This book presents mathematical tools to solve partial differential equations, typical of physical problems. It explains in a detailed manner the process of solving the problems that typically arise in the context of physics. Although there are a large number of textbooks on this topic, few go so deep into the topic. One of the original and unique features of this book is emphasis on the mathematical formulation of the problems, as well as the analysis of several alternative ways to solve them. Importantly, the book provides a graphical analysis of the results when appropriate. It describes a wide scope of the problems, with detailed solutions and the methods involved, ranging from cases in one to three dimensions, from Cartesian to polar, cylindrical, and spherical coordinates and includes properties and applications of the Fourier transform to solve partial differential equations.