Page 459 - Handbook Of Integral Equations

Chapter 8

Methods for Solving Linear Equations

x
of the Form K(x, t)y(t) dt = f(x)
a

8.1. Volterra Equations of the First Kind

8.1-1. Equations of the First Kind. Function and Kernel Classes
In this chapter we present methods for solving Volterra linear equations of the first kind. These
equations have the form
x

K(x, t)y(t) dt = f(x), (1)
a
where y(x) is the unknown function (a ≤ x ≤ b), K(x, t) is the kernel of the integral equation, and
f(x) is a given function, the right-hand side of Eq. (1). The functions y(x) and f(x) are usually
assumed to be continuous or square integrable on [a, b]. The kernel K(x, t) is usually assumed
either to be continuous on the square S = {a ≤ x ≤ b, a ≤ t ≤ b} or to satisfy the condition
b b
2
2
K (x, t) dx dt = B < ∞, (2)
a a
where B is a constant, that is, to be square integrable on this square. It is assumed in (2) that
K(x, t) ≡ 0 for t > x.
The kernel K(x, t) is said to be degenerate if it can be represented in the form K(x, t)=
g 1 (x)h 1 (t)+ ··· + g n (x)h n (t).
The kernel K(x, t) of an integral equation is called difference kernel if it depends only on the
difference of the arguments, K(x, t)= K(x – t).
Polar kernels
L(x, t)
K(x, t)= + M(x, t), 0 < β < 1, (3)
(x – t) β
and logarithmic kernels (kernels with logarithmic singularity)

K(x, t)= L(x, t) ln(x – t)+ M(x, t), (4)
where L(x, t) and M(x, t) are continuous on S and L(x, x) /≡ 0, are often considered as well.
Polar and logarithmic kernels form a class of kernels with weak singularity. Equations containing
such kernels are called equations with weak singularity.
The following generalized Abel equation is a special case of Eq. (1) with the kernel of the
form (3):
x
y(t)

dt = f(x), 0 < β <1.
(x – t) β
a
In case the functions K(x, t) and f(x) are continuous, the right-hand side of Eq. (1) must satisfy
the following conditions:

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 441