Entry requirements

120 credits including 60 credits in mathematics, including Several Variable Calculus, Linear Algebra II, and Transform Methods.

Learning outcomes

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.

Content

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.

Instruction

Lectures and problem solving sessions.

Assessment

Written examination at the end of the course combined with assignments given during the course.