SECTION 12 - Green's Function

Abstract

This chapter discusses the boundary value problem for nonhomogeneous linear equations. The function that depends on the parameter ξ is denoted by G(x, ξ) and is called Green's function. In the case where the equation has regular singularities at one or both endpoints of the interval, the boundary conditions have to be modified because of the behavior of the solution near the boundary points. In the case, one of the independent solutions is finite at each of the endpoints, while the second includes a logarithmic term. The notion of Green's function can be generalized in two directions. First, G(x, ξ, λ) can be defined for arbitrary linear equations of degree n and for more general boundary conditions. Second, a generalized function G(x, ξ, λ) can be defined for λ = λn so as to possess similar properties to the ordinary Green's function and to reproduce the sulution of the nonhomogeneous equation by integration. The chapter discusses a necessary and sufficient condition for the existence of solutions of nonhomogeneous systems.