Solution of integral equations by hybrid computation

Abstract

The mathematical description of many problems of engineering interest contains integral equations. Typical of a large class of such problems is the Fredholm integral equation of the second kind, y(x) = f(x) + λ ∫ba K(x,t) y(t) dt (1) where f(x) and the kernel K(x,t) are given functions, a and b are constants, λ is a parameter and y(x) is to be found. From a computational point of view, equations of this type may be considered as problems in two dimensions, where one dimension (t) is the dummy variable of integration. For digital computer solution, both variables must be discretized. For analog computer solution, it is possible to perform continuous integration with respect to the variable t for a fixed value of x and perform a scanning process to obtain step changes in the second variable. In either case, the solution is iterative and results in a sequence of functions {yn(x)}, n=1, 2,... which, under certain conditions, converge to the true solution y(x) as n increases.