Page 466 - Handbook Of Integral Equations

8.4-5. Reduction to Ordinary Differential Equations

Consider the special case in which the transform of the kernel of the integral equation (1) can be
represented in the form
M(p)
˜
K(p)= , (20)
N(p)
where M(p) and N(p) are some polynomials of degrees m and n, respectively:
m n
k k
M(p)= A k p , N(p)= B k p . (21)
k=0 k=0

In this case, the solution of the integral equation (1) (if it exists) satisfies the following linear
nonhomogeneous ordinary differential equation of order m with constant coefficients:
m n
(k) (k)
A k y (x)= B k f x (x). (22)
x
k=0 k=0
We can rewrite Eq. (22) in the operator form

d
M(D)y(x)= N(D)f(x), D ≡ .
dx
The initial data for the differential equation (22), as well as the conditions that must be imposed on
the right-hand side of the integral equation (1), can be obtained from the relation

m k–1 n k–1
k–1–s (s) k–1–s (s)
A k p y (0) – B k p f (0) = 0 (23)
x
x
k=0 s=0 k=0 s=0
by matching the coefficients of like powers of the parameter p.
The proof of this assertion can be given by applying the Laplace transform to the differential
equation (22) followed by comparing the resulting expression with Eq. (3) with regard to (20).

8.4-6. Reduction of a Volterra Equation to a Wiener–Hopf Equation

A Volterra equation of the first kind with difference kernel of the form
x

K(x – t)y(t) dt = f(x), 0 < x < ∞, (24)
0
can be reduced to the following Wiener–Hopf equation of the first kind:

∞

K + (x – t)y(t) dt = f(x), 0 < x < ∞, (25)
0
where the kernel K + (x – t)isgiven by

K(s) for s >0,

K + (s)=
0 for s <0.
Methods for solving Eq. (25) are presented in Chapter 10.
•
References for Section 8.4: G. Doetsch (1956), V. A. Ditkin and A. P. Prudnikov (1965), M. L. Krasnov, A. I. Kiselev,
and G. I. Makarenko (1971), V. I. Smirnov (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. D. Gakhov and
Yu. I. Cherskii (1978).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 448