Types of integral equation

INTEGRAL EQUATIONS
We shall illustrate the principles involved by considering the differential equation
y (x) = f(x, y),
(23.1)
where f(x, y) can be any function of x and y but not of y (x). Equation (23.1)
thus represents a large class of linear and non-linear second-order differential
equations.
We can convert (23.1) into the corresponding integral equation by first integrating with respect to x to obtain
x
f(z, y(z)) dz + c1 .
y (x) =
0
Integrating once more, we find
x du
y(x) =
0
u
f(z, y(z)) dz + c1 x + c2 .
0
Provided we do not change the region in the uz-plane over which the double
integral is taken, we can reverse the order of the two integrations. Changing the
integration limits appropriately, we find
x
x
y(x) =
f(z, y(z)) dz
du + c1 x + c2
(23.2)
z
0 x
(x − z)f(z, y(z)) dz + c1 x + c2 ;
(23.3)
=
0
this is a non-linear (for general f(x, y)) Volterra integral equation.
It is straightforward to incorporate any boundary conditions on the solution
y(x) by fixing the constants c1 and c2 in (23.3). For example, we might have the
one-point boundary condition y(0) = a and y (0) = b, for which it is clear that
we must set c1 = b and c2 = a.
23.2 Types of integral equation
From (23.3), we can see that even a relatively simple differential equation such
as (23.1) can lead to a corresponding integral equation that is non-linear. In this
chapter, however, we will restrict our attention to linear integral equations, which
have the general form
b
K(x, z)y(z) dz.
(23.4)
g(x)y(x) = f(x) + λ
a
In (23.4), y(x) is the unknown function, while the functions f(x), g(x) and K(x, z)
are assumed known. K(x, z) is called the kernel of the integral equation. The
integration limits a and b are also assumed known, and may be constants or
functions of x, and λ is a known constant or parameter.
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