The first chapter introduces PDEs and makes analogies to familiar ODE concepts, then strengthens the connection by exploring the method of separation of variables. Chapter 2 examines the “Big Three” PDEs-- the heat, wave, and Laplace equations, and is followed by chapters explaining how these and other PDEs on finite intervals can be solved using the Fourier series for arbitrary initial and boundary conditions.
Chapter 5 investigates characteristics for both first- and second-order linear PDEs, the latter revealing how the Big Three equations are important far beyond their original application to physical problems. The book extends the Fourier method to functions on unbounded domains, gives a brief introduction to distributions, then applies separation of variables to PDEs in higher dimensions, leading to the special funtions, including the orthogonal polynomials.
Other topics include Sturm-Liouville problems, adjoint and self-adjoint problems, the application of Green’s functions to solving nonhomogeneous PDEs, and an examination of practical numerical methods used by engineers, including the finite difference, finite element, and spectral methods.
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