In order to pass the course (grade 3) the student should be able to
describe the most common partial differential equations that appear in problems concerning e.g. heat conduction, flow, elasticity and wave propagation;
give an account of basic questions concerning the existence and uniqueness of solutions, and continuous dependence of initial and boundary data;
solve simple first order equations using the method of characteristics;
classify second order equations;
solve simple initial and boundary value problems using e.g. d'Alembert's solution formula, separation of variables, Fourier series expansion or expansion in other orthogonal systems;
describe, compute and analyse wave propagation and heat conduction in mathematical terms;
formulate maximum principles for various equations and derive consequences.
Introduction of some common partial differential equations, physical background and derivation from physical principles. First order partial differential equations: characteristics, linear, quasilinear and general nonlinear equations. Classification of second order partial differential equations in two variables. The one-dimensional wave equation, Cauchy's problem, d'Alembert's formula, the nonhomogeneous wave equation. Separation of variables, the heat and wave equations. The energy method, uniqueness. Sturm-Liouville problems and eigenfunction expansions. Elliptic equations. Dirichlet's problem, harmonic functions, the maximum principle. Poisson's formula. Green functions and integral representations. The heat kernel. Partial differential equations in higher dimensions.
Lectures and problem solving sessions.
Written examination at the end of the course combined with assignments given during the course.
week 24, 2013
An introduction to partial differential equations