ABSTRACT
For many years the subject of functional equations has held a prominent place in the attention of mathematicians. In more recent years this attention has been directed to a particular kind of functional equation, an integral equation, wherein the unknown function occurs under the integral sign. Such equations occur widely in diverse areas of applied mathematics and physics. They offer a powerful technique for solving a variety of practical problems. One obvious reason for using the integral equation rather than differential equations is all of the conditions specifying the initial value problem or boundary value problem for a differential equation. In the case of PDEs, the dimension of the problem is reduced in this process so that, for example, a boundary value problem for a practical differential equation in two independent variables transforms into an integral equation involving an unknown function of only one variable. This reduction of what may represent a complicated mathematical model of a physical situation into a single equation is itself a significant step, but there are other advantages to be gained by replacing differentiation with integration. Some of these advantages arise because integration is a smooth process, a feature which has significant implications when approximate solutions are sought. Whether one is looking for an exact solution to a given problem or having to settle for an approximation to it, an integral equation formulation can often provide a useful way forward. For this reason
In 1825 Abel, an Italian mathematician, first produced an integral equation in connection with the famous tautochrone problem . The problem is connected with the determination of a curve along which a heavy particle, sliding without friction, descends to its lowest position, or more generally, such that the time of descent is a given function of its initial position.
