Applied Mathematics for Science and Engineering

7Analytic Solution of Partial Differential Equations

Introduction

Many of the phenomena that are of interest to us in engineering and the applied sciences are modeled with partial differential equations (PDEs). Fluid flow, heat transfer, and mass transfer are prime examples, but problems in gravitation, electrostatics, and quantum theory all give rise to similar equations. The purpose of this chapter is to provide the reader with some basic skills, enabling him/her to find analytic solutions for many commonly encountered PDEs.

Several valuable references will be provided as we move through this material, but at the outset, we want to point out that there are two uniquely important monographs devoted to the analytic solution of PDEs: The Mathematics of Diffusion, Second Edition, by Crank (1975), and Conduction of Heat in Solids, Second Edition, by Carslaw and Jaeger (1959). These two books are known to nearly every worker in applied mathematics. Both are incredibly useful as guides to the solution of practical problems where diffusional (molecular) transport processes are dominant. Practitioners in this field are often heard to say, “I found a similar problem in Crank” or “I verified my solution with Carslaw and Jaeger.” Anyone wishing to become adept with the subject matter of this chapter simply must own both of these books.

Classification of Partial Differential Equations and Boundary Conditions

We have to be able to recognize and classify PDEs to attack them successfully; a book ...