This introductory text on partial differential equations interates modern and classical techniques for solving PDEs, at a level suitable for undergraduates. The author complements the classical topic of Fourier series with modern finite element methods. The result is an up-to-date, flexible approach to solving PDEs, which both faculty and students should find rewarding. Linear algebra is a key component of the text, providing a framework for both computing solutions and for understanding the theoretical basis of the methods. While techniques are emphasized over theory, the methods are presented in a mathematically sound fashion, developing a strong foundation for further study. Numerous exercises and examples involve meaningful experiments with realistic physical parameters, allowing students to use physical intuition to understand the qualitative features of the solutions.