Chapter 7 - Separation of Variables in Cartesian Coordinates

Abstract

This chapter concentrates on solving partial differential equations that involve the Laplacian. The chapter analyzes the three prototypical equations—the heat equation, the wave equation, and Laplace's equation—in significant detail. The chapter considers four techniques of solving partial differential equations: separation of variables, the Fourier transform, the Laplace transform, and Green's functions. The chapter solves each of these equations in Cartesian coordinates by separation of variables. The chapter considers the case of Laplace's equation in two variables. One sees in all of the examples of the chapter that the resulting ordinary differential equations are familiar and elementary to solve. The chapter analyzes Laplace's equation in three variables. The chapter gives a detailed description of the heat equation in one dimension and also studies the wave equation in one dimension.