You are here

An Introduction to Partial Differential Equations

Daniel Arrigo
Publication Date: 
Number of Pages: 
Synthesis Lectures on Mathematics and Statistics
[Reviewed by
Bill Satzer
, on
This textbook for undergraduates takes a promisingly original approach to introducing partial differential equations (PDEs). Many comparable texts have a tendency to present an intimidating picture right from the beginning. This book offers a bit more relaxed approach but still provides a solid introduction.
The first section connects with students’ experience in ordinary differential equations, and explains some of the differences. Nowhere here is there anything like a discussion of existence and uniqueness of solutions; the author concentrates instead on the basics. An early chapter discusses the big PDEs: the wave, diffusion, Laplace, and advection equations. This offers the student both motivation and context: this is why you’re studying PDEs.
Following this is a treatment of first-order PDEs; the author does this to parallel students’ prior experience with ordinary differential equations. He carefully discusses PDEs with constant coefficients, linear PDEs and the method of characteristics before turning to quasilinear and higher dimensional PDEs, and finally nonlinear first order PDEs.
With this background students are then introduced to second-order PDEs, the method of separation of variables, and then Fourier series. The latter fits naturally with a discussion that follows of finding solutions of the heat equation.
Two chapters follow. One treats the Fourier transform, and a second touches on higher dimensional problems (e.g., the heat and diffusion equation in two dimensions plus time, and Laplace’s equation). A final short section briefly describes special functions - mostly Bessel and Legendre functions.
This is a plain but appealing text that would work well in courses for mathematics majors as well as science and engineering majors. It introduces the relevant subjects simply and clearly. Carefully chosen exercises come in each chapter with solutions in an appendix. This is one book that would work reliably for self-study. Prerequisites include only a solid calculus course and elementary differential equations.
Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems end celestial mechanics.