A First Course in Partial Differential Equations

This second edition to A First Course in Partial Differential Equations provides a clear, rigorous, and student-friendly introduction to the core theory and solution techniques for partial differential equations (PDEs), making it an ideal text for upper-level undergraduates in mathematics, physics, engineering, and the applied sciences.

This volume builds on the strengths of the first edition by integrating new topics that bridge classical theory with modern applications. In addition to comprehensive treatments of standard second-order linear PDEs — the heat equation, wave equation, and Laplace's equation — this edition includes substantial new content:

- A new chapter on the Fourier Transform, providing students with powerful tools to analyze PDEs in the frequency domain, along with practical examples relevant to physics and engineering.

- A new chapter on Green's Functions, illustrating their construction and use in solving nonhomogeneous boundary value problems, thereby deepening understanding of linear operators and solution representations.

- Expanded content on nonhomogeneous equations and boundary conditions, with methods such as Duhamel's principle fully developed.

- Enhanced coverage of numerical methods, especially finite difference approximations, to offer a practical introduction to computational approaches in solving PDEs.

- More than 400 new exercises, now organized by section, promoting targeted practice and easier integration into coursework.

- Many chapters conclude with open-ended explorations and project suggestions, making the text ideal for undergraduate theses, research projects, or independent study.

Core topics also include first-order linear and nonlinear PDEs arising in the physical and life sciences, Fourier series, Sturm–Liouville problems, and special functions of mathematical physics. Appendices review essential background in complex analysis and linear algebra, ensuring accessibility for students from a broad range of STEM disciplines.

With its flexible structure, this textbook supports both one- and two-semester courses, and provides a solid foundation for students preparing for graduate-level PDE courses. It is equally valuable as a reference text for researchers and practitioners seeking practical methods for solving PDEs in scientific and engineering contexts.

Dr J Robert Buchanan is a Professor of Mathematics at Millersville University of Pennsylvania (Millersville, PA) where he has been since 1995. Dr Buchanan received a BS in Physics from Davidson College (Davidson, NC) and a MS and PhD in Applied Mathematics from North Carolina State University (Raleigh, NC). He is also the author of An Introductory Textbook on Financial Mathematics, An Undergraduate Introduction to Financial Mathematics, published by WSPC. Dr Buchanan's mathematical interests include differential equations, financial and actuarial mathematics.

Dr Zhoude Shao retired from the position of Professor of Mathematics at Millersville University of Pennsylvania (Millersville, PA) in 2022 after a 28-year career. Dr Shao received his BS and MS in Mathematics from Shandong University (PR China) and PhD in Mathematics from the University of Minnesota (Minneapolis, MN). His academic career saw the publication of numerous peer-reviewed journal articles, presentations at international conferences, and the supervision of many student research projects and theses. His research interests focused on dynamical systems, Navier–Stokes equations, reaction diffusion equations, and stability of solutions of delay and functional differential equations.