CHAPTER 6 - APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS

Abstract

The applications of integral equations are not restricted to ordinary differential equations. 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 the equations of elliptic type can be reduced to Fredholm integral equations, while the study of parabolic and hyperbolic differential equations leads to Volterra integral equations. This chapter examines the linear partial differential equations of the elliptic type, specifically to the Laplace, Poisson, and Helmholtz equations, wherein the most interesting and important achievements of the theory of integral equations lie. Three types of boundary conditions arise in the study of elliptic partial differential equations. The first type is the Dirichlet condition. The second type is the Neumann condition. The Green's function is an auxiliary function that plays the same crucial role in the integral-equation formulation of partial differential equations as it plays in the case of ordinary differential equations. This function depends on the form of the differential equation, the boundary condition, and the region.