Syllabus academic year 2008/2009
(Created 2008-07-17.)
 APPLIED MATHEMATICS FMA021

Higher education credits: 7,5. Grading scale: TH. Level: A (Second level). Language of instruction: The course will be given in Swedish. FMA021 overlap following cours/es: FMA022, FMA020, FMA022, FMA020 och FMA022. Compulsory for: F2, Pi2. Optional for: E2, N3, N3nel, N3nf. Course coordinator: Director of Studies Lars-Christer Böiers, Lars_Christer.Boiers@math.lth.se, Matematik. Prerequisites: FMA430 Flerdimensionell analys. Recommended prerequisits: FMA450 Systems and Transforms. Assessment: Written test comprising theory and problem solving. Computer work. A voluntary test at the middle of the course provides an opportunity to collect credits for the final exam. Home page: http://www.maths.lth.se/matematiklth/vitahyllan/vitahyllan.html.

Aim
In engineering sciences the term "continuous system" means a system whose state space is described by a continuous family of parameters. Continuous systems occur frequently in physics and other natural sciences, in mechanics, electricity and other engineering sciences, in economical sciences, etc. To describe a continuous system one is generally led to partial differential equations (pde).

One aim of the course is to provide mathematical tools and the ability to use them for the whole chain of model building - analysis - interpretation av solutions to pde:s appearing in such systems. Another aim is the opposite: by showing how abstract mathematical concepts such as Hilbert space are used in concrete applications, to lay a foundation for general competence in mathematics useful as well in further studies as in professional activities. A further aim is that the student should make acquaintance with the use and usability of software programs for computation and simulation.

Knowledge and understanding
For a passing grade the student must

be able to show capability to draw up mathematical models for phenomena in heat conduction, diffusion, wave propagation, and stationary heat conduction/diffusion/electrical fields.

be able to show capability to physically interprete mathematical models with different boundary conditions for the three basic types of pde:s: the heat equation, the wave equation and the Laplace/Poisson equation, and to understand the characteristics of their solutions.

be able to show capability to use spectral methods (Fourier) and source function methods (Green) to solve problems for the three basic equations in simple geometries.

be able to show capability to interpret functions as abstract vectors in Hilbert space, and to use for functions concepts such as norm, distance, scalar product.

be able show capability to decide whether an operator is symmetric, and capability to identify Sturm-Liouville operators.

be able to show capability to find eigenfunctions and eigenvalues for some types of Sturm-Liouville operators, in particular those connected to the Laplace operator in one, two and three dimensions.

be able to show capability to explain the projection formula and to use it to solve minimisation problems using the least squares method.

have some experience and understanding of mathematical and numerical software in simple situations.

Skills and abilities
For a passing grade the student must

be able to show capability independently to choose appropriate methods to solve the three basic types of partial differential equations, och to carry out the solution essentially correct.

be able to show capability to use theoretical tools from areas such as Hilbert space, special functions, distribution theory, Fourier and Laplace transforms and Green functions to solve the three basic pde:s in simple geometries.

in connection with problem solving be able to show capability to integrate knowledge from the different parts of the course.

with proper terminology, in a well structured way and with clear logic be able to explain the solution to a mathematical problem within the course.

Contents
Physical models. Fourier's method, series expansions and integral transforms. Green functions. Wave propagation. Function spaces and function norms. Hilbert space. Sturm-Liouville operators. Special functions, e.g. Bessel, Legendre, spherical harmonics. Distributions. The Fourier and Laplace transforms. Something about numerical solving of partial differential equations.

Literature
Sparr, G & Sparr, A: Kontinuerliga system. Studentlitteratur 2000. ISBN 91-44-01355-8