Page 505 - Handbook Of Integral Equations

9.9-2. A Formula for the Resolvent

The resolvent of the integral equation (1) is determined via the iterated kernels by the formula
∞

R(x, t)= K n (x, t), (6)
n=1

where the convergent series on the right-hand side is called the Neumann series of the kernel K(x, t).
Now the solution of the Volterra equation of the second kind (1) can be rewritten in the traditional
form
x
y(x)= f(x)+ R(x, t)f(t) dt. (7)
a
Remark 2. In the case of a kernel with weak singularity, the solution of Eq. (1) can be obtained
by the successive approximation method. In this case the kernels K n (x, t) are continuous starting
1
from some n.For α < , even the kernel K 2 (x, t) is continuous.
2
•
References for Section 9.9: W. V. Lovitt (1950), V. Volterra (1959), S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev,
and G. I. Makarenko (1971), V. I. Smirnov (1974).

9.10. Method of Quadratures

9.10-1. The General Scheme of the Method
Let us consider the linear Volterra integral equation of the second kind

x
y(x) – K(x, t)y(t) dt = f(x), (1)
a
on an interval a ≤ x ≤ b. Assume that the kernel and the right-hand side of the equation are continuous
functions.
From Eq. (1) we find that y(a)= f(a). Let us choose a constant integration step h and consider
the discrete set of points x i = a + h(i – 1), i =1, ... , n.For x = x i , Eq. (1) acquires the form

x i
y(x i ) – K(x i , t)y(t) dt = f(x i ), i =1, ... , n. (2)
a
Applying the quadrature formula (see Subsection 8.7-1) to the integral in (2) and choosing x j
(j =1, ... , i) to be the nodes in t, we arrive at the system of equations

i

y(x i ) – A ij K(x i , x j )y(x j )= f(x i )+ ε i [y], i =2, ... , n, (3)
j=1

where ε i [y] is the truncation error and A ij are the coefficients of the quadrature formula on the
interval [a, x i ] (see Subsection 8.7-1). Suppose that ε i [y] are small and neglect them; then we
obtain a system of linear algebraic equations in the form
i

y 1 = f 1 , y i – A ij K ij y j = f i , i =2, ... , n, (4)
j=1

where K ij = K(x i , x j ), f i = f(x i ), and y i are approximate values of the unknown function y(x)at
the nodes x i .

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 487