Applications to Partial Differential Equations

Abstract

The applications of integral equations are not restricted to ordinary differential equations. In fact, the most important applications of integral equations arise in finding the solutions of boundary value problems in the theory of partial differential equations of the second order. The boundary value problems for equations of elliptic type can be reduced to Fredholm integral equations, whereas the study of parabolic and hyperbolic differential equations leads to Volterra integral equations. We confine our attention to the linear partial differential equations of the elliptic type, specifically, to the Laplace, Poisson, and Helmholtz equations wherein lie the most interesting and important achievements of the theory of integral equations.