Beginning Partial Differential Equations

Beginning Partial Differential Equations provides a challenging yet accessible introduction to partial differential equations for advanced undergraduate and beginning graduate students. Features include: * A discussion of first order equations and the method of characteristics for quasi-linear first order PDEs * Canonical forms of second order PDEs * Characteristics and the Cauchy problem * A proof of the Cauchy-Kowalevski theorem for linear systems * A self-contained development of tools from Fourier analysis * Connections between the mathematics and physical interpretations of PDEs * Numerous exercises, many with solutions provided * Experimental, computer-based exercises designed to develop lines of inquiry. The treatment of second order PDEs focuses on well-posed problems, properties and behavior of solutions, existence and uniqueness of solutions, and techniques for writing representations of solutions. Techniques include the use of characteristics, Fourier methods, and, for the Dirichlet problem, Green's function and conformal mappings. Also included are the Kirchhoff/Poisson solution of the wave equation, Huygens's principle, and Lebesgue's example of a Dirichlet problem with no solution.