Page 538 - Handbook Of Integral Equations

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7 . The solution of the dual integral equation of the first kind

∞
g(t)P 1 (cosh x)y(t) dt = f(x) for 0 < x < a,
– +it
0 2
(32)
∞
t tanh(πt)P (cosh x)y(t) dt =0 for a < x < ∞,
1
– +it
0 2
2
where P µ (x) is the Legendre spherical function of the first kind (see Supplement 10), i = –1, and
g(x) is a given function, is determined by the formula
a
y(x)= cos(xt)ϕ(t) dt, (33)
0
and the function ϕ(x) is to be found from the Fredholm equation (20) of the second kind in which

∞

K(x, t)= [1 – g(s)]{cos[(x + t)s] + cos[(x – t)s]} ds,
0
√ x
2 d f(s) sinh s
ψ(x)= √ ds. (34)
π dx 0 cosh x – cosh s
On the basis of relations (34), we can readily see that the kernel K(x, t) is symmetric.

8 . The solution of the dual integral equation of the first kind
◦

∞
tg(t)P – +it (cosh x)y(t) dt = f(x) for 0 < x < a,
1
0 2
(35)
∞
tanh(πt)P 1 (cosh x)y(t) dt =0 for a < x < ∞,
– +it
0 2
2
where P µ (x) is the spherical Legendre function of the first kind (see Supplement 10), i = –1, and
g(x) is a given function, is determined by the formula
a
y(x)= sin(xt)ϕ(t) dt, (36)
0

and the function ϕ(x) is to be found from the Fredholm equation (20) of the second kind in which

∞
K(x, t)= [1 – g(s)]{cos[(x – t)s] – cos[(x + t)s]} ds,
0
√ x
2 f(s) sinh s
ψ(x)= √ ds. (37)
π
0 cosh x – cosh s
On the basis of relations (37), we can readily see that the kernel K(x, t) is symmetric.

•
References for Section 10.6: E. C. Titchmarsh (1948), I. Sneddon (1951), Ya. S. Uflyand (1977), F. D. Gakhov and
Yu. I. Cherskii (1978).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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