Many of the fundamental laws of nature are formulated as partial differential
equations. Their qualitative behavior is as varied as the world around us.
This course introduces broad areas and their mathematical treatment. Further
analytical development comes in the two successor courses, which are independent
of each other.
Topics will include the method of characteristics, weak solutions, shocks,
and brief applications; formulation of equations from physical laws;
the notion of a well-posed problem; solution of the wave equation in
1,2, or 3 dimensions, domain of dependence, energy conservation;
solution of linear equations with Fourier series and integrals; Laplace's
equation, the maximum principle, integral representations, Green's functions,
eigenvalues of Laplacian; the fundamental solution of the heat equation,
properies of solution, scaling and similarity; some fundamental ideas of
numerical methods for PDE's; wave motion in several dimensions; introduction
to generalized derivatives and Sobolev spaces.
The most important prerequisite is real analysis
(or "advanced calculus") in several variables as in an undergraduate
course for math majors; good understanding of ordinary differential
equations is also important. Familiarity with Fourier series and integrals
is desirable.
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