Unable to find a course book that provided all the topics needed for an introductory PDE course, the author pursued this book, which covers all of the essential topics. The needed foundation and theory is complimented with tangible applications in physics and other disciplines. Since many practical applications are non-linear, numerical solution techniques are required. Consequently, the book introduces this topic in a general way before providing the necessary details. As for an introduction to a specific method, the finite difference method is the natural place to begin. With this approach, readers more clearly understand the notations of order and convergence as well as explicit and implicit methodologies. Later, readers are introduced to the finite element method in such a way that it is seen as essentially a sub-space approximation technique. Finally, the finite analytic method is introduced, where readers are presented with the application of the Fourier Series methodology to linearized versions of non-linear PDEs. In terms of theory, material on linear PDEs reinforces the important concept of inner product spaces introduced in a linear algebra course, especially those of infinite dimension. Further, it introduces the concept of completeness, thereby introducing readers to Hilbert Spaces. Past experience with ordinary differential equations is called upon to understand the solution process for Sturm-Liouville boundary value ODE problems, which leads to an infinite-dimensional basis for an inner product space, and ultimately, a Fourier Series representation of the solution of an initial boundary value problems. Computer algebra resources such as MapleTM, Mathematica®, and MATLAB® can be used to aid in understanding and applying the solution techniques to interesting problems. This can begin as soon as the theoretical work is in Sturm-Liouville problems and Fourier series is covered. Later on, it is used to apply numerical solution methods to various applications.
