Page 676 - Handbook Of Integral Equations

Chapter 14

Methods for Solving

Nonlinear Integral Equations

14.1. Some Definitions and Remarks

14.1-1. Nonlinear Volterra Integral Equations

Nonlinear Volterra integral equations can be represented in the form
x

K x, t, y(t) dt = F x, y(x) , (1)
a

where K x, t, y(t) is the kernel of the integral equation and y(x) is the unknown function (a ≤ x ≤ b).
All functions in (1) are usually assumed to be continuous.
The form (1) does not cover all possible forms of nonlinear Volterra integral equations; however,
it includes the types of nonlinear equations which are most frequently used and studied. A nonlinear
integral equation (1) is called a Volterra integral equation in the Urysohn form.
In some cases, Eq. (1) can be rewritten in the form
x

K x, t, y(t) dt = f(x). (2)
a
Equation (2) is called a Volterra equation of the first kind in the Urysohn form. Similarly, the
equation
x

y(x)– K x, t, y(t) dt = f(x), (3)
a
is called a Volterra equation of the second kind in the Urysohn form.
By the substitution u(x)= y(x)– f(x), Eq. (3) can be reduced to the canonical form

x

u(x)= K x, t, u(t) dt, (4)
a

where K x, t, u(t) is the kernel* of the canonical integral equation.

The kernel K x, t, y(t) is said to be degenerate if
n

K x, t, y(t) = g k (x)h k t, y(t) .
k=1
* There are another ways of reducing Eq. (3) to the form (4) for which the form of the function K may be different.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 659