Obtaining an integral equation from a differential equation

23
Integral equations
It is not unusual in the analysis of a physical system to encounter an equation
in which an unknown but required function y(x), say, appears under an integral
sign. Such an equation is called an integral equation, and in this chapter we discuss
several methods for solving the more straightforward examples of such equations.
Before embarking on our discussion of methods for solving various integral
equations, we begin with a warning that many of the integral equations met in
practice cannot be solved by the elementary methods presented here but must
instead be solved numerically, usually on a computer. Nevertheless, the regular
occurrence of several simple types of integral equation that may be solved
analytically is sufficient reason to explore these equations more fully.
We shall begin this chapter by discussing how a differential equation can be
transformed into an integral equation and by considering the most common
types of linear integral equation. After introducing the operator notation and
considering the existence of solutions for various types of equation, we go on
to discuss elementary methods of obtaining closed-form solutions of simple
integral equations. We then consider the solution of integral equations in terms of
infinite series and conclude by discussing the properties of integral equations with
Hermitian kernels, i.e. those in which the integrands have particular symmetry
properties.
23.1 Obtaining an integral equation from a differential equation
Integral equations occur in many situations, partly because we may always rewrite
a differential equation as an integral equation. It is sometimes advantageous to
make this transformation, since questions concerning the existence of a solution are more easily answered for integral equations (see section 23.3), and,
furthermore, an integral equation can incorporate automatically any boundary
conditions on the solution.
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