INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS

Second cycle degree in MATHEMATICS

Campus: PADOVA

Language: English

Teaching period: First Semester

Lecturer: FABIO ANCONA

Number of ECTS credits allocated: 8


Syllabus
Prerequisites: Differential and integral calculus.
Elementary theory of ordinary differential equations.
Basic theory of complex analysis (functions of complex variables, holomorphic and analytic functions).
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.
Course unit 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, Harnack's inequality, maximum principle. Poisson equation. Green's function and Poisson's representation formula of solutions.
- 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.