CHAPTER IV - BOUNDARY VALUE PROBLEMS

Abstract

This chapter boundary provides an overview of boundary value problems for both ordinary differential equations and partial differential equations and presents a systematic exposition of these problems. It reviews Green's function for a linear second order equation and symmetry of Green's function. The chapter discusses eigenvalues and eigenfunctions of a boundary value problem. A theory can be constructed for the heat conduction equation, analogous to the potential theory for Laplace's equation and hence, boundary value problems for the heat conduction equation to integral equations can be reduced. A Green's function can be formed for the heat conduction equation in precisely the same way as for Laplace's equation. Green's function can be used to form the solution of the nonhomogeneous heat conduction equation satisfying a homogeneous initial condition and homogeneous boundary conditions. The chapter presents the solution of the boundary value problems for equations of the elliptic and parabolic types by making use of potential theory, the whole of the working being based on a singular solution of the corresponding differential equation. This method of potential theory cannot be used for equations of the hyperbolic type. It is only in the one-dimensional case for the telegraphist's equation that the fundamental idea of this method can be used to reduce a boundary value problem to a Volterra integral equation.