This book teaches the basic methods of partial differential equations and introduces related important ideas associated with the analysis of numerical methods for those partial differential equations. Standard topics such as separation of variables, Fourier analysis, maximum principles, and energy estimates are included. Numerical methods are introduced in parallel to the classical theory. The numerical experiments are used to illustrate properties of differential equations, and a theory for finite difference approximations is developed. The text would be suitable for advanced undergraduate and graduate courses in mathematics and engineering. Necessary prerequisites for this text are basic calculus and linear algebra. Some elementary knowledge of ordinary differential equations is also preferable.