This book is written to meet the needs of undergraduates in applied mathematics physics and engineering studying partial differential equations. It is a more modern comprehensive treatment intended for students who need more than the purely numerical solutions provided by programs like the MATLAB PDE Toolbox and those obtained by the method of separation of variables which is usually the only theoretical approach found in the majority of elementary textbooks. This will fill a need in the market for a more modern text for future working engineers and one that students can read and understand much more easily than those currently on the market.