This text is aimed at graduate students in engineering, physics, and applied mathematics. I have included four essential topics: Green's functions, integral equations, Fourier transforms, and Laplace transforms. As background material for understanding these topics, a course in complex variables with contour integration and analytic continuation and a second course in differential equations are assumed. One may point out that these topics are not all that advanced – the expected advanced-level knowledge of complex variables and a familiarity with the classical partial differential equations of physics may be used as a justification for the term “advanced.” Most graduate students in engineering satisfy these prerequisites. Another aspect of this book that makes it “advanced” is the expected maturity of the students to handle the fast pace of the course. The fours topics covered in this book can be used for a one-semester course, as is done at the Illinois Institute of Technology (IIT). As an application-oriented course, I have included techniques with a number of examples at the expense of rigor. Materials for further reading are included to help students further their understanding in special areas of individual interest. With the advent of multiphysics computational software, the study of classical methods is in general on a decline, and this book is an attempt to optimize the time allotted in the curricula for applied mathematics.

I have included a selection of exercises at the end of each chapter for instructors to choose as weekly assignments.