Integral representations involving Laplace transforms developed for use in solutions of radial heat equatio

Bragg, L. R.

English

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NASA CONTRACTOR 
REPORT 
THE RADIAL HEAT EQUATION 
AND LAPLACE TRANSFORMS 
by Louis R. Bragg 
Prepared under Grant No. NsGd544 by 
CASE INSTITUTE OF TECHNOLOGY 
Cleveland, Ohio 
f OY 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION - WASHINGTON, D. C. 
https://ntrs.nasa.gov/search.jsp?R=19660002965 2020-03-16T22:15:58+00:00Z
NASA CR-337 
THE RADIAL HEAT EQUATION AND LAPLACE TRANSFORMS 
By Louis R. Bragg 
Distribution of this report is provided in the interest of 
information exchange. Responsibility for the contents 
resides in the author or organization that prepared it. 
Prepared under Grant No. NsG-544 by 
CASE INSTITUTE OF TECHNOLOGY 
Cleveland, Ohio 
for 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
For sole by the Clearinghouse for Federal Scientific and Technical Information 
Springfield, Virginia 22151 - Price $1.00 
THE RADIAL HEAT EQUATION AND LAF'LACE TRANSFORMS 
L. R. Bragg, Case Institute of Technology 
1. Introduction. Let P be a real parameter and let Au denote 
PT1 the radial Laplacisn operator AP 3 Df + rDr. In some recent papers, 
the author [l] and D. T. Haimo [4] have been concerned with the study of 
expansions of solutions of the radial heat equation 
(1.1) & uht) = APu(r,t) 
in terms of the radial heat polynomials {RT(r,t)]@& and their Appell 
transforms C;;jWr,t)l~=, when p > 1 (the author's notations differ). The 
elements of these sets are defined by 
0.2) 
(a) Rr(r,t) = j!(4t)jLj 
(&l) 
(-r2/4t) 
(b) T(r,t) = t -2jSP(r,t)Ri(r,-t) 
in which L('-l)(x) 5 
is the generalized Laguerre polynomial of degree j 
and index 5-l and SP(r,t) is the source solution (4nt) -p/2e-r'/4t . 
Underlying the development of these expansion theorems is a pair of in- 
tegral representations of solutions of (1.1). The first representation 
yields a solution of (1.1) subject to the initial condition 
ub-, 0) = q(r) and is given by 
(1.3) u(r,t) = J Kp(r,5;t)dS)dS 
0 
in which 
(1.4) 
1 
and I denotes the modified Bessel function of index 5-l . The 
-- 
; iI. 
second representation for a solution of (1.1) for large t(t > CY 2 0) is 
given by 
b5) 
in which 
s 
co 
u(r,t) = C ,(rrS;t)+(5)dS 
0 
(1.6) <,(t,S;t) = (2rr)+12rl-P~2~'/2P (rs)e-12t 
-- 21 
and ~(5) is an entire function of growth (1,~) in s2. Although these 
representations are quite useful in many situations, the involvement of the 
Bessel functions in these integrals generally lead to complications in most 
applications. Moreover, a number of results of theoretical significance are 
not immediately evident from (1.3) or (1.5). 
In this paper, we develop representations alternative to (1.3) and 
(1.5) that involve Laplace transforms and their inverses. The importance 
of these alternate forms lie in the fact that .(i) the elements of distri- 
bution theory can be more readily fitted into the study 
properties of solutions of (1.1) and (ii) the extensive 
tables relating to Laplace transforms can be brought to 
utions of (1.1) in applications. We obtain in 4 3, for 
suit that the radial function $(r,t) is a solution of 
of solutions and 
literature and 
bear on the sol- 
instance, the re- 
(1.1) defined by 
(1.3) that corresponds to a point distribution involving r2-%(j)(r2). 
This permits another characterization for expansions of solutions of (1.1) 
in terms of the set {?!(r,t)}m . Finally, we note that inverse La- 
j=O 
place transforms often lead to convolution type integrals, some of which 
diverge in the ordinary sense. One can, nevertheless, attach meanings to 
such integrals through the use of finite and logarithmic parts of divergent 
2 
integrals [23. We give an illustrative example of one such integral in g 4. 
2. -lace Transform Representations. In this section, we - 
state and develop the integral representations alternative to (1.3) and 
(1.5). The proof of only the first of these will be undertaken since the 
second result follows by a similar line of reasonAng. It will also follow 
that if suitable data is supplied on the positive time axis, rather than on 
the r-axis, then we can obtain a solution of (1.1) by calculating the in- 
verse Laplace transform of a function involving this data. 
Theorem 2.1. Let u(r,t) be the solution of (1.1) defined by (1.3) 
that corresponds to the initial data u(r,O) = cp(r). Let 
(2.1) 
Then 
(2.2) u(r,t) = TT 
in which the variable in this inverse Laplace transform is replaced by 
r2/16t2. 
Proof. If we select a = r2/16t2 and m&e use of (1.3) and (1.4), 
we have 
W(r,t) = (4t) '1/2er2/4t(r2/16t ) 2 P/*-l 45 t) 
co 
= 2 s VW5 1"/2e-s2/4tap/4-l+2, 0 ; l(&%S -- 
= I- (p(x1/2)xp/2-1e-x/4t a p/4-1/2I c;;] (2@& , 
0 $1 
this last following from the change of variables s2 = x. 
3 
Then 
(2.3) 
sp{W(r,t)] =sa emp%(r,t)da 
0 
= 
s 
o~~(xl/2)~'/2-le-" 
IJ 
f-i 
aempa[E] 1 
X 
0 $1 
s 
CD 
I cp(X 
l/2)xU/2-le-x/4t 1 
p 
ex/'dx 
Then solving for u(r,t), we obtain Theorem 2.1. The validity of the inter- 
change of the orders of integration from the second to the third member in 
(2.3) follows by absolute integrability while the value of the bracketed 
term in the third member of (2.3) is tabulated in [3]. 
Corollary 2.1. Let u(r,t) be a solution of (1.1) that corres- 
ponds to u(O,t) = f(t) and suppose that this u(r,t) can be represented 
by (1.3) for some q(r). Then u(r,t) has the representation 
(2.4) u(r,t) = e -r2/4t(r2/16t2) 
in which the variable in this inverse Laplace transform is replaced by 
r2/16t2. 
Proof. We need only identify the expression TV(p,t) in (2.1) for 
this. It follows from our hypotheses, (1.3), and (1.4) that for s > 0, 
(2.5) 
f(s) = u(o,s) = lim u(r,s) = 1 mc11-1e-52/4s cp&)dS 
r-+0 s 2~-ls%(5) 0 
1 
= 2'ls'l/2$) 0 s 
O3 xp/2-le-x/4s 
,pd2)~ > 
this last step following from the change of variables 52 = x. We now make 
the identification & = & - $ so that s = pt/(p-4t). Upon substituting 
this into the first and last members of (2.5), we get 
T (p,t) '= CI (4pt)'/21(~/2)(p-4t)-'/2 
The stated result follows upon substituting this into (2.2). 
Theorem 2.2. Let u(r,t) be a solution of (1.1) defined by the 
integral representation (1.5). L&t 
-It 
(2.6) 
+ $}x 
xd2-1J1 (x1/2)& . 
Then 
(2.7) u(r,t) = $,; Qp+i2 $(p,t)l 
in which the variable in this inverse transform is replaced by r2/4. 
3. Generalized Functions. We now relate the above theorems to dis- 
tribution theory in so far as it applies to the expansions of solutions of 
(1.1). Our first result relates the radial function %(r,t) to the dis- 
tribution r 2-&i)@) while the second result relates R:(r,t) to this 
ssme distribution. Our final result of this section gives a distribution 
characterization for expansions of solutions of (1.1) in 'terms of the set 
Theorem 3.1. Let the initial data cp(r) in Theorem 2.1 be selected 
to be r2%(')(r2) where 6(x) denotes the usuaJ. delta distribution. Then 
(3.1) u(r,t) = 
(&p/2 
42jl?(j + 5) 
i+,t) . 
Proof. If we use the fact that 
s 
holds for 
0 
the delta function, then we get 
- 
5 
p-'1/2 TP(p,t) = (4t)-j(p-4t)jp 
-'($ + j ) 
. 
But (p-4t)j = Zp{e4at8(j)(a)] 
Using the convolution theorem for Laplace transforms along with the 
properties of the Dirac distribution, we obtain 
1 
s 
r2/16t2 r2 
=(4t)jr(j + 5) 
( -- s> 
j +5-l 
e45t8(j)(<)d< 
0 16t2 
(-l)j Djie45t(r2 _ 
= (4t)jr(j + 5) I 16t2 
%)j + 142-l 3 s-0 
.r2/4t j+2 > 
cl- 1 
= (4t) j+p/2-+-(j + !i) 
D; 
I 
ema0 
I a=r2/4t ' 
r2 this last following from the change of variables 0 = 4t - 45t. If we now 
make use of the Rodrigues!. formula for the Laguerre polynomials 
(p.84 of Csl) and the definitions (1.2), it follows that the u(r,t) de- 
fined by (2.2) for this q(r) is given by (3.1). 
l3y similar reasoning, we can prove 
Theorem 3.2. Let the function g(r) in Theorem 2.2 be selected 
to be b(r) = r %-+$+,2)~ Then 
(3.2) 
On the basis of Theorem 3.1, Theorems 5.3 and 5.4 of [l], and 
Stirling's formula, we have the following characterization for expsnsions 
in terms of the set {gy(r,t)]y=o . 
6 
Theorem 3.3. Let [a.)" 
J j=O be a number sequence satisfying the 
condition lim sup jajl l/j = bu and let cp(r) = r2-' "c a.s(j)(r2) in 
j*- j=O J 
Theorem 2.1. Then the corresponding function u(r,t) given by 
u(r,t) = TN'/;! "c 
j=O 
converges absolutely to a solution 
(-l)ja 
j ?Jr,t) 
42jY(j + 5) 
of (1.1) in the half-plane t 7 u 2 0. 
Thus, the generalized function cp(r), in this case, acts as a potential 
function concentrated at the origin that gives a meaningful effect only 
after Q units of time have elapsed. From Theorem 3.2 and the results in 
B 5 of c11, an analogous theorem can be formulated for expansions in terms 
of the radial heat polynomials in the strip ItI < Q by regarding the 4(r) 
in Theorem 2.2 as a Potential function concentrated at r = 0 in the half 
planet7a. We omit its formal statement here, however. 
4. Finite and Iogarithmic Parts. In determining the inverse La- 
place transforms in (2.2), say, it is natural, in many situations, to make 
use of the convolution integral and in a variety of ways. It is clear 
from (2.1) that, in general, lim TCL(p,t) a 0 wQGi,!Le lim p -'j2T (p,t) = 0 
P+= Pdco 
and 170. In this case, one can write p -'"12T (p t) = CL ' {p 
-($ "E) 
I- 
{p-eTp(p,t)} for some E with 0 < E < CL/~. With this decomposition and = 
the abbreviation a = r2/16t2, it follows by the convolution theorem that 
k-l,, -1 
(4.1) 'Gepl{p'/2T,(p,t)] = ' a2 
r<g 4) 
*%tp b-ETp(p,t)3 - 
The choice of E will, of course, depend upon p and TII(p,t) as well a 
the particular form of the inverse transform one seeks. If, however, 
@ < 0, any integral of the form (4.1) diverges in the ordinary sense. It = 
7 
is still possible to associate meanings with such integrals by the use of 
the finite and logarithmic parts (pf and&) of divergent integrals. This 
procedure thus gives us a method for continuing solutions. For notations 
and basic definitions connected 
to [2]. 
We illustrate the method 
with this method, the reader is referred 
for only one example, namely q(r) = r2+. 
For this, Tp(p,t) = 4pt(p-4t) -1 andp -'12T (p,t) = (4t)p 
-($l) 
CI (p-4+. 
If I.r= 2, this transform can be inverted directly to give 
gp1{p-'/2T,(p,t)]= 4te4at. If 1-1 7 2, we can write 
(4.2) g;1[p-'/2T,(p,t)] = 4t {ad2-2 * e4"t] . 
r($- 1) 
Although this integral is improper if 2 < p < 4, it converges nonetheless. 
Suppose, however, that CL 2 2 in (4.2). Then that integral is meaningless 
in the ordinary sense. For the purpose of attaching a meaning to this in- 
tegral, we examine separately the cases CL not an even integer and ~1 
an even integer. It is convenient to first rewrite the solution u(r,t) 
given by (2.2) and (4.2) for p 7 2 by making the change of vakirbles 
4aS=u. Then 
(4.3) u(r,t) = 
pP s r2/4t ,3d2-2e-o& . r($- 1) 0 
Case (i) ~1 not an even integer. Then the solution of (1.1) 
is given by 
u(r,t) = 
pJ r2/4t 
r($-1) 
Pf o~/2-2e-o~ . 
Through an integration by parts, we can calculate u(r,t) explicitly to be 
8 
m 
u(r,t) = Z r 
2-11(r2,4t)V/2-l+je-r2/4t 
j=O 
r(c1/2+3 1 
$-P $ r2/4t + r($+m) ’ ,P/2-l+me-odo 
where m is the least positive integer such that ILL/~ + m > 0. Clearly, 
lim u(r,t) = r2+ for this. 
t -3o+ 
Case (ii) CI and even integer. In this situation, a solution may 
be defined by 
u,bA = 
i 
c2 PQ. V r2/4t cr-le-udu if CL=2 0 1 
c r2-& 
is 
r2/4t 
2m Pf u 
m-2e-u da if CL= 2m, mS 0. 
0 
The constants C 23 here are so selected that lim u(r,t) = r2-2j t+ o+ 
and replace the reciprocal of the gamma function in (4.3) which vanishes. 
Using the technique for evaluating these, we finally obtain explicitly, 
\ 
1 
se i 
-?/4t ln(r2/4t) + 
s 
r2/4t 
(lrm)e-"du 
j 
, p = 2 
0 
u(r,t) = 
\ 
(-1)-m+3( -m+l)! 
r;(l) 1 
-m+l l-1) j r2 -cm+2-j3 2-a -r2/l+t $+-) r e 
c 
j=l ( -*1)(-m) . . . ( -m+2-j) 
[ + ,*. e-r2/4tln(r2/4t)+ hGJr2'4!ln3)eu&j 
0 
9 
-. 
REFERENCES 
1. L. R. Bragg,"The Radial Heat Polynomials and Related Functions," 
Trans. Amer. Math. Sot. (To appear). 
2. F. J. Bureau,"Divergent Integrals and Partial Differential 
Equations," Comm. Pure Appl. Math 8, 143-202, (1955). 
3. R. V. Churchill, Modern Operational Mathematics in Engineering, 
McGraw-Hill Co., New York, 1944. 
4. D. T. Haimo,"Functions with the Huygens Property," Bull. Amer. 
Math. Sot., ~01 71, no. 3, May, 1965. 
5. W. Magnus and F. Oberhettinger, Formulas and Theorems for Functions 
of Mathematical Physics, Chelsea Pub. Co., New York, 1949. 
10 
NASA-Langley, 1965 CR-337 


The radial heat equation and Laplace transforms

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