Introduction, PDEs as mathematical models, first order PDEs: method of characteristics, quasilinear first order PDEs.
Distributions, distribution solutions and weak solutions.
Classification of second order PDEs, Laplace equation: fundamental solution, mean-value formulas, The maximum principle, Poisson equation, properties of harmonic functions, Green’s functions, energy methods.
Heat equation: diffusion and Brownian motion, Fourier transforms, fundamental solution, the maximum principle, energy methods
Wave equation: one-dimensional wave equation, d'Alembert’s formula, higher-dimensional wave equation, energy estimates
Fourier series, boundary value problems, separation of variables
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