ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXII (1981) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXII (1981)
J
a nM
u s i a l e k(Krakôw)
On the Green function for the heat equation and for the plane rectangular wedge
1. In this paper we shall construct the Green function G for the parabolic equation
(1) (D2 x + D2 + D2- D t) u ( X , t ) = 0, X = (x,y,z) in the domain
D = {(X , t): x > 0, y > 0, \z\ < a, t > 0},
where a is a positive constant satisfying the homogeneous boundary conditions of the Dirichlet type.
2. Let X s D x = {X : x > 0, y > 0, \z\ < a}. Let Y = ( p , q , r ) denote an arbitrary point of the space £ 3.
Let
G i(x, t , p , s) = (t - s ) 1/2
G2(y, t, q, s) = (t - s ) 1/2 exp
exp
& 1 LJ __ 1
— exp (p + x)2
— 4 ( t - s ) —4(t —s) (q - y ?
— exp ( q+y)2 _ —4 ( t - s ) _ i 1 1 05 Let us consider the series
G3( z , t , r , s ) = g0( z , t , r , s ) + X (- 1 )" l9n (z> t, r, s) + g2(z, t, r, s)], n=l
where
g i i Z ' t ^ r ^ ) = ( t - s ) 1/2 exp (4 )2
- 4 ( t - s ) (i - 1 , 2 ; n = 1,2, 3,...), ( r - z ) 2
g0( z , t , r , s ) = gl0(z, t , r , s ) = ( t - s ) 1/2 exp
- 4 ( t - s ) ( i = U2)
302 J. Mu s i al e k
and
4 = r - z \ (i = 1,2; n = 0, 1,2,...),
z\n = 4an + z, *2n = —4 an + z, z2n + 1 = —4an — 2a — z, z 2 n + i — 4an — z + 2a (n = 0, 1,...).
Let
G( X, t, Y, s) = Gt (x, t, p, s)G2{y, t, q, s)G3(z, t, r, s).
3. Now we shall give some lemmas which we shall use in the sequel.
L emma 1. I f t > s ^ 0, then the function Gx satisfies the equation (la) { D i - D , ) G 1( x , t , p , s ) = 0
and the function G2 satisfies the equation
(lb) ( D * - D t) G 2(y, t, q, s) = 0.
We omit the simple proof.
L emma 2. I f t > s ^ 0 and the function G3 satisfies the equation (le) ( D Î - D , ) G 3( z , t , r , s ) = 0,
then the function G satisfies equation (1).
P roof. By Lemma 1 we have
(Dj + D,2 + D f - D ,)G (X , t , Y, s) = G2G3(D2- D ,)G ,+
+ G1G3(£>2-Z>,)G2 + G1G2(B2-Z ),)G 3 = 0.
L emma 3. I f t > s ^ 0, z e( — a, a) and r e f —a, a}, then the functions gl„ (i = 1,2; n = 0, 1,2,...) satisfy equation (lc).
We omit the simple proof.
L emma 4. Let t > s ^ 0, z e (- fl,a ), r e f —a, a}. Then Mil ^ c ( n - 1) ({ = 1,2; n = 2, 3, 4,...), where C is a convenient positive constant.
We omit the simple proof.
Let us consider the functions
(2) Sl(au a2) = (t - s ) ei № )"2 exp
0d'n) i\2
■4(t —s)
(i = 1,2; n = 2, 3,4,...), where t > s ^ 0, (alt a2)e{{%, 0), (§, 0), (f, 1), (f, 2)} (i = 1, 2; n = 2, 3,...).
Now we shall prove the following
Green function fo r the heat equation 303
L emma 5. I f r e ( — a, a>, z e( — a,a) and t > s ^ 0, then
\SUai,a2)\ ^ C t (2t)ai + a2(n—4)~2 (i = 1,2; и = 2 ,3,...), where Cx is a convenient positive constant.
P roof. The function Si„(al , a2) given by formula (2) may be written in the form
Si„(a1,a2) = ( t - s ) a i +a2 m 2 4 (r -s )
2 a \ + a 2
