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.