Chapter 5 - Three Important Equations

Abstract

Partial differential equations (PDEs) are extremely important in both mathematics and physics. This chapter provides an introduction to some of the simplest and most important PDEs in both disciplines, and techniques for their solution. The chapter focuses on three equations—the heat equation, the wave equation, and Laplace's equation. Following the nomenclature of the geometrical figures, if B2−4AC<0 the partial differential equation is said to be parabolic; if B2−4AC=0 the equation is elliptic; and if B2−4AC>0 the equation is hyperbolic. Thus, the heat equation is the prototypical parabolic equation, the wave equation is the prototypical wave equation, and Laplace's equation is the prototypical elliptical equation. It is shown that for the heat equation the initial conditions diffuse in time, whereas in the wave equation initial conditions are propagated, changing position but not shape.