Select Ideas in Partial Differential Equations

This text provides an introduction to the applications and implementations of partial differential equations. The content is structured in three progressive levels which are suited for upper–level undergraduates with background in multivariable calculus and elementary linear algebra (chapters 1–5), first– and second–year graduate students who have taken advanced calculus and real analysis (chapters 6-7), as well as doctoral-level students with an understanding of linear and nonlinear functional analysis (chapters 7-8) respectively. Level one gives readers a full exposure to the fundamental linear partial differential equations of physics. It details methods to understand and solve these equations leading ultimately to solutions of Maxwell’s equations. Level two addresses nonlinearity and provides examples of separation of variables, linearizing change of variables, and the inverse scattering transform for select nonlinear partial differential equations. Level three presents rich sources of advanced techniques and strategies for the study of nonlinear partial differential equations, including unique and previously unpublished results. Ultimately the text aims to familiarize readers in applied mathematics, physics, and engineering with some of the myriad techniques that have been developed to model and solve linear and nonlinear partial differential equations.

Peter J. Costa, is Principal Applied Mathematician at Hologic Incorporated in Marlborough, MA. Dr. Costa is the co-creator of MATLABs Symbolic Math Toolbox. He has developed mathematical methods for the diagnosis of cervical cancer, tracking of airborne vehicles, the diffusion of nonlinear optical systems, and the transmission of infectious diseases throughout a population. His research interests include mathematical physics and mathematical biology. He received the PhD in Applied Mathematics, specializing in nonlinear partial differential equations, from the University of Massachusetts at Amherst.