The course aims to give a general overview of the theoretical aspects of the most important partial differential equations arising as fundamental models in the description of main phenomena in Physics, Biology, economical/social sciences and data analysis, such as diffusion, transport, reaction, concentration, wave propagation, with a particular focus on well-posedness (i.e. existence, uniqueness, stability with respect to data). Moreover, the theoretical properties of solutions are studied in connection with numerical approximation methods (e.g. Galerkin finite dimensional approximations) which are studied and implemented in the Advanced Numerical Analysis and Scientific Computing courses.
First order partial differential equations : Transport equation, Method of Characteristics. Introduction to Calculus of Variations and Hamilton-Jacobi equations. Introduction to Scalar Conservation laws. Second order partial differential equations : heat equation, Laplace equation, second order parabolic equations, second order hyperbolic equations, wave equation. Introduction to Semigroup theory.
|D. Gilbarg - N. S. Trudinger||Elliptic Partial Differential Equations of Second Order||Springer||1998||3-540-13025-X||Revised printing|
|Evans, L. C.||Partial Differential Equations (Edizione 1)||American Mathematical Society||1998||0821807722|
|András Vasy||Partial Differential Equations - An Accessible Route through Theory and Applications||American Mathematical Society||2015||978-1-4704-1881-6|
|S. Salsa||Partial Differential Equations in Action||Springer Verlag Italia||2008||978-88-470-0751-2|
The assesment is based on an oral presentation of selected topics of the course program together with an individual project on PDE modelling in open form to be agreed with course instructors.
The aim is to evaluate the skills of the students in understanding what are the appropriate mathematical tools and techniques, among those studied in the course, that have to be used to effectively solve problems arising as PDE modelling of different phenomena.