Partial Differential Equations and Applications

Introduction: Multi-index notation, examples. Transport Equation: homogeneous and non-homogeneous. Laplace equation: Fundamental solution; Average value; Green's Property, Function; Energy methods. Heat equation: Fundamental solution; Duhamel's principle; Average value formula; Solution properties; Energy Methods; Wave equation: Solution by spherical means (n = 1 and n = 3); Duhamel's principle; Energy method. Separation of variables (Fourier series). Plane and Traveling Waves (group speed and dispersion). Fourier transform. Distribution space and Fourier transform.

#### Basic Information

Workload

45 hours

Requirements

Functional Analysis: Fundamentals

**Mandatory: **

- Lawrence C. Evans; Partial Differential Equations; Springer-Verlag.
- John, Fritz (1982); Partial Differential Equations; Springer-Verlag.
- Iório, Rafael & Iório, Valéria; (1988); Equações Diferenciais Parciais: Uma Introdução. IMPA.

**Complementary: **

- Brezis, Haim (1993). Analyse Fonctionelle, Theorie et Applications Masson.
- Figueredo, Djairo G. (1987); Análise de Fourier e EDP; IMPA.
- Gustafson, Karl E. (1980); Partial Differential Equations and Hilbert Spaces Methods; John Wiley & Sons.
- Smoller, Joel; Reaction Diffusion Equations; Springer-Verlag.
- Trudinger, N.; Gilbarg, D. (1983); Elliptic PDE of Second Order; Springer-Verlag.