Solution of an Integro-differential Equation with Dirichlet Conditions using Techniques of the Inverse Moments Problem

Abstract

It will be shown that finding solutions from some integro-differential equation under Dirichlet conditions is equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain E = {(x, t), 0 > x > L; t > 0}. We will see that an approximate solution of the equation integro-differential can be found using the techniques of generalized inverse moments problem and bounds for the error of the estimated solution. First the problem is reduced to solving a hyperbolic or parabolic partial derivative equation considering the unknown source. The method consists of two steps. In each one an integral equation is solved numerically using the two-dimensional inverse moment problem techniques. We illustrate the different cases with examples.