Partial Differential Equations - M5M3
An introduction to partial differential equations - M4M3
1) Basic concepts: PDEs, linearity, superposition principle. Boundary and Initial value problems.
2) Gauss Theorem: gradient, divergence and rotational. Main actors: continuity, heat or diffusion, Poisson-Laplace, and the wave equations.
3) Linear and Qasilinear first order PDEs in two independent variables. Well-posedness for the Cauchy problem. The linear transport equation. Upwinding scheme for the discretization of the advection equation.
4) A brief introduction to conservation laws: The traffic equation and the Burgers equation. Singularities.
5) Derivation of the heat equation. The boundary value problem: separation of variables. Fourier Series. Explicit Euler scheme for the 1d heat equation: stability.
6) The Cauchy problem for the heat equation: Poisson’s Formula. Uniqueness by maximum principle.
7) The ID wave equation. D’Alembert Formula. The boundary value problem by Fourier Series. Explicit finite difference scheme for the 1d wave equation: stability.
8) 2D and 3D waves. Casuality and Energy conservation: Huygens principle.
9) Green’s functions: Newtonian potentials. Dirichlet and Neumann problems.
10) Harmonic functions. Uniqueness: mean property and maximum principles.
An introduction to partial differential equations - M3M3