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Basic PDE

Basic PDE – 60650

The goal of this course is to teach the basics of Partial Differential Equations (PDE), linear and nonlinear. It begins by providing a list of the most important PDE and systems arising in mathematics and physics and outlines strategies for their “solving.” Then, it focusses on the solving of the four important linear PDE: The transport, the Laplace, the heat, and the wave equations, all in several space dimensions. In addition to deriving solution formulas, it also introduces “energy” methods involving L2-norms, which foreshadow modern theoretical developments for studying PDE. Next, it continues with the solving of nonlinear first-order equations (like the Burgers equation) by using the method of characteristics. Also, it defines weak solutions, introduces several nonlinear concepts and phenomena, and demonstrates the point that nonlinear PDE are much more difficult to solve than linear PDE. The first (and major) part of the course concludes with the presentation of other ways for representing solutions, including the separation of variables, traveling waves and solitons, similarity and scaling, Fourier, Radon and Laplace transform methods, converting nonlinear PDE to linear by using transformations like the Cole-Hopf, asymptotics, power series methods and the Cauchy-Kovalevskaya Theorem. The second part of the course introduces function spaces like Hӧlder and Sobolev spaces together with their properties including, approximation, extensions, traces, important inequalities, and compactness. These spaces are used for designing appropriate solution spaces for a given PDE problem (initial or boundary value) inside which the solution resides. Proving this may require sophisticated theories that could involve many branches of mathematics as well as physics. Such theories are presented in the later part of the textbook, for which there is no time available in this course. However, this course will prepare the interested student to continue learning more about PDE and their applications to mathematics, the sciences, engineering and economics.

Prerequisites: Basic Analysis I and concurrently enrollment in Basic Analysis II. Some familiarity with Ordinary Differential Equations is desirable but not a prerequisite. Exposure to an undergraduate PDE course is helpful but not required.

Textbook: Lawrence C. Evans, Partial Differential Equations: Second Edition. American Mathematical Society, Graduate Studies in Mathematics, Vol. 19, Providence RI, 2010

References

  • Fritz John, Partial Differential Equations
  • David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order
  • Lars Hӧrmander, The Analysis of Linear Partial Differential Operators (4 volumes)
  • Felipe Linares and Gustavo Ponce, Introduction to nonlinear dispersive equations
  • Ivan Georgievich Petrovsky, Lectures on Partial Differential Equations
  • Walter Strauss, Partial Differential Equations: An Introduction
  • Michael E. Taylor, Partial Differential Equations (3 volumes)