This volume provides an introduction to the analytical and numerical aspects of partial differential equations (PDEs). It unifies an analytical and computational approach for these, with the qualitative behavior of solutions established using classical concepts such as maximum principles and energy methods. Notable inclusions are the treatment of irregularly shaped boundaries, polar coordinates, and the use of flux-limiters when approximating hyperbolic conservation laws. The numerical analysis of difference schemes is rigorously developed using discrete maximum principles and discrete Fourier analysis. A novel feature is the inclusion of a chapter containing projects, intended for either individual or group study, that cover a range of topics such as parabolic smoothing, traveling waves, isospectral matrices, and the approximation of multidimensional advection-diffusion problems.