Introduction to partial differential equations and their classification. Classical solutions and well-posedness. Deduction of the wave and diffusion equations from physical models. The wave equation on the real line: d’Alembert formula. The heat equation on the real line: the fundamental solution. Wave and heat equation on an interval: maximum principle for the heat equation, boundary conditions, separation of variables, autofunctions and eigenvalues ​​of the Laplacian on an interval, the Fourier series. Laplace equation and harmonic functions: divergence theorem, Green identities, maximum principle for harmonic functions, representation formula for the harmonic functions, Green function. Variational properties of the solutions of the Dirichlet and Neumann problem. Wave equation in several dimensions: Kirchhoff's formula, method of descent, Huygens principle. Introduction to the theory of distributions: convolution of distributions, Fourier transform of distributions, fundamental solutions for the Laplace, heat, and wave equation.