 1950.  Quart.  L o t k i n , The C a l c u l a t i o n o f Heat Flow i n M e l t i n g S o l i d s , Quart. J . A p p l . Math., V o l . 18, pp. 79-85, 1960.  J.C. Mason & I . F a r k a s , Continuous Methods f o r Free Boundary Problems, Proceedings IFIP Congress 1971, Numerical Math. S e c t i o n , pp. 61-65, L j u b l j a n a , Y u g o s l a v i a , 1971. V.G.  Melamed, Stefan's Problem Reduced to a System of O r d i n a r y D i f f e r e n t i a l E q u a t i o n s , I z v . Akad, Nauk SSSR Ser. G e o f i z . , pp. 848-869, 1958.  E.A.  M u l l e r & K. Matschat, Uber das A u f f i n d e n von Ahnlichkeitslosungen p a r t i e l l e r D i f f e r e n t i a l gleichungssysteme unter Benutzung von Transformationsgruppen, mit Anwendungen auf Probleme der Stromungsphysik, M i s z e l l a n e e n der Angewandfcen Mechanik, B e r l i n , pp. 190-221, 1962.  L. N i r e n b e r g , A Strong Maximum P r i n c i p l e f o r P a r a b o l i c E q u a t i o n s , Comm. Pure and A p p l i e d Math., V o l . 6, pp. 167-177, 1953. L.V.  Ovsjannikov, Gruppovye Svoystva D i f f e r e n t s i a l n y U r a v n e n i , N o v o s i b i r s k , 1962. (Group P r o p e r t i e s of D i f f e r e n t i a l E q u a t i o n s , t r a n s l a t e d by G.W. Bluman, 1967).  L.I.  R u b i n s t e i n , On the d e t e r m i n a t i o n of the p o s i t i o n of the boundary which separates two phases i n the oned i m e n s i o n a l problem of S t e f a n , Doklady Akad. Nauk SSSR, V o l . 58, pp. 217-220, 1947.  L.I.  R u b i n s t e i n , The S t e f a n Problem, AMS T r a n s l a t i o n s , V o l . 27, P r o v i d e n c e , Rhode I s l a n d , 1971.  R.W.  Sanders, T r a n s i e n t Heat Conduction i n a M e l t i n g F i n i t e S l a b : An Exact S o l u t i o n , ARS J . , V o l . 30, pp. 1030-1031, 1960.  J.  - 155 -  M.H.  S c h u l t z , S p l i n e A n a l y s i s , P r e n t i c e - H a l l , Inc Englewood C l i f f s , New J e r s e y , 1973.  P.V. S o l o v i e v , F o n c t i o n s de Green des Equations P a r a b o l i q u e s , Comptes Rendus (Doklady) de l'Academie des Science de l'URSS, V o l . 24, No pp. 107-109, 1939. '  APPENDIX A  Lemma 1.1  addition,  for  all  (Friedman Let  £>Cfc)  let  set)  [l?])• be continuous  s a t i s f y the f o l l o w i n g  t,,t €C°>°"3  and some constant  4  \  1  Lipschitz  M  .  S  Then d e f i n i n g  3C-4Ct)  K  1  Ct) KCx,ti 5<r),r)  we have -1  C>t X-S-SOW-O  /5  (t) ,  X  Proof. B e f o r e proceeding we make the f o l l o w i n g d e f i n i t i o n : r  I(*U  t  XVCT) t * ' - * » K C / t ^ c y ) , r ) dg ( Y  t  -  \ J \ r ) L l i i l l - i i l l l K (S(-l), t ; 5Ct),t> d r J , 2Ct-t)  -  156  -  In  condition  C K ( X , t : S(T),T^J d T  /MT)  r  Co, ?! .  on the i n t e r v a l  -  where  ^  show  that  157 -  % e (o,t).  i s any c o n t i n u o u s f u n c t i o n and  Now i f  we c a n  (A-l) r  Ji—.  | Kp^  + L  + -L/>Ok)| <•  as  %->o .  where. i  _  ^P(t)  (x-j(y))K(x,t-.iWl^df  UW.-3»-tt Hit-*)  J  o  then to  the statement  zero.  Further  K  ( J i t ) , tj5C*VT)d2•  o f t h e lemma f o l l o w s note  that  by a l l o w i n g  < B  to  tend  since  (A-2)  (A-l)  follows  if  we c a n show  x->stt)-o  that  2  %->o  I IC«)| < O U )  To s e e  this  we f o r m t h e e x p r e s s i o n  p < * ) i p C i v - (pti^-^o <r>)  (A-3)  -  and s u b s t i t u t e  it  into  To e s t a b l i s h  1(0  = I,  +  (A-l),  (A-3)  158  -  to  we  obtain  write  I.  where  and c o n s i d e r le  , - .(g..." M .5_J  P»l  S i n c e -SOf)  c  for  *  all  is  real,  ^  To e s t i m a t e  L i p s c h i t z c o n t i n u o u s and we  obtain  (A-4)  I  (  define  -  and  159  -  consider  (A-5)  *-*  {lx -5tr»*-{x - 4 «»*j /V (t-  If we  take  and  "M"  we one  see that the e x p r e s s i o n since  <:  Then u s i n g the  K\ IX.-.SC-OI 2  +  M  1  &  & $  ]  inequalities  I t - <c~*l <  and  i n the e x p o n e n t i a l can be bounded by  3 ^  for  \£) $ I  160  Z  ¥  (A-5) becomes-  IT,-1,1* [S fA S * ^ . 8 To e v a l u a t e  ^T,  n''*.  x-> , s t V > - o ,  as  (A-6)  we l e t  Then  where  $> "  /(x-SCtn •  From (A-7) we conclude that  cj  To complete imply  T, =  - -jj •  the proof of (A-.3) we note that  ( _ ) A  (A-6, 7, 8)  8  - 161 -  (A-9)  Hence w r i t i n g  1 -  1 I, + i i I  we o b t a i n  11(01$ ft M 5 * ' V  2 Tr"*  +. JL  .2  . .  Thus  x->Jtt>-o  Ti*a  and the proof i s complete.  X->JC«-e>  (A-10)  APPENDIX B  Proposition^^ If  with  1.1  The (weak) maximum  vu*,-t)  principle.  satisfies  .St/O a p o s i t i v e continuous f u n c t i o n and /UCx.t) 6 C C - B )  -M^U.t), V U,t) ^ C ( «£> U  t  where  i s the c l o s u r e of  S  T  then  t  wU,t)  boundary  - I <*,T)  (g^)  and.  : O < X < 6 C.T)|  a t t a i n s i t s maximum and minimum v a l u e on the data - iQy  .  (See F i g . 1.0 Chapter I)  Proof!" For  any  € > o define  v<*,-t)= v. Ix,-U - •£ t •  vttXj-fc)  satisfies  vt*,t)  assumes i t s maximum v a l u e at a p o i n t  (  1  )  <> 2  the hypotheses o f p r o p o s i t i o n  See H e l l w i g F o r more  [  19 ]  general  -  P-  -  1.1. I f  Cx.,i,> e °& U  @  T  47.  results  162  where  see L . N i r e n b e r g  [  27  ].  -  then  v  (»,t)  x<  -  163  is  defined  h  and  and c o n t i n u o u s on  5  C  ^ 'Sj  where  for  some p o s i t i v e  and  hence  and  hence  (.x, .jt.Y 5,,- € . V^Cx.-t)  cannot  is  attain  its  continuity  the  all  C*, ,!,')»< o of  existence of I s  vt^l-*, !') -  a  <t,-fc,i,7  maximum v a l u e  on  S € CO,K) Hence  ^  that  e do©~<g>  r  Now s u p p o s e  odU*S  i.e.  By t h e  for  £  vtx.t)  Furthermore,  we e s t a b l i s h  *.x,,\>$ - /z  such t h a t  and  K«  l  t>*,t<,V  such  The maximum v a l u e  i s not  JJd'&j  at  v<C.x 't)  T  %  Then  attains  its  maximum v a l u e  that  attained  on t h e  data  boundary  on  -  164  -  i Thus  the  maximum v a l u e  of  v(*,i)  is  attained  at  Cx,,!,), - t , < i . „  However,  in  particular *w.C*,,t,) >, - u C X o , ! ^  -«r(te-l,)  and h e n c e  £>o  for  but  otherwise  and ' a l l o w i n g  f>©  which  is  c o n c l u s i o n of  Proposition  tend  to  that  t|<"t  0  for  z e r o We 'have  b,  the the  f o r e g o i n g argument  to  -u()r,-fc)  we  obtain  proposition.  1.2.  If (0.3,a,  to  Noting  a contradiction. Applying  the  f  arbitrary.  c,  (v(.,,s) d,  e)  is then  a s o l u t i o n of vt cscu,t^ x  the  system of  equations  >o  Proof.  (3)  F o r more g e n e r a l  results  see Friedman  [l6J  - 165 -  If exists some  X  f o r some 0  & .»  6 C Now  s  "t , -w (stt>,ii < o u  c  h  that  note t h a t  if  v\(x  e )  there  that  is  * >o  then the Maximum  c  exists a  h >o  for  Principle  "tv(S>) $ t!  such  (0.3b, d)  implies  that  v^(o,t(6))- &• > © • To  and  there  $  ( P r o p o s i t i o n 1.1), together with c o n d i t i o n s that  i)^  (0.3b)  define  i€D».T] t  and  then by  x  show that t h i s  -u. ( o , "t (t)) - £  note that  i s impossible,  define  f o r any  i s the maximum v a l u e o f  £^£0,%]  -vi.(.* i.) l  on S  Then s i n c e  £  -  jCx.t): Of x j 5 C U , o j - t i i ( t l  ( o + , •fc') - M  X  U ^ o + , t t n ) c M (ov.Ho) { O x  o  we conclude  for a l l  that  ^c-Co.J,! ,  Thus  - 166 -  that  is  "U. (o, t(.b)) .< O  which  is  a contradiction. (sct»,ti z,o  and t h u s  v\  arguments  we c a n show  x  (Note  Hence the that  u ( o , i ) *o  original  by e x t e n d i n g for  all  assumption i s the  above  L>A3  ).  false  APPENDIX C  To show that as (1.1)  write  (1.1)  «f-?o  becomes ( 1 . 2 ) ,  we  as  where  V  =. ^ V v ( ^ v-£) < 5 ( x t +  t  (  (  ;?)  t - f ) d£;  (c  o  Then c o n s i d e r i n g  V,, v  ,Vj  a  s e p a r a t e l y we take the l i m i t as  £-> o . Since at  T=t  ,  G (x, t •  has an i n t e g r a b l e  +  we see immediately o f V  3  r  that  - VV^SC^t) o  To f i n d  efCwv V,  we w r i t e  -  167  singularity  -  G ^ x . t ; SC*),*)  .  (C  - 168 -  where  o  .sen V," - ^ j > < ^ ) -"U.tO] G*0,t s-,c) d£ , :  V/ : - ^  G\x,t;^,o)  M  From w h i c h we  .  have  G^x.tjf.o) j J ( f ) .  Since we s e e  for  fixed  is  G (x, +  ^ , 7:)  is continuous  at  r= O  that  3l £->0 It  t.>o,  easy  to  see  v.'-O .  that  (C-3)  -  169  iv,"l 5 -*"P  1 "K(£,d>  ~-U (£)\ 0  hence  V/' r O ,  Caf  Finally  for  fixed  where  -S I f ) $ * £ b  (C-3,  V,'"  4,  c a n be w r i t t e n  from which  V,"' - o  cj w v ^  Combining  i > o,  (C-4)  5)  we  €->©  it  as  c a n be e a s i l y  seen  that  .  (C-5)  have  (C-6)  o  Now f o r  V,  we f i x  S f (o,x) n ( o , i t t - n - a n d  wr i t e  >  as  £  tends  to  zero  the  first  and l a s t  integral,  as w e l l  as  the  -  K ( x j ' t j - f -t-£) contribute  170 -  part of the second  integral  do not.  since  -3T as  £->©  .  Hence we a r e l e f t  ^ -u.C;5;i-.£) l<  C  *  with  .  ,  .  x-* V-  A f t e r making the change o f v a r i a b l e  Villi) i n  (C-7), we  consider  f  V  " V  c / v i — \ \ \ e" u(.x-at' 'v i-cUlvI '«->© L TT '* ) -J V  /  >  Since write  i s continuous  A*Cx,"k)  in  - v<.W,i) .  and hence  oQ  £  , we can  (A-8) as  f o r some  x  (.*-$>,x+  Letting  S  tend  Combining  (C-2, 6, 9) we see that as  to z e r o , we concluc  £->o  (1.1) becomes  4 ^ vVl-S«t>.t) G ^ t X j t - . - K * ) , * ) At  APPENDIX D  Lemma 1.4  (The equivalence  of the d i f f e r e n t i a l  and i n t e g r a l  systems). •If  *u(.t) i s a s o l u t i o n of (1.4), where  by  (1.5).., then  by  (1.5)) forms a s o l u t i o n o f (0.3,a,b,c,d , e ) .  <x*,s)  C/M,S> (  )  JC-tvis given  d e f i n e d by (1.2) and -sc-f)  i s a s o l u t i o n o f (0. 3 , a,b , c , d , e) then  defined  Conversely i f v>(t^ = V (•sci^t) K  is a s o l u t i o n of (1.4).  Proof. By  c o n s t r u c t i o n any s o l u t i o n  defines a s o l u t i o n  V -  Conversely then i n t h e r e g i o n  u„i-sit\t)  i f viC-tV oQ  ("u.s) of (0. 3 , a ,b , c , d, e)  of (1.4).  i s a s o l u t i o n o f (1.4), (1.5)  the i n t e g r a l s o f (1.2) a r e r e g u l a r .  they can be d i f f e r e n t i a t e d d i r e c t l y to show that by  (1.2)) s a t i s f i e s  G (.x,tj +  T}  (0.3) i n <=&  .  v a ^ t ) (defined  Furthermore, s i n c e  i s an even f u n c t i o n of x. we see t h a t c o n d i t i o n  (0.3c) i s s a t i s f i e d  and d i f f e r e n t i a t i n g  (0.3e) h o l d s . ;  Hence  Evaluating  c  b  - 171 -  (1.5) we have that c o n d i t i o n  - 172 -  shows that c o n d i t i o n condition (defined  (0.3d) i s s a t i s f i e d .  To demonstrate  (0.3b) , holds , we note that we have shown that by (1.2)) s a t i s f i e s  (0.3,c,d,e) on JQ  i n t e g r a t e Green's I d e n t i t y w i t h wU,t) ( d e f i n e d and €->© are  use c o n d i t ions and s u b t r a c t i n g  left  (0.3,c,d,e). (1.2)  .  that •u.c.x.'O  Hence we can  by (1.2)) over  A f t e r taking  the l i m i t as  from the r e s u l t i n g e x p r e s s i o n  we  with  O  6  -  \  G^- (x, tyJt*),T)d»-  o  using  the r e l a t i o n  O-  Of*  t  xY~ - G Cx;t; x  w.e have.  2- \ v<C5«),t) G " ( x,t; 3tt»,T)cl2) o  dx  (D-l)  + ^ •uC.Strj.t)  Letting  x->JCt>-o  ic«r)  G Cx,t3 5ltj,r) d >  and u s i n g  4  Lemma 1.1 (D-l) becomes  O - vt-C-SCt^t) + X } u ( s t r \ , t )  o  Since  [G'CSCDjt^itt),^)  s$ Ct) i s L i p s c h i t z c o n t i n u o u s , from (D-2)  Ivastt),-fc)|  satisfies  the i n e q u a l i t y  (D-2)  we deduce that  -  VM\  173  -  $ • ^ l-ul * ' d* o  where  ^"  inequality  has at most an i n t e g r a b l e s i n g u l a r i t y . o f the G r o n w a l l t y p e  u. (ic-t^-fc) u.o  ( 1 )  where  we conclude that  and thus the lemma i s proven.  If  °  then  Using an  APPENDIX E  THE  F I X E D BOUNDARY  We wish to s o l v e  SOLUTION  the f o l l o w i n g  x « Co.il  system of equations.  1 >o  o  To o b t a i n where  a s o l u t i o n of ( E - l ) we l e t  wlx,Vi  satisfies  W 0,"k^Ox  V >o  - 174 -  v.(< t)= w < *,t W v (x, (  -  and  v(*,t)  175  -  satisfies  V  (o,t^ o  x  The  W  \ .  ^ >o  s o l u t i o n of  (E-2)  can  be w r i t t e n  (E-3)  as  tx  where  (E-4) 4-<5  To t r a n s f o r m of  construct the  the  equations  s o l u t i o n of (E-3)  =  we  take the  Laplace  obtain  (E-5)  -U„ (o,p) * e • ^ x 0,p)  and  (E-3)  HCp)  -  176  -  where  The s o l u t i o n  *U(*,p)-  of  H (p) 4p  and h e n c e  the  solution  vff,tv-  (Er5)  -i-  c a n be w r i t t e n - as  Co<V\Jp_x.  5w>.\\Tp  of  (E-3)  is  \  M ( pVP  P  given  t  by  (E-6)  c P - S ^ ^ P K), d p  V-toa  Finding  the  inverse  by e x p a n d i n g can  the  L a p l a c e Transform of  same f o r  b e e x p r e s s e d as  the  large  cas^n)  we s e e t h a t  p  convolution  - £  \  )  V(*,t)  integral  || ( e " °  + W /  e  *  ( t  -»  +  e  -  ( , + a w +  * * c . » ) ) | 6\ f c  (E-7)  It the  is  easy  domain  v^c^so  to  see  that  (E-7)  | U A V . x t (t>,A , t >t)|  satisfies as w e l l  and ••v„('b,t)-0 .  the as  heat the  equation  on  conditions  To s e e t h a t  v„ U,t)= V\(fc)  -  we  note  177  -  that  *:t"«  For  fixed  that  (\-x)  f> C V -x > / c  we  f o r any  can  take  \l-xl  prescribed  £>o.  small  enough so  Hence  xJt"«  •n*k  where  € >'« , S> > o  a  "d  x->i  *f*i  1  M l W m - K U ) | U G > V ,I  a c o n t i n u i t y p o i n t of  Y\(i)  we  see  that  .  I f *t  -  Hence the s o l u t i o n as  o  (the  small time s o l u t i o n ) .  178  -  u.(«,t)  of ( E - l ) can be  written  APPENDIX F  THE ASYMPTOTIC EVALUATION OF  FOR SMALL t To obtain an asymptotic approximation of +1)^+ t ) x  w  e  d i f f e r e n t i a t e (3.14) with respect to  and evaluate v'„(x,t)  = ^  'W'CJt.At;)  G„(«,t ;> ic  el?  where  t)  GO (F-l)  -e  asymptotically f o r small  -t  at  - 179 -  x*  -^-t.  -  We h a v e ,  H e n c e we must  Upon m a k i n g  obtain  the  Substituting  up t o  the  180  -  exponential  an a s y m p t o t i c  substitution  terms,  expression  , J £ (•" C  ( ~ ) F  expression  •f o  into  (F-3)  and u s i n g W a t s o n ' s  Lemma we  have  2  for  becomes  /  - 181  3¥  and hence  +  OCO.  

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