Radial heat polynomial transformations in series expansions of heat equatio

Bragg, L. R.

English

NASA Technical Reports Server

.. 
+ 
L. R. Bpagg, Oakland U n i v e r s i t y  
1. In tFodzc t ion .  I n  r e c e n t  papers,  c o n a i t i o n s  have been e s t a b l i s h e d  
( [ 13, [ 3 ] ,  [43 ) under  which an arialytic s o l u t i o n  of t h e  radial  heat 
equa t ion  
has v a l i d  s e r i e s  expansions of' the form 
co .-u 
C b n n  R ' ( r , t )  . 
n=O 
I n  chese,  t h e  RE(r,t) denote  t h e  rad ia l  heat polynomials while  the 
RK( r, t ) denote  t h e i r  corresponaing Appel l  t ransforms [ 13. 
t h i s  paper  we assume t h a t  p > 1. T;2e resLl t s  show that; u ( r , t )  has  
t h e  r e p r e s e n t a t i o n  (1.2a) v a l i d  i n  t h e  tine s t r i p  it1 < i 4 a provided 
there e x i s t s  a n  e n t i r e  f u n c t i o n  $(r  ), of growth (1 ,a)  i n  r , such 
r\, 
Throughout 
2 0 2 
tha '1; 
Moreover, u( r , O )  = +( r2).  S imi l a r ly ,  u( r, t) has t'ne r e p r e s e n t a t i o n  
( 1 . 2 b )  i n  the  haif  p lane  t > a - > 0 provided that u ( r , t )  s a t i s f i e s  a 
iiuygen' s p r i n c i p l e  as w e l l  as the i n t e g r a b i l i t y  cond i t ion  
- 
* The r e s e a r c h  on t h i s  paper  was suppor ted  i n  p a r t  by the Na t iona l  
Aeronaut ics  and Space Adminis t ra t ion Grant Number NSG-544. 
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I n  t h i s  chse ,  t n e r e  e x i s t s  an e n t i r e  f u n c t i o n y ( r  2 ), of g r o w t h . ( l , a )  
2 i n  r , sccn t h a t  
The c o e f f i c i e n t s  i n  (1 .22 )  and (1 .2b )  are given,  r e s p e c t i v e l y ,  by 
n = 0, 1, 2, ... 
There e x i s t  s o l u t i o n s  or' (1.1) t h a t  have expansions of one of t h e s e  
types  but n o t  the o t h e r .  
Tinere remain a number of ques t ions  p e r t a i n i n g  t o  these expan- 
s i o n s .  For example, i f  a s o l u t i o n  of (1.1) i s  t o  s a t i s f y  the 
c o n d i t i o n  u( r , O )  = d (  r2), i s  the re  a s imple  t e s c  to determine i f  
u ( r , t )  has the  expansion (1.2b)  wi thout  e v a l u a t i n g  (1 .3)  and 
va l idaTing  cond i t ion  (:.4)? If  so ,  can the c o e f f i c i e n t s  bn i n  this 
expansion be determined from t he  data? It i s  e v i d e n t  t h a t  the data 
should provide  a l l  r equ i r ed  informat ion .  Next, suppose t h a t  t h e  
expansion ( 1 . 2 b )  converges t o  an a n a l y t i c  s o l u t i o n  of (1.1) f o r  
t > G > 0. Is i t  p o s s i b l e  t o  cont inue t h i s  s o l u t i o n  across  t h e  
boundary t = a t o  d e f i n e  an expansion of t he  form ( 1 . 2 a ) ?  
does t h e r e  e x i s t  an expansion of the form (1.2a) hold ing  i n  any 
time s t r i p ?  
If no t ,  
Answers t o  these and s i m i l a r  q u e s t i o n s  can be provided by 
o b t a i n i n g  connect ions between the d u a l  Tunctions d ( r  2 ) a n d y  ( r  2 ) -
def ined  above. It w i l l  b e  shown i n  s e c t i o n  2 t h a t  bo th  expansions 
( 1 . 2 )  ho ld  i f  there e x i s t s  a p a i r  of symmetrical  i n t e g r a l  r e l a t i o n s h i p s  
! 
, i . .. 
. 3 
between these d u a l  func-cions. Tne r e l a t i o n s h i p s  resemble Hankel 
transforms. In t roduc ing  Laplace transforms, necessz ry  and s u f f i c i e n t  
c o n d i t i o n s  are  ob ta ined  ( i n  s e c t i o n  3 )  i n  o r d e r  t h a t  an e n t i r e  
f u n c t i o n  d ( r  ) d e f i n e  an  e n t i r e  dua l  f u n c t i o n  v/(r2). 
d ( r  ) ( o r y ( r  ) )  may f a i l  t o  e x i s t  or e l s e  ex is t  on ly  a s  a d i s t r i -  
b u t i o n .  I n  t he  former case ,  we re fer  to t = 1/40 as a n a t u r a l  
boundary of u ( r J t ) .  
( 1 .5 )  lead t o  the same s e r i e s  expansions.  
to i l l u s t r a t e  theorems. Tle connect ion between the above problem 
and D. V. Widder 's  s t u d i e s  on the resemblance between c l a s s i c a l  
2 The d u a l  of 
2 2 
I n  the  l a t t e r  case ,  t he  i n t e g r a l s  (1 .3 )  and 
Examples are provided 
f u n c t i o n  theo ry  and c l a s s e s  of solutions of t h e  s t anda rd  heat 
equa t ion  [7] i s  e v i d e n t .  
2. 
e n t i r e  f u n c t i o n s  t h a t  s a t i s f y  the fo l lowing  ,pair of r e l a t i o n s  
Symmetrical I n t e g r a l  Re la t ionsh ips .  Le t  d (  r2) and \y( r2) be 
Tnroughout t h i s  s e c t i o n ,  w e  assume that, x ' / ~ ~ ( x ~ )  and X ~ / ~ ~ ( X ~ )  
E L ' ( 0 , w ) .  
Theorem 2 .1 .  
(1 .5)  reduces t o  t h e  i n t e g r a l  (1.3).  
Proof. Upon s u b s t i t u t i n g  ( 2 . l a )  i n t o  ( l . 5 ) ,  w e  f i n d  
Under these condi t ions ,  we have 
Let v( r2) be def ined  by ( 2 .  l a ) .  Then the. i n t e g r a l  
A 
. !>. . .. . 
4 
m 2 2  
0 c 
= r  L p / 2  J x ~ / ~ + ( x ~ ) ( &  e -(r tx ) /4 t  IU/2-1 (zq 2 t  dx. 
The l a s t  member of t h i s  i s  p r e c i s e l y  ( 1 . 3 ) .  
o r d e r s  of i n t e g r a t i o n  f r o x  t h e  seco3d t o  t h e  t h i r d  member fo l lows  
f r o n  the a b s o l u t e  i n t e g r a b i l i t y  o f  xp/2d(x2). 
the  inf ier  i n t e g r a l  i n  the  t h i r d  meniber i s  g iven  i n  [5]  (p.29, 
secor,d fo rmula ) .  
Tne in t e rchange  of 
me e v a l u a t i o n  of 
By a s imi la r  argumenc, w e  ob ta in  
‘Pleorern 2 .2 .  
,cL’(O,m). 
L e t  $ ( r  2 ) b e  deTined b y  ( 2 . l b )  where X ’ / ~ Y ( X ~ )  
Tnen t h e  i n t e g r a l  ( 1 . 3 )  reduces t o  the i n t e g r a l  ( 1 . 5 ) .  
From these theorems, w e  see t h a t  a p a i r  of f u n c t i o n s  re la ted 
by ( 2 . l a )  or ( 2 . l b )  d e f i n e  a s o l u t i o n  f u n c t i o n  of (1.1) t h a t  has 
exparisions of both  types  indicated.  i n  ( i . 2 ) .  The reg ions  i n  which 
these d i r ‘ fe ren t  expansions a r e  v a l i d  need noT; be adj’acent.  
2 2 . Then + e-2r 2 Example 1. S e l e c t  i-1 = 2 and G(r ) = e-r 
y ( r2 )  = -rr/2je -r2/4 + ~ e  -r2/8]([5], p .  137). It fo l lows  t h a t  
bo th  types  of expansions ( 1 . 2 )  hold Tor the corresponding u ( r , t ) .  
The f i r s t  expansion holds  for It1 < 1/8 whi le  the second expansion 
i s  val id .  for t > 1/4 (Tneorem 5.1 and tneorem 7.3 of [l]). 
3.  Laplace Transform C r f  t e r i o n .  The i n t e g r a b i l i t y  requirements  
on y ( x  ) and d(x  ) i n  s e c t i o n  2 are  more s t r i n g e n t  than are necessa ry  2 2 
t o  o b t a i n  both  types  of expansions ( 1 . 2 ) .  The f u n c t i o n  d ( r  2 ) may 
have an o s c i l l a t o r y  c h a r a c t e r  and l e a d  to a c o n d i t i o n a l l y  convergent  
i n t e g r a l  i n  ( 2 . l a ) .  
i s  convenient  to r e l a t e  4 and y.‘ by Laplace t ransforms.  
For t he  pmpose of t r e a t i n g  such f u n c t i o n s ,  i t  
Following 
I 
5 
:he T o r m ~ a - ~ i o n  i t 2 1  ( s e e  7;i-ieoreix 2 . 2 ) ,  we have 
'LneDF?::. 3.1. Let  r- 
CO 
(3.1) T (s) = J e - X S x P / 2 -  1 Q(x)dx. 
0 IJ. 
Tien 
( 3 . 2 )  %'(r 2 ) = - 1 ( 2T)v/2r2-P- L - 1 {p-'12T ( l /p ) )  
4 P CL 
i n  which t h e  v a r i a b l e  i n  chfs inve r se  Laplace transforin i s  rep laced  
By in t roduc ing  the  nethoa or" proof O f  theorem (14.a), p.96 of  
[ 6 3 ,  w e  r e a d i l y  obta in :  
f o p  Is] > c w i t h  u > - 1 I-( v+n+l) n+l  
CO 
L ~ , . J ~  ^ X  3.1. L e t  ~ ( s )  = z an 
a, - -sx v and e a pGsiLive conscant .  Tien s 'f'(s) = e x d(x)dx where 
+ ( x )  i s  the e n t i r e  funcLion C anxn. 
1?=G S 
03 0 
n=G 
I f  w e  make use  of S z i r l i n g ' s  f o m u i a ,  i t  follows t h a t  the above 
ser ies  r'or f ( s )  convepges for Is] > c i f  and on ly  i f  t h e  se r ies  
03 ann!  
c -  converges for 1s 1 > e .  Upon combining t h i s  observa t ion  n +1 
;T,= 0 s 
w i t h  c o r o l l a r y  (14.~)~ p.  96 of [6] and us ing  theorem 3."l, w e  f i n d  
Theorem 3 .2 .  L e t  IJ. > 1 and l e t  d ( r  2 ) be an e n t i r e  f u n c t i o n  of r2 or' 
A CO 
o r d e r  G(eErC) as r + co for a r b i t r a r y  E > 0. L e t  F ( s )  = J e-sxd(x)dx. 
0 
Then t h e  dua l  f u n c t i o n y  ( r  2 ) of d ( r  2 ) i s  e n t i r e  i n  r2 or' o r d e r  
O(  e E*r' 
S CL/2-2F(l/s) a r e  both  e n t i r e  func t ions  of l/s f o r  Is] > c .  
* 
) Tor  a r b i t r a r y  E > 0 as r + Q) 11' and only  i f  F ( s )  and 
Tnis theorem shows chaz t h e  dua l  of t h e  e n t i r e  f u n c t i o n  d ( r  2 ) 
T w ( s )  i n  theorem 3.1 invo lves ,  a t  
These can on ly  l i e  i n  the l e f t  h a l f  
4 2 -  1 i s  e n t i r e  i f  t he  func t ion  s 
wors t ,  p o l e s  o r  branch p o l e s .  
p l ane  or along the imaginary a x i s  w i t h  In( s )  4 0. 
cond i t ions ,  t h e  corresponding u ( r , t )  has expansions of both types  
i n d i c a t e d  i n  ( 1 . 2 ) .  This a l s o  shows t h a t  i f  one of t he  represen-  
Under t h e s e  
I 
. 
4 
c 
6 
t Z t 2 O r i S  ( 1 . 2 s )  or (i.2G) is g l v c n ,  t hen  :he corresponding d(r  2- ) 
? or jj((rL) can be deternrineG. 
s e r i e s  rep-esenta t ion  then reduces t o  t h e  e x i s t e n c e  o f  an e n t i r e  
d u a l  f u n c t i o n .  
%?e e x i s t e n c e  o f  t he  o t h e r  type of 
Note t h a t  if T, ( s )  f a l l s  t o  e x i s t  f o r  0 < Re(s) < a ,  
4 
then u ( i - , t )  corresponding t o  d ( r  2 ) has  t n a t u r a l  boundary a t  
Y’,r 1 2  ) ape in te rchanged  i n  theoren 3 .1  acd ( 2 7 ~ ) ’ ~ ~  i s  r ep laced  by 
t = 1/(&a). T A i s  fo i iows  f r ~ x  theorein 2 . i  of [ 21. If d (  r2) and 
(27r)-’12 i n  ( 3 . 2 ) ,  we o b t a i n  an analogous theorem. However, i n  
chis s i x a t i o n  i f  T ( s )  f a i l s  t o  e x i s t  T o r  0 < Re(s) < a, t?,en 
~ ( r ,  t ) correspondixg 50 ‘?( r ) has a n a t u r a l  boundary at t; = a 
(cheorern 2 .2  of [ 2 ] ) .  
Z x a q l e  2.  
P 
2 
Let  F = 2 and d(r 2 ) = s i n  r 2 . By theorem 3.1, we 
, b2n+l = 0 as - ( -1)”/(2n+l)!  and b2n = ( - - l ) ? T  
4qn(  2n)  ! 2 n t l  - 
a 
c o e f f i c i e n t s  i n  ( 1 . 2 a )  a d  (1.213). ~n this example, 
2 2 /  2 
u ( r , t )  = (1+16c 2 )’-e 7 - 4 r  t/(i+l6t )isin f LCt 
i+16t2 
Exsnple 3 .  L e t  p > 1 and cons ide r  t he  expansion 
cn % $(r, t) .  From (1.6b) w e  f i n d  y ( r  2 ) = C ( 
n=O 
p ( n + l  2 n=O 
0 ,  
t he  
cos 
2 n  4 r  ) /?(rl+1/2) = 
7T 
A s  r t ends  t o  i n f i n i t y ,  Y(T*) behaves a s  7r-’I2 + 2re  4 r 2  . 
c l e a r  from theorem 3.1 ( w i t h  4 and y i n t e rchanged)  t h a t  
no corresponding dua l  f u n c t i o n  (j( r 2 ) . Since t h e  i n t e g r a l  (3.1) 
f a l l s  t o  e x i s t  for s < 4, The s o l u t i o n  Li(r,t) de f ined  by the  given 
expansion has a n a t u r a l  boundary a t  t = 4. 
It i s  
there ex i s t s  
! 
_ -  
e 
4 7 
4 . -:r .3 -A ai izecl  %mi F - x z 5 f o ~ s .  TY.3 dud.  of' an e n t i r e  f u n c t i o n  
+(.-j cay  f a i l  to be c a t i r e  eit>lei- because T ( s )  i-r: theorern 3 .1  
does n o t  e x i s t  for some s wizh Ze(s)  > 0 or because t h e  c o n d i t i o n s  
or' tkeoiier;: 3 . 2  are  n c t  szLuisr'led. If, however, q ( r  ) i s  7;he product  
of a polynomial i n  p2 by a bomeeci e n t i r e  f u n c t i o n  of r , a s i n p l e  
3 
IJ. 
1 2  
2 
c a l c u l a t i o n  i n  tneoreni 3 . i  shows t h a t  
ip-~/2Tw(p-l)l - pi J j ['/*]+l for >I a p o s i t i v e  c o n s t a n t  and 
r )  -'J-/2T (fl) Re suf 'f ' icierl t ly large. Lnder tkese circumstances,  p 
i s  tiie Laplace t rL i?S?OTX of a d i s t r i b u t i o n  bounded on t h e  l e f t  at 
< = 13 (2 .  236, [ 8 ] ) .  Depem2ing upon the choice  of d (  r ), t he  
s.c;port ~f cha d i s t r l b x t i o n  Tiajr l i e  t o  t h e  r i g h t  of 5 = 0. Tne 
r'ui-,ction d ( r  ) can graw nore r a p i d l y  than  a polynomial b u t  less 
r ~ 2 i c l y  than a f u n c t i o n  boundez by fde , for a > 0 ,  0 < f3 < 1, 
and s t i l l  g i v e  r ise  to a g e n e r s l i z e d  dcal y ( r  ) .  
r.. e ;Fi 2 1- u~ 9 -I 
s e e  i8], p .  258. Tne twa-sided Laplace t r m s f o r m  i s  needed t o  
ci 
2 
2 
B ar 
2 
2 
For t he  most 
zonditicns on '7, ( s )  i n  oixier ckae y ( r  ) be a d i s t r i b u t i o n ,  5
!-- 
t x a t  cnis  case .  
2 I n  o r 6 e r  t h a t  u ( r , t ) ,  corresponding to an e n t i r e  c$(r ), have 
i t  i s  necessa ry  t h a t  t h e  dua l  alTi e x 2 a s i o n  i n  t he  s e t  )3j(r,tgJ=o! 
of $(r ) be e n t i r e  i n  r . 
2, d u a l  of G(r 
C Y ,  f \y 
2 2 The above d i s c z s s i o n  shows t h a t  i f  the 
i s  a g e n e r a l i z e d  f u n c t i o n  c h a t  i s  n o t  e n t i r e ,  then 
v a l i d  for a l l  r > - 0 u ( r , t )  ;-.,as expansions i n  trle s e ~  {E;( 2 ,  t )1  J n=O - 
2 2 and a i l  c .  
s p x e t r i c a l ,  we see Lhat if 'che d u a l  o f  y ( r  ) i s  a non-en t i r e  
g e n e r a l i z e d  fur?ction, then u (  ;?, t) has v a l i d  expansions i n  the 
Since the  r e l a t i o n  b e w e e n  d ( r  ) and y ( r  ) i s  
2 
- 
\> CJ r, t g  n=O for t > 0 > 0, o a c o n s t a n t .  The g e n e r a l i z e d  
! 
. 
# 
. 
.- 
I 
n <<I ‘ ,2 ) dxi..l 0- , 
the r-&xis ( s e e  theoren 3.3, [2]). 
, acts as a p o c e ~ t i z 1  f-mc‘sion concentrated along 
h m 2 I . e  L L .  S e l e c t  = 2 an2 d ( ~ - ~ )  =d-(r 2 2  -a ), a > 0 .  From 
i’ I 2 T Lheorern 3.1, we fk .6  thar; ) = - 2 (ar). Using (1.6), we obtain 
b 
2 0  
= -  7: . a2”/[2 4n (E!) 2 ] A’or = - t ne  c o e f f i c l e n z s  in (1.2b). “??at n 2  
expansion h o l d s  f o r  t > 0. The corresponding u ( r , t )  in this case 
is (8t)-‘e 
I 
. t  
-A. . b. 
J 
* 
I 9 
He ?e-ences 
1. L. 3. Eragg, The k d i a l  Heat Polynomials and Related Funct ions,  
m--- iiciris. h e r .  Math. Soc . ,  v o l .  119 (1969, 273-290. 
2.  , The Radial  Fea t  Equat ion and Laplace 
?rarisfoTmS, S ian  Jour. RAp?l. I k t h . ,  li ( ~ 9 6 6 ) ~  986-993. 
4. , Zxgansicas i n  Terms  of General ized Keat 
Tolgnomials a d  or' s h e i r  Appell Transforms, J o u r .  Math. blech., 15 
(1966 1 J 735-758. 
5. W. !<agnus and F. Oberhet t inger ,  Forinulas and Theorems for 
F c r c t i o n s  02 Methematical Physics, Chelsea,  N e w  York, 1949. 
7. , Some Analogies from C l a s s i c a l  Analysis  
i n  the Theory of Beat Conduction, Arch. f o r  R a t .  Mech. and Anal.,  
21 (19661, 108-120. 
8. .A. 3. Zemanian, 34str ibuLion Tleo;ny E,:;? Transform Azalgsis, 
XeSraw-Bill, N e w  York ,  1965. 


Dual Functions and Heat Expansions

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