The student will learn mathematical tools to tackle and solve different types of PDEs, appearing in fundamental problems of Mathematical physics.
Knowledge of topics of mathematical courses of class L-35 (first degree).
Functions of a complex variable.
Fourier and Laplace transforms.
Green function and Sturm-Liouville problems.
Classification of second order PDEs.
Fundamental equations of mathematical physics: Laplace equation, heat equation, wave equation.
Functions of a complex variable: Cauchy-Riemann conditions, types of singularities, residues and integration, Laurent series.
Fourier and Laplace transforms: definitions and basic properties, transforms of the fundamental functions, transform of Dirac delta, inverse transforms.
Introduction to the Green function and to Sturm-Liouville problems.
Classification of linear second order PDEs with two independent variables; Cauchy problem.
Fundamental equations of mathematical physics: Laplace equation, heat equation, wave equation (physical derivation, mathematical properties, methods of solution).
F. Gazzola, F. Tomarelli, M. Zanotti, Analisi complessa - Trasformate - Equazioni Differerenziali, Esculapio, Milano.
S. Salsa, Equazioni a derivate parziali, Springer, Milano.
G. Spiga, Problemi matematici della Fisica e dell'Ingegneria, Pitagora, Bologna.
A. N. Tichonov, A. A. Samarskij, Equazioni della fisica matematica, MIR, Mosca.
During class lectures, the topics will be proposed from a formal point of view, and equipped with meaningful examples and applications.
The examination is based on an oral discussion, where the level of knowledge and understanding of the topics is valued, as well as the mathematical accuracy of exposition.