Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 17Integral Equations

Differential equations have been extremely useful in describing physical processes. They are composed of the derivatives of the unknown function. Since derivatives are defined in terms of the ratios of differences in the neighborhood of a point, differential equations are local equations. In our mathematical toolbox, there are also integral equations, where the unknown function appears under an integral sign. Since the integral equations involve integrals of the unknown function over a domain, they are global or nonlocal equations. In general, integral equations are much more difficult to solve. An important property of the differential equations is that to describe a physical process completely, they must be supplemented with boundary conditions. Integral equations, on the other hand, constitute a complete description of a given problem where extra conditions are neither needed nor could be imposed. Because the boundary conditions can be viewed as a convenient way of including global effects into a system, a connection between differential and integral equations is to be expected. In fact, under certain conditions integral and differential equations can be transformed into each other. Whether an integral or a differential equation is more suitable for expressing laws of nature is still an interesting problem, with some philosophical overtones that Einstein himself once investigated. Sometimes the integral equation formulation of a given problem ...