Chapter 13 - Solving PDEs With Green's Functions

Abstract

This chapter solves the partial differential equations with Green's function. The chapter constructs the Green's function for the heat equation using the Dirac-δ function. The chapter helps in finding the Green's function for several forms of the heat equation. There are a few factors that can complicate a particular form of the heat equation, including boundary terms, initial conditions, and existence of a heat source or sink. When complicating factors are present, it is sometimes advantageous to separate the problem into pieces, each of which contains one of the complicating factors. The solutions to each of the pieces are added together to give the solution to the original problem. The method of images is a technique for solving heat equation–type problems on a bounded interval or semiinfinite interval. It uses imaginary heat sources or sinks at points outside the interval to obtain the desired boundary conditions.