Internat. J. Math. & Math. Sci.
VOL. ii NO. 2 (1988) 393-400
393
ON GENERALIZED HEAT POLYNOMIALS
C. NASIM
Department of Mathematics and Statistics
The University of Calgary
Calgary, Alberta T2N 1N4 Canada
(Received March 18, 1987)
We consider the generalized heat equation of n
ABSTRACT.
@__u
th
@2u
@u
+ n-I
r @r
then the heat
If the initial temperature is an even power function
@t
order
2
a
-Ur
transform
the source solution as the kernel gives the heat polynomial. We discuss various
Also, we give series
properties of the heat polynomial and its Appell transform.
with
representation of the heat transform when the initial temperature is a power function.
KEY WORDS
heat
AND
Generalized heat equation, source solution, heat transform,
function,
transform, generating
Lauerre polynomial,
PHRASES.
Appell
polynomial,
hypergeomet ric function.
SUBJECT CLASSIFICATION:
35A05
I.
INTRODUCTION.
In this paper we shall establish various properties of the polynomial solutions
th
order,
and its Appell transforms of the generalized heat equation of the n
2
@u
a
@u
n-1
+
br
2u
where r
2
2 +
Xl
2 +
x2
+ x2
n
--r
r
Also
we
--u
r
shall
give
a
series
expansion
of
the
temperature in terms of Laguerre polynomials and confluent hypergeometric
funct ions. Most of the results derived here are similar to the ones found in [4 & 5],
which are for the less general equation
general ized
@2u
-
x @x
which in turn is a generalization of the ordinary heat equation, [7]
@2u
a-x
@u
These known results can be considered as special cases of our
0 and n
when a
2.
I.
PRELIHINARY RESULTS.
Consider the equation
more
4 F(r,S)
n
where r
2
2
x + x
I
+
+ x 2 and 8
n
tan- l(r/Xn)
Then we have
general
results,
394
C. NASIM
1
sinn-20
r2s inn.- 20
Suppose the solution is of the type
r(r,S)
u(r,t)p(S),
then
p(S).
Letting
1
n--2.dp
d
2
p()sin_2 [stn ]
(2.l)
we finally have
2
-u 2u @u a
-u
r @r
.--+
or
r
where n
2u + I, the generalized heat equation.
d
l
n-2 0 dp
}
sinn-2s [sin
d2p
or
Let
]
+ (n-2)cot S dp
cos O, then from above, we obtain
d2p
(1-r2)---
(n-1) dp
(2.2)
Now from (2.1), we have
._2p(O)
-
2p
2
-a p
d
which has a solution
1
(_)-P’ (),
p()
where
m
l(n_3), a2
kind, [2,p.122].
(u-m)(u+m+l)
and
m
Pu()
is the Legendre function of the first
Also by elementary methods [cf. 6], we
can
find
the
solution
of
(2.2) as
u(r,t)
U(s,r: t)u(s,O)ds,
O
where
U(s,r:t)
+ a2
where
kind.
(2.2).
-u e (s2+r2)
u+
1
s
r
[sr]
I/[],
(2.3)
and I (z), the usual modified Bessel function of the first
We shall call the functioon U to be the source solution of the heat equation
If U is considered as the kernel, then for a suitable f, its heat transform F
is defined by
where k
-
+ 1
rkF(r’t)
u and F(r,O)
fO U(s’r:t)skf(s)ds’
f(r), the initial temperature.
of the heat transform have been given in [6].
rkf(r)
Numerous
properties
We note.that its inversion is given by
U(s,ir:t)(s/i)kF(is,t)ds.
(2.4)
0
Suppose now that the initial temperature is the power function
f(r)
r
m real and positive, then from (2.3), its heat transform,
P
m,
y(r,t)
f
U(s,r,:t)sk+mds
0
r(
+
+ 1)
/’(.u + 1)
(2.5)
1
(4t)mrk IF1 -m;
GENERALIZED HEAT POLYNOMIALS
-I,
t
> O,
[8 p.394].
Hypergeometric function
Thus
giving
solution
a
395
of
(2.2)
involving
the
As a special ease, if
1F1
2n, n
m
0,1,2,...
then
P2n,/(r,
t)
r.+p+l)r"+n+l) } [x2
n
(4t)nrkLt-r2/4t)n
n!
(4t)nrkp=oZ
p
(2.6)
[I
defining the heat polomial of degree 2n in r and of degree n
in t, nvolvin
O, we have the special case given in [4].
Next we define the Apll trsfo of P
m real d positive
Laerre
the
If we let k
polial.
,(r,t),
Wm,(r,t) lp[,(r,_m t)]
1
the Green’s fction, is defined by
where H
U(s,r:t)
and
s
+u+ r/t)k(s,r’t),=
s2+r 2
tk-I
M (s,r:t)
4
e
[s]
I
2(sr)
(2.7)
It c be seen readily that
N (0 r’t)t -k
Wm,(r,t)
1
P,(r,-t).
(.a)
-r/4t
1
therefore we c write
W
t-(m+p+l)e -r2/4t
i
(r,t)
P
(r,-t),
I
In
thi
Pn,(r,t)
ectin
11
e
etblih
nd it lppeII trf
reult
vri
2n,(r,t).
invling the fctin
ging the
is an ey matter to calculate the folling estites:
P2n,(r,t)
P2n (r,t)=
L 1.
For 0
x
< m,
t
2n+k)
r
O(r
0[]
n
> O,
k+2n
0
Usin the
ove
(2 6) the definition of
P2n,
PF.
2n,
estimates, note that the
twice, we have,
f
(s,r:t)P2n (s,-t)ds
n
(_4t)n-p
E r(,+n+l)
F(y+p+l)
p=O
n
2
lntesral converses.
f("+n+l) (-4t)n-P
(p+p+ 1
p=O F
U(s’r:t)sk+2Pds
0
[;1 P2p,
p
(r,t)
N, usin$
c. NASIM
396
Z
(-4t)
p=O /’(#/P+]
(-I)
N conJ.der the
nne
r (#+m+l
r(+n*l)(4t)n-
n
s s 0 f
k+2n
reduces to r
d heBce
The
_
0 d I if
0
(3.1)
equation
For 0
(3.2)
.e.
0
P2n,p(r,t)
< =,
<
f m
Thearefore
n.
t
< =,
y
<
(3.2),
k+
r
inversion foula of (2.5) with m
gives us
x
ti+mJ
U(s,r:t)P2n ,( s,-t)
derive a generating function for
LEMA 2.
(-1) p
r
i=O
nner
desired.
r
8
p=m
Thus the
(4t)
2n.
We now
,
1
2
Z Y
n=O
Let t
PR(X)F.
>
n
P2n y(r,t)
O.
Z
n=O
r
ry
k
I
e
l-4yt w-
,k=-u+.
Using (2.5) and (2.3), we have
n
Y P2n,(r’t)
Z Y
n=O
oU(S,r: t)sk+2nd
s
(R)
U(s,r’t)s k Z
n=O
U(s,r:t)
/
(s2y) n
n:
ds
skes2y
0
[8,p.394] as required. The interchange of summation and integration is valid since
y)
-y)
-s
-s
k+v+
2(
2(
0
If t
0
O, the result can easily be computed, since
P2n,y(r,O)
k+2n
r
<
For t
O,
we use the fact that
P2n,(r,-t)
from its representation given in (2.6).
for the case t > O.
Now we give a generating
function
i
2n-k
The lemma is then proved on the same lines as
for
W2n,/a(r,t),
P2n,u(r, t).
LEMA 3.
For t_> O, Iz[
<
t, and k
tt- u +
,
the
Appell
transform
of
k
n
Z
n=O
(3.3)
P2n,g(ir,t)
W2n,(r,t)
H(O,r’t+4z),
(3.4)
GENERALIZED HEAT POLYNOMIALS
Note that
P)OF.
t.
abso utely when [z[
z
W2n,/( r
0
,
as n
teJ
and hence the series converges
Using (2.8), we have
H(0,r’t)t-k n=0 1
W2n’(r’t)
n=0
t)
37
(z/t)
P2n,(r’-t)
r
l
2
k
H,(0,r:t+4z),
due to Lena 2 d ming e of the definition of H
given by (2.9).
If we expd the right hd side of (3.4) by Taylor series in
pers
of
z,
we
have
[]k H(O,r:
r
t+4z)
n=02
(4z)n
n
[]n [[]kH(O
r: t)
On co=paring this series with the series on the left hand side of (3.4), we tain
-
n
Hg(0,r:t)]
1
22"+Ir(+1)
r(+)
0
(-lr(+l))n22nintegral representation for
giving
trsfo
1
u
J(ru) u2n++le_t
r
du,
(3.5)
W2n,(r,t).
Also we give other generating fctions
Appell
_r2/4t
e
for
the
fction
P2n,(r,t)
d
its
We shall simply write d the results, which c be
W2n,y(r,t).
proved folling a similar alysis as ed for the L 2 d 3 ove.
L 4. For
< t < d all complex z,
1
4tz 2
z 2n
z
r
I (2xz).
e
Z nr(n+l)
(r,t)
n=0
P2n,
5.
<
For
t
<
-
d all cplex z,
2n
k
n=O nr(n+l)
iortt property of the sets of fctions
N we shall prove
d sh tt they fo a biorthogonal syst.
W2n,(r,t)
TO.
For t
> O,
P,(’-t)n,. (x’-t)
ere
i tge irdelt
P.
gin (.8),
r(l)’ n
P2n,(r,t)
d
398
C. NASIM
10 H(O,x
t)t -2n-"
P2n,/a (x,-t)P2m,(x _t)x2Udx
tm-n-/a-ln’m"(-4)m+n x2/a+l e -x2/4t
1
L/afX2]L/a[]dx,
ntl’J
f0
22/a+Ir(/a+l)
x
2
m
(3.6)
due to (2.6).
The integral on
written as, [3,p. I88].
the
right
yp
22/a+ 1 t/a+ 1
0
handside of (3.6) with a change of variable can be
(+./ ) t+
22/a+l rr(n+l)
e-YL/an (y) L/am (y) dy
6
Hence the right hand side of (3.8) gives,
r (/a+n+ 1) tm-n
m.’ (-4)
"r(l)
r (trtn+ 1 n; 42n
-r(u,1)
as required.
Next
P2m,/a (x,
we
shall
t)W2n,/a(x,
LI 6.
establish
For x, y, s and t
generating
function
for
the
>
[z2t[
0 and
<
biorthogonal
Note that the series converges
estimates of the functions
and
P2n,/a
W2n,/a
<
s,
the
using
(-1)n y
2mn’.r(/a+n+l)
:Z
n=O
due to (3.5),
1
-
u/a+l
2y
1
2 +l
z-/a(xy)
1
(xy)
2t
e
P2n’/a(x’ t)
-su
u
f0(2/a)/a+2n+l
e
-su2
J/a(yu)du,
I
J (yu)du 2
’n’:(+l)
/a
n=O
ue
-u2(s+z2t)
J/a(xuz)J
P2n,/a (x’t)
(yu)du
y2+x2z2
z-/a e4(
I
"
]k (xz’y:s+z2t)’
ts+z2t H/a
[l,p.511.
s+-- 2t)
fxy
due to the definition in (2.7).
Now two results on finite sums involving the functions
For t
n
Z
m-O
> O,
> O,
/a
By (2.5),
z
tn-m
(-I
z
m=O
0
m
k
m
P2n,/a(r,t)
fn+.] z2n,
m tn-mj
(-l)m
m=O
P2n,/a
and
and a complex z,
(-l)m fn+/a]
*’:’
n
PF.
asymptotic
therefore,
P2n,/a (x t )W2n,/a (y,s)
n=O n.’ (r(+n+l)
1
+z
Iz2t
for
,y:s+z2t).
2n
z
LE 7.
set
s,
xy
P2n,/a(x,t)W2n /a(y,s)/= 4z2t
n=O n.’r(/a+n+l)
PROOF.
a
t).
in-m]
z
r (1-4tz)
r, t
m
U(s
n
m
m=O
m
r’t)sk+ds
n-m
2
nLnftl-4ztJ
zr
W2n,/a.
GENERALIZED HEAT POLYNOMIALS
U(s,r:t)
0
sk LY(zs2)ds
n
2
-r
1
-[ r
r
399
s/+I
e
zr2
k( l-4tz) n L"[
n "’’t,-zJ
sr
I
(zs2)ds
L
[I, p.43],
as required.
A similar result can also be proved involving
LEMMA 8.
> 0,
For t
n
Z (-1)m
’&’;
m:O
>
[n+"]zm W2m,a(r’
tn-mj
0 and a complex z,
k
t
t)
241(+i)
W2n,.
n++ I
(t+4z) nL
t(t/4z)
SERIES REPHESENTATION
In this section we shall establish a series representation of the heat transform
F(r,t) in terms of Laguerre polynomials and confluent hypergeometric functions.
As mentioned earlier, for a suitable f, its heat transform F is given by
4.
where F(r,O)
f(r)
THEAOREM:
rkF(r,t)
emd rkF(r,t)
If f(x)
has a growth
I,
u
;0U(s,ir:-t)(s/i)kf(is)ds,
0U(s,r:t)skf(s)ds,
>
0, then
0
<
t
<
0
<
t
<
o
I
u + ][
/a
PIKX)F.
If 0
<
t
<
6, we have
rkF(s,t)
Z a s n ds
fo U(s,r:t)sk n=O
n
a
n=O n
Z a
n:O n
due to (2.5).
JO
Pn,ta(r,t),
; [U(s,r:t)sk+n[ds ; t(s+r)2
<
<
t
e
s k++n+l/2 ds
0
<
.
< 0,
U(s, ir:_t)(s/i)kf(s)ds
0
;
U(s,ir:_t)(s/i)k
0
n-O
Z
a i
n
a
n= 0
due to (3.3).
"| U(s,r’t)sk+n ds
The interchange of summation and integration is valid since
0
Also, if-6
> 0,
t
is a solution of the generalized heat equation
Z a
n=O n
rkF(r,t)
where k
U(s,r:t)skf(s)ds,
0
Hence the result
Furthermore, for 0 < ltl < 8,
n
n-k
_ Z an(is)nds
nO
f U(s,ir:-t)sk+nds
, n-k P n, (ir,-t)
Z a P
(r,t),
n=O n n,
c. NASIM
00
rkF(r,t)
a
n=O
P
n
n,
(r,t).
Or
rkF(r’t)
n=O
a2n
P2n (r,t)
+
n=O
a2n+l
P2n+l,(r,t).
Now making use of the definitions given in (2.5) and (2.6), we obtain
n+
r(.+n+)
n
n=O
giving
r
t
n=O
us a representation involving Laguerre polynomial and confluent hypergeometric
function.
If we set
0 i.e. g
u
1
and k’= O, throughout, most of the results derived
here, reduce to known results given in [4] and [5].
n
Further, if we
set
O,
u
i.e.
I, the results coincide with those derived in [7].
This research
ACKNOWLEDGEMENT.
is
partially
supported by
a
grant
Natural
from
Sciences and Engineering Research Council of Canada.
et
al. "T__le__s_of I.n_tegral__Transformsj Vol. 2", Mc Graw Hill, Toronto,
I.
^1954.Erdelyi
2.
A. Erdelyi et sl. "Hi_her Transcendental Fun_c.t_io_n_s, Vol. i", McGraw Hill,
1954.
3.
^1954.Erdelyi
4.
D.T. Haimo, "Expansions in terms of generalized heat polynomials and
Appell transforms", J. Math. & Mech. Vol. I5, No. 5, (1966), 735-758.
5.
D.T.
Toronto,
et al. "H_h__er_.Tr_an.scenden_t@l Functions, Vol. 2", McGraw Hill, Toronto,
Haimo,
"Series
expansion
and
integral
representation
of
of
their
generalized
temperataures", Illinois J. Math. Vol. 14, No. 4, (1970), 621-629.
equation",
Proc.
Edinburg
6.
C. Nasim and B.D. Aggarwala, "On the generalized heat
Math. Soc. (1984), 27, 261-273.
7.
P.C. Rosenbloom and D.V. Widder, "Expansions in tearms of heat polynomials and
associated functions", Trans. amer. Math. Soc., Vol. 92 (1959), 220-266.
8.
G.N. Watson, "_Theor
Cambridge, 1966.
of
Bessel
functions",
Second Edition,
University
Press,