Second Level Degree in Mathematics
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INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS - 8 CFU
Teacher
Laura Caravenna
Scheduled Period
I Year - 1 Semester | 04/10/2021 - 15/01/2022
Hours: 64 (32 esercitazione, 32 lezione)
Prerequisites
Differential and integral calculus:basis on integration and differentiation, explicit integrals and derivatives of elementary functions, the fundamental theorem of calculus, basics on curves and surfaces. Green-Gauss-Stokes theorems and the theorems concerning limiting procedures in integrals are important but they will be revised at an essential level.
Elementary theory of ordinary differential equations and on the Cauchy problem. Gronwall estimates and classical well posedness for ODEs will be recalled at an essential level.
Basic theory of complex analysis: what are functions of complex variables, holomorphic and analytic functions, very essential properties as Cauchy–Riemann equations.
Target skills and knowledge
Being able to: - explain basic notions of the classical theory of linear partial differential equations; - solve problems analogous for type and difficulty to the ones done during the year, a bit more for full grade; -recognise applications to other disciplines and thier historical impact. It's a fundamental course suggested to students with interests both in pure and in applied mathematics, and in particular to students with a curriculum in analysis.
Examination methods
The exam consists of a final oral examination on the topics treated in class. There will be both theoretical questions and the discussion of some exercise to solve. The final exam could be reduced via in itinere activities.
Assessment criteria
The evaluation criteria will be the following:
- coherence and rigor in the exposure of statements and theorems
- thoroughness and adherence to the topics of discussion
- ability to use the acquired knowledge to solve problems and exercises.
Course contents
Didactic plan:
- First order PDEs: transport equation with constant coefficients, conservation laws (classical and weak solutions, Rankine-Hugoniot conditions, Riemann problem).
- Wave equation: existence of solutions, D'alembert formnula, method of spherical means, Duhamel's principle, uniqueness, finite speed of propagation.
- Laplace equation: fundamental solution, harmonic functions and main properties, mean value formulas, Liouville's Theorem, Harnack's inequality, maximum principle. Poisson equation. Green's function and Poisson's representation formula of solutions. Basic notions of the theory of distributions.
Weak solutions of Laplace equations on bounded domains are harmonic functions.
- Heat equation: fundamental solution, existence of solutions for the Cauchy problem and representation formula. Uniqueness and stability of solutions. Mean value formulas, maximum principle, Hopf's maximum principle.
Planned learning activities and teaching methods
The methodology of teaching used will be the classical taught lesson (with tablet).
Small active-learning moments will be proposed.
Additional notes about suggested reading
They include:
Salsa, Sandro, Partial differential equations in actionfrom modelling to theorySandro Salsa, Cham [etc.], Springer, 2015.
L.C. Evans, Partial Differential Equations, 2nd edition, Providence, Rhode Island, American Mathematical Society, 2010.
W. A. Strauss, Partial Differential Equations. An Introduction, New York, Wiley, 1992.