Students will know basic of properties (existence, uniqueness, etc.) and techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic PDEs (conservation laws, heat, Laplace, wave equations).
Linear first order PDE's. Method of characteristics. The Burgers' equation. Shocks and rarefaction waves. Riemann problem for scalar conservation laws. Partial differential equations of second order. Well posed problems, IBV problems. The heat equation. Derivation, maximum principle, fundamental solution. The Fourier transform method. Laplace's and Poisson's equations. Maximum principle, fundamental solution and Green's functions. The wave equation. One dimensional equation, fundamental solution and D'Alembert formula. Fundamental solution in three dimension and strong Huygens' principle, Kirchoff's formula.
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