Numerical Analysis of Integral Equations

Integral equation problems come in a variety of forms. Some are reformulations of ordinary or partial differential equations, for example, equations from mechanics, hydrodynamics, heat transfer, geophysics, electrostatics, and acoustic and electromagnetic scattering. One illustrative such topic is potential theory; and an example of it is to determine the charge density on a surface that leads to a given electrostatic field. Other integral equations are direct formulations of a physical problem, for example, the radiosity equation in computer graphics. The radiosity equation models the illumination of all parts of a surface when given the sources of illumination. Another major area of application is integral equations that model how populations change with time. This includes the dynamics of population growth and the spread of disease.

In this article numerical methods are discussed for solving some of the most important forms - Fredholm integral equations of the first and second kind, Volterra integral equations, and Cauchy singular integral equations. Other integral equations that have a smaller number of applications have been omitted, for example Wiener-Hopf equations and Abel integral equations. Also, most nonlinear integral equations have been omitted. There is a large literature on the theory and application of integral equations; for example, see