Integral - Equation Formulation of Two-Parameter Eigenvalue Problems

Abstract

This chapter discusses the integral-equation formulation of two-parameter eigenvalue problems. The chapter considers a two-parameter eigenvalue problem in differential equations associated with which are the three-point boundary conditions. It shows how the problem can be transformed into an integral-equation problem. This procedure is carried out to result in an integral equation whose nucleus is a Green's function (G) or Neumann's function (N). The integral equations in which the nucleus is an analogue of the kernel function, is focused. Such a nucleus is difficult to construct, in principle, then the Green's or Neumann function, because it arises as a solution of a partial differential equation; but it has two compensating advantages. First, nucleus has smoothness properties that the Green's Function lacks, and second, it is less stringently dependent on the boundary conditions. As a result, integral equations that are satisfied by integral relations are obtained. The kernel function is of more recent origin than the Green's and Neumann's functions. However, in certain respects it provides a more effective tool for the solution of boundary value problems.