Page 476 - Handbook Of Integral Equations

Example. Consider the following integral equation of the first kind with variable lower limit of integration:
∞

e a(x–t) y(t) dt = A sin(bx), a >0. (5)
x
According to (3) and (4), we can write out the expressions for ˜ f(p) (see Supplement 4) and ˜ K(–p),
Ab ∞ 1
˜ f(p)= , ˜ K(–p)= e (p–a)z dz = , (6)
2
p + b 2 0 a – p
and the solution of Eq. (5) in the form
1 c+i∞ Ab(a – p)
y(x)= e px dp. (7)
2
2πi c–i∞ p + b 2
Now using the tables of inverse Laplace transforms (see Supplement 5), we obtain the exact solution
y(x)= Aa sin(bx) – Ab cos(bx), a >0, (8)
which can readily be verified by substituting (8) into (5) and using the tables of integrals in Supplement 2.

8.8-2. Reduction to a Wiener–Hopf Equation of the First Kind

Equation (1) can be reduced to a first-kind one-sided equation

∞
K – (x – t)y(t) dt = –f(x), 0 < x < ∞, (9)
0
where the kernel K – (x – t) has the following form:

0 for s >0,
K – (s)=
–K(s) for s <0.
Methods for studying Eq. (9) are described in Chapter 10.

•
References for Section 8.8: F. D. Gakhov and Yu. I. Cherskii (1978), A. D. Polyanin and A. V. Manzhirov (1997).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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