Beginning Partial Differential Equations

Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addressing more specialized topics and applications.

Maintaining the hallmarks of the previous edition, the book begins with first-order linear and quasi-linear PDEs and the role of characteristics in the existence and uniqueness of solutions. Canonical forms are discussed for the linear second-order equation, along with the Cauchy problem, existence and uniqueness of solutions, and characteristics as carriers of discontinuities in solutions. Fourier series, integrals, and transforms are followed by their rigorous application to wave and diffusion equations as well as to Dirichlet and Neumann problems. In addition, solutions are viewed through physical interpretations of PDEs. The book concludes with a transition to more advanced topics, including the proof of an existence theorem for the Dirichlet problem and an introduction to distributions.

Additional features of the Second Edition include solutions by both general eigenfunction expansions and numerical methods. Explicit solutions of Burger's equation, the telegraph equation (with an asymptotic analysis of the solution), and Poisson's equation are provided. A historical sketch of the field of PDEs and an extensive section with solutions to selected problems are also included.

Beginning Partial Differential Equations, Second Edition is an excellent book for advanced undergraduate- and beginning graduate-level courses in mathematics, science, and engineering.

Peter V. O'Neil, PhD, is Professor Emeritus in the Department of Mathematics at The University of Alabama at Birmingham. Dr. O'Neil has over forty years of academic experience and is the recipient of the Lester R. Ford Award from the Mathematical Association of America. He is a member of the American Mathematical Society, the Society for Industrial and Applied Mathematics, and the American Association for the Advancement of Science.