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

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NUMERICAL ALGORITHMS FOR THE SOLUTION OF A SINGLE PHASE ONE-DIMENSIONAL STEFAN PROBLEM by F a u s t o M i l i n a z z o B . S c , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the I n s t i t u t e o f A p p l i e d Mathematics rtrvd S t a t i s t i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH A p r i l , 1974 COLUMBIA In presenting th is thesis in par t ia l fu l f i lment of the requirements f o r an advanced degree at the Univers i ty of B r i t i s h Columbia, I a g r e e t h a t the L ibrary sha l l make it f ree ly ava i lab le for reference a n d study I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department n r by h is representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of Aymlipr) I" a -h>iPTr);T+. -j n « rmrl fit.gtistins The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date June I> . ~i 97/ ABSTRACT . A o n e - d i m e n s i o n a l , s i n g l e phase S t e f a n Prob lem i s c o n s i d e r e d . T h i s p rob lem i s shown to have a u n i q u e s o l u t i o n which depends c o n t i n u o u s l y on the boundary d a t a . In a d d i t i o n two a l g o r i t h m s a re f o r m u l a t e d f o r i t s approx imate n u m e r i c a l s o l u t i o n . ' . The f i r s t a l g o r i t h m ( the S i m i l a r i t y A l g o r i t h m ) , which i s based on S i m i l a r i t y , i s shown to c o n v e r g e w i t h o r d e r o f c o n v e r g e n c e between one h a l f and o n e . M o r e o v e r , n u m e r i c a l examples i l l u s t r a t i n g v a r i o u s a s p e c t s o f t h i s a l g o r i t h m a re p r e s e n t e d . In p a r t i c u l a r , m o d i f i c a t i o n s to the a l g o r i t h m which a r e s u g g e s t e d by the p r o o f o f c o n v e r g e n c e a r e shown to improve the n u m e r i c a l r e s u l t s s i g n i f i c a n t l y . F u r t h e r m o r e , a b r i e f c o m p a r i s o n i s made between the a l g o r i t h m and a w e l l - k n o w n d i f f e r e n c e scheme. The second a l g o r i t h m (a C o l l o c a t i o n Scheme) r e s u l t s from an a t t e m p t to r e d u c e the problem to a 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 e q u a t i o n s . I t i s o b s e r v e d tha t t h i s 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 e q u a t i o n s i s s t i f f . M o r e o v e r , n u m e r i c a l examples i n d i c a t e t h a t t h i s i s a h i g h o r d e r scheme c a p a b l e o f a c h i e v i n g v e r y a c c u r a t e a p p r o x i m a t i o n s . I t i s o b s e r v e d t h a t th ( i i ) a p p a r e n t s t i f f n e s s o f the system o f 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 r e n d e r s t h i s second a l g o r i t h m r e l a t i v e l y i n e f f i c i e n t . TABLE- OF CONTENTS ABSTRACT TABLE OF CONTENTS L I S T OF TABLES L IST OF FIGURES ACKNOWLEDGMENTS < . INTRODUCTION CHAPTER I The P r o b l e m E x i s t e n c e and Un iqueness o f the S o l u t i o n The N u m e r i c a l S t e f a n Prob lem N o n - d i m e n s i o n a l i z i n g EXISTENCE AND UNIQUENESS The I n t e g r a l E q u a t i o n s E x i s t e n c e and Un iqueness o f the S o l u t i o n f o r S m a l l Time CHAPTER I I THE SIMILARITY METHOD L i e Group o f T r a n s f o r m a t i o n s I n v a r i a n c e ( i ) ( i i i ) ( v i ) ( v i i ) ( v i i i ) 1 2 4 5 7 11 11 18 E x i s t e n c e and Un iqueness of the S o l u t i o n f o r A l l t 6 O i T l 29 32 33 35 The Most G e n e r a l m-Parameter L i e Group o f T r a n s f o r m a t i o n s L e a v i n g I n v a r i a n t (2 .1 ) 37 Reduc ing the Number o f V a r i a b l e s 42 T h e , C l a s s i c a l Group o f the Heat E q u a t i o n 44 ( i i i ) ( i v ) CHAPTER I I I A USEFUL SIMILARITY SOLUTION OF THE HEAT EQUATION FOR.THE STEFAN PROBLEM 46 CHAPTER IV THE SIMILARITY ALGORITHM 60 CHAPTER V THE CONVERGENCE OF THE SIMILARITY ALGORITHM 65 C o n t i n u o u s Dependence f o r S m a l l Time 65 C o n t i n u o u s Dependence f o r A l l Time 80 Convergence o f the S i m i l a r i t y A l g o r i t h m 82 Order o f Convergence 83 CHAPTER VI THE SIMILARITY ALGORITHM - NUMERICAL RESULTS 86 : N u m e r i c a l Examples 86 O p t i m i z a t i o n o f the S i m i l a r i t y A l g o r i t h m 95 O r d e r o f Convergence o f the S i m i l a r i t y - A l g o r i t h m 98 The S m a l l Time V e r s u s the L a r g e Time R e p r e s e n t a t i o n o f the S i m i l a r i t y S o l u t i o n 102 Compar ison o f 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 112 CHAPTER VI I A COLLOCATION SCHEME 115 The L a g r a n g i a n E q u a t i o n s f o r Heat C o n d u c t i o n 115 A G a l e r k i n Scheme 127 A C o l l o c a t i o n Scheme 135 N u m e r i c a l R e s u l t s 142 CHAPTER V I I I CONCLUSIONS 150 BIBLIOGRAPHY 152 APPENDIX A . APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F (v) 156 162 167 171 174 179 LIST OF TABLES TABLE 6.0 Errors Versus Time Increment (Boundary Data (6.0)) TABLE 6.1 Error Versus Time Increment Ai. Using Modifications I and I I TABLE 6.2 Observed Order of Convergence TABLE 6.3 Approximate Operation Count TABLE 6.4 S i m i l a r i t y Algorithm Versus Lotkin's Difference Scheme TABLE 7.0 Errors in UU,t),W^' 1t\M l w(« 1t)and JCtV (Exact Solution (7.25)) TABLE 7.1 Errors in v*w,t>, u. and J t t ) (Exact Solution (7.26)) 88 97 101 109 1.13 146 147 (vi) LIST OF FIGURES FIGURE 0.1 FIGURE 1.0 FIGURE 4 .0 FIGURE 6 .0 FIGURE! 6.1 FIGURE 6 .2 FIGURE 6 .3 FIGURE 6 . 4 A M e l t i n g S l a b 3 The (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 13 The S i m i l a r i t y A l g o r i t h m 62 Approx imate Temperature D i s t r i b u t i o n s a t T =.40 f o r the Boundary Da ta (6 .0) 90 A p p r o x i m a t i o n s to the P o s i t i o n o f the Boundary >5C-i)up to T=.4 f o r the Boundary 91 Data (6 .0 ) Comparing H(10 and Vi("t) f o r the Boundary Data (6 .0 ) 92 The Approx imate Temperature D i s t r i b u t i o n f o r "t Between 0 and .1 f o r the Boundary Data (6 .1) 94 Comparing the G i v e n Heat F l u x w i t h t h a t Genera ted by the S i m i l a r i t y A l g o r i t h m U s i n g M o d i f i c a t i o n I I 97 (vii) ACKNOWLEDGMENTS I am v e r y g r a t e f u l to my s u p e r v i s o r , Dr . George B luman , f o r h a v i n g s u g g e s t e d the t o p i c and f o r h i s . p a t i e n t , e n c o u r a g i n g g u i d a n c e d u r i n g the c o u r s e o f the work . I am g r a t e f u l to D r . James Varah f o r many hours o f h e l p f u l and u s e f u l d i s c u s s i o n s . I thank the N a t i o n a l Research C o u n c i l o f Canada and the U n i v e r s i t y o f B r i t i s h Co lumbia Mathemat ics Department f o r t h e i r f i n a n c i a l a s s i s t a n c e . I a l s o e x p r e s s my a p p r e c i a t i o n to my w i f e , B e v e r l y , f o r c a r e f u l l y t y p i n g the m a n u s c r i p t . F i n a l l y , I thank my f r i e n d s i n the I n s t i t u t e o f A p p l i e d Mathemat ics and S t a t i s t i c s , p a r t i c u l a r l y H a r t Katz and E u s e b i u s D o e d e l ; f o r each one has c o n t r i b u t e d s i g n i f i c a n t l y to whatever I have a c h i e v e d . ( v i i i ) INTRODUCTION T h i s T h e s i s i s c o n c e r n e d w i t h the d i f f u s i o n o f heat th rough a medium which i s e x p e r i e n c i n g a change o f p h a s e . C h a r a c t e r i s t i c a l l y such problems i n v o l v e a "mov ing" s u r f a c e made up o f p o i n t s at w h i c h . o n e phase changes to a n o t h e r . I f the p o s i t i o n o f the s u r f a c e i s g i v e n as a f u n c t i o n o f t i m e , the p r o b l e m , known as the I n v e r s e S t e f a n P r o b l e m , r e d u c e s e s s e n t i a l l y to one o f s o l v i n g a p a r a b o l i c d i f f e r e n t i a l e q u a t i o n w i t h a s s o c i a t e d boundary c o n d i t i o n s on an i r r e g u l a r d o m a i n . E v i d e n t l y i f the d i f f e r e n t i a l system i s l i n e a r then so i s the I n v e r s e S t e f a n P r o b l e m . However , when the p o s i t i o n o f t h i s s u r f a c e i s no t g i v e n a p r i o r i , the p r o b l e m , r e f e r r e d to as a D i r e c t S t e f a n o r F r e e -Boundary P r o b l e m , , becomes one o f f i n d i n g s i m u l t a n e o u s l y the t e m p e r a t u r e d i s t r i b u t i o n o f the medium and the p o s i t i o n o f the "moving" s u r f a c e . As can be seen r e a d i l y the D i r e c t S t e f a n Prob lem i s n o n - l i n e a r . A l t h o u g h F r e e Boundary Problems da te back to a work o f « G . Lame and B . P . C l a p e y r o n p u b l i s h e d i n 1831 and to s e v e r a l - 1 - - 2 - papers o f J . S t e f a n wh ich appeared i n 1889, not u n t i l the n i n e t e e n t h i r t i e s d i d work on such problems b e g i n i n e a r n e s t . D u r i n g the p a s t twenty y e a r s a c o n s i d e r a b l e amount has been p u b l i s h e d document ing the a n a l y t i c p r o p e r t i e s o f o n e - d i m e n s i o n a l S t e f a n P r o b l e m s . In a d d i t i o n , a number o f schemes have been d e v e l o p e d f o r t h e i r n u m e r i c a l s o l u t i o n . The Prob lem We c o n s i d e r a p a r t i c u l a r o n e - d i m e n s i o n a l s i n g l e phase S t e f a n P r o b l e m . More p r e c i s e l y , we w i s h to d e s c r i b e the m e l t i n g o f a homogeneous s l a b , wh ich i n i t i a l l y o c c u p i e s the space between ^= 9 and ^ * £ Q , and whose i n i t i a l t empera tu re d i s t r i b u t i o n i s 3© ^ ) • W e assume t h a t the t empera tu re d i s t r i b u t i o n , ^\ytZ) > obeys the heat e q u a t i o n i n t e r i o r to the s l a b f o r T > o . F u r t h e r m o r e , we assume the s l a b to be i n s u l a t e d a t o , w h i l e a t > the p o s i t i o n o f the " m o v i n g " boundary a t t ime 2" > a h e a t f l u x H 0 ( f ) causes an i s o t h e r m a l phase c h a n g e . By h a v i n g the mel t removed immed ia te ly upon f o r m a t i o n , we r e s t r i c t our a t t e n t i o n to the s o l i d phase o n l y . The f o l l o w i n g e q u a t i o n s govern the tempera tu re d i s t r i b u t i o n i n s u c h a s l a b . - 3 (1) J-Crf Cr;, r) J 7-^ JyiOyZ) - O r e c o u p ) , r t t o , T F ) , W»(r> (0 .1 ) r « C o . T p ) , ( 0 . 1a ) (0 .1b ) ( 0 . 1 c ) ( O . l d ) ( O . l e ) F i g . 0.1 A M e l t i n g S l a b ( 1 ) Here X T iV.r) , J>, ( y , * ( y , * ) denote p a r t i a l d e r i v a t i v e s o f JT(y;r ) a n d J ' t i d_ ['jS'tJ:)] - 4 - H e r e , - the m e l t i n g t e m p e r a t u r e , c " t n e s p e c i f i c h e a t , p - the d e n s i t y , H - the c o n d u c t i v i t y , and .\» - t h e l a t e n t h e a t o f f u s i o n a r e c h a r a c t e r i s t i c o f the m a t e r i a l and a re t a k e n to be c o n s t a n t . M o r e o v e r , we r e s t r i c t our a t t e n t i o n to the c a s e X * 3-**. , H „ ( T ) f c 6 . A l t h o u g h the g e n e r a l problem ( 0 . 1 , a , b , c , d , e ) can be d e a l t w i t h n u m e r i c a l l y , th roughout the a n a l y s i s we w i l l assume t h a t t h e h e a t f l u x , V-i 6 (r) » i- s s u f f i c i e n t to m a i n t a i n m e l t i n g , i . e . .J'Gs'Or), ~) » t n e t empera ture o f the s l a b at the m e l t i n g boundary i s n e v e r a l l o w e d to f a l l below vTwv ~ the m e l t i n g t e m p e r a t u r e . Whether f o r a r b i t r a r y J* 0 and M 6 t h i s can be g u a r a n t e e d a p r i o r i w i l l be d i s c u s s e d b r i e f l y a t a l a t e r t i m e . Under the above a s s u m p t i o n , c o n d i t i o n ( O . l e ) becomes the S t e f a n o r F r e e Boundary c o n d i t i o n and de te rmines the a l l o c a t i o n o f a v a i l a b l e energy to the d i f f u s i o n and the m e l t i n g p r o c e s s e s . The comple te s o l u t i o n o f ( 0 . 1 , a , b , c , d , e ) i s then the p a i r o f f u n c t i o n s ( T ^ Z ) , »SHr>) , E x i s t e n c e and Un iqueness o f the S o l u t i o n The e x i s t e n c e and u n i q u e n e s s o f s o l u t i o n s to S t e f a n Prob lems a r e e s t a b l i s h e d by one o f s e v e r a l methods . U s u a l l y the s o l u t i o n i s e x p r e s s e d i n terms o f a s e t o f c o u p l e d V o l t e r r a - 5 - • . Integral Equations, then the proof proceeds by either using the Maximum P r i n c i p l e or a fixed point argument. Cannon and Denson H i l l £ 7 ]J use the Strong Maximum P r i n c i p l e together with a retarded argument approach to es t a b l i s h the existence and uniqueness of the solution of the problem they consider. Priedman £ 17 3» refining the work of Rubinstein |~ 29 ]],' treats the same problem using a fixed point argument. ' Using methods as outlined i n Friedman f 17 1 we es t a b l i s h the existence, uniqueness (Chapter I) and continuous dependence (Chapter V) on the boundary data y H» (r^ J"0 (*)| of the solution to the system of equations (0.1,a,b,c,d,e). The Convergence of the S i m i l a r i t y Algorithm (see Chapter IV) then follows from the continuous dependence of (0 .1 ,a,b ,c ,d,e) on Vl0l^» The Numerical Stefan Problem Usually numerical schemes dealing with Free Boundary Problems are p a r t i c u l a r to the boundary conditions being considered. For instance, the f i n i t e difference scheme developed by Douglas and G a l l i e £ 11 ] uses two boundary conditions to es t a b l i s h an i t e r a t i o n which at each step i n time locates the po s i t i o n of the "moving" boundary. S i m i l a r l y the continuous methods of Mason and Farkas £ 24 J r e l y on the appearance of ^{X) twice in the system of equations so that again an i t e r a t i o n to the solution can be - 6 - established. Several authors, following the lead of Landau [ 22 ], make the transformation X s ¥/jS'ltr) , then construct approximating schemes for the r e s u l t i n g system on the fixed space i n t e r v a l £o,lJ. For example Lotkin £ 23 ~\ uses this transformation to obtain a f i n i t e difference approximation for (0.1 ,a ,b ,c ,d , e) . One a l t e r n a t i v e to difference schemes on the fixed space i n t e r v a l [o,lJ has been to reduce the Stefan Problem to a countable set of ordinary d i f f e r e n t i a l equations. This was f i r s t achieved by Melamed £ 25 J by expressing the temperature d i s t r i b u t i o n , "J(X#<r) as an appropriate Fourier Series with time dependent c o e f f i c i e n t s . The system (0.1, a ,b,,c ,d , e) then yields a set of ordinary d i f f e r e n t i a l equations for the Fourier Coe f f i c i e n t s and the po s i t i o n of the boundary. We propose several schemes. The f i r s t scheme, referred to as the S i m i l a r i t y Algorithm (Chapter IV) is based on an exact s o l u t i o n of the Inverse Stefan Problem obtained through the S i m i l a r i t y Method (Chapter I I ) . That i s , solutions of the Inverse Stefan Problem are pieced together in such a way as to given an approximate solution of the Direct Stefan Problem. In Chapter VI we give numerical examples i l l u s t r a t i n g the S i m i l a r i t y Algorithm. ~ - 7 - The second and t h i r d schemes (Chapte r V I I ) a r e c l o s e l y r e l a t e d , and a r i s e f rom a t tempts to reduce the D i r e c t S t e f a n Problem ( 0 . 1 , a , b , c , d , e ) to a c o u n t a b l e system o f 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 . Hence they can be l o o k e d upon as e x t e n s i o n s o f the method o f Melamed. However , i n s t e a d o f t a k i n g as a b a s i s , f u n c t i o n s which a re g l o b a l on £ o , l ] ( s u c h as the T r i g o n o m e t r i c f u n c t i o n s ) we adopt a f i n i t e e lement a p p r o a c h . That i s , we approx imate vT(v'T^ by a f i n i t e l i n e a r c o m b i n a t i o n (wi th t ime dependent c o e f f i c i e n t s ) o f f u n c t i o n s which have s u p p o r t i n a s u b i n t e r v a l ( f i n i t e e lement ) o f £ o , l ] . A system o f d i f f e r e n t i a l e q u a t i o n s f o r the c o e f f i c i e n t s and rt'{Z) can be o b t a i n e d i n s e v e r a l ways by u s i n g e q u a t i o n s (0.1,e). We w i l l d e r i v e two systems o f e q u a t i o n s . The f i r s t i s known as a c o n t i n u o u s G a l e r k i n sys tem w h i l e the second i s r e f e r r e d to as a C o l l o c a t i o n s y s t e m . Some n u m e r i c a l r e s u l t s a r e g i v e n . N o n - d i m e n s i o n a l i z i n g B e f o r e p r o c e e d i n g f u r t h e r we n o n - d i m e n s i o n a l i z e the system o f e q u a t i o n s ( 0 . 1 , a , b , c , d , e ) by i n t r o d u c i n g the f o l l o w i n g - 8 - variables: (2) X= >Vet , where " a " is a characteristic length T 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 v The characteristic length "a" can be taken to be the i n i t i a l length of the slab - 9 <*,*V= M f c U , * ) O < X < 4 C t ) < J C « J s b I « C o,T^ J J ( T ) - b„>o (0 .3 ) i<t>* o U ( o j l j ( 0 .3a ) U U I D ^ J - O t 6 ( o , T ) , (0 .3b) M R t « > * ) - © .' t t ( O j T ) , ( 0 . 3 c ) U t x . o ) - . X t C«, (0 .3d) i t ! ) s « * | M « U U ^ - 0 - h(t \J t « C o , T K (0 .3e ) We n o t e t h a t (0 .3e ) w i t h j(o); b can be w r i t t e n e q u i v a l e n t l y as ( 0 . 3 f ) We now s e e k the s o l u t i o n ( -u. (.*, i), j ct>) to the sys tem o f equa t ions ( 0 . 3 , a , b , c , d , e ) . More p r e c i s e l y , we take V . U ' ) e C'Co.b] w i t h v . l O i © on [ 0 , b ] , A0(»)= U 0 ( b ) r o and \-»Ct) c o n t i n u o u s but f o r a f i n i t e number o f jump . d i s c o n t i n u i t i e s on [o , l]J. Then we l o o k f o r a s o l u t i o n (*u. j ct)). o f ( 0 . 3 , a , b , c , d , e ) s a t i s f y i n g the c o n d i t i o n s : (a) M x x t * , t ) , M lw,t) 6 C t>,.scu), t <s < O , T ) ; (b) v.(.x,t̂  <S. c r o ^ c t ) ! , t C C o . T ) ; (c ) U „ C*,t> € C C o . i t i ^ t e Co,T); We b e g i n i n C h a p t e r I by showing t h a t the s y s t e m e q u a t i o n s ( 0 . 3 , a , b , c , d , e ) has a un ique s o l u t i o n . CHAPTER I EXISTENCE AND UNIQUENESS In this chapter we show that the system of equations (0.3 ,a ,b ,c ,d ,e) has a unique solution (14,5) for a l l , <£) s j U,t) i o< x < 4<tl , o < t < T To accomplish this we follow the lead of Friedman [17 J , by constructing an equivalent system of Coupled Volterra Integral Equations and showing that there exists a C>© such that for a l l "K< 9- , these i n t e g r a l equations have a unique s o l u t i o n . We then show that this procedure can be repeated to y i e l d existence and uniqueness of the solution of the system (0.3,a,b,c,d,e) for the i n t e r v a l of time ( o > T ) . The Integral Equations .. Before constructing the int e g r a l equations, we state several useful lemmas. The f i r s t , due to Friedman [ 17 ], is a working lemma used extensively throughout the construction of the in t e g r a l equations; while the other two es t a b l i s h properties of -SCt.̂  (the po s i t i o n of the free boundary) and vt>x (..St-Û -fĉ  (the - 11 - - 12 - amount''of. heat a l l o c a t e d to the d i f f u s i o n p r o c e s s ) r e s p e c t i v e l y . D e f i n i n g K i x , - t ; ^ t t ) to be the u s u a l s o u r c e s o l u t i o n o f the hea t e q u a t i o n , tha t i s , - C x - g ) * we have the f o l l o w i n g lemma. Lemma 1 . 1 . L e t <fit^i -set) be c o n t i n u o u s f u n c t i o n s on the i n t e r v a l . I n a d d i t i o n , l e t je t ) s a t i s f y the L i p s c h i t z c o n d i t i o n f o r some c o n s t a n t M . Then f o r a l l "k £ i0,0"! we have sctw© Ox ^ P r o o f . The p r o o f , g i v e n i n Append ix A , c o n s i s t s o f showing - 13 - r 1 In order to establish the next two lemmas we use the following a u x i l i a r y propositions whose proofs are given i n Appendix B. Proposition 1.1 (The (weak) Maximum P r i n c i p l e ) . Suppose "UAx^) s a t i s f i e s with <St-t) a pos i t i v e continuous function and -uCx,-t> e C ) , 'M„U,t>, vi^ <.«,*> £ C (•& V <8X ) where .£) is the closure of ̂  and F i g . 1.0 The (weak) Maximum P r i n c i p l e on a Non-rectangular Domain - 14 - then v . U,t) assumes i t s maximum and minimum v a l u e s on the d a t a boundary - <Sy • P r o p o s i t i o n 1.2 (The n e c e s s a r y c o n d i t i o n f o r m e l t i n g ) . I f (-u,s) i s a s o l u t i o n o f the system ( 0 . 3 , a , b , c ,d ,e) then Vj.uui . t ^ j o f o r a l l t e CO,T3 . We a re now ready to e s t a b l i s h the f o l l o w i n g p r o p e r t i e s . Lemma 1 . 2 . ( 0 . 3 , a , b , c , d , e ) t h e n .SCiV s a t i s f i e s the L i p s c h i t z c o n d i t i o n I f SC*.) i s a s o l u t i o n o f the system of e q u a t i o n s (1) f o r a l l t., ,t j . €. O . T ] . P r o o f . C o n s i d e r c o n d i t i o n ( 0 . 3 f ) « o at t, and i i Then N o t a t i o n : l t * i l , - where o o S i n c e A' - - H, (Jlt.+j),!,^) and /Wo) = o , P r o p o s i t i o n 1.2 i m p l i e s t h a t ^ ^ ^ 5 > <3*°* T n u s t, and hence ••Uct,..>-s(i,)| > «<*«miT » t ( - t , l . Lemma 1 . 3 . I f ("U,5) i s a s o l u t i o n o f the system o f e q u a t i o n s ( 0 . 3 , a , b , c , d ,e) then ' M * C*,tV s a t i s f i e s f o r a l l (x,t} C c5b , p r o v i d e d vt^t^-O s a t i s f i e s the h y p o t h e s e s o f P r o p o s i t i o n 1 . 1 . P r o o f : t r i v i a l . H a v i n g e s t a b l i s h e d t h e s e p r e l i m i n a r y r e s u l t s we can c o n s t r u c t the i n t e g r a l e q u a t i o n s wh ich w i l l u l t i m a t e l y a l l o w us to e s t a b l i s h e x i s t e n c e and u n i q u e n e s s o f the s o l u t i o n to the system o f e q u a t i o n s (0 »3 , a ,b , c ,d ,e) . To t h i s end. we i n t r o d u c e the G r e e n ' s f u n c t i o n s f o r the h a l f p l a n e 6 * C i , t ; 5 , T)t K ( * , t i ^ t ) + K U , i ; - 5 , : r ) , - 16 - and n o t e t h a t any s o l u t i o n v . C ^ , t ) o f the heat e q u a t i o n s a t i s f i e s G r e e n ' s I d e n t i t y § L f G + £ M - ^ 3 6 + l - &L f ( 5 * v l = 6 (1 .0 ) Sk 5 f l - a r t J i n the domain • O8£ = JC?,^- o<^,<s(rv > o<f<T<t-f^ where £ >o. I n t e g r a t i n g (1 .0) o v e r and u s i n g c o n d i t i o n s ( 0 . 3 b , c ) and 6 ^ t * , i ; o,2*V~ O we o b t a i n r ° O- ^ vt(^,£) G + CK lt;^ <r)c| <F .sen + \ vUE;,t-0 G + (*,t;^, W ^ g ; (1 .1 ) In Append ix C we show t h a t as f - > o (1 .1) becomes (1 .2 ) - 17 - D i f f e r e n t i a t i n g (1 .2) w i t h r e s p e c t to x and a p p l y i n g Lemma 1.1 as x-»J<-k ) -o we f i n d tha t -o l - O 2 -un c 5 t t i s a t i s f i e s •--vm* * . } * 0 < 3 > G £ U t t i , t ; e , o ) d £ (1 .3 ) t ^ vcr> G* W(t> , t i 3(*^r) d r . S i n c e <S^.(*,i>? to)*- <»*<*•,t;£,o) , G~(x, t • o,o)s o and - u 0 t » = o , a f t e r mak ing the a p p r o p r i a t e s u b s t i t u t i o n , we i n t e g r a t e by p a r t s the f i r s t i n t e g r a l e x p r e s s i o n o f (1 .3 ) to o b t a i n ° (1 .4 ) i t • © M o r e o v e r , as does F r i e d m a n £ 17 } we i n t e g r a t e the c o n d i t i o n (0 .3e) and f i n d t h a t .sm- b + \ (v(rl-h(r)) dt. (1 .5 ) © We have t h a t W-t) s a t i s f i e s (1 .4) where s(t) i s g i v e n by ( 1 . 5 ) , hence we r e f e r to v ( t ) as the s o l u t i o n o f ( 1 . 4 ) , ( 1 . 5 ) . F u r t h e r m o r e , we have the f o l l o w i n g e q u i v a l e n c e between v ( t ) ( t h e s o l u t i o n , o f ( 1 . 4 ) , ( 1 . 5 ) ) and ( v , s ) ( the s o l u t i o n o f (0 .3 , a , b , c ,d , e ) ) - 18 - Lemma 1 .4 (The equivalence of the d i f f e r e n t i a l and int e g r a l sys terns). If v(t) is a solution of (1.4), where s(t> is given by (1.5), then (v , s ) ( v < ( * , t ) defined by (1.2) and j t i ) defined by (1.5)) forms a solution of (0.3,a,b,c,d,e). Conversely i f ( M t s ) is a so l u t i o n of (0.3,a,b,c,d,e), then \> (t> * H» U t t ) , t ) is a solution of (1.4). Existence and Uniqueness of the Solution for Small Time From the equivalence 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 of coupled i n t e g r a l equations ((1.4), (1.5)), we see that showing that the former has a unique s o l u t i o n reduces to demonstrating that the l a t t e r has a unique s o l u t i o n . To e s t a b l i s h that ((1.4), (1.5)) has a unique s o l u t i o n we make the following d e f i n i t i o n s . D e f i n i t i o n 1.1: The proof is standard and is given in Appendix D. the set of bounded continuous functions on |0,tf" D e f i n i t i o n 1.2: the closed M-sphere in C r . - 19 - D e f i n i t i o n 1 . 3 . D e f i n e • H"* to be the t r a n s f o r m a t i o n g i v e n , by ( 1 . 4 ) , ( 1 . 5 ) , t h a t i s , + 2 \ v t * v G;csto,t;5«*i,t;H-t where I t i s easy ,to see t h a t M o r e o v e r , we have the f o l l o w i n g theorem. Theorem 1 . 1 . There e x i s t s a <r>o such t h a t Hf1 : C ^ M - > CO-.M where P r o o f . - 20 - Suppose V« CO-,M then U v l l ^ x< M and hence Let s a t i s f y the following inequality 0" 5 "<v^ L~N- Since from ( 1 . 4 ) we conclude that Writing (1.6) where N*\\WI|T. Then for a l l 1 C- [ o ^ ] which i n turn implies that v< *ft) t 1 b f o r a l l * * < r - (1.7) \\T^1| * a n i A + M ^ ^ ^ ^ M ^ • . (1 .8 ) where G, = \\K - 21 - we estimate & i and G a i n turn. Noting that J(t) is L i p s c h i t z continuous we fi n d that \ G,l i -?L? , ( ^ ^ N ) -t"« . To estimate S, we use (1.7) to obtain (1.9) . A(2) 16,1 «: a «*&e(-^. i) « (1.10) we Now applying the inequality etftcfO,* — • JL to (1.10) have IGJ* J. / t » . (1.11) Combining (1.9) and (1.11) we see that (1.8) becomes H T v l l t f 2 » * 0 r t ' b + M ( o ( ' ( M 4 W ) + J.) t V a . ty'/j b Hence the conclusion of the theorem follows i f we i n s i s t that o- also s a t i s f y the in e q u a l i t y C- J X ! . (1.12) (2) „ , t \ -** v ' We use the notation erucvM- — Vc At , - 22 - The f o l l o w i n g theorem shows t h a t we can f u r t h e r r e s t r i c t t h e s i z e o f <r so t h a t T 1 i s a c o n t r a c t i o n mapping on ^ C , M a n d hence a l l o w s us to c o n c l u d e tha t ( 1 . 4 ) , (1 .5 ) has a u n i q u e s o l u t i o n i n C f o r a s m a l l t i m e . Theorem 1 . 2 . There e x i s t s a C>o such tha t i s a c o n t r a c t i o n mapping o n C g - | ( ^ f o r a l l t 6 Co, «rl,. P r o o f . I n i t i a l l y l e t c be such t h a t <r i «r«0 % where c e s a t i s f i e s ( 1 .6 ) and ( 1 . 1 2 ) . I f ^ 1 i t ) , v> Ct> C- CVtf^ l e t V< fc>, 5 Ct> s a t i s f y ( 1 . 5 ) w i t h -o'Ct) and - J C t ) r e s p e c t i v e l y and d e f i n e S i n c e t-fe) ^ v'tt^ <c we have f « f o r a l l • . From (1 .5 ) we have the f o l l o w i n g i n e q u a l i t i e s : | sit) - j'(t\V * e «•* (1 .13) (1 .14) (1 .15) - 23 - and as before V $ 5lkV, S»<*>-.$ 3 b . for a l l r,t « Z*,<*1 . (1.16) Now consider T v T v ' = v, - V, where We can write V, - V,' + V," where V> 2 ̂ o (%)[wCS(^^o)-W (5 ' ( t)^;^ Jo)]a^ % o v; . , s -2^v.^) [KUrt ) , t r f ,o ) -K(s , a) J t i -^o) ] d | . o Applying the Mean Value Theorem and the inequality (1.16) to V,' we obtain IVI * £ nv. i l . (1 .17) To estimate V,' we assume that .s'ct) > .S ("fc) and consider the possible cases: 24 Case I: o< b * AU>< V<t > i | b > Case I I : o < h * * x< * | b , Case I I I : o < | * 5It) < «'<t) .< b . C o n s i d e r i n g v; s ^^C5>[K_<SW , t 3 f I o>-KCS , <i) > t ; f ,o)]e»5 i n Case I, i n Case I I , © Jt-fc) i n Case I I I . and a p p l y i n g the Mean V a l u e Theorem an a p p r o p r i a t e number of t i m e s , we a r r i v e a t the e s t i m a t e s : Case I: IV.'W i . II • «t't "* Case II: W\\ J nv . i l ^ 1t " * ^ (1 .18) Case I I I : I v'. I * ^ •<* t"' Combin ing (1 .18) w i t h (1 .17) we see t h a t W,l S t l f M ^ A * ' V (1 .19) where 25 To e s t i m a t e we w r i t e V * = ^ [ \ v l * > ^ l ± i + i £ * l > K(s<t),t--s «->,*> d r o *Ct-r) S i n c e J i t ) i s L i p s c h i t z c o n t i n u o u s we see t h a t 17 "* (1 .20) A p p l y i n g the Mean Va lue Theorem to I (w> ( i " *> J i we o b t a i n - 26 - lw*J $ t"* . (1 .21) ft"*- W r i t i n g W j as _ ( s t t w w f w . _1_ \ -u'trvU'ttr- &'ĉ ) e «<fc-*> f i we see t h a t the l a s t term can be e s t i m a t e d as f o l l o w s \ (S'CO- -S'ttV)*- <SU>- 3 < 2 . ' (M+A/ ) S U'til - Hence t a k i n g a- to f u r t h e r s a t i s f y 3 M (ot^ C M +//) o- J i (1 .22) and u s i n g the i n e q u a l i t i e s I I ~ «~& | $ 3 l ^ l . ( l ^ l j i) and (1 .15) we f i n d t h a t Wv,» i 3 M (+* + t 3 ' 2 . (1 .23) To comple te the e s t i m a t e o f Vg , we w r i t e as - 27 - where L = 9 \ (•-Qtt)-i>'<*»(.-SC-fc.)-«-3(tn e d t The e s t i m a t i o n o f U, i n v o l v e s a s t r a i g h t f o r w a r d a p p l i c a t i o n o f ( 1 . 7 ) , (1 .10 ) and y i e l d s * 3£.t>, (1 .24) Tr"a b To o b t a i n an e s t i m a t e f o r L 2 , the Mean V a l u e 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 must be a p p l i e d to the f u n c t i o n C*-*^)e (a -any n o n - z e r o c o n s t a n t ) . A s i m p l e c a l c u l a t i o n then l e a d s to the e s t i m a t e VLJ < » * « (1 .25) U s i n g (1 .25 ) and (1 .24) we see tha t | V ' | < £ (lVM* '+3) t ' ' 1 (1 .26) Hence ( 1 . 2 0 ) , ( 1 . 2 1 ) , (1 .23) and (1 .26) imply tha t s a t i s f i e s 28 - I V a l < J. T HM** +1 +• 3 M ( K f ( M 4 « ) ) ' + - ^ W + i v ) ] t * . (1.27) Now combining the estimates (1.19) and (1.27) we see that where >A is a constant dependent only on the data Taking O" to further s a t i s f y A o-"» < \ , the conclusion of tire Theorem follows. Theorems 1.1 and 1.2 imply that for o->o(given in Theorem 1.2) (1.4), (1.5) has a unique solution for a l l t < <r in Co-,*"1 where M - a«"u,t|̂  + 1 . Note that o~ depends only on the data (1.29) . To complete the proof of uniqueness of the so l u t i o n of (1.4), (1.5) we must show that any solution of (1.4), (1.5), i r r e s p e c t i v e of whether i t belongs to - C,. M (where <r is the " <r" of Theorem 1.2), must coincide with the fixed point of H"1 i n C <j-#!̂  say -o , i n th e i r common in t e r v a l of existence. (1.28) (1.29) (1.30) - 29 - I f v(tV i s a n o t h e r s o l u t i o n of ( 1 . 4 ) , (1 .5) on the i n t e r v a l C°,?l then we must show tha t v<-k)«v<i* on Co,5-] where J : •MW|»,»| ) the common i n t e r v a l o f e x i s t e n c e . Note tha t when i n Theorems 1.1 and 1 .2 : M i s r e p l a c e d by M*= -w\«-y ^ n-Gw«tfA^ we have t h a t v,-C a re b o t h f i x e d p o i n t s o f T 1 i n C<J.',M' where i n g e n e r a l a*<?. Hence we c o n c l u d e t h a t "v<^=v(t> on the i n t e r v a l Now i f or, (<r, < 5) i s such tha t -0<*l=O(i) on O J O then i t i s c l e a r from the i n t e g r a l e q u a t i o n s ( 1 . 4 ) , (1 .5) tha t v <«•,)! ••*>< «-,) ,• Hence i f \ V 15 <rj} S IT, , ^ u t <r,) , s < <M] a r e the t e m p e r a t u r e d i s t r i b u t i o n and p o s i t i o n s o f the boundary at "t r <f, c o r r e s p o n d i n g to v(U,tf(l) r e s p e c t i v e l y then v< (5,0-,) s ^-C .̂o-^ t >= si*-,}. S h i f t i n g the o r i g i n o f t ime i n Theorems 1.1 and 1.2 to we~'can a g a i n c o n c l u d e t h a t t h e r e e x i s t s an f>o such t h a t vt^ivU^ on Co, 0-, + 4) , S i n c e the o n l y r e s t r i c t i o n on c( was tha t i t s a t i s f y c-, < 5- we c o n c l u d e t h a t •u<^ s v t t ^ on L"«>»3 t h e i r common i n t e r v a l o f e x i s t e n c e . E x i s t e n c e and Uniqueness of the S o l u t i o n f o r a l l 1 6 Q . T 1 L e t o - W ) s a t i s f y ( 1 . 6 ) , ( 1 . 1 2 ) , (1 .22) and ( 1 . 3 0 ) ; .then t h e r e e x i s t s a u n i q u e s o l u t i o n o f ( 1 . 4 ) , (1 .5) f o r t< Mov ing the o r i g i n o f t ime to i s ^ we can f i n d a s u c h t h a t the - 30 - s o l u t i o n o f ( 1 . 4 ) , (1 .5 ) e x i s t s and i s u n i q u e f o r a l l t i o*1^ C o n t i n u i n g i n d u c t i v e l y we see tha t we can g e n e r a t e a sequence _ ** ^ ô *̂ ._ such t h a t ( 1 . 4 ) , (1 .5 ) has a u n i q u e s o l u t i o n f o r a l l 1 j. i *" . I f we can show t h a t t h e r e e x i s t s a 0" V such f o r each o-0) > 0 - ° (1 .31) then we c a n c o n c l u d e tha t f o r some A/ and hence ( 1 . 4 ) , (1 .5 ) has a un ique s o l u t i o n f o r a l l t C- <°,T). However , t h i s i s immediate i f we can f i n d g l o b a l u p p e r bounds f o r F o r then o-° de te rmined by the i n e q u a l i t i e s ( 1 . 6 ) , ( 1 . 1 2 ) , (1 .22 ) and (1 .30) w i t h M r e p l a c e d by and r e p l a c e d by T* s a t i s f i e s ( 1 . 3 1 ) . S i n c e " M K U , ! ) i s c o n t i n u o u s on ob we see t h a t Lemma 1.3 i s a p p l i c a b l e and hence T h e r e f o r e we have t h a t ( 1 . 4 ) , (1 .5) and hence ( 0 . 3 , a , b , c , d , e ) - 31 - has a u n i q u e s o l u t i o n f o r a l l "t-* ( 0 , T V - p r o v i d e d i s bounded .on £O,T] and v0<x) i s u n i f o r m l y bounded on £o,bJ. In Chapte r I I we w i l l o u t l i n e the S i m i l a r i t y Method which w i l l be used to d e r i v e the S i m i l a r i t y S o l u t i o n ( C h a p t e r I I I ) upon w h i c h i s based the S i m i l a r i t y A l g o r i t h m (Chapte r I V ) . CHAPTER II THE SIMILARITY METHOD The a l g o r i t h m to be i n t r o d u c e d i n C h a p t e r IV i s based on p a r t i c u l a r s o l u t i o n s o f the d i f f u s i o n e q u a t i o n found by the S i m i l a r i t y M e t h o d . The f o l l o w i n g p r o v i d e s the t h e o r e t i c a l b a s i s as w e l l as the p r o c e d u r e f o r c o n s t r u c t i n g such s o l u t i o n s o f d i f f e r e n t i a l e q u a t i o n s . A common method of s o l v i n g d i f f e r e n t i a l e q u a t i o n s i s to change v a r i a b l e s i n o r d e r to t r a n s f o r m the e q u a t i o n to one whose s o l u t i o n i s more e a s i l y o b t a i n a b l e . The t r a n s f o r m a t i o n s which g i v e r e s u l t s a re o f t e n those which e x p l o i t a symmetry o f the o r i g i n a l s y s t e m . The S i m i l a r i t y Method p r o v i d e s a s y s t e m a t i c r e c i p e f o r f i n d i n g such t r a n s f o r m a t i o n s u s i n g L i e ( c o n t i n u o u s ) G r o u p s . Sophus L i e showed t h a t i n v a r i a n c e o f an 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 under a one parameter c o n t i n u o u s group o f t r a n s f o r m a t i o n s l e a d s d i r e c t l y to a r e d u c t i o n by one i n the o r d e r o f an 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 . He showed how to f i n d the " L i e " Group o f t r a n s f o r m a t i o n s l e a v i n g i n v a r i a n t an o r d i n a r y - 32 - - 33 - d i f f e r e n t i a l equation^^ and found a subgroup of the f u l l group of (2) the heat equation. However, i t remained for authors of more recent years to show how to use continuous groups of transformations to reduce the number of variables, and hence find p a r t i c u l a r (3) solutions, of p a r t i a l d i f f e r e n t i a l equations. The major contributions i n this regard have come from Ovsjannikov £ 28 J , Matschat and Muller £ 2^]» and Bluman [ 2 J . More recently, Bluman £33»L"̂3 has applied the S i m i l a r i t y Method to boundary value problems. (4) L i e Group of Transformations Central to the theory is the concept of a L i e Group of Transformations. D e f i n i t i o n 2.1: -(-a L i e Group of Transformations). A one parameter family of transformations For a treatment of the S i m i l a r i t y Method applied to ordinary d i f f e r e n t i a l equations see Bluman and Cole Part I. (2) Lie did not see how to use invariance to construct p a r t i c u l a r solutions to p a r t i a l d i f f e r e n t i a l equations. (3) For a thorough treatment of the S i m i l a r i t y Method as applicable 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) Since we are interested only i n a p a r t i a l d i f f e r e n t i a l equation involving one dependent and two independent variables we r e s t r i c t our attention to this case. - 34 - where and forms a L i e Group o f T r a n s f o r m a t i o n s w i t h parameter £ i f : (a) ( A s s o c i a t i v e P r o p e r t y ) t h e r e e x i s t s a f u n c t i o n w i t h f o r a l l a.bjCCQ such t h a t , f o r x * x * * j X e S s a t i s f y i n g (b) ( I d e n t i t y E lement ) t h e r e e x i s t s an €„£Q such t h a t f o r a l l x € S ; (c) ( I n v e r s e Element) f o r every £ £ Q t h e r e e x i s t s an € Q such t h a t We n o t e tha t c o n d i t i o n s ( a ) , ( b ) , (c) make the f a m i l y a group o f t r a n s f o r m a t i o n s , w h i l e the c o n t i n u i t y c o n d i t i o n s on • J ' C X J O , make i t a L i e Group o f t r a n s f o r m a t i o n s . We - 35 - remark that by a suitable rep-arameterization, the i d e n t i t y element can be assumed to be zero. To apply the S i m i l a r i t y Method to a second order p a r t i a l d i f f e r e n t i a l equation we consider the following Lie Group of transformations: X *= X<U,X,t;£) > (2 .0 ) where -u. is the dependent variable and X, "t are the independent vari a b l e s . Invariance. A p a r t i a l d i f f e r e n t i a l equation G ( ' U ^ > M x t , M t t , v < k , v < t , - w < > x l - f c ) = o ( 2 < 1 ) together with the boundary conditions By(-u«,v4 t l u,x , -b)so (2 .1a) on the boundary curves W v t x , t > = o Y= ' . • • • , p (2 .1b) is invariant under (2*0) provided - 36 - (2 .2 ) and (2 .2a ) on the boundary c u r v e s (2 .2b) h o l d whenever ( 2 . 1 , a , b ) h o l d . That i s , the g o v e r n i n g d i f f e r e n t i a l e q u a t i o n , the boundary c u r v e s and the boundary c o n d i t i o n s on t h e s e c u r v e s take the same form i n bo th t r a n s f o r m e d and o r i g i n a l v a r i a b l e s . S i n c e a 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 se ldom has a group r i c h enough to l e a v e i n v a r i a n t boundary d a t a such a s . ( 2 . 1 , a , b ) , we seek a group l e a v i n g i n v a r i a n t o n l y the g o v e r n i n g d i f f e r e n t i a l e q u a t i o n ( 2 . 1 ) . What boundary c o n d i t i o n s cannot be l e f t i n v a r i a n t can f r e q u e n t l y be s a t i s f i e d by s u p e r p o s i t i o n ( c f . Bluman and C o l e [] 6 J P a r t I I C h a p t e r 11 ) . In a d d i t i o n we can c o n s t r u c t n u m e r i c a l 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 c e r t a i n boundary c o n d i t i o n s ' ^ F u r t h e r , u s e f u l p a r t i c u l a r s o l u t i o n s o f (2 .1 ) may be o b t a i n e d by f o r m u l a t i n g boundary c o n d i t i o n s i n terms o f the g e n e r a t e d by the S i m i l a r i t y A l g o r i t h m i s o b t a i n e d by l e a v i n g i n v a r i a n t ( 0 . 3 , a , b , c ) , by s a t i s f y i n g (0 .3d) by s u p e r p o s i t i o n and by " a l m o s t " s a t i s f y i n g ( 0 . 3 e ) . (5) The approx imate s o l u t i o n o f ( 0 . 3 , a , b , c , d , e ) invariants of the group leaving invariant ( 2 . 1 ) . The Most General m-Parameter Lie Group of Transformations Leaving Invariant (2 .1) First 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 is necessary to reformulate invariance in a more useful way. To this end, we introduce the infinitesimal transformations. Noting that T J ' C H ^ t - O , X(-*,wt\,i) , T (M ,x,t ;<r) 6 C ° ° ( / / ? 3 , ) we expand about f = o (the identity) to obtain (2 .0 ) in infinitesimal form, where 38 » i ^ v , ^ « l > T ( u ) » , t i O l The transformations (2.3) induce transformations on the deri v a t i v e s , i . e . , and where S i m i l a r l y we o b t a i n where To o b t a i n the second e x t e n s i o n s we w r i t e - 40 - i . e . 7x* £ 5 * \S*4v< 8K'/M< C>*» * Ve>~1 ,3*0.*/ * * dv. 3R <JW V Now (2.2) can be expanded about f r o to y i e l d - 41 - G IK-*-. K- e» *Vt»» • **V»**>**) = ° + f G C^x J«,UMt l v » i B - t l v«,M t ,w y i c l-t) . +• ot£») where we have introduced the f i r s t order d i f f e r e n t i a l operator It can be seen that invariance of (2.1) under (2.0) is equivalent to X G = 0 (2.4) whenever G = O , With t h i s formulation of invariance we are prepared to f i n d the most general m-parameter group leaving (2.1) invariant Substituting f,?>r, *?t , 9?xt ,r?it into (2.4) and using the r e l a t i o n G~0, we obtain the determining equations for by setting equal to zero the c o e f f i c i e n t s of the independent derivative terms (^KI )• We are 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 equations for - 42 - Reducing the Number of Variables To every L i e Group of transformations there corresponds a set of Canonical Coordinates, in which the group is a t r a n s l a t i o n of one of the v a r i a b l e s . Using these Canonical Coordinates i t can be shown that i f the translated v ariable is an independent v a r i a b l e then invariance of a p a r t i a l d i f f e r e n t i a l equation under a one parameter L i e Group of transformations leads to a reduction by one i n the number of independent variables provided the s o l u t i o n i s unique (cf. Bluman and Cole [^6j Part II Chapter 3). number of independent variables by one leaves us with an ordinary d i f f e r e n t i a l equation. Suppose (2.1,a,b) is invariant under (2.0), whose i n f i n i t e s i m a l transformations are given by (2.3). If w* toe*,*) is a solution of (2.1) then both v«s &ix*,t') and vv."*U(©.**'tiO are solutions of (2.2). Now assuming (2.2) has a unique solution then v = v^*' . Expanding V, w.v about £zo and gathering-terms in powers of <f we find that It should be noted that in this instance reducing the (2.5) (The Invariant Surface Condition) must be s a t i s f i e d i f v i v * and - 43 - conversely. The general solution of (2.5) can be found by solving the c h a r a c t e r i s t i c equations c\x s . - d ® > (2.6) • g<G.«,±Y • (6) If % / t is independent of © , then the general solution of (2.6) takes the form © = €(*,±> \, where f s $ ( * | t ) (the S i m i l a r i t y Variable) and are the two constants generated by-solving (2.6). Substituting © into (2.1) and using the r e l a t i o n $ V*,*) we obtain an ordinary d i f f e r e n t i a l equation for ^C^) Hence the number of variables has been reduced by one. The complete solution of (2.1) can be found by solving the ordinary d i f f e r e n t i a l equation for £t0. However, i f a two parameter Lie Group of transformations leaves (2.1,a,b) invariant and the invariants (the s i m i l a r i t y variables) (8) associated with the two parameters are f u n c t i o n a l l y independent If 2*/T depends on © then the general solution of (2.6) is of the form O = «&<€>.*, t;i,V CO) and $-^(G>,.*,tK The boundary conditions (2.1 a,b) become boundary, conditions to be s a t i s f i e d by ?̂ V>- (Q\ Two invariants are f u n c t i o n a l l y independent provided th e i r respective 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 functions ( c f . Bluman and Cole [6^ Part II Chapter 8). - 44 - then the solution of (2.1) can be found d i r e c t l y using the invariants without recourse to (2.1) ( c f . Bluman and Cole [ 6 ] Part I I Chapter 8). In general, i f an m-parameter L i e Group of transformations leaves invariant a p a r t i a l d i f f e r e n t i a l equation with accompanying boundary conditions, i t i s necessary that the associated invariants ( s i m i l a r i t y variables) be f u n c t i o n a l l y independent before we are assured that the number of variables can be reduced by m. (9) The C l a s s i c a l 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 six parameter group: 7 c w u e [ . * \ j \ | ^ | x * i ] • A Ct> j (2.7) (9) See Bluman and Cole \j>~\- Here «<, v̂ 'V, 6, K, \ a re the parameters o f th.e group w h i l e • i s an a r b i t r a r y s o l u t i o n o f the heat e q u a t i o n . The group (2 .7 ) i n the (x,t) p l a n e i s a subgroup o f (10) the e i g h t parameter p r o j e c t i v e g r o u p . The parameters <*, K r e p r e s e n t t r a n s l a t i o n s i n the i.. and x d i r e c t i o n s r e s p e c t i v e l y ; "O r e p r e s e n t s a s t r e t c h i n g or s i m i l i t u d i n u o u s t r a n s f o r m a t i o n ; w h i l e ^ i s a s s o c i a t e d w i t h the G a l i l e a n t r a n s f o r m a t i o n . To f i n d the form of the g l o b a l t r a n s f o r m a t i o n a s s o c i a t e d w i t h V we s o l v e , the s e t o f c h a r a c t e r i s t i c e q u a t i o n s The r e s u l t i n g t r a n s f o r m a t i o n s are g i v e n by In the nex t c h a p t e r a subgroup o f (2 .7) w i l l be used to c o n s t r u c t the s i m i l a r i t y s o l u t i o n c e n t r a l to the S i m i l a r i t y A l g o r i t h m . See Bluman and C o l e [^6] P a r t I Chapte r 7 . CHAPTER III A USEFUL SIMILARITY SOLUTION OF THE HEAT EQUATION FOR THE STEFAN PROBLEM In this 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 velocity. We proceed as follows. Given set) , the system (0.3,a,b,c,d) reduces to the Inverse Stefan Problem: (3.0) i^Co . T ) - , (3.0a) u(ito,t)-o 't«Co/r>, (3.0b) MK<»,±)^ o (1) tCCo.r), (3.0c) *A(*,O\- V i „ ( V > xeC«iCl. (3.0d) ^ The methods o f t h i s chapter may be used to deal w i t h the boundary c o n d i t i o n s vWo,t>- PCt) or uK(o,t>-RC-O • - 46 - - 47 - We w i l l show using the S i m i l a r i t y Method that for a member of.a two parameter family of curves, the system (3.0,a,b,c,d) has a closed form analytic s o l u t i o n . For convenience the group (2.5) together with a f i r s t extension Is given below, «t + 2M t + V t l %<*>±\-- K+• St f»x + Yxt (3-1) where FCX , + , ) = -Y[|\ J ] - | * +- A dx V ^x/ -a ' * a* We consider (3.1) with ^cx,t)5o and note that i f the boundaries x=o, x.-sit) are invariant under (3.1) and l " ^ 6 ' ^ 1 °, then C 3 . 0 , a , b , c ) is l e f t invariant by (3-1). The condition i f ' ' ^ " » o is s a t i s f i e d provided & = o ; x-o is invariant under (3.1) i f and only i f x*"= o whenever X J O , i . e . , S*K=o . The invariance ofx=^Ct) under (3.1) implies that s a t i s f y the d i f f e r e n t i a l equation - 48 - ^"(.SCt),*) «. i r t ) 3T(k) . (3 .2 ) •When combined w i t h .S<<rt = c (3 .2) i m p l i e s tha t Hence the t h r e e parameter subgroup o f (3 -1 ) l e a v i n g i n v a r i a n t ( 3 . 0 , a , b , c ) whenever Jt-kJ«- ( c ' u v t + y t 1 ) ^ (3-3) i s g i v e n b y : ? ( < , t ) t x ( v f V t ) f ( 3 .4 ) U s i n g (3 .4 ) the S i m i l a r i t y S o l u t i o n o f the system ( 3. 0 , a ,b , c , d) c o r r e s p o n d i n g to the most g e n e r a l "mov ing" boundary (3 .3) can be c o n s t r u c t e d ( c f . Bluman [ 4 ] ) . However , f o r our purposes we o n l y c o n s i d e r a subgroup o f (3 .3 ) to o b t a i n the S i m i l a r i t y S o l u t i o n o f the sys tem ( 3 . 0 , a , b , c , d ) c o r r e s p o n d i n g to the "moving" boundary - 49 - L e t t i n g v--/3c > V - ^ i * (3.4) r educe s t o + + > ^ | ^ (3-5) where The i n f i n i t e s i m a l s (3.5) y i e l d the set of c h a r a c t e r i s t i c equations From the f i r s t equality of (3.6) we obtain the S i m i l a r i t y Variable V = x / ( c y 3 i ) » where K = o <=> $ = O, Integrating the second equality of (3.6) along the s i m i l a r i t y curves ^ = constant we obtain the solution surface (3.7) of (3.6). Here ^((ijjx) must s a t i s f y a c e r t a i n ordinary d i f f e r e n t i a l equation together with the boundary conditions -^--•jx)-- X^Oj/*) - o for a l l ^ (3.8) i f v-CXjij^*) is to be a solution of (3.0,a,b,c) with eyai', To derive the d i f f e r e n t i a l equation s a t i s f i e d by we write vet*,!) (the solution of (3.0,a,b ,c ,d) with iCt)-c-/3± ) as a superposition of functions, £i j^O, and substitute the re s u l t i n g expression into (3.0). Introducing the variables into (3.7) we obtain v^tj^-.vU.t^)-. ^ J F ^ p e *tr+/U>c/(*,P> <P*. • (3.9) The form of (3.9) suggests we take - 51 - (3 .10) S u b s t i t u t i n g (3 .10) i n t o (3 .0 ) and t a k i n g i n t o account the boundary c o n d i t i o n s ( 3 . 8 ) , ve see t h a t c ^ f ( ^ ; j > ) must s a t i s f y ^ -As*1 The s o l u t i o n o f (3 .11 ) i s where we have i n t r o d u c e d $ C ^ ) - Sc e ^ v . <«^> (3 .11) (3 .12) S u b s t i t u t i n g (3 .12) i n t o (3 .10) and e v a l u a t i n g the L a p l a c e - 52 - Transform by closing the contour in the l e f t half plane we obtain /3-aL <*=> - u£ t . ( 2 ) U ( * , 0 » J ^ e r t e y t i J ^ T . e « 7 y i U co«(w H.a_) (/S) ( 3 . 1 3 ) where © If we evaluate the Laplace Transform by expanding for large p we obtain the small time representation of (3 .13) UCx,-t>-- ^ ^ ( S p G C x / t ^ ) d S - , (3 .14). where n r x* f a G(M-^)--T=Ue- ' J E(-») V + <? ( 2 ) _ The expression %*. C*, ) of (3 .7 ) can be substituted d i r e c t l y into ( 3 . 0 ) . The result when combined with the boundary conditions (3 .8 ) is a Regular Sturm-Liouville System for the eigenfunctions ^-?((il**M}. > '•N* st * - i[hr . The solution (3 .13) i s then obtained as a li n e a r combination of the eigenfunctions - 53 - We n o t i c e t h a t when y3-ot ( 3. 0 , a , b , c , d ) reduces to the u s u a l f i x e d boundary prob lem and (3 .13) reduces to the w e l l - k n o w n F o u r i e r S e r i e s S o l u t i o n . C l e a r l y the s e r i e s (3 .13) t o g e t h e r w i t h a l l i t s d e r i v a t i v e s converges u n i f o r m l y on t ^ C y ^ t ] f o r any f i x e d "fc>0. I f "U^txi € c 'CojCl and "u c ( O =, i u e l c) ~° then we w i l l show tha t (3 .13 ) i s u n i f o r m l y c o n v e r g e n t on O . c l at -t - c , and tha t v „ U , i : o ) converges to' vve C O . S i n c e the f u n c t i o n s jc»s(i«3„)f){ s a t i s f y a r e g u l a r S t u r m - L i o u v i l l e Prob lem on C©. »"V > f o r any f u n c t i o n ^<JO C C* C°I»"} we can w r i t e T"(x)= ZZZzos^x)} fct*0 C « J ( W ^ , 1 ^ , (3 .15) I f the sum i n (3 .15) i s u n i f o r m l y c o n v e r g e n t we a l s o have V 'OO (W AK) ̂  V c ^ i c o s K ^ «J.»j . (3 .16) We c l a i m the f o l l o w i n g . Lemma 3 . 1 . L e t ^Cx) 6 C* Co,."] w i t h £ c n = V ' < o V - o then the sums i n (3 .15) and (3.1-6) converge u n i f o r m l y on C°, i l to tCx) and V (x ) r e s p e c t i v e l y . - 54 - P r o o f . L e t J (x) be the f o l l o w i n g e x t e n s i o n to C - -? ) 2 ! o f (l) 5co* *M * s C ° - . » l , (2) § ^ = $(->o x eC-a,*7, (3) i ( « + i ) s - i <x-i) x e C*l,.ll. ' C l e a r l y ^ ( 0 i s p i e c e w i s e C* on we c l a i m A € C * C-2, i l , On ly the p o i n t s x-o, 1 need be checked to e n s u r e t h i s r e s u l t . X = o : £(0-)- cfv^. I <-> o L © - C j « ? ( o + ) =. - +"'(0+) = O ; c->o L «r J CfM)- l J c ->•© L S i n c e $"00 i s p i e c e w i s e C* as w e l l as C we have t h a t -55 - 0 0 «l 4 where the sura is uniformly convergent on C©,il ( c f . Courant and H i l b e r t [8] Chapter 2) . That the sum in (3.15) converges uniformly on C°> i l follows t r i v i a l l y since § <*> G C* C-*,*J ( c f . Courant and Hil b e r t ^8] Chapter 2), and hence the lemma is proven. Applying Lemma 3.1 to vt6(.e*> we see that - * € V c ZT t^K^) ) . « ' cci(w,«l^ (3.17) and e - 2 ^ S ^ K f ) \ e % , . ( ^ - , C w ^ U a (3.18) where both sums converge uniformly on C°itl. It should be noted that the results (3.17) and (3.18) are e s s e n t i a l since the Fourier Series Expansions for v * , U , t > - 56 - and U „ Cx* i ( t ) , - t ) a r e used near i - o i n the S i m i l a r i t y A l g o r i t h m . F u r t h e r n o t e tha t the h y p o t h e s e s o f Lemma 3.1 a r e s a t i s f i e d a t each s t e p o f the S i m i l a r i t y A l g o r i t h m a f t e r the i n i t i a l s t e p (see Chapte r I V ) . The i n i t i a l c o n d i t i o n "HeCx) need no t s a t i s f y the hypotheses o f Lemma 3 . 1 , tha t i s , n 0 <x) may no t s a t i s f y ( 0 . 3 b , c ) . I f V„(x5o ) ^ o the S i m i l a r i t y A l g o r i t h m i s no t a f f e c t e d , t h a t i s , the sum i n (3 .17) s t i l l converges u n i f o r m l y . on [ o,C ] , w h i l e the sum i n (3 .18) o n l y converges u n i f o r m l y on [ S , c ] f o r any b>o . However i f v , t t ^ o ( the s l a b i s h o t p r e p a r e d to m e l t ) then we must m o d i f y the a l g o r i t h m and use the u s u a l F o u r i e r S e r i e s s o l u t i o n o f ( 3 . 0 , a , c , d ) and (we r e f e r to t h i s as the f i x e d boundary s o l u t i o n ) w i t h \$(i}=o u n t i l \ * . ( c ) t ) = o ( the s t a n d a r d d e r i v a t i o n o f t h i s s o l u t i o n i s g i v e n i n A p p e n d i x E ) . We see t h a t t h e r e i s some d i f f i c u l t y w i t h t h i s a l g o r i t h m i f a t any t ime the hea t f l u x i s i n s u f f i c i e n t to m a i n t a i n m e l t i n g . S i n c e we p r o v i d e no mechanism f o r f r e e z i n g , we m u s t , a t each t ime s t e p , d e t e r m i n e i f the hea t f l u x i s s u f f i c i e n t to m a i n t a i n m e l t i n g , - 57 - i . e . , determine the sign of S(t;),t;) - W\ (!«•)), * ; 6 ^ ~ . ( c f . Chapter IV) I f A > o we u t i l i z e the fixed boundary solution u n t i l the heat flux i s again s u f f i c i e n t to maintain melting. In what follows we w i l l assume the heat flux to always be s u f f i c i e n t to maintain melting. Returning to the group (3 .1) we note that s e t t i n g Y 5 ° aid proceeding as above we generate the p a r t i c u l a r solutions of (3.0,a,b,c,d) given by Sanders £3l] . Because of t h e i r r e l a t i v e s i m p l i c i t y , the Trigonometric functions lend themselves much more ea s i l y to numerical c a l c u l a t i o n than "do the Confluent Hypergeometric functions which are the basis of Sanders' solutions. Apart from the inherent d i f f i c u l t i e s in c a l c u l a t i n g with the Confluent Hypergeometric functions, there are also convergence questions which would place i n doubt the u t i l i t y of such a scheme. For much the same reasons, an algorithm, s i m i l a r to that given in Chapter IV, based on the most general "moving boundary" would encounter d i f f i c u l t i e s from the onset, as here too the solution of (3.0,a,b ,c,d) is expressed as a sum of Confluent Hypergeometric functions. - 58 - / O \ We remark that in 1939 Huber [20^ proposed e s s e n t i a l l y the algorithm of Chapter IV to approximate solutions of c e r t a i n one-dimensional two-phase Stefan Problems. Huber's s o l u t i o n , however, is not based on a s i m i l a r i t y s o l u t i o n . He eliminates the i n t i a l condition by introducing the usual source solution of the heat equation (J*C (*,i:,i;,r.) of .Ch.ap.ter ..I.)., then uses a set of Appell Transformations to transform X-JC-t)=. c.-y3t to the fixed boundary y- X , while leaving invariant the heat equation. The solution i s then given as a sum of a source term plus a complicated convolution i n t e g r a l . Huber's so l u t i o n is. too unwieldy for numerical purposes. Recently Rubinstein [30j has suggested that Huber's method can be s i g n i f i c a n t l y s i m p l i f i e d by using a Green's function on the domain x e ( 0 , cys-fc)^ t <c (o.,i)j f i r s t derived by Soloviev [33]. By so doing the need for the complicated Appell transformations i s eliminated and the solution can be given d i r e c t l y ( 3 ) Recently A. Fasano and M. Primcerio [ l 4 ] have demonstrated convergence of Huber's Method for a one-dimensional single phase Stefan Problem. (4) The representation suggested by Rubinstein is *u<*,tU %cC$)r/rrx,t-,5,.w •*•<*,t',§>}«l$ where .5'(*,f,^ ) • is the Green's function of Soloviev, and v , , ^ ) i s the i n i t i a l condition. We remark that this i s b a s i c a l l y the representation given by (3.14). However, the Green's function GCx.-t-,̂ ) in (3.14) i s given in a more compact form than is the Green's function, K(x,t'>5,o) + ^ ( x , . - 59 - The calculations involved in using Ruber's Method, with Rubinstein's s i m p l i f i c a t i o n are very sim i l a r to those involved i n the S i m i l a r i t y Algorithm i f we were to use the small time representation (3.14) of the S i m i l a r i t y s o l u t i o n . We w i l l argue in Chapter VI that the large time representation (3.13) is more p r a c t i c a l than i t s small time counterpart, (3.14), and hence that the S i m i l a r i t y Algorithm i s more useful than Huber's Algorithm. ! CHAPTER IV THE SIMILARITY ALGORITHM H a v i n g d e r i v e d the S i m i l a r i t y S o l u t i o n (3 .13) we are now ready to o u t l i n e the S i m i l a r i t y A l g o r i t h m . Suppose t h a t we a r e g i v e n the p a i r o f f u n c t i o n s C"VA,S) s a t i s f y i n g the system ( 0 . 3 , a , b , c , d , e ) and we wish to f i n d an a p p r o x i m a t i o n ( £ , $ ) to ( v i s i o n I[O,T]. We proceed by p a r t i t i o n i n g the t ime i n t e r v a l (not n e c e s s a r i l y a u n i f o r m p a r t i t i o n ) and e s t i m a t e >5("t) on £ t o , \ , by Sit)'- C e - /3„ ( t - t o ) , t 0 < t < t , where As we have s e e n ( c f . C h a p t e r I I I ) , f o r 5 C t ) so d e f i n e d S i m i l a r i t y Methods y i e l d a c l o s e d form s o l u t i o n o f ( 0 . 3 , a , b , c , d ) . We denote - 60 - - 61 - t h i s s o l u t i o n by V ' t K , i ' t , ) and remark t h a t i t i s v a l i d on the domain To extend our e s t i m a t e to (̂ »;̂ al w e d e f i n e |<t>* <t-f >, t, < t 5 - t 4 where and /3, i s 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 ) to o b t a i n /3,« *'< Wet,) - U „ * t , ) ) . Now c o n s i d e r i n g as the o r i g i n o f t ime i n (0 .3 , a ,b , c ,d) and Vt°Cx, ^ i | ) as the i n i t i a l c o n d i t i o n U e t O w e c a n , as b e f o r e genera te a s o l u t i o n M.' (*) •i.-t,) o f (0 .3 , a , b , c ,d) v a l i d on the domain C o n t i n u i n g i n d u c t i v e l y we o b t a i n the approx imate s o l u t i o n on the i n t e r v a l ( t^ i i+ j^ by d e f i n i n g where - 62 - We obtain "U* Ix, i - t ^ ) by taking to be the o r i g i n of time i n (0.3,a,b,c , d ) and the i n i t i a l condition Uc Lx) to be w**'( x> • Again the S i m i l a r i t y Method yield s "u^x, t - t { ) on F i g . 4 .0 • - The S i m i l a r i t y Algorithm Time - t The approximate solution C t S) i s then taken to be We prove i n Chapter V that as we refin e the p a r t i t i o n in such a manner that -w\ <x̂  J4,t^ —> O the approximation C"M.j5 ) converges in the supremum norm to (\*ts) - 63 - That i s , g i v e n £ > o t h e r e e x i s t s a S>e such t h a t w\ew» < & i m p l i e s t and . A (1) ? In p a s s i n g we remark tha t h e r e we have c o n s i d e r e d the c a s e where the f l u x c o n d i t i o n (0 .3e) i s s a t i s f i e d at the e n d p o i n t s t-i o f the s u b i n t e r v a l s C "t«^ . I*1 f a c t , we c o u l d have s a t i s f i e d (0 .3e ) anywhere i n the s u b i n t e r v a l . However we see no p a r t i c u l a r advan tage i n d o i n g s o , w h i l e , as w i l l become a p p a r e n t , the c h o i c e o f a p o i n t i n s i d e the s u b i n t e r v a l c o m p l i c a t e s the a c t u a l n u m e r i c a l p r o c e d u r e . A. Note t h a t a heat f l u x V* Ct ) i s i n d u c e d by the a p p r o x i m a t i o n {y^tt) i . e . , one can c a l c u l a t e the hea t f l u x which p roduces the m e l t i n g d e s c r i b e d by T h i s heat f l u x s a t i s f i e s f o r a l l "kj € 7T , and as w i l l be shown i n C h a p t e r V I , can be used as a i n d i c a t o r o f the e r r o r s ^ N o t a t i o n : g i v e n 4~Cx,t,> > <^t* (t) and e ! t t ) ^ 0 then « » i - . « - j t - - . « i U W M : f 4 « k i \ , * i * ' * , - ' S , K A , ' l J - - 64 - and ( SCt), 3 (tl) i . e . , the q u a n t i t y \ U U > - K C T > | 4* (4 .0 ) can be computed and used to de te rmine whether 77* s h o u l d be r e f i n e d f u r t h e r . A l t h o u g h i n p r i n c i p l e (4 .0 ) can be computed , i t r e q u i r e s a good d e a l o f l a b o u r , and i n s t e a d we d e f i n e ECi) = «<* \ (van- wunar c and c a l c u l a t e C 1 / C i > 1 0 ^ o o J (a r e l a t i v e l y i n e x p e n s i v e c a l c u l a t i o n ) . We see t h a t £ C t ) g i v e s an i n d i c a t i o n o f how c l o s e l y ( 0 . 3 f ) i s s a t i s f i e d by C v i J ) M o r e o v e r , i f £ ( t ) i s s m a l l f o r a l l Co,T] we expec t ( v l j i ) to be a good a p p r o x i m a t i o n to Cvt.s) . We remark t h a t tu,s) p r o v i d e s an a n a l y t i c a p p r o x i m a t i o n to the s o l u t i o n < v\,s) ° f (0 .3 , a ,b , c ,d ,e ) . That i s j s a t i s f i e s ( 0 . 3 , a , b , c , d ) e x a c t l y w h i l e i t " a l m o s t " s a t i s f i e s ( 0 . 3 e ) . We w i l l use t h i s p r o p e r t y i n C h a p t e r V to demonst ra te the c o n v e r g e n c e o f the S i m i l a r i t y A l g o r i t h m . CHAPTER V THE CONVERGENCE OF THE SIMILARITY ALGORITHM Since the pair of functions generated by the S i m i l a r i t y Algorithm is an exact solution of (0 .3 ,a,b ,c ,d, e) A with V^tt) replaced by the interpolate, V\(t) , i t follows that the convergence of the S i m i l a r i t y Algorithm is equivalent to the continuous dependence of the solution 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 £ 0 ; T * 3 as we r e f i n e the p a r t i t i o n ( c f . Chapter IV) such that v^»^ 1 A t ; | - > o , Continuous Dependence for Small Time r We proceed by demonstrating that the system of equations (0.3,a,b,c,d ,e) i s continuously dependent on the boundary data j b , Vl cl .x) , M t l j for a small time o - > o . More p r e c i s e l y , given C M , S ) and s a t i s f y i n g (0.3 , a,b , c ,d, e) with H(t) Co, b,] j (5.0) K,U)! THx"* on In fact we w i l l show that <V,A> depends continuously on the boundary data | b, "ucu>, W t-uj . - 65 - and W(-M- R(t) (5.1) V.OO* on Co,bal (2) respectively, where W, > bj and j (T> - b, -r(T), we have the following Theorem. Theorem 5.1. If U,s), s a t i s f y (0 .3 ,a ,b ,c ,d ,e) with (5.0), (5.1) r e s p e c t i v e l y , then there exists a c->o such that the following i n e q u a l i t i e s hold: r ^ - s i i ^ j : d>)v>;-fcai + o-Vuv- 1 ^ A 3 c-|lH-R« t (5.2) where cl -r^l^. ̂  T l i ) , s t t ^ and the constants depend only on data £ T ; H < « , R ' t ^ u ^ ^ ^ o o , b ^ b ^ d f o ) , ^ • (5.4) That i s , the system of equations (0.3,a,b,c,d,e) is continuously (2) The functions H(t^, R(t^ are taken large enough to sustain melting. - 67 - dependent on the boundary d a t a | b )M 0 C x ^ h ( U j . In o t h e r w o r d s , t h e , s y s t e m ( 0 . 3 , a , b , c , d ,e) i s ,s.t a b l e .w i th r e s p e c t to v a r i a t i o n s i n the boundary d a t a , . P r o o f . W i t h the d e f i n i t i o n s we n o t e t h a t the f o l l o w i n g e q u a t i o n s h o l d ( c f . e q u a t i o n s ( 1 . 4 ) , ( 1 . 5 ) ) (5 .5 ) X (5 .5a ) and o (5 .6 ) c .t (5 .6a ) o - 68 - In a d d i t i o n , s u b t r a c t i n g (5 .6a ) f rom (5 .5a ) we f i n d t h a t « l b , - b 4 J + v - > i ¥ - - t + K '-rw-'RU^t . (5 .7 ) To o b t a i n the i n e q u a l i t y (5 .2 ) we f i r s t s u b t r a c t (5 .6 ) from ( 5 . 5 ) . a n d w r i t e the r e s u l t i n g e x p r e s s i o n as where and S i n c e Ĝ -Ot,t; r \ - - <5*<*,t; £ and NKb^- 6 " < * , t ; o j 0 ^ O, the e x p r e s s i o n f o r I can be r e w r i t t e n i n the form where r * © 69 o It is easy to see that V/f s a t i s f i e s w,\ * " y - ^ ' i b , . (5.8) To estimate Vj we write where o Proceeding, as i n Theorem 1.2 with v̂ ' we obtain |v;i v< A « i \ ,. (5.9) while applying the Mean Value Theorem to v̂ " and evaluating the r e s u l t i n g integral we can show that t V4" | N< JL l i i l ^ Hs-*H, . ( 5 . 1 0 ) - 70 - From (5.9) and (5.10) we conclude that s a t i s f i e s I V.I $ 1 lla- . (5.11) F i n a l l y we estimate by writing where V^, -z) *>(?>[ <ii!!if> >] AS,. Substituting v* (Sl*)-%)/ztv* into Vj and using the inequality Iwle $1 we obtain \VJl * ftJPflv,, (5.12) T*» t. S i m i l a r l y we f i n d that V" s a t i s f i e s Tt"t t The estimates (5.12) and (5.13) imply that s a t i s f i e s - 71 - 7T '* t Combining (5.8), (5.11) and (5.14) we find that \1\< ±. W"LH*->H. where We now estimate 7" by writing w t * 2 J ) | vet) isa^sm KU<t),t-jStr> tti 3(t-t)  J where 72 The term WJ can be e x p r e s s e d as the sum W, = VA/,' + Wj" + W ( M I 2 V -tSiS (»(t)»»(«> J ,< (.set,, t • jew, r) S i n c e .5 <t.V i s L i p s c h i t z c o n t i n u o u s we see t h a t iw/ix< »HJ!T II^.^II t " » . ( 5 . i 6 ) Then n o t i n g t h a t and - 73 - we find that \ Vct1-5<f)X _ .(t(t>-r(t)H < o t i t f t y - ^ H . + itH-nirl (5 17) and hence that w," s a t i s f i e s Iw,"!* Q('u^*»i; [ l \ v - x n t + l > H - ^ n T ] t"*. (5.18) F i n a l l y to estimate VA/,'" we f i r s t apply the Mean Value Theorem for a function of two variables (.x,̂ ) to K (*, t; ̂ f, t ) and use the fact that V-CtV is L i p s c h i t z continuous. To the res u l t i n g expression — * J Ct-*> U-ti I (where X is between -cctN and 4<t^ and Cj is between + U ) and H z ) ) we apply the i n e q u a l i t i e s | v | e * I and (5.17) to obtain l w , ' " | 5 ll/^i^nftll^. ^Ilv-,*.!^ 4 | I H - R I T | t . (5.19) - 74 - T o g e t h e r ( 5 . 1 6 ) , (5 .18) and (5 .19) imply tha t Ti"t ( Now-we w r i t e VA/^ as where (5 .20) - K(f<t\,t;-*<*),7U dr S i n c e b, >*(t^ s(Vt > b* we o b t a i n (5 .21) and - 75 - |W;w ajMI4U3-**. \ 1 ± * > A 7 (5.22) To estimate ŵ'" we apply the Mean Value Theorem for a function two variables (*,£) to K(*,t;-f, <r) and obtain .A. A (where if is between 5 ft* and r(t> and «̂  is between s m and •+(*) ) . Since * b« $ 2+ a t>, w e have tw^'l* C V ^ ' * * d r . (5.23) o Combining (5.21), (5.22) and (5.23) we see that W A satisfies **** 1 (t-t>*4 J Making the substituting v = W/(t-*) v * i t i s e a s Y t o s e e that - 76 - and hence u s i n g the i n e q u a l i t i e s «t^e _L J. and ™ 3 <2 J ~ w e o b t a i n r ( 5 * 2 4 ) fit — <''4><V)! Thus -X s a t i s f i e s Combined w i t h ( 5 . 1 5 ) , (5 .25) i m p l i e s k i l l (5 .25) (5 .26) (5 .27) - 77 - where C, i s a c o n s t a n t dependent o n l y on the d a t a ( 5 . 4 ) . Now u s i n g (5 .26) and (5 .7 ) we o b t a i n n v  s< Cz ̂  1 I «,,. ̂  , + ^ II V- v« b j + t ' S l V v i A i ^ + t v * IIH-RH T^ t where i s a c o n s t a n t dependent o n l y on the d a t a ( 5 . 4 ) . Now take «j?c(<v) and l e t c-> o s a t i s f y C J » - , / * s I-y . F o r i « ( o , » ) we see from (5 .27) tha t and hence u s i n g (5 .7 ) we have fc*-*^ j A , lV>,-b2l + ^ , * , ' 4 | ^ - v f | l | , 1 1 A j » ' l , H ' f i l l T . That i s , (5 .2 ) h o l d s . To o b t a i n (5 .3 ) we n o t e t h a t eu,U"> w ( « , i ) s a t i s f i e s t h e heat e q u a t i o n i n the r e g i o n j C'f.tV: o<-t<-r| . Hence by the Maximum P r i n c i p l e ( P r o p o s i t i o n 1.1) we see t h a t the - 78 - maximum and minimum values of 9Cx,i.) occur at o , x - o or x - d ( t ) . There are three possible cases. Case I: ) e ( * , i ) | attains i t s maximum value at 1 = 0 , here II6C -,a-> II < || Vii. , (5.29) Case I I : l e ( x , i>| takes on i t maximum value at * d Ci) (say for fixed "fc e U t ) - •*•(+.) ) , here where ^ € Cx, + t t ^ and hence xe(©,5(t.) t€<o,T) Using Lemma 1.3 we obtain n e t ' , * " > t l d(0 5 ™**{ B H!V, M* r tk,| l U ' ^ V • (5.30) Case I I I : . takes on i t s maximum value at -X=o , here we use the i n t e g r a l equation (1.2) to obtain where - 79 - o I t i s easy to see t h a t and hence tha t - 80 - U s i n g ( 5 . 2 9 ) , ( 5 . 3 0 ) , ( 5 . 3 1 ) , (5 .28) and (5 .2) i t i s easy to see t h a t C o n t i n u o u s Dependence f o r A l l Time ^ £°> T3 I f i n Theorem 5.1 we r e p l a c e ftVllu by rtv(-,tH!3(t, ft^rl^ by u W ( . , t)H . ... <*>H. by ft-u, (-.t^,,., H VII ^ by « ^ . ( - j t i H ^ , then we can f i n d a new t \ say and hence a new C " say CT such t h a t (5 .2 ) and (5 .3) h o l d at any t ime y i , C C°. T- <rc] w i t h S>(^)= w ( ^ , ) , - 81 - That i s |j(t)-v<t)| (5 .32) V + o-„ HH-ftll T } d ( i 1 + cr.) < G, ISCV>-V<i,)\ +• T ^ l l V i C - . t . l - v A / t e . l l l j ^ (5 .33) f o r new c o n s t a n t s ^ At , A^ , A j , Q, , B, j 84^ . Now (5 .32) and (5 .33) imply tha t ( 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 d a t a | b, "M6<*I( V>(t)| f o r a l l t ime *t € C o i T J ^ s i n c e we can take A> such t h a t '//j < ^ < y^N^ and a p p l y (5 .32) and (5 .33) s u c c e s s i v e l y at the t imes ^ i * o , <r0, a <r0l.. •, Vi(.*/-/)J to o b t a i n M-+»IT \ / Ai'- A, <r/» A 3o; \ / li>,-ij \ IIH- Rll, - 8 2 - Hence ( 0 . 3 , a , b , c , d ,e) i s c o n t i n u o u s l y dependent on the i n i t i a l d a t a \b, M 0oo ,M t t | fo r a l l i» <c O ,T3 . We remark t h a t f o r n u m e r i c a l c o n s i d e r a t i o n s the s i z e o f the c o n s t a n t s Q t } i* •» *j 3 i s o f c o n s i d e r a b l e i m p o r t a n c e . That they appear to be l a r g e , we f e e l i s a f a i l i n g o f the. method o f p r o o f and no t c h a r a c t e r i s t i c . o f the a c t u a l s y s t e m . Convergence o f the S i m i l a r i t y A l g o r i t h m I f ( v i , 4 ) ( c f . Chapte r IV) i s the s o l u t i o n o f ( 0 . 3 , a , b , c , d , e ) g e n e r a t e d by the S i m i l a r i t y A l g o r i t h m w i t h V\tt> ( the i n d u c e d h e a t f l u x ) and (vi.,s) i s the exac t s o l u t i o n o f ( 0 . 3 , a , b , c , d , e ) we see tha t fts-slb > \ U V - l - ^ U - u ( • , 1 • t-« E»'.*rJ ( •*wu^l.j J 4>i (where T = - \ " T J T ^ and T s a t i s f i e s J ( . T ) - b ^ ) depend <v A o n l y on 1\*A»H\\ — # Hence we must show tha t V\ t ends to as vv̂ ft̂ t <̂ -t̂  _ > e ( c f . Chapter IV) i n o r d e r to p r o v e t h a t the S i m i l a r i t y A l g o r i t h m c o n v e r g e s . Suppose "tj i s a p o i n t o f the p a r t i t i o n 7T then we w i l l show To a c c o m p l i s h t h i s we r e t u r n to the e x p r e s s i o n (3 .14) f o r the S i m i l a r i t y S o l u t i o n . S i n c e we want *u.*K ( S ( i ^ t ^ i ) f o r s m a l l "t we d i f f e r e n t i a t e (3 .14) and e v a l u a t e "M. * ( s ( - v -i), t ) - 83 - a s y m p t o t i c a l l y f o r s m a l l "t ( the d e t a i l e d c a l c u l a t i o n i s g i v e n in A p p e n d i x F ) , To f i r s t o r d e r we o b t a i n "II 4 I f K tx) i s c o n t i n u o u s f rom the r i g h t a t a l l p o i n t s o f the p a r t i t i o n 7t we have h C t i + t > * h t t ^ 4 o(i) as t - > o and hence f o r i <f ( ©, ,) ©<* ( K l i i + t ) * h + Thus V\(t} tends to on £ O ^ T ] as max ^1- —> o and hence we have shown tha t the S i m i l a r i t y A l g o r i t h m c o v e r g e s in the s e n s e o f C h a p t e r IV . Order o f Convergence H a v i n g shown t h a t the S i m i l a r i t y A l g o r i t h m c o n v e r g e s we t u r n our a t t e n t i o n to the r a t e at wh ich i t c o n v e r g e s , i . e . i t s o r d e r o f c o n v e r g e n c e . - 84 - I f (u,s), (u ,S ) a r e as a b o v e , then we say tha t Cltx, tW> and j ttl -> Sit) w i t h o r d e r o f c o n v e r g e n c e and p: r e s p e c t i v e l y p r o v i d e d and r e s p e c t i v e l y as (<vtt) —> d . To e s t a b l i s h the o r d e r o f c o n v e r g e n c e o f the S i m i l a r i t y A l g o r i t h m we assume t h a t W (t) s a t i s f i e s WUi+t)* K(ti)-+O.C.t.) (5.34) f o r a l l p o i n t s "fc.̂  o f the p a r t i t i o n I f , then we have t h a t « W - U l T < 1 ^ ^ ^ M ^ C c ; j < i t i ) - - / J . < ' ( c ^ t < . U ( ^ t ; + ()' /i + 0 ( * t S i n c e V .Cx. ' t ) s a t i s f i e s the hea t e q u a t i o n at x=.S( i) the e x p r e s s i o n tL P vi. ( s t i X t l l - o i m p l i e s t h a t ett J and hence t h a t - 85 - Thus we have Hence i f V\ Ct) s a t i s f i e s ( 5 . 3 4 ) , we have t h a t the o r d e r o f c o n v e r g e n c e o f 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 . S i n c e i n p r a c t i s e i s s m a l l we a s s e r t tha t the e f f e c t i v e o r d e r o f c o n v e r g e n c e o f the S i m i l a r i t y A l g o r i t h m i s between one h a l f and o n e . CHAPTER VI THE SIMILARITY ALGORITHM NUMERICAL RESULTS In this chapter the result s of our numerical experiments with the S i m i l a r i t y Algorithm are given. We present several examples i l l u s t r a t i n g the properties of the algorithm, including i t s order of convergence, and suggest two ways of increasing i t s accuracy. In addition, we attempt to j u s t i f y the use of the large time representation (3.1.3) rather than the small time representation (3.14) i n the S i m i l a r i t y Algorithm. We conclude the chapter by comparing the S i m i l a r i t y Algorithm with Lotkin's Difference Scheme. Numerical Examples By presenting the following numerical examples, we attempt to bring to l i g h t the advantages as well as the d i s - advantages of using the S i m i l a r i t y Algorithm. We f i r s t consider (0.3,a,b,c,d,e) with - 86 - - 87 - 3 / (6.0) For the data (6.0) Sanders [ 31 ] has given the exact solution m « ( t ) ' J t J - " ( i - a ) on o < * •< .sc-n * 0 - * t) v*. Comparisons of the exact solution with the approximating solutions are- summarized by Table 6.0 and Figures 6.0, 6.1, 6.2. Here six terms of the S i m i l a r i t y Solution (3.13) and equal time increments ( AI^ - a t for a l l i ) are used. In each case the approximation is used to 80% of the t o t a l melting time, i . e . T=.4. In what follows we use the notation e^(Ty = I U - J « T , If more than two or three terms of the series (3.13) are used, we have found Filon's Rule for integrating S* * co i eU ( K a real number) (cf. 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 generating the c o e f f i c i e n t s ^ n(/3) (cf. Chapter I I I ) . - 88 - where ^ u. > £ C*,*), J C-O, .r Ci)^ £ Ct) j are defined as i n Chapter IV. Table 6.0 Errors Versus Time Increment (Boundary Data (6.0)) % % •<wt g w( .4) Error (.4) Error e w (.4) .200 . 8 7 ( - l ) ( 2 ) 11 .89(-l) 20 .96(^1) .05 .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) From Table 6.0 and our 'numerical examples i t seems that an accuracy of one to f i v e percent .is e a s i l y obtained.. However, higher accuracy i s d i f f i c u l t to achieve. For instance, we see that with -6.t =.01 the algorithm requires 40 time steps and leads to errors ^ ( . 4 ) , 6 S(.4) which.are smaller than 1% and 27„ r e s p e c t i v e l y . However, for accuracy better than ? ^ ( . 4 ) = .006 (,757o) and e s(.4> = .003 (.6%) more than 400 time steps are necessary. The r e l i a b i l i t y of the l a s t column of Table 6.0, © ^ C T ) t as an indicator of the errors CT) and 6 a CT) i s d i f f i c u l t to assess. However, the solution generated by the (2) v\ Here we introduce the notation ek.C>M= a. x \o a € (0,1), vi an integer, - 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) exactly. Hence, for the S i m i l a r i t y Algorithm (,TV-> O is a necessary and s u f f i c i e n t condition for the convergence of the algorithm. In addition, a rough c a l c u l a t i o n shows that i f i ' ' > i ' then vv% »-y /i I- —> <j 0 ' A where V\ CIO i s the given heat flux and \-»(t) i s the heat flux generated by the algorithm. Hence we consider ^j) to be a "rough" indicator of the errors S^CT) a«d es (.T) • Moreover, ve take the order of convergence of Q. (y)-> o * A to be an estimate of the orders of convergence of © ^ C T i - ^ o and C -> O . We remark that for any scheme leading to an approximate solution of (0.3,a,b,c,d,e), the quantity &̂  (T) can be c a l c u l a t e d . However, i n these cases ( T) -> u would not in general be equivalent to convergence of the corresponding scheme. For instance, i n the cases of f i n i t e difference schemes and the Collocation scheme (c f . Chapter VII) C?w K T ) i s only an indicator of the truncation errors of the schemes. - 90 - R U ( X . T ) VS X TIME=G.40 - 91 - F i g . 6 .1 A p p r o x i m a t i o n s to the P o s i t i o n o f the Boundary Stt) up to T=.4 f o r the Boundary Data (6 .0) Fig. 6.2 - 92 - Comparing >̂ (V> and Mi.) for the Boundary Data (6.0) HEAT FLUX VS TIME - i — a.is -r-I I - 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 Algorithm depends on how c l o s e l y the generated heat f l u x , W("t) , approximates the given heat flux, V% tt} . This i l l u s t r a t e s the proof of convergence. For the S i m i l a r i t y Algorithm to be p r a c t i c a l we should have to use at most six .to eight" terms ,o,f~ the ser.i.es (3) (3.13) during most of the c a l c u l a t i o n . Experimentally, fo^ smooth i n i t i a l temperature d i s t r i b u t i o n s , such.as the one given in (6.0), we find that six to eight terms is more than adequate for results s i m i l a r to those given in Table 6.0. However, more terms- of the series i n (3.13) are necessary when the i n i t i a l temperature d i s t r i b u t i o n is r i c h in the higher frequencies. The number of required terms is governed by how cl o s e l y (3.13) evaluated at " t 5 0 reproduces the i n i t i a l condition. Although i n i t i a l l y a r e l a t i v e l y large number of terms may be required, the following example (see F i g . 6.3) shows that during a r e l a t i v e l y short i n i t i a l period of time (short compared to the t o t a l melting time) the higher frequencies are largely attenuated. This is a consequence of the d i s s i p a t i v e character of the.heat equation. Hence only the f i r s t few terms (3) A bound on the error made in truncating the series in (3.13) w i l l be given l a t e r in this chapter. - 94 - o f (3 .13) need be r e t a i n e d f o r most o f the c a l c u l a t i o n . As an example , we c o n s i d e r ( 0 . 3 , a , b , c , d , e ) w i t h ~ u 0 u \ - u~>) e (t) - '°/3 J> = 4 and use the S i m i l a r i t y A l g o r i t h m w i t h e q u a l t ime i n c r e m e n t s •oA =.001 and f i f t e e n terms o f the s e r i e s i n (3 .13) to o b t a i n an approx imate s o l u t i o n . F i g . 6.3 The Approximate Temperature Distr.ibu'tion for t. B'etw.een 0 .and .1 (6.1) for the Boundary Data (6.1) - 95 - F i g u r e 6.3 shows the approx imate tempera ture d i s t r i b u t i o n a t t =0, . 0 0 1 , .01 f o r the boundary d a t a ( 6 . 1 ) . I t can be seen t h a t a t t. =.01 (about f i v e p e r c e n t o f the t o t a l m e l t i n g t ime) the h i g h f r e q u e n c y components o f the i n i t i a l t empera tu re d i s t r i b u t i o n have been s i g n i f i c a n t l y damped. Hence by t h i s t ime fewer than f i f t e e n terms o f (3 .13) a r e needed i n the c a l c u l a t i o n . F u r t h e r m o r e , our n u m e r i c a l exper iments i n d i c a t e t h a t the i n i t i a l t empera ture d i s t r i b u t i o n i s o f impor tance o n l y i n i t i a l l y d u r i n g the c a l c u l a t i o n . To a l a r g e e x t e n t the l o n g term b e h a v i o u r o f the s o l u t i o n o f ( 0 . 3 , a , b , c , d , e ) seems to be independent o f the shape o f the i n i t i a l t empera ture d i s t r i b u t i o n . O p t i m i z a t i o n o f the S i m i l a r i t y A l g o r i t h m So f a r we have made no at tempt to o p t i m i z e the a c c u r a c y o f the a l g o r i t h m . T h i s can be a c c o m p l i s h e d by c h o o s i n g an o p t i m a l p a r t i t i o n 7T a n d , o r an o p t i m a l v a l u e at each t ime ~t± o f IT . I t i s c l e a r t h a t any s t r a t e g y aimed at o p t i m i z i n g t h e s e c h o i c e s s h o u l d be gu ided by a d e s i r e to have V\ (t 1 approx imate ta<i} c l o s e l y ( the p r o o f o f c o n v e r g e n c e ) o r at l e a s t tha t j K i r ) , !? approx imate ) M t W r w e l l f o r o o a l l i. . Below we i n t r o d u c e two m o d i f i c a t i o n s to the S i m i l a r i t y - 96 - Algorithm as a step towards optimization. We f i r s t note that the S i m i l a r i t y Algorithm provides an exact solution when the boundary moves at a constant speed. Hence when the boundary moves at a slowly varying speed, i . e . |i*Ct)| small, the straight l i n e approximation to the boundary should lead to better resu l t s than when \ ^ C t l \ i s large. Thus we concentrate points of the p a r t i t i o n II during periods of time when \ J C--k\\ i s large, which, for the most part, corresponds to periods of time when V»(t) ± s changing most r a p i d l y . As an example, for the data (6.0), the points 4.̂  of 77* can be taken to s a t i s f y I: j U(r).lT= C • ti for an appropriate constant C>0 . The next modification i s motivated by the proof of convergence and i s aimed at optimizing the choice of on the time i n t e r v a l £t^v"^<+\) f ° r a given p a r t i t i o n 7T The strategy i s to add i n the time i n t e r v a l r^o^< + i) a portion of the heat which was missing i n the previous time i n t e r v a l E^N'-I, "tj ) • More p r e c i s e l y , we choose /3(- o n t * i , tj-M^ t o s a t i s f y I I : / 3 . - M « | M V ) + * - ( W ( i i O - £ < t ; - ) ) - < " f C ; , * t ; ) j - 97 - for some >,«? (see F i g . 6 .4 ) . 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 Algorithm Using Modification II Time - t We remark that both modifications I and II 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 of the above modifications, we consider (0.3,a,b ,c,d,e) with the data (6 .0) . With C= )K(Mc!r and V, - \ for a l l i we obtain the following r e s u l t s . Table 6.1 Error Versus Time Increment -<=>*t Using Modifications I and II (a) fit ( .4) • - Error i n JCt) Unaltered Algorithm Modifications i II. I and II .05 . -27(-l) 6% T22(-l) 57. .13C-1) 37. .99(-2) 27. .01 •75(-2) 1.77. •65(-2) 1.5% •40(-2) .9% •36(-2) .8% .005 •58(-2) 1.3% .44<-2) 1.07. • .31(-2) .7% .29(-2) .77. - 98 - (b) e..(.4) - Error in v. l*,t) x Unaltered Algorithm .05 -25(-l) 31 .01 .84(-l) IS .005 .72(-l) .9% .22(-l) 31 .80(-2) 1% .70(-2) .9% Modifications II .13(-1) 2% .68(-2) .9% .64(-2) .87. I and II . l l ( - l ) 1% .66(-2) .8% .64(-2) .8% (c) gh(.4) - Error i n HCA) A ! .05 .01 .005 Unaltered Algorithm .30(-l) .86(-2) .58(-2) I •30(-l) -76(-2) .53 (-2) Modifications II •15(-1) .48(-2) .39(-2) I and II .14(-1) .44(-2) •36(-2) As is to be expected the modifications are most e f f e c t i v e for the larger time increments. However, even for the shorter time steps the improvement i n accuracy i s s i g n i f i c a n t . Modification II proves to be very useful, reducing ^-u. («4)> Q_5(.4) and e^(.4) from ten to f i f t y percent. We remark that Modification I increases the accuracy of the approximations although WCti of (6.0) i s actually slowly varying for t e [o.,.4] ( h(0)=5.33, h(.4)=7.36). Order of Convergence of the S i m i l a r i t y Algorithm In Chapter V we showed that the order of convergence - 99 - of the S i m i l a r i t y Algorithm i s one h a l f . That i s , we showed that for a l l T less than the t o t a l melting time © 4 C T ) " O U ^ * * as w i e-y -a "t • —> i Pt where ^ j , : ^ : . This i s important i n that i t . explains our observation that the S i m i l a r i t y Algorithm should be used only to obtain coarse accuracy. In Chapter V we observed that the c o e f f i c i e n t multiplying the order one ha l f term of the error expansions of ? U("T) and & S { T ) is usually small, hence, the e f f e c t i v e values of and are larger than one h a l f . Hence the S i m i l a r i t y Algorithm should be s i g n i f i c a n t l y better than an order one h a l f scheme. Here we give some numerical examples which support that -claim. For exact solutions we use those given by Sanders £ 31 In p a r t i c u l a r , for the data fact (6, - 100 - Sanders gives the solutions J 2 J 0-**t) (6.3) on o < A < ittr- V \-«A t) * . Here M ( n ; t;^) are the Confluent Hypergeometric Functions, and arid A are related by the condition that: \,a is the smalle.s„t positive root of the equation M (-X0 ^ - i ; ^ ) - o The data (6.0) corresponds to A =.5, A 0 = l . We also consider the data i with various i n i t i a l temperature data •U, Ur- x ' - l ) (6.3a) Ue0O = (x-Oe . (6.3b) V0«,XN-- x- I •. (6.3c) To the data (6.2) and (6.3) we apply the Similarity Algorithm with equal time steps, A "t , varying from five to thirty percent of the total melting time. For the data (6.2) we are able to estimate the order of convergence of - 101 - CT) -50, However, for the data (6.3) we must s e t t l e for the order of convergence of since the corresponding exact solutions are not a v a i l a b l e . In each case the algorithm is used to approximately f i f t y percent of the t o t a l melting time. Table 6.2 provides a summary of the r e s u l t s . Table 6.2 Observed Order of Convergence, D a t a g,(T)->o <=>V4CT)->o 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 - 102 - T a b l e 6.2 s u p p o r t s our c l a i m t h a t the o r d e r o f c o n v e r g e n c e o f the S i m i l a r i t y A l g o r i t h m i s between one h a l f and o n e . M o r e o v e r , i t p r o v i d e s some e v i d e n c e t h a t the o r d e r o f c o n v e r g e n c e o f ;T> - > o i s c l o s e l y r e l a t e d to the o r d e r s o f c o n v e r g e n c e o f f>jCT)->o and _> 0 The S m a l l Time V e r s u s the Large Time R e p r e s e n t a t i o n o f the S i m i l a r i t y S o l u t i o n In t h i s s e c t i o n we g i v e o p e r a t i o n counts f o r one i t e r a t i o n (one t ime s t e p , i . e . t-i to t ^ , ) o f the S i m i l a r i t y 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 (3 .13) ( l a r g e t ime) and (3 .14) ( s m a l l t ime) r e s p e c t i v e l y . M u l t i p l i c a t i o n s , d i v i s i o n s and a d d i t i o n s are c l a s s i f i e d as e q u i v a l e n t o p e r a t i o n s , w h i l e e x p o n e n t i a t i o n s and square r o o t s a re taken to be e q u i v a l e n t to twenty and f i v e o p e r a t i o n s r e s p e c t i v e l y . One i t e r a t i o n o f the S i m i l a r i t y A l g o r i t h m u s i n g the l a r g e t i m e r e p r e s e n t a t i o n (3 .13) i n v o l v e s a p p r o x i m a t e l y 40 + 50 m + 3 n + 4 m n o p e r a t i o n s , where n terms o f the s e r i e s i n (3 .13 ) a r e r e t a i n e d and the n e c e s s a r y q u a d r a t u r e s a r e p e r f o r m e d by means' o f a 2 m - p o i n t F i l o n I n t e g r a t i o n R u l e . On the o t h e r h a n d , one i t e r a t i o n o f the S i m i l a r i t y A l g o r i t h m u s i n g the s m a l l t ime r e p r e s e n t a t i o n (3 .14) r e q u i r e s 2 30 + 70 J + 90 JK + 40 J + 80 J K o p e r a t i o n s , where 2 + K terms o f the s e r i e s f o r G C v , t - ^ ) ( s e e (6 .5 ) ) a re r e t a i n e d and - 103 the i n t e g r a l i n (6 .5 ) i s e v a l u a t e d u s i n g a j node q u a d r a t u r e r u l e . To i n i t i a t e the c o m p a r i s o n we w r i t e (3 .13) ( "u.'^ ( 3 . 1 3 ) ) and (3 .14) ( u * 4 <—> (3 .14 ) ) as (6 .4 ) s&n(P<y- ^i$C' & S,«t;) c o s ( w ^ « l . and (6 .5 ) where T^S X / C , ^ , } i A c ( C ; 4 L and the n o t a t i o n o f C h a p t e r s I I I and IV has been u s e d . Our aim i s to compare the number o f o p e r a t i o n s n e c e s s a r y to c a l c u l a t e u * ( C ^ 4 ( ̂ , - a l ± + l ) to a g i v e n a c c u r a c y 104 u s i n g (6 .4) and (6 .5) r e s p e c t i v e l y . However , the -term ? i n both (6 .4) and (6 .5) m o t i v a t e s us to c o n s i d e r i n s t e a d , the number o f o p e r a t i o n s n e c e s s a r y to c a l c u l a t e •̂c»+>\> A"^t-n ^ e *** ^ t o a p r e s c r i b e d a c c u r a c y , say s 0' ^ . In each c a s e the e r r o r e n t e r s from two s o u r c e s - the e r r o r made by t r u n c a t i n g the s e r i e s and quadrature , e r r o r made i n e v a l u a t i n g the i n t e g r a l s . We f i r s t c o n s i d e r the l a r g e t ime r e p r e s e n t a t i o n (6 .4 ) by w r i t i n g I n t e g r a t i n g the e x p r e s s i o n f o r -^>^(j^-^ t w i c e by p a r t s we o b t a i n " U , ( f t > l 5 -L-l ^ ' ^ ( c " / J . C ^ V , ( c , 3 i a i ) ) c e , ( ^ ) c l ^ ^ 0 c ^ Hence i t i s easy to see tha t 0 0 , , where 105 - Evaluating the above integral by parts we have \ R ^ S -TT*(2n- l>H t (6.6) Moreover, i f , TT, (2^,-0 ̂  * is. lftfge, enp.ugh we,, obtain the. asymptotic estimate le^(oK< ^ - \ H L 4 e _ . (6.6a) The expressions (6.6,a) give us a "rough" estimate of the number of terms, n , of the series (6.4) necessary to achieve a pres'crrbed -accuracy for a given We now focus our attention on the quadrature error a r i s i n g from the c a l c u l a t i o n of ^ ^ ^ ^ j ^ ' 1 ! ' " ] * 1 by a 2 m - point F i l o n Integration Rule. Suppose -Sv-(p') is the F i l o n approximation to y^^.(^-), then we can write where o<t^<i (cf. Davis and Rabinowitz [ 9 ] p. 64). If we take w large enough so that v-j^ / % ^ .$ "V^ t then the t o t a l - 106 - error due to the quadratures, c a l l i t , can be seen to s a t i s f y where Evaluating the above trigonometric sum we a r r i v e at the estimate Since j ^ i x j j x for x >, o we can write (6 Hence we have Now turning our attention to the small time representation (6.5), we write 107 *3 <3>s e ' ^ C < , 4 , S U * ( C ; 4 | j ; * t i + 1 ) 11 • ^ 4-. + e S i n c e we have 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- W R , V *«- where K . = 0<H <• » (6.8) To i n v e s t i g a t e the q u a d r a t u r e e r r o r i n v o l v e d i n - 108 - e v a l u a t i n g the i n t e g r a l i n ( 6 . 5 ) , we no te t h a t the dominant term o f G ( C - 4 l ̂ , A t ; + 1 j c-^) i s a lways Q ' ^ & ' & ^ I . That i s to s a y , s i n c e a l l o t h e r terms o f G ( Ĉ , ̂ , , \ ^i^) a r e w e l l - b e h a v e d f o r ° * 3 > * ' > t n e s o u r c e term l a r g e l y d e t e r m i n e s the r e q u i r e d number of q u a d r a t u r e n o d e s , J Hence we choose J l a r g e enough to e v a l u a t e to 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 J to be r e p r e s e n t a t i v e o f the number o f nodes n e c e s s a r y to e v a l u a t e to the same a c c u r a c y . To compare the o p e r a t i o n counts we note t h a t \TT js. 1 . Hence we s e t j — = i and f i n d n , m and K so t h a t l ^ C ^ l ) o f ( 6 . 6 ) , ) c ^ | o f ( 6 .7 ) and I e j | o f (6 .8) a r e each l e s s than 10*e' f o r g i v e n v a l u e s o f . d 5 L a and K, - 109 - For values of S; we take .001, .01 , . 1 , 1. (a typ i c a l range over which varies during the course of a ca l c u l a t i o n ) . Since K, has very l i t t l e influence on the magnitude of K ( X =0,1,2) we set V<j=l. Furthermore, to assess the e f f e c t of the magnitudes of L.( and on the operation count, for*, t-hse* larger time., representation-, we3 vary both and between 1 and 100. Table 6.3 provides a summary of the r e s u l t s . Table 6.3 Approximate Operation Count (a) *d -=4 .001 .01 .1 1. 0 1 0 1 1 2 ,(4) 25+ 25+ 15 15 5 5 # of Operations Small Time S o l u t i o n n m 27,000 75,000 9 9 11,000 28,000 4 4,000 2 6,000 1 of Operations Large Time S o l u t i o n 1,000 700 300 200 14 38 5 19 2 10 1 10 # o f Operations Large Time . S o l u t i o n L, : I , : 100 7,000 1,700 700 700 (4) Here we have used a Gaussian quadrature scheme (cf. Isaacson and K e l l e r [ 2 l J p. 327) to evaluate the i n t e g r a l in (6 .5 ) . - 110 - (b) cl =6 r .001 .01 .1 K l l l 2 J 25+ 15 10 5 t of Operations Small Time Solu tion n 75,000 28,000 13,000 6,000 14 36 5 18 3 18 1. 9 # of Operations Large Time Solution 5,000 2,000 1,500 700 n m 17 114 6 52 3 52 2 29 # of Operations Large Time Solution 21,000 5,000 4,000 2,000 We remark that the number of nodes given in Table 6.3 for Filon's Integration Rule is larger than was used for any of our numerical experiments (usually m was taken between 10 and 6 20). Moreover, we note that 10 is normally the l i m i t of accuracy which one would want, since we are wasting computing time i f we try to make the truncation and quadrature errors s i g n i f i c a n t l y smaller than the inherent error i n the actual approximation to (0.3,a,b,c,d,e) generated by the S i m i l a r i t y Algorithm. Although (3.14) is the small time representation of the S i m i l a r i t y Solution, Table 6.3 shows that i t i s numerically impractical to use i f the i n t e g r a l i s calculated by a - U l - conventional quadrature r u l e . That i s , because the dominant term of G ^U,y C;^) for small behaves l i k e a delta function in about ^ , a conventional quadrature scheme requires a r e l a t i v e l y large number of nodes, covering the whole of the i n t e r v a l [ 0 , l j , to achieve the necessary accuracy. Moreover, we remark- that i n estimating J we assumed u v C,-̂ , <&X;)<2 * a to be a constant. Hence the number of operations given in Table 6 .3 for the small time representation could be an underestimate. Furthermore, since ^ (̂ >,")j are independent of <it- and ^. , the cosine, terms which enter Filon's Rule need be calculated only once during the entire c a l c u l a t i o n . At each step these constitute the weights of the F i l o n Quadrature Rule. This i s in sharp contrast with the c a l c u l a t i o n of the i n t e g r a l appearing in the small time representation (3 .14) where at each time step " i ; } G ( C;+) ^ t ^ , ' C;vj\ must be calculated at a l l quadrature points i n ^ and in ^ . That i s , i f . a r e the nodes of the quadrature scheme then at each time "t̂ ' G C C u , ^ , ^ . - / ^ ^ ) must be calculated. These resu l t s indicate that the large time represent- ation (3 .13 ) is better than the small time representation (3 .14) for the numerical solution of (0.3,a,b,c,d,e) using the - 112 - S i m i l a r i t y Algorithm. Table 7.3 also provides us with an estimate for the number of required terms for -the large time representation (3.13). We can see that unless -u.*"' (C; ̂  > o,*t; ) i s exceptionally misbehaved (reflected i n the values of L, and L i ) at most twenty terms of the series i n (3.1,3) need b.e us.ed. For reasonably behaved functions three to ten terms are adequate. Our numerical experiments support these statements. Comparison of the S i m i l a r i t y Algorithm with Lotkin's Difference Scheme We conclude this chapter by comparing the S i m i l a r i t y Algorithm with Lotkin's Difference Scheme. Lotkin [ 23 ] transforms to a fixed boundary by making the transformation ^ = X/.SCi) i n (0.3 ,a ,b ,c , d , e) . Then he employs centered difference approximations (cf. Isaacson and Kell e r [ 21 J p . 445) for the s p a t i a l derivatives appearing i n the r e s u l t i n g d i f f u s i o n equation, together with backward > difference approximations ( c f . Isaacson and K e l l e r [ 21 ] P. 445). for both -u..C»,"t) and i(fc) in the transformed version of (0.3e). The resulting non-linear system of difference equations is solved i t e r a t i v e l y . I f a uniform mesh is taken i n ^ , then the above scheme i s second order accurate i n space and f i r s t order - 113 - accurate i n time. In comparing the schemes, we use the data (6 .2 ) with A = . 5 , .85403 , and 1. In each case the approximations are employed to approximately ninety percent of the t o t a l melting time. In the S i m i l a r i t y Algorithm three terms of the series in (3 .13) are used and the errors given are »?j(T) and 6vi. CT) • For Lotkin's Scheme nine i n t e r i o r mesh points are used and the errors given are the maximum absolute errors at these mesh points. The result s are summarized by Table 6 . 4 . Table 6 .4 S i m i l a r i t y Algorithm Versus Lotkin's Difference Scheme S i m i l a r i t y Algorithm L o t k i n ' s D i f f e r e n c e Scheme T=.45 A=.5 T=.22 A=l: .01 .005 T=.26 A-.85403 .01 .005 .01 .005 .32C-1) .32C-1) -(5), Computer ' E r r o r i n E r r o r i n Computer g» CT) Time(Sec) uU.-t) J C t ) Time(Sec) .62(-2) .69(-2) ,57(-2) .45(-2) .24(-l) .10 ( -1 ) ,24(-l) .79(-2) .12C-1) •91(-2) .14 .24 .07 .16 .06 .12 .70(-2) .97(-2) .08 •37(-2) .55(-2) .12 .45(-l) •24(-l) .89(-l) •58(-l) •16(-1) •96(-2) •22(-l) •13(-1) .06 .09 .05 .09 (5) A l l calculations were done on the IBM 370 /168 . - 114 - I t can be seen f o r these examples , t h a t L o t k i n ' s Scheme and the S i m i l a r i t y A l g o r i t h m g i v e comparable a c c u r a c y f o r a p p r o x i m a t e l y the same amount o f comput ing t i m e . We remark tha t i f g r e a t e r a c c u r a c y i s r e q u i r e d , then L o t k i n ' s Scheme i s the more e f f i c i e n t a l g o r i t h m . CHAPTER VII A COLLOCATION SCHEME In this chapter we consider (0v3 ? ,a ,b;,c ,d , e) from a v a r i a t i o n a l point of view i n order to develop algorithms for approximating i t s s o l u t i o n . Our ultimate aim i s to achieve a f i n i t e element formulation of ( 0 . 3,a,b,c ,d , e) . The Lagrangian Equations for Heat.Conduction To i n i t i a t e a variational-formuTation of (0 .3,a,b,c, d,e) we follow the lead of Biot £ .1 ] by defining §f-C*it), referred to as the heat displacement f i e l d , to be the time i n - tegral of the rate of heat flow across a unit cross se c t i o n a l area of a given slab. With t h i s d e f i n i t i o n the equation of heat conduction can be written as | M ) * 5 L [ § C * ( t>}v ~ U y U , t ) . ( 7 . 0 ) In addition, the law of conservation of energy is expressed by the r e l a t i o n I x M : - «*HU , t ) . ( 7 . 1 ) - 116 - To o b t a i n the L a g r a n g i a n e q u a t i o n s f o r heat c o n d u c t i o n as d e r i v e d by B i o t [ 1 ] we f i r s t l e t "uC*,"i\ and §(x,t) be the tempera tu re d i s t r i b u t i o n and heat d i s p l a c e m e n t f i e l d r e s p e c t i v e l y a s s o c i a t e d w i t h ( 0 . 3 , a , b , c ,d ,e) . Then we c o n s i d e r a r b i t r a r y v a r i a t i o n s o f the hea t d i s p l a c e m e n t f i e l d wh ich a r e c o n s i s t e n t w i t h the c o n s e r v a t i o n o f energy r e l a t i o n (7 .1 ) and the boundary c o n d i t i o n s ( 0 . 3 , c , e ) , i . e . Su(«,u-- §K<«,t) and F o r -any i n t e r v a l (a,.b) a l o n g the s l a b , (7 .0 ) i m p l i e s t h a t ^ b . (1) ° - ) [ ^ ^ x ^ . t H § U,t>] S f U . t W U . ( 7 . 3 ) a Upon i n t e g r a t i n g by p a r t s the f i r s t term o f (7 .3) and u s i n g the c o n s t r a i n i n g r e l a t i o n s (7 .2) we o b t a i n (7 .2 ) 4) x-a. where S i n c e (7 .3) must be s a t i s f i e d f o r a l l t i m e , the l i m i t s o f i n t e g r a t i o n a and b can be taken to be f u n c t i o n s o f t i m e . In f a c t , we w i l l take e s : o ) b - - i ( t ) ( ( /u ,s ) s a t i s f y i n g ( 0 . 3 , a , b , c , d , e ) ) . - 117 - The v a r i a t i o n a l p r i n c i p l e (7 .4 ) l e a d s to a s e t o f e q u a t i o n s r e f e r r e d to by B i o t ^ 1 ] as the L a g r a n g i a n e q u a t i o n s for h e a t c o n d u c t i o n , i f we assume t h a t $ ( * , i > can be e x p r e s s e d as a g i v e n f u n c t i o n o f x and %. and a t most a c o u n t a b l e s e t o f i ndependent parameters ( g e n e r a l i z e d c o o r d i n a t e s ) ^ ? « ^ ^ | - » i . e . $(«,tu $<•$,,... 'hen f o r a r b i t r a r y v a r i a t i o n s ^ ^ ^ ' ^ ^ j . * n t ^ i e P a r a m e t e r s \ V^\- c o n s i s t e n t w i t h ( 7 . 2 ) , &$(x,i), the it* v a r i a t i o n o f the hea t d i s p l a c e m e n t f i e l d i s g i v e n by In a d d i t i o n we have the r e l a t i o n and h e n c e *J . (7 .6 ) M o r e o v e r , s i n c e V(«i,b^"fcV i s a l s o a g i v e n f u n c t i o n o f the parameters ^ C t ) | % we have - 118 - (7 .7 ) I n t r o d u c i n g (7.5), (7 .6 ) and (7 .7) i n t o the v a r i a t i o n a l p r i n c i p l e (7 .4) we o b t a i n f °<M i?* F + ^ r? ^ ° i (7 8) where we have i n t r o d u c e d the d i s s i p a t i o n f u n c t i o n S i n c e the parameters | Ctlj, can be v a r i e d i n d e p e n d e n t l y , (7 .8 ) i m p l i e s tha t **« * l . ^ « ( } - the L a g r a n g i a n E q u a t i o n s f o r heat c o n d u c t i o n . In o r d e r to use ( 7.9) we take (w,i) to be the s o l u t i o n o f the s y s t e m of e q u a t i o n s ( 0 . 3 , a , b , c , d , e ) and l e t i v ; t , c M . be a s e t o f b a s i s f u n c t i o n s f o r L lT«, i"] ( the s e t o f s q u a r e i n t e g r a b l e f u n c t i o n s on C°» l l , ) w i t h the p r o p e r t y \>; Co\~ -v-(I) = ° f o r a l l i * v t l t . . . , F u r t h e r m o r e we assume t h a t 5 ( « | t ) can be w r i t t e n as the sum - 119 ^;(5t,). (7.10) The e q u a t i o n s (7 .9 ) then become the d e t e r m i n i n g e q u a t i o n s f o r * T h a t i S ' W e t a k S C k : o i ' 3 : S ( t l ; s u b s t i t u t e Vfo . i t t^t^ O <°> -s<*»;t) i n t o (7 .9 ) and n o t e t h a t 5 J t) = O, S2?te,t)=o to o b t a i n S*<t) C,^(*j • ( A,- i(.tu(t) Q,) ^<t> = o (7 .11) where o .1 « o f ( 0 = . . ( , | , C « , . . ; ; ^ ( « , . . . ) T . M o r e o v e r , i f (-u.s) i s to s a t i s f y (0 .3d) then |^<t) must s a t i s f y an a p p r o p r i a t e i n i t i a l c o n d i t i o n . I f 5 (*<°)= °<* $0**) then we take ^o") to s a t i s f y ^ [ f o ^ ' I ^ ^ ^ C ^ l ^ C t l ^ x ^ f o r a l l y . l J ? J . . . ; t h a t i s where - 120 - < 2 > and r 1 L f - V } $ 0 U b ) ^- txUx.- Roughly speaking, ^i(o) is the projection of 58<x) onto ^ - ( V j j ) • F i n a l l y the Stefan Condition, (0.3e), becomes (3) where (7 i C© )=b Even i f the i n i t i a l value problem (7.11), (7.12), (7.13) has a sol u t i o n i t would be d i f f i c u l t to obtain because of the coupling of the derivative terms -i(t), »0 in (7.11) and (7.13)* Hence we seek ways of reformulating (7.11) and (7.13) in order to avoid this d i f f i c u l t y . (2) For a given 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) , is not uniquely determined.. Since we are interested in *u. O,-t,) and M^(v;-t) only we take § U,-tl- - 5 ^^."t' 1 c ^ • As noted by Biot £ 1 ] the determination of JCt) is not part of the v a r i a t i o n a l procedure but merely another ordinary d i f f e r e n t i a l equation added to. the Lagrangian equations - 121 - I f i n the S t e f a n C o n d i t i o n (0 .3e) we use ~ <s<*\J^ f o r ©('v^Utt), t> i n s t e a d o f - § U c u , - t ) then (7 .13) becomes and the i n i t i a l v a l u e problem to be s o l v e d becomes - 0 where (7 . J C o ) - b Another way by which we can e l i m i n a t e the d i f f i c u l t i e s o f (7 .11) and (7 .13) i s s u g g e s t e d by the work o f B i o t [ 1 ] . I n s t e a d o f e x p r e s s i n g 5 ( . * » t ) as the l i n e a r c o m b i n a t i o n (7.10) , we w r i t e E p i t t e d - */ J ( t ) ) . 4Ct> is, *w> d e f i n e o and use (7 .1 ) to o b t a i n 122 5 jf <Pi<ky^ P r o c e e d i n g as b e f o r e we o b t a i n the i n i t i a l v a l u e prob lem * n S*(t> CjiplO + ( A a - iit>.sit> QA) J i t ) * O -SU>= - $ -f £ ft.(t)v?(o) + hCt)7 C 4pto)= J. .S<.o)-W where ^ 0 - 123 - I t i s i n t e r e s t i n g to no te tha t by t a k i n g V»v\ (x^* - (,v%-^)TTi i uw-fC'n" j)TT xj 4 (7 .15) becomes the system o f e q u a t i o n s o b t a i n e d by V . G . Melamed ( c f . R u b i n s t e i n £ 3 0 ] Chapte r 8) Melamed reduces ( 0 . 3 , a , b , c , d , e ) to a denumerable system o f d i f f e r e n t i a l e q u a t i o n s by assuming t h a t oo ^ (4) • U U A ) * X A ^ l O e o i f V A - x j - T r . M o r e o v e r , he has shown that- i f an a p p r o x i m a t e s o l u t i o n i s o b t a i n e d by c o n s i d e r i n g the f i r s t - ,V A/ c o e f f i c i e n t s ] Ct) ( then as / V - > c o the a p p r o x i m a t i o n c o n v e r g e s . In o t h e r w o r d s , he takes f o r an approx imate b a s i s the f i r s t A/ f u n c t i o n s | cos j[(vv-l)'TrxJ | and o b t a i n s an a p p r o x i m a t i o n s o l u t i o n to ( 0 . 3 , a , b ,c . ,d ,e) by s o l v i n g the a p p r o p r i a t e l y t r u n c a t e d v e r s i o n o f ( 7 . 1 5 ) . I n s t e a d o f p r o c e e d i n g as a b o v e , i . e . u s i n g , f o r an a p p r o x i m a t e b a s i s f u n c t i o n s which a re g l o b a l on £ o , l ] , we p r o p o s e a f i n i t e e lement f o r m u l a t i o n o f (7 .14) and ( 7 . 1 5 ) . We w i l l s t a r t w i t h an a p p r o x i m a t e b a s i s , c o n s i s t i n g o f f i n i t e e l e m e n t s . F o r c o n v e n i e n c e t h e s e b a s i s f u n c t i o n s w i l l be l a b e l l e d so t h a t the systems we o b t a i n w i l l be a p p r o p r i a t e l y t r u n c a t e d v e r s i o n s o f (7 .14) and ( 7 . 1 5 ) . In p a r t i c u l a r we w i s h to e x p r e s s uCK,t) as a l i n e a r (4) T- T h i s m o t i v a t e d us to w r i t e g(.x,"t) i n the form g i v e n by ( 7 . 1 0 ) . 124 - c o m b i n a t i o n o f p i e c e w i s e c u b i c H e r m i t e p o l y n o m i a l s . To t h i s end we d e f i n e a p a r t i t i o n 77" o f the i n t e r v a l [ o , l ] : and l e t where 9^-tlx) i s the c u b i c H e r m i t e p o l y n o m i a l d e f i n e d by w i t h x % Jf, s VCx) = and we h a v e i n t r o d u c e d the n o t a t i o n - 125 * vs > ( c f . s c h u l t z [ 32 ] Chapte r 3). I t . i s . w e l l known.that fp„r v f ixed; . "fc a», fu.nct.ion• on £o , l ] , wh ich has s u f f i c i e n t l y w e l l behaved d e r i v a t i v e s , can be a p p r o x i m a t e d a r b i t r a r i l y w e l l i n m e a n ^ ^ by a l i n e a r c o m b i n a t i o n o f f u n c t i o n s i n ^/^(Tf**), i . e . f o r a s e t o f f u n c t i o n s > G 'U,i>*r E 9^<*) ~> g<*,*^ i n mean f o r f i x e d t as V\ —> o . A g a i n , f o r ^ t *> '^ s u f f i c i e n t l y smooth i t i s known t h a t , f o r f i x e d "t , the d e r i v a t i v e s G^ U,t} ; ( *, t)t G ? IK, * > tend to the d e r i v a t i v e s ^ t / x . - t ' i , Q*« (*,i) , ̂ w (v,t) r e s p e c t i v e l y i n mean Hence t a k i n g j v . o r t.VjU) | ^ . . W j •• •. <^W*! the sys tems (7.14) and (7.15)7 become "* The sequence of f u n c t i o n s V,*. ( O G L*C°,»] -r\z\,3t, i s s a i d to c o n v e r g e in mean to a f u n c t i o n •^"(,0 6 L* C«*, il p r o v i d e d S - V W}'<1* -> o as v\ -> «o • - 126 - and s'cof] Q(i) + (tf,-i<tmty3,) QCt) = o 5 (7 .16) (7 .17) r e s p e c t i v e l y . H e r e | > /^*" \ > ' " 1 ' * a re the a p p r o p r i a t e l y t r u n c a t e d v e r s i o n s o f | C ̂  , A« , | , i - 1> 2. r e s p e c t i v e l y and - 127 - We see t h a t and f"̂  a r e n o n s i n g u l a r s i n c e they are Gram M a t r i c e s o f l i n e a r l y independent f u n c t i o n s . Thus b o t h (7 .16) and (.7...17) can be s o l v e d as i n i t i a l v a l u e p r o b l e m s . The q u e s t i o n o f c o n v e r g e n c e o f the schemes o u t l i n e d by (7 .16) and (7.17) i s u n r e s o l v e d . A G a l e r k i n Scheme We n o t e t h a t , because the b a s i s f u n c t i o n s I ^ P w ^ * * | each have s u p p o r t ' C X K . , , x K + ,] ( X o - o , X W 4 ?= i ) t H*\,..., A/+ / y j P, , a/,; ̂  (| are b l o c k t r i d i a g o n a l m a t r i c e s , w h i l e ^ f \ } °it } f^*-1 a v e m a t r i c e s . H o w e v e r , the e s t i m a t e f o r • u x c i ( t » , t ) u s e d i n - 128 - (7 .17) i s b e t t e r than the e s t i m a t e used i n ( 7 . 1 6 ) , s i n c e i n the former we u s e the a p p r o x i m a t i o n w h i l e i n the l a t t e r we u s e tke estimate v.x.W.tt\ We would l i k e to a c h i e v e a f o r m u l a t i o n which combines the b e t t e r a p p r o x i m a t i o n o f -u x (s (0 , - r^ i n (7 .17) w i t h the c o m p u t a t i o n a l advantage o f the s p a r s e m a t r i c e s o f ( 7 . 1 6 ) . How to p r o c e e d becomes apparen t i f we c o n s i d e r the f i r s t e q u a t i o n s o f the systems (7 .14 ) and (7 .15) , . .r.espect i v e l y , i . e . . and M r 1 " 1 I n t e g r a t i n g by p a r t s the m i d d l e t e r m , t h e n s u b s t i t u t i n g the a p p r o p r i a t e e x p r e s s ions f o r 5 ("i"^ > 5XK C><i't\, we o b t a i n 129 and r e s p e c t i v e l y . That i s , the L a g r a n g i a n e q u a t i o n s do no more than f o r c e the heat d i s p l a c e m e n t f i e l d j>(x,"fc) to s a t i s f y the heat e q u a t i o n i n a weak s e n s e . T h i s s u g g e s t s t h a t we f o r e g o the i n t r o d u c t i o n o f S§C*i"t} and i n s t e a d work d i r e c t l y w i t h v<Cx,-fc) . That i s we w r i t e where now i s a b a s i s f o r L*C«>(il w i t h x>J(o) =• v . O V - o ia-i, a, • • . and the p a r a m e t e r s , are to be de te rmined by the G a l e r k i n c o n d i t i o n s 0 0 I 1 ^ ( S V< V-(X) dJt ) cl; <o) r ̂  V o ( V > 0 v In o t h e r w o r d s , we f o r c e vt(.x,-t) ' t o s a t i s f y the hea t e q u a t i o n i n a weak s e n s e . The system ( 0 . 3 , a , b , c , d , e ) then becomes where - 130 - * i U t ) C efct) 4 (A--i<k)i«t)Q)d(i)so o o M ; > = W«>vj<*>^> (7 <2>: .«=.»,-a, Now to e x p r e s s t h i s i n terms o f the f i n i t e e lement a p p r o x i m a t i o n we take Hence (7 .18) becomes ict)5U)- « * (,&A/*»,»<t) - JU)K(il) ^ (7 TOCo):' O c b where - 131 - 0 ((SC, C ^ i l , » " r > ' - 132 - fckls* W*U ) V ° c U s ^ a,...,*/ i * o, D(i)~ (0,,u>, 0A,ct),..., D^c-t), D„4,,x(t»T = ( ., c l 3 A /(t)) T and we have i n t r o d u c e d the n o t a t i o n a - 133 - The approximation TJU,^). t o V-(x,i) c a n be written as UC*,t> o„ct> Hence the variational principle (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 spatial 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 M„ <t},-t) of (7.17). Moreover, unlike (7.16) and (7.17), the solution of (7.19) yields directly and hence most accurately approximations to the quantities of interest ^ u.Cx,-t^, U* U ,-k>̂  . It should be noted that i f "ue OO <j C ' O i ^ l j 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 "ÛCx̂  at the points \ X. { of the partition T T " , i.e., O^, (oV- "U/b*^ , Oj, loi= b -U0(bXj) ^=',* instead of projecting "U0(>o into j^^ilr") . The order of accuracy of the approximation U"lx,t) to \*. (x|t) is not affected. Since is a Gram Matrix of S. A/ linearly - 1 3 4 • independent functions over [ u , l ] i t is i n v e r t i b l e and hence the system (7.19) is solvable l o c a l l y in time. In fact (7.19) has a unique solution for a l l i . such that •S(\t>c, To show t h i s , we define and note that which upon integration by parts becomes ' ° o that i s , Hence multiplying the f i r s t equation of (7.19) by D T(t) we obtain i[6 T(t)TD(t>l 4<tW cU(t). f D T ( t ) ^ 6(t)l - - 1 0(t) ' dt cli 1 J i(t) or cL ns<t ) -"5 T<t>^D(i>l - - z D T C t ) 6 ( 5<t). c l i J JCi) Since ,S(M>© and «< is po s i t i v e d e f i n i t e , we have - 135 - 3<t) 0 T l t>Y *D l t> $ biSJVo,, , We see t h a t i f (.*)-><*? as t i n c r e a s e s then D c t ) —> o . T h i s l e a d s to a c o n t r a d i c t i o n s i n c e the second e q u a t i o n o f (7 .19) then i m p l i e s t h a t <3(t)$o' as t i n c r e a s e s s i n c e V\ (A) >, o, Hence -S(i) remains bounded . I f D It) -> °o then we must have tha t J ct) —> o . Hence we a re a b l e to c o n c l u d e t h a t (7 .19) has a u n i q u e s o l u t i o n f o r a l l i. such t h a t j t t ) > 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 o f \ > , (7 .19 ) i s a c o n v e n i e n t sys tem to a n a l y z e . However t h e r e i s a n o t h e r ( C o l l o c a t i o n ) f o r m u l a t i o n c l o s e l y a s s o c i a t e d w i t h the c o n t i n u o u s G a l e r k i n system (7 .19) which i s more c o n v e n i e n t f o r n u m e r i c a l c o m p u t a t i o n . To i n t r o d u c e the C o l l o c a t i o n scheme we w r i t e the G a l e r k i n C o n d i t i o n s (7 .19) as and i n t e g r a t e the m i d d l e term by p a r t s to o b t a i n - 136 - - A / 4 / , rf= 2 , U s i n g a two p o i n t G a u s s i a n scheme to e v a l u a t e the i n t e g r a l over • C*v\, Xw+il we o b t a i n A/ 2 O 'S, *t hHi) w t < ' ? « • t , - w > " f*»Si*>- ? « * « > * < t > w,(v£t>? ^ W.jjv, where Now i f - 137 - then the f i r s t e q u a t i o n o f (7 .19) - i s s a t i s f i e d to OC^M I t h a s . b e e n shown t h a t a c o n t i n u o u s G a l e r k i n scheme, u s i n g c u b i c Hermi te p o l y n o m i a l s , p roduces a t b e s t O ( W ¥ ) a p p r o x i m a t i o n s to s o l u t i o n s o f f i x e d boundary p a r a b o l i c systems ( c f . Douglas and Dupont [ 11 ] ) hence g r e a t e r a c c u r a c y i n e v a l u a t i n g the above- i n t e g r a l - i s - in* a? s/en-s-e; " w a s t e d " . Thus we have the f o l l o w i n g C o l l o c a t i o n scheme f o r ( 0 . 3 , a , b , c , d , e ) i t t V i C t V = ( w y O j t i - J-lt)W itv) y (7 . w i ^ , o ) * u e cb^) ^ i , . . . ^ *S<©)= b I t i s c l e a r tha t (7 .20) can be w r i t t e n as the system - 138 - •sVt)^J C K ^ + + i<usct> B) Bet) 0 J 1 3(o^ - b where A- 'f% e, \ ft, (6) F o r a c o l l o c a t i o n f o r m u l a t i o n b a s e d on C u b i c S p l i n e s see Doede l [ l O ] . 139 / V - A : a; 6" »K K M , - 140 - A K A/- ; (51 e: >?,K<,07l) \ To show t h a t the system o f e q u a t i o n s (7 .21) can be s o l v e d l o c a l l y i n t ime we must show tha t i s i n v e r t i b l e . The f o l l o w i n g argument , due to Douglas and Dupont £ 12 In [ 12 J i t was demonst ra ted t h a t c o l l o c a t i o n schemes s u c h as (7 .21) p r o v i d e O t V\*) a p p r o x i m a t i o n s to s o l u t i o n s o f a l a r g e c l a s s o f second o r d e r n o n - l i n e a r p a r a b o l d i f f e r e n t i a l e q u a t i o n s . - 141 - demonst ra tes t h i s . The p r o o f p roceeds by c o n t r a d i c t i o n . Suppose t h e r e e x i s t s a n o n t r i v i a l v e c t o r such t h a t b - o . (7.22) Upon c o n s t r u c t i n g the p i e c e w i s e c u b i c p o l y n o m i a l " i .we-see ..that (7.22) . i m p l i e s tha t 2<>?J>-0 , ^ t ^ . . . ^ / . (7.23) S i n c e 3J ( l ) *o we see t h a t £ (x) v a n i s h e s a t t h r e e p o i n t s on L^A/>YA/-*»1 • Hence e i t h e r I : ? ( x ) * 0 o n C X ^ I ] , o r II:' < o . S i n c e the p i e c e w i s e c u b i c 2(^)€C,C°/l] > i t i s c l e a r t h a t I t o g e t h e r w i t h (7.23) i m p l i e s t h a t j?(x) = 0 on [ o , l ] . S i n c e la^o t h i s i s i m p o s s i b l e and hence I I h o l d s . - 142 - From Z(X*) Z'(*~)<o and ? (J?,"*'^ £ C1?'') - O we have t h a t the p i e c e w i s e q u a d r a t i c must v a n i s h i n l"') a n d ^ f ' j ' h e n c e i f c f o l l o w s t h a t C o n t i n u i n g i n d u c t i v e l y we o b t a i n t h a t * ! ( O £ Y o ) < 0 which i s -a . , con t r a d i t Ion s i n c e . Hence the o r i g i n a l a s s u m p t i o n i s f a l s e and i s n o n s i n g u l a r . We remark t h a t t h e r e i s l i t t l e d i f f i c u l t y i n g e n e r a l i z i n g (7 .21 ) to i n c l u d e more g e n e r a l one d i m e n s i o n a l s i n g l e phase S t e f a n p r o b l e m s . However even f o r the s i m p l e s t ease o f (7.-21) the c o n v e r g e n c e q u e s t i o n i s a d i f f i c u I t -one . The d i f f i c u l t y a r i s e s from the extreme s t i f f n e s s o f the system o f e q u a t i o n s ( 7 . 2 1 ) . M o r e o v e r , the non-1 i n e a r i t y i s such t h a t the methods o f Douglas and Dupont £ 12 J a re not e a s i l y a p p l i c a b l e . We c o n c l u d e t h i s s e c t i o n w i t h the remark t h a t the p i e c e w i s e C u b i c H e r m i t e f u n c t i o n s were chosen f o r c o n v e n i e n c e . No doubt a w e a l t h o f systems s i m i l a r to (7 .21) can be o b t a i n e d by c h o o s i n g bases c o n s t r u c t e d from o t h e r f i n i t e e lement f u n c t i o n s . N u m e r i c a l R e s u l t s Here we g i v e n u m e r i c a l r e s u l t s f o r the C o l l o c a t i o n scheme (7.21) only. We do this since for parabolic equations on f i x e d s p a t i a l domains, Collocation and Galerkin schemes, based on Piecewise Cubic Hermite Polynomials, have the same order of convergence, provided we collocate at the Gaussian points ' S i . n c e we have no reason to believe that the Galerkin scheme pro videos a., bj&tker. e;s4t,ima>t;e> fo.r "Uk^A-^'h , and hence i t t ) , than does the Collocation scheme, we adopt the l a t t e r on the basis of computational ease. Computationally the Collocation scheme (7.21) is much easier to implement than the Galerkin scheme (7.19), since the former requires function evaluations where the l a t t e r requ-ires quadratures. Moreover, the "bandwidth oJf the Collocation Matrices ^&><&J * s t o u r while that of the Galerkin Matrices ^V^^C^y^! is s i x . Hence the system (7.21) is computationally more e f f i c i e n t than the system (7.19). To solve the i n i t i a l value problem (7.21), we must contend with i t s s t i f f n e s s . That i s , the condition number of the matrix "ô "'̂  , a r i s i n g in the Galerkin system (7.19), increases as , hence we see that the time constants present in the s o l u t i o n of (7.19) have r a d i c a l l y d i f f e r e n t magnitudes for large N . The intimate r e l a t i o n s h i p between the Galerkin system (7.19) and the Collocation system (7.21) leads us to - 144 - suspect that the system (7.21) is also s t i f f . This is substantiated by the following numerical experiment. We employ two numerical procedures to solve (7.21). The f i r s t is an Adams-Basford-Moulton Multistep Predictor- Corrector Method ( c f . Isaacson and K e l l e r [ 21 ] p. 388) while the second is a Multistep Predictor-Corrector Method due to Gear £ 18 J constructed s p e c i f i c a l l y for s t i f f systems. In a l l cases tested, we have found that the time step required to maintain a given accuracy in the solution of (7.21) was much larger for Gear's Algorithm. Consequently Gear's Algorithm was •found to execute f i v e to ten times faster than the "A'dams'-B'asTord Moulton Algorithm. We conclude from this that the system (7.21) is indeed s t i f f . In what follows we adopt the notation: where {M>0 is the solution of (0.3 ,a ,b ,c , d , e) , (U>?) is the - 145 - s o l u t i o n o f (7 .21) and i t i s u n d e r s t o o d t h a t v. (.*,t) = M W ) t (x,"D ' o f o r J< > *<.i,\ . To i l l u s t r a t e tha t the C o l l o c a t i o n Scheme p r o v i d e s a c c u r a t e a p p r o x i m a t i o n s to the s o l u t i o n o f ( 0 . 3 , a , b , c , d , e) , we s o l v e the inhomogeneous hea t e q u a t i o n ••"M̂  cx /O + * " l x » t ) o < x < 4tt> y \ >o (7 .24) t o g e t h e r w i t h the boundary c o n d i t i o n s (0 .3 , a ,b , c ,d , e) . We o b t a i n our f i r s t example by s e t t i n g b = ) j and p i c k i n g "f"l*,i.)> W (i), and w0 (*) so t h a t the s o l u t i o n becomes ( (7 .25) We do t h i s f o r A =10, 20 and 5 0 . In each case we take u n i f o r m s p a c i n g and use the a p p r o x i m a t i o n to T = . 4 . T a b l e 7 .0 p r o v i d e s a summary o f the r e s u l t s . To d e a l w i t h the i n h o m o g e n e i t y , V***"!) > t n e o b v i o u s changes a r e made to ( 7 . 2 1 ) . - 146, - T a b l e 7.0 E r r o r s i n l U x . i ) , "Uy (,x,-t), TAX« and JCt) (Exac t S o l u t i o n (7 .25 ) ) N e ,( .4) e , ( . 4 ) e ^ 4 l A=10 3 • 3 0 ( - l ) .30 (-1) 5 .2 (0 ) • 58 ( -3 ) 5 . 2 0 ( - 2 ) . 95 ( -2 ) 3 .5 (0 ) . 8 0 ( - 5 ) 7 • 55 ( -3 ) .44 (-2.) 2 .3 (0 ) . 6 0 ( - 5 ) 9 . 1 3 ( - 3 ) . 14 ( -2 ) 1 .4(0) •12 ( -4 ) Observed O r d e r o f Convergence 4 .9 2 .7 1.2 A=20 3 .25(0) .30(0 ) 6 .5 (0 ) . 4 0 ( - l ) 5 • l l ( - l ) .22 (-1) 9 .5 (0 ) . 1 9 ( - 4 ) 7 . 2 4 ( - 2 ) . 1 4 ( - 1 ) 7 .1 (0 ) . 2 0 ( - 5 ) 9 . 8 0 ( - 3 ) . 7 3 ( - 2 ) 4 .8 (0 ) • 25 ( -4 ) Observed Order o f Convergence 5 .2 3 .3 1.2 A=50 4 .90(0) 1 .10(0) 18 .3 (0 ) .15 (0 ) 5 .25(0) .38(0) 18 .2 (0 ) . 4 1 ( - 1 ) 7 .16C-1) • 3 0 ( - l ) 24 .2 (0 ) . 6 2 ( - 3 ) 8 . 9 4 ( - 2 ) • 3 2 ( - l ) 22 .6 (0 ) • 36 ( -3 ) Observed Order o f Convergence 6.9 6.5 -- From T a b l e 7.0 one can see tha t good a c c u r a c y i s o b t a i n e d i n s p i t e the l a r g e v a l u e s o f the s p a t i a l d e r i v a t i v e s o f the tempera ture d i s t r i b u t i o n n e a r x - o . - 147 - Next we consider (7.24) with b - l » a ^ * = l and •^tx,*.); W i t ) t u , lx> chosen so that the solution becomes V U x / k ) * c o s x ( x * - Approximations to the, above solutio.n- are obtained- fo,-r. Ŝ. =100, 500 and 10,000. We are interested in the accuracy of the Collocation approximation when \j(t)| is lar g e s t , hence we employ the approximation u n t i l approximately 20% of the slab remains. In each case the 'partition Tf" i s taken to *be •"uniform. The results are summarized by Table 7.1. Table 7.1 Errors i n UCx.-tl^.tx.-t), u«,(x,t) and si-t) (Exact Solution (7.26)) N e B(T) e t( T) e ?(T) e.,(T) B=100 3 .13(-3) .80(-3)~ .11(0) .96(-4) 5 .19(-4) . l l ( - 3 ) .43(-l) .16(-4) .14(-1) .60(-5) 1.9 (7.26) 9 .40(-5) .24(-4) Observed Order ^ 3 2 of Convergence - 148 - B=500 3 . 1 4 ( - 3) • 14 ( - 2) .12(0) . 9 7 ( - 4 ) 4 . 3 9 ( - 4) . 4 8 ( - 3) • 6 7 ( - l ) •23 ( -4 ) 5 . 1 3 ( - 4) . 1 6 ( - 3) • 4 3 ( - l ) . 8 0 ( - 5 ) 6 . 1 8 ( - 4) . 9 0 ( - 4) • 3 0 ( - l ) . 8 0 ( - 5 ) Observed Order o f Convergence 4 .6 4.1 2 .0 B=10,000 3 . 1 7 ( - 3) . 8 4 ( - 2) .17(0) • 80 ( -4 ) 4 , 4 6 ( - 4) . 2 5 ( - 2) • 6 7 ( - l ) . 2 2 ( - 4 ) 5 . 2 5 ( - 4) . 1 1 ( - 2) . 4 7 ( - l ) . 1 0 ( - 4 ) 6 . 1 6 ( - •4) .45 (- 3) . 3 1 ( - 1 ) ..60.(-.5) Observed Order o f Convergence 3 .4 4 .2 2 . 4 T a b l e 7.1 shows t h a t good a p p r o x i m a t i o n s f o r b o t h J ( i . ) and \*<.*,*) a re o b t a i n e d a l t h o u g h i s r e l a t i v e l y l a r g e d u r i n g the p e r i o d s o f t ime under c o n s i d e r a t i o n . T a b l e s 7.0 and 7.1 i n d i c a t e t h a t the C o l l o c a t i o n Scheme i s o f h i g h o r d e r , however , the r e s u l t s c o n c e r n i n g the o r d e r o f c o n v e r g e n c e a re s c a t t e r e d and do not g i v e a good e s t i m a t e f o r the a c t u a l o r d e r o f c o n v e r g e n c e . T h i s i s not s u r p r i s i n g s i n c e the o r d e r o f c o n v e r g e n c e i s a c h a r a c t e r i z a t i o n o f the a s y m p t o t i c b e h a v i o u r o f the e r r o r . Hence a c c u r a t e e s t i m a t e s f o r the o r d e r o f c o n v e r g e n c e can u s u a l l y o n l y be o b t a i n e d when f i n e p a r t i t i o n s 77^ o f [o,i] a re t a k e n . - 149 - The above results indicate that the Collocation Scheme (7.21) can be used to obtain accurate approximations to the solution of (0.3,a,b,c,d ,e) . However, i t should be emphasized that the s t i f f n e s s of (7.21) makes i t a numerically i n e f f i c i e n t scheme. Hence u n t i l a method i s devised to solve (7.21) e f f i c i e n t l y , the u t i l i t y of this scheme, is in dqu,bt. CHAPTER V I I I CONCLUSIONS T h i s - thes- is has; pr-esent.ed two- a l g o r i t h m s - f o r the n u m e r i c a l s o l u t i o n o f the S t e f a n Prob lem (0 .3 , a ,b , c ,d , e) . We have seen tha t the S i m i l a r i t y A l g o r i t h m p r o v i d e s 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 method o f o b t a i n i n g " r o u g h " a p p r o x i m a t i o n s to the s o l u t i o n o f ( 0 . 3 , a ,b , c , d , e ) . M o r e o v e r , f o r the p r a c t i c a l s i t u a t i o n o f bo th a smooth i n i t i a l telnp'ter'a'tu're d i s t r i b u t i o n and a c o n s t a n t hea t f l u x , the. S i m i l a r i t y A l g o r i t h m promises to be an e f f i c i e n t a l g o r i t h m . F u r t h e r m o r e , the heat f l u x g e n e r a t e d by the a l g o r i t h m can be used bo th to improve the a c c u r a c y and to e s t i m a t e the e r r o r o f the a p p r o x i m a t i o n . We o b s e r v e t h a t i f a v e r y a c c u r a t e a p p r o x i m a t i o n i s r e q u i r e d the S i m i l a r i t y A l g o r i t h m i s not an e f f i c i e n t a l g o r i t h m . The l a r g e number o f terms and the s m a l l t ime i n c r e m e n t r e q u i r e d to a c h i e v e t h i s a c c u r a c y r e s u l t s i n l o n g c o m p u t a t i o n t i m e s . The S i m i l a r i t y A l g o r i t h m i s the d i r e c t r e s u l t o f a p p l y i n g the S i m i l a r i t y Method to a system o f d i f f e r e n t i a l - ' 1 5 0 - - 151 - equations. The above procedure i l l u s t r a t e s how the S i m i l a r i t y Method can be used e f f e c t i v e l y to obtain approximate numerical solutions of non-linear problems. While the S i m i l a r i t y Algorithm gives "rough" accuracy, the Collocation scheme i s capable of achieving high accuracy. Although both, the. S.imilar.ity,. Al gpr;i£"hm> and .Lp.tltin-.'s . d i f f.er.ence scheme execute faster than the Collocation scheme, we have found that 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" , the C o l l o c a t i o n scheme achieves accuracies which the other schemes cannot "attain. We have seen that the apparent s t i f f n e s s of the system of ordinary d i f f e r e n t i a l equations (7.21") is 'the caus'e'of th'e i n e f f i c i e n t performance of "the Collocation scheme. We conjecture, that the simple form of the non- l i n e a r i t y appearing i n (7.21) w i l l allow us to construct a scheme which deals with the s t i f f - ness of the equations in an e f f e c t i v e way. We also conjecture that this Collocation Method can be used to deal with Stefan Problems involving both more general boundary conditions and more general governing parabolic d i f f e r e n t i a l equations. BIBLIOGRAPHY M.A. Biot, V a r i a t i o n a l P r i n c i p l e s In Heat Transfer, Oxford University Press, Ely House,London W.l, 1970. G.W. Bluman, Construction of Solutions to P a r t i a l D i f f e r e n t i a l Equations by the Use of Transformation Groups, Ph.D. Thesis, C a l i f o r n i a I n s t i t u t e of Technology, 1967. G.W. Bluman, S i m i l a r i t y Solutions of the one-dimensional Fokker-Planck equation, Int. J. Non-1 i n . Mech., Vol. 6, pp. 143-153, 1971. G.W. Bluman, Applications of the General S i m i l a r i t y Solution of the Heat Equation to Boundary Value . Problems, Quart, of A. Ma., Vol. 31, No. 4, pp. 403-415, 1974. G.W. Bluman & J.D. Cole, The General S i m i l a r i t y Solution of the Heat Equation, J. of Math, and Mech., Vol. 18, No. 11, pp. 1025-1042, 1969. . G.W. Bluman & J.D. Cole, S i m i l a r i t y Methods for Ordinary and P a r t i a l D i f f e r e n t i a l Equations, Springer-Verlag, New York, Inc., New York, 1974. J.R. Cannon & C. Denson H i l l , Existence, Uniqueness, S t a b i l i t y , and Monotone Dependence in a Stefan Problem for the Heat Equation, J . Math. Mech., Vol. 17, No. 1, pp. 1-19, 1967. R. Courant & D. H i l b e r t , Methods of Mathematical Physics Vol. I, Interscience Publishers, Inc., New York, 1953. P.J. Davis & P. Rabinowitz, Numerical Integration, B l a i s d e l l Publishing Co., Waltham,Mass., 1967. E.J. Doedel, A Collocation Method with Cubic Splines, Manuscript, University of B r i t i s h Columbia, Vancouver, B.C., 1973. - 152 - - 153 - Douglas, J r . , & T. Dupont, Galerkin Methods for Parabolic Equations, SIAM J, Numer. Anal., Vol. 7, No. 4, pp. 575-626, 1970. Douglas, Jr., & T. Dupont, A F i n i t e Element Collocation Method for Non-linear Parabolic Equations, Manuscript, University of Chicago, I l l i n o i s , 1973. Douglas, J r . , & T.M. G a l l i e , J r . , On the Numerical Integration of a Parabolic D i f f e r e n t i a l Equation SubjecJ: to a a Mo>v.in<g> B.ouj^da^y. Condition, Duke, Math. . J . , Vol. 22,' pp. 557-571, 1955. Fasano & M. Primicerio, Convergence of Huber's Method for Heat Conduction with Change of Phase, ZAMM, Vol. 53, pp. 341-348, 1973. N.G. F i l o n , On a Quadrature Formula for Trigonometric Integrals, Proc. Roy. Soc. Edinburgh, Vol. 49, pp. 38-47, 1928. Friedman, Remarks on the Maximum P r i n c i p l e for ••Bar-abol i c •Equations and Its Applications, P a c i f i c J . Math., Vol. 8, pp. 201-211, 1958. Friedman, Free Boundary Problems for Parabolic Equations I. Melting of Sol i d s , J . Math. Mech., Vol. 8, No. 4, pp. 499-517, 1959. W. Gear, Numerical I n i t i a l Value Problems in Ordinary 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 Jersey, 1971. Hellwig, P a r t i a l D i f f e r e n t i a l Equations An Introduction, B l a i d e l l Publishing Co., New York, 1964. Huber, Hauptaufsatze liber das Fortschreiten der Schmelzgrenze i n einem Linearen L e i t e r , ZAMM, Vol. 19, pp. 1-21, 1939. Isaacson & H.B. K e l l e r , Analysis of Numerical Methods, John Wiley & Sons, Inc., New York, 1966. - 154 - H.G* Latidau, Heat Conduction in a Melting S o l i d , Quart. J. Appl. Math., Vol. 8, pp. 81-94, 1950. M. Lotkin, The Calculation of Heat Flow in Melting S o l i d s , Quart. J. Appl. Math., Vol. 18, pp. 79-85, 1960. J.C. Mason & I. Farkas, Continuous Methods for Free Boundary Problems, Proceedings IFIP Congress 1971, Numerical Math. Section, pp. 61-65, Ljubljana, Yugoslavia, 1971. V.G. Melamed, Stefan's Problem Reduced to a System of Ordinary D i f f e r e n t i a l Equations, Izv. Akad, Nauk SSSR Ser. Geofiz., pp. 848-869, 1958. E.A. Muller & K. Matschat, Uber das Auffinden 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, Miszellaneen der Angewandfcen Mechanik, B e r l i n , pp. 190-221, 1962. L. Nirenberg, A Strong Maximum P r i n c i p l e for Parabolic Equations, Comm. Pure and Applied Math., Vol. 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 Uravneni, Novosibirsk, 1962. (Group Properties of D i f f e r e n t i a l Equations, translated by G.W. Bluman, 1967). L.I. Rubinstein, On the determination of the po s i t i o n of the boundary which separates two phases in the one- dimensional problem of Stefan, Doklady Akad. Nauk SSSR, Vol. 58, pp. 217-220, 1947. L . I . Rubinstein, The Stefan Problem, AMS Translations, Vol. 27, Providence, Rhode Island, 1971. R.W. Sanders, Transient Heat Conduction i n a Melting F i n i t e Slab: An Exact Solution, ARS J . , Vol. 30, pp. 1030-1031, 1960. - 155 - M.H. Schultz, Spline Analysis, P r e n t i c e - H a l l , Inc Englewood C l i f f s , New Jersey, 1973. P.V. Soloviev, Fonctions de Green des Equations Paraboliques, Comptes Rendus (Doklady) de l'Academie des Science de l'URSS, Vol. 24, No pp. 107-109, 1939. ' APPENDIX A Lemma 1.1 (Friedman [ l ? ] ) • Let £>Cfc) be continuous on the in t e r v a l Co, 1 ?! . In addition, l e t se t ) s a t i s f y the following L i p s c h i t z condition for a l l t,,t 4 €C°>°"3 and some constant M . Then defining \ /MT) C K ( X , t : S(T),T^J d T r 1 S K 3C-4Ct) Ct) KCx,ti 5<r),r) we have C>t - 1 /5 ( t ) , X -S-SOW-O X Proof. Before proceeding we make the following d e f i n i t i o n : r t I ( * U XVCT) t * ' - * ( Y » K C / t t ^ c y ) , r ) dg - \ J \ r ) L l i i l l - i i l l l K (S(-l), t; 5Ct),t> d r J , 2 C t - t ) - 156 - - 157 - where ^ i s any c o n t i n u o u s f u n c t i o n and % e (o,t). Now i f we can show t h a t ( A - l ) r J i — . | Kp^ + L + -L/>Ok)| <• as % - > o . where. i _ ^P ( t ) ( x - j ( y ) ) K ( x , t - . i W l ^ d f UW.-3»-t t K (Jit) , tj5C*VT)d2• J Hit-*) o then the s ta tement o f the lemma f o l l o w s by a l l o w i n g <B to tend to z e r o . F u r t h e r note that s i n c e (A-2) ( A - l ) f o l l o w s i f we can show tha t To x->stt)-o 2 %->o (A-3) I IC«)| < OU) see t h i s we form the e x p r e s s i o n p < * ) i p C iv - (pti^-^o <r>) - 158 - and s u b s t i t u t e i t i n t o ( A - l ) , to o b t a i n where To e s t a b l i s h (A-3) we w r i t e 1(0 = I , + I . and c o n s i d e r c S i n c e -SOf) i s L i p s c h i t z c o n t i n u o u s and le , - .(g..." M .5_J f o r a l l r e a l , we o b t a i n P»l * ^ ( A - 4 ) To e s t i m a t e I ( d e f i n e - 159 - and consider * - * (A-5) {lx -5tr»*-{x - 4 «»*j /V (t- If we take and "M" we see that the expression in the exponential can be bounded by one since <: K\ IX.-.SC-OI + M 1 & $ ] 2 & Then using the i n e q u a l i t i e s I t - <c~*l < 3 ^ for \£) $ I and 160 Z ¥ (A-5) becomes- IT,-1,1* [S fA S * ^ . (A-6) 8 n''*. To evaluate ^T, as x-> , stV>-o, we l e t Then where $> " /(x-SCtn • From (A-7) we conclude that c j T, = - -jj • ( A_ 8) To complete the proof of (A-.3) we note that (A-6, 7, 8) imply - 161 - (A-9) Hence writing 1 - 1 I, + i i I we obtain 11(01$ ft M 5V* ' +. JL . . (A-10) 2 Tr"* .2 Thus x->Jtt>-o X->JC«-e> Ti*a and the proof is complete. 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 . If vu*,-t) s a t i s f i e s with .St/O a po s i t i v e continuous function and /UCx.t) 6 C C - B ) t -M^U.t), V tU,t) ̂  C ( «£> U (g )̂ where i s the closure of and. ST - I <* ,T) : O < X < 6 C.T) | then wU,t) attains i t s maximum and minimum value on the data boundary - iQy . (See F i g . 1.0 Chapter I) P r o o f ! " For any € > o define v<*,-t)= v. Ix,-U - •£ t • where vttXj-fc) s a t i s f i e s the hypotheses of proposition 1.1. If vt*,t) assumes i t s maximum value at a point Cx.,i,> e °& U @ T ( 1 ) See H e l l w i g [ 19 ] P- 4 7 . <2> F o r more g e n e r a l r e s u l t s see L . N i r e n b e r g [ 27 ] . - 162 - - 163 - then v x < (»,t) i s d e f i n e d and c o n t i n u o u s on 5 C ^ 'Sj where f o r some p o s i t i v e h and K « F u r t h e r m o r e , C*, , ! , ' )»< o and hence (.x, .jt.Y 5,,- € . By the c o n t i n u i t y o f vt^l-*, -!') and hence V^Cx.-t) we e s t a b l i s h the e x i s t e n c e o f a S € C O , K ) such t h a t *.x,,\>$ -  £/z f o r a l l I s < t , - f c , i , 7 Hence and vtx.t) cannot a t t a i n i t s maximum v a l u e on ^ t h a t i s e do©~<g>r Now suppose v<C.x l't) a t t a i n s i t s maximum v a l u e on o d U * S T a t t>*,t<,V such tha t i . e . The maximum v a l u e i s not a t t a i n e d on the d a t a boundary JJd'&j % Then - 164 - i Thus the maximum v a l u e o f v ( * , i ) i s a t t a i n e d at C x , , ! , ) , - t , < i . „ However , i n p a r t i c u l a r *w.C*,,t,) >, - u C X o , ! ^ - « r ( t e - l , ) and hence f o r £>o but o t h e r w i s e a r b i t r a r y . N o t i n g tha t t|<"t0 f o r f>© and ' a l l o w i n g f to tend to ze ro We 'have which i s a c o n t r a d i c t i o n . A p p l y i n g the f o r e g o i n g argument to -u()r,-fc) we o b t a i n the c o n c l u s i o n o f the p r o p o s i t i o n . P r o p o s i t i o n 1 . 2 . I f (v(.,,s) i s a s o l u t i o n of the system o f e q u a t i o n s ( 0 . 3 , a , b , c , d , e) then v t x c s c u , t ^ > o P r o o f . (3) For more g e n e r a l r e s u l t s see Fr iedman [ l6J - 165 - If for some "t , -wx(stt>,ii < o then by (0.3b) there exists X0 6 C s u c h that v \ ( x e ) i ) ^ h > o for some & .» Now define i€D».T] t $ and note that i f * c > o then the Maximum P r i n c i p l e (Proposition 1.1), together with conditions (0.3b, d) implies that there exists a "tv(S>) $ t! such that that i s v^(o,t(6))- &• > © • To show that this is impossible, define for any £ ^ £ 0 , % ] and note that -u. (o, "t (t)) - £ is the maximum value of -vi.(.*li.) on S £ - jCx.t): Of x j 5 C U , o j - t i i ( t l Then since ( o + , •fc') - MX o we conclude that U x ^ o + , t t n ) c M (ov.Ho) { O for a l l ^c-Co.J,! , Thus - 166 - t h a t i s "U. (o, t(.b)) .< O which i s a c o n t r a d i c t i o n . Hence the o r i g i n a l a s s u m p t i o n i s f a l s e and thus v\x (sct»,ti z,o (Note tha t by e x t e n d i n g the above arguments we can show u (o , i ) *o f o r a l l L>A3 ) . APPENDIX C To show that as «f-?o (1 .1 ) becomes ( 1 . 2 ) , we write ( 1 .1 ) as where V t =. ̂  Vv(^ ( v-£) <5 + (x (t ; ? ) t-f) d£; o (c Then considering V,, v a ,Vj separately we take the l i m i t as £-> o . Since G + ( x , t • has an integrable s i n g u l a r i t y at T=t , we see immediately that o f V 3 r - V V ^ S C ^ t ) G^x . t ; SC*),*) . (C o To find efCwv V, we write - 167 - - 168 - where o .sen V," - ^ j > < ^ ) -"U.tO] G*0,t :s-,c) d£ , V/ M : - ^ G\x,t;^,o) . From which we have G^x.tjf.o) j J ( f ) . S i n c e f o r f i x e d t .>o, G + (x , ^ , 7:) i s c o n t i n u o u s at r= O we see tha t 3l v . ' - O . (C-3) £->0 I t i s easy to see t h a t - 169 iv,"l 5 -*"P 1 "K (£ ,d> ~-U0(£)\ hence Caf V/' r O , (C-4 ) F i n a l l y f o r f i x e d i > o, V,'" can be w r i t t e n as where -S If) $ * £ b f rom which i t can be e a s i l y seen t h a t cj w v ^ V,"' - o . (C-5) Combin ing ( C - 3 , 4 , 5) we have €->© o Now f o r V , we f i x S f (o,x) n ( o , i t t - n - a n d (C-6 ) wr i t e > as £ tends to z e r o the f i r s t and l a s t i n t e g r a l , as w e l l as the - 170 - K(xj'tj-f -t-£) part of the second integral do not. contribute since -3T as £->© . Hence we are l e f t with ^ -u.C;5;i-.£) l< C * . , . x-* After making the change of variable V- Villi) in (C-7), we consider f V " V c / v i — \ \ \ e" Vu(.x-at' /'v >i-cUlv I - v<.W,i) . '«->© L TT '* ) -J Since A*Cx,"k) is continuous in and hence oQ£ , we can write (A-8) as for some x (.*-$>,x+ Letting S tend to zero, we concluc Combining (C-2, 6, 9) we see that as £->o (1.1) becomes 4 ^ vVl-S«t>.t) G ^ t X j t - . - K * ) , * ) At APPENDIX D Lemma 1.4 (The equivalence of the d i f f e r e n t i a l and in t e g r a l systems). •If *u(.t) i s a solution of (1.4), where JC-tvis given by (1.5).., then C/M,S> ( ) defined by (1.2) and -sc-f) defined by (1.5)) forms a solution of (0.3,a,b,c,d ,e). Conversely i f <x*,s) i s a solution of (0. 3 , a,b , c , d , e) then v>(t^ = V K (•sci^t) is a s o l u t i o n of (1.4). Proof. By construction any solution ("u.s) of (0. 3 , a ,b , c , d, e) defines a s o l u t i o n V - u„i-sit\t) of (1.4). Conversely i f viC-tV is a solution of (1.4), (1.5) then i n the region oQ the integrals of (1.2) are regular. Hence they can be d i f f e r e n t i a t e d d i r e c t l y to show that v a ^ t ) (defined by (1.2)) s a t i s f i e s (0.3) in <=& . Furthermore, since G+(.x,tj T} i s an even function of x. we see that condition (0.3c) is s a t i s f i e d and d i f f e r e n t i a t i n g (1.5) we have that condition (0.3e) holds. ; Evaluating cb - 171 - - 172 - shows that condition (0.3d) is s a t i s f i e d . To demonstrate that condition (0.3b) , holds , we note that we have shown that •u.c.x.'O (defined by (1.2)) s a t i s f i e s (0.3,c,d,e) on JQ . Hence we can integrate Green's Identity with wU,t) (defined by (1.2)) over and use condit ions (0.3,c,d,e). After taking the l i m i t as €->© and subtracting (1.2) from the re s u l t i n g expression we are l e f t with O 6 - \ Ĝ - (x, tyJt*),T)d»- o using the r e l a t i o n Of* txY~ - Gx Cx;t; w.e have. O - 2- \ v<C5«),t) G"( x,t; 3tt»,T)cl2-dx ) o + ^ •uC.Strj.t) ic«r) G4Cx,t3 5ltj,r) d > L e t t i n g x->JCt>-o and using Lemma 1.1 (D-l) becomes O- vt-C-SCt^t) + X } u ( s t r \ , t ) [G'CSCDjt^ i t t ) ,^ ) o (D-l) (D-2) Since s$ Ct) is L i p s c h i t z continuous, from (D-2) we deduce that Ivastt),-fc)| s a t i s f i e s the inequality - 173 - V M \ $ • ̂  l-ul * ' d* o where "̂ has at most an integrable s i n g u l a r i t y . Using an inequality of the G r o n w a l l t y p e we conclude that u. (ic-t^-fc) u.o and thus the lemma is proven. ( 1 ) If where ° then APPENDIX E THE FIXED BOUNDARY SOLUTION We wish to solve the following system of equations. x « Co.il 1 >o o To obtain a solution of (E-l) we l e t v.(<(t)= w < *,t W v (x, where wlx,Vi s a t i s f i e s W x0,"k^O- V >o - 174 - - 175 - and v(*,t) s a t i s f i e s V x ( o , t ^ o ^ > o \ . (E - 3 ) The solution of (E-2) can be written as W t x where 4-<5 (E-4) To construct the solution of (E-3) we take the Laplace transform of the equations (E-3) and obtain -U„ (o,p) * e • ^ x 0,p) = HCp) (E-5) - 176 - where The s o l u t i o n o f (Er5) can be w r i t t e n - as *U(*,p)- H (p) Co<V\Jp_x. 4 p 5w>.\\Tp and hence the s o l u t i o n o f (E -3 ) i s g i v e n by v f f , t v - - i - \ M ( pVP P t c P - S ^ ^ P K ) , d p (E -6 ) V-toa F i n d i n g the i n v e r s e L a p l a c e T r a n s f o r m of \ c a s ^ n ) ) by expand ing the same f o r l a r g e p we see t h a t V(*,t) can be e x p r e s s e d as the c o n v o l u t i o n i n t e g r a l - £ || e ( e " ° + W / * ( t - » + e - ( , + a w + * * c f c . » ) ) | 6\ (E-7 ) I t i s easy to see tha t (E -7 ) s a t i s f i e s the heat e q u a t i o n on the domain | U A V . x t (t>,A , t >t)| as w e l l as the c o n d i t i o n s v ^ c ^ s o and ••v„('b,t)-0 . To see t h a t v„ U,t)= V\(fc) - 177 - we note that *:t"« For fixed f > C V - x > we can take \ l - x l small enough so that (\-x) / c for any prescribed £>o. Hence x- Jt"« •n*k x->i 1 * f * i where € >'« , S> > o a " d M l W m - K U ) | . If *t UG>,VI a continuity point of Y\(i) we see that - 178 - as Hence the solution u.(«,t) of (E-l) can be written o (the small time s o l u t i o n ) . APPENDIX F THE ASYMPTOTIC EVALUATION OF FOR SMALL t To obtain an asymptotic approximation of +1)^+ t ) w e differentiate (3.14) with respect to x and evaluate v'„(x,t) = ^ ' W ' C J t.At;) G„(«,t i c;> e l? where GO t) (F-l) - e asymptotically for small -t at x* -^-t. - 179 - - 180 - We h a v e , up to e x p o n e n t i a l t e r m s , Hence we must o b t a i n an a s y m p t o t i c e x p r e s s i o n f o r Upon making the s u b s t i t u t i o n , J C £ (•" ( F ~ 2 ) becomes S u b s t i t u t i n g the e x p r e s s i o n •f o i n t o (F-3) and u s i n g W a t s o n ' s Lemma we have / - 181 3¥ a n d h e n c e + OCO.

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