Applied Partial Differential Equation with Fourier Series and Boundary Value Problems 5th Edition

Abstract

This text discusses partial differential equations in the engineering and physical sciences. It is suited for courses whose titles include “Fourier series,” “orthogonal functions,” or “boundary value problems.” It may also be used in courses on Green’s functions, transform methods, or portions on advanced engineering mathematics and mathematical methods in the physical sciences. It is appropriate as an introduction to applied mathematics. Simple models (heat flow, vibrating strings, and membranes) are emphasized. Equations are formulated carefully from physical principles, motivating most mathematical topics. Solution techniques are developed patiently. Mathematical results frequently are given physical interpretations. Proofs of theorems (if given at all) are presented after explanations based on illustrative examples. Over 1000 exercises of varying difficulty form an essential part of this text. Answers are provided for those exercises marked with a star (∗). Further details concerning the solutions for most of the starred exercises are available in an instructor’s manual available for download through the Instructor Resource Center at PearsonHigherEd.com. Standard topics such as the method of separation of variables, Fourier series, orthogonal functions, and Fourier transforms are developed with considerable detail. Finite difference numerical methods for partial differential equations are clearly presented with considerable depth. A briefer presentation is made of the finite element method. This text also has an extensive presentation of the method of characteristics for linear and nonlinear wave equations, including discussion of the dynamics of shock waves for traffic flow. Nonhomogeneous problems are carefully introduced, including Green’s functions for Laplace’s, heat, and wave equations. Numerous topics are included, such as differentiation and integration of Fourier series, Sturm–Liouville and multidimensional eigenfunctions, Rayleigh quotient, Bessel functions for a vibrating circular membrane, and Legendre polynomials for spherical problems. Some optional advanced material is included (for example, asymptotic expansion of large eigenvalues, calculation of perturbed frequencies using the Fredholm alternative, stability conditions for finite difference methods, and direct and inverse scattering). Applications briefly discussed include the lift and drag associated with fluid flow past a circular cylinder, Snell’s law of refraction for light and sound waves, the derivation of the eikonal equation from the wave equation, dispersion relations for water waves, wave guides, and fiber optics. The text has evolved from the author’s experiences teaching this material to different types of students at various institutions (MIT, UCSD, Rutgers, Ohio State, and Southern Methodist University). Prerequisites for the reader are calculus and elementary ordinary differential equations. (These are occasionally reviewed in the text, where necessary.) For the beginning student, the core material for a typical course consists of most of Chapters 1–5 and 7. This will usually be supplemented by a few