APPROXIMATE SOLUTION OF RANDOM DIFFERENTIAL AND INTEGRAL EQUATIONS

Abstract

In the recent past, random differential equations and random integral equations have been used to model many problems of interest in the real world. Such diverse phenomena as the scattering of tsunamis, electrical circuit output, population growth, fluid flow, stream pollution, and the twinkling of a star have all lent themselves to more accurate description when random, rather than deterministic, equations were used. The main drawback to the use of random equations, however, has been that they are in general more difficult to solve than their deterministic counterparts. While much progress has been made on existence, uniqueness theory for random equations, the derivation of usable closed form solutions has proved impossible in most cases. Thus, a meaningful use of random equations in applications has come to depend upon the effectiveness of techniques to approximate their solutions. This chapter discusses simple perturbation techniques using an example. Hierarchy techniques are a class of approximation methods that have found favor with a number of physicists.