Page 529 - Handbook Of Integral Equations

10.5. The Carleman Method for Equations of the
Convolution Type of the First Kind

By the Carleman method we mean the method of reducing an integral equation to a boundary
value problem of the theory of analytic functions, in particular, to the Riemann problem. For
equations of convolution type, this reduction can be performed by means of the integral transforms.
After solving the boundary value problem, the desired function can be obtained by applying the
inverse integral transform.

10.5-1. The Wiener–Hopf Equation of the First Kind

Consider the Wiener–Hopf equation of the first kind

1 ∞
√ K(x – t)y(t) dt = f(x), 0 < x < ∞, (1)
2π 0
which is frequently encountered in applications. Let us extend its domain to the negative semiaxis
by introducing one-sided functions,

y(x) for x >0, f(x) for x >0,
y + (x)= f + (x)= y – (x)=0 for x >0.
0 for x <0, 0 for x <0,
Using these one-sided functions, we can rewrite Eq. (1) in the form

1 ∞
√ K(x – t)y + (t) dt = f + (x)+ y – (x), –∞ < x < ∞. (2)
2π –∞
The auxiliary function y – (x) is introduced to compensate for the left-hand side of Eq. (2) for x <0.
Note that y – (x) is unknown in the domain x < 0 and is to be found in solving the problem.
Let us now apply the alternative Fourier transform to Eq. (2). Then we obtain the boundary
value problem
+
1 – F (u)
+
Y (u)= Y (u)+ . (3)
K(u) K(u)
If σ is the order of K(u)atinfinity, then the order of the coefficient of the boundary value problem
at infinity is η = –σ < 0. The general solution of problem (3) can be obtained on the basis of
relations (65) from Subsection 10.4-7 by replacing P h–m–1 (z) with P ν–n+η–1 (z) there. The solution
of the original equation (1) can be obtained from the solution of problem (3) by means of the
inversion formula
∞
1 + –iux
y(x)= y + (x)= √ Y (u)e du, x > 0. (4)
2π –∞
+
Note that in formula (4), only the function Y (u) occurs explicitly, which is related to the
–
function Y (u) by (3).

10.5-2. Integral Equations of the First Kind With Two Kernels

Consider the integral equation of the first kind

∞ 0
1 1
√ K 1 (x – t)y(t) dt + √ K 2 (x – t)y(t) dt = f(x), –∞ < x < ∞. (5)
2π 0 2π –∞

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 511