28 - Solutions of Elliptic, Parabolic, and Hyperbolic Equations

Abstract

This chapter focuses on the solutions of elliptic, parabolic, and hyperbolic equations. The various boundary value problems related to the elliptic equations (the Laplace equation) are reviewed. The initial boundary value problems of various parabolic equations (the heat or diffusion equation) are discussed. The chapter also discusses the hyperbolic equations (wave equation). The D’Alembert solution concerns traveling waves and is of fundamental importance, because it shows how initial conditions specified at t = 0 on the infinite initial line influence the solution of a wave equation.