(Created 2011-09-01.)
 APPLIED MATHEMATICS - PARTIAL DIFFERENTIAL EQUATIONS FMAF15
Credits: 7,5. Grading scale: TH. Cycle: G2 (First Cycle). Main field: Technology. Language of instruction: The course will be given in Swedish. FMAF15 overlaps following cours/es: FMA062 and FMA435. Alternative for: M3, W3. Optional for: B4, K4. Course coordinator: Director of Studies Anders Holst, Anders.Holst@math.lth.se, Mathematics. Recommended prerequisits: Basic university studies in calculus and linear algebra. The course might be cancelled if the number of applicants is less than 12. Assessment: Written test. Some computer work. Home page: http://www.maths.lth.se/matematiklth/vitahyllan/vitahyllan.html.

Aim
The aim of the course is to treat such mathematical concepts and methods in vector analysis and partial differential equations above the basic level that are important for further studies within e.g. mechanics, solid mechanics, fluid mechanics, ecology, electrical engineering and for further professional activities.

Knowledge and understanding
For a passing grade the student must

be able to state the important definitions and theorems in three-dimensional vector analysis, and understand their interpretation in the applications.

have a good knowledge of Fourier series and their application for solving model problems in partial differential equations, and have knowledge of the Fourier transform.

be able to show good understanding of concepts such as initial conditions and boundary conditions.

be able to demonstrate an ability to draw up and interpret mathematical models with different boundary conditions for the three basic types of differential equations: the heat/diffusion equation, the wave equation, the Laplace equation.

have some experience and understanding of mathematical and numerical software for differential equations.

Skills and abilities
For a passing grade the student must

be able to demonstrate an ability to identify problems which can be modelled with the concepts introduced.

be able to show an ability to use the concepts introduced in the course in connection with modelling and problem solving.

with proper terminology, suitable notation, in a well-structured manner and with clear logic, be able to explain the solution to a problem.

Contents
Vector analysis: Scalar and vector fields. Gradient, divergence, rotation. Conservative fields, potential. Curve integrals and surface integrals. Gauss' and Stokes' theorems. The continuity equation.

Fourier series and partial differential equations: Fourier series. Half period expansions. The Fourier transform. Step and impulse functions.

The heat conduction and diffusion equation. The wave equation. Method of separation of variables. The Laplace equation.

Literature
Persson, A. & Böiers, L.C.: Analys i flera variabler, Chapter 10. Studentlitteratur 2004. ISBN 91-44-03869-0.
Sparr, A.: Tillämpad matematik 1. KF-Sigma.
Supplementary literature from the department.