The aim of the course is to develop a good knowledge of fundamental methods in the solution of elementary problems in partial differential equations
teacher profile teaching materials
Heat equation: deduction from a random walk on the line. Solution of the heat equation on the line. Maximum principle. Applications of the maximum principle to unicity theorems. Heat equation in a interval using the separation of variables method. Different generalizations of this problem. Introduction to the elliptic equations. Laplacian in spherical and cylindrical coordinate systems. Representation formula via Green’s formula. Properties of harmonic fuc- tions. Maximum principle. Unicity results for the inner problems. Problem of existence for a circular domain. Poisson formula. External problems. Unicity theorems in the plane and in the space. External problem for the circle. Green functions: appllications to the sphere case and in the semispace case.Potential theory. Introduction to Quantum Mechanics. Schroedinger equa- tion. Separation of variables. Free particle in a interval. Potential barrier. Harmonic oscillator. Hydrogen atom.
Programme
Classification of the semilinear partial differential equations in any dimension. Classification in 2 dimensions and canonical forms. Wave equation in an interval using the separation of variables method: homogeneous case, general case. Wave equation on the line: D’Alambert’s solution. The half line case. Wave equation in R3.Heat equation: deduction from a random walk on the line. Solution of the heat equation on the line. Maximum principle. Applications of the maximum principle to unicity theorems. Heat equation in a interval using the separation of variables method. Different generalizations of this problem. Introduction to the elliptic equations. Laplacian in spherical and cylindrical coordinate systems. Representation formula via Green’s formula. Properties of harmonic fuc- tions. Maximum principle. Unicity results for the inner problems. Problem of existence for a circular domain. Poisson formula. External problems. Unicity theorems in the plane and in the space. External problem for the circle. Green functions: appllications to the sphere case and in the semispace case.Potential theory. Introduction to Quantum Mechanics. Schroedinger equa- tion. Separation of variables. Free particle in a interval. Potential barrier. Harmonic oscillator. Hydrogen atom.
Core Documentation
A.N. Tichonov, A.A. Samarskij Equazioni della Fisica Matematica Edizioni MIRType of delivery of the course
Classrooms and exercisesType of evaluation
There are also two intermediate tests