(Created 2011-09-01.)

APPLIED MATHEMATICS | FMA021 |

**Aim**

In the 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 in general 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: to lay a foundation for a general competence in mathematics, useful in further studies as well as in professional activities, by showing how abstract mathematical concepts, such as Hilbert spaces, may be used in concrete applications. A further aim is that the student should become acquainted 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 demonstrate an ability to physically interpret 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 demonstrate an ability to use spectral methods (Fourier) and source function methods (Green) to solve problems for the three basic equations in simple geometries.

be able to demonstrate an ability to interpret functions as abstract vectors in a Hilbert space, and to use for functions concepts such as norm, distance, scalar product.

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

be able to demonstrate an ability 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 demonstrate an ability to explain the projection formula and to use it to solve minimization 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 demonstrate an ability to use theoretical tools from areas such as Hilbert space theory, 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 demonstrate an ability 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 of 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 spaces. Sturm-Liouville operators. Special functions, e.g. Bessel, Legendre, spherical harmonics. Distributions. The Fourier and Laplace transforms. Something about numerical solution of partial differential equations.

**Literature**

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