APPLIED MATHEMATICS | FMA021 |

**Aim**

The course treats mathematical concepts and methods for partial differential equations. The intention is to make the student acquainted with the whole chain from the drawing up of a model, the theoretical analysis of the model up to a numerical solution. One aim is also to give the students ability to read and judge mathematical reasoning, to give ability in problem solving and training in accounting of mathematical discussions. The main stress is on calculations with paper and pencil, but also on providing via computer exercises acquaintance with mathematical and numerical software.

*Knowledge and understanding*

For a passing grade the student must

be able to show capability to physical interpretation of mathematical models with different boundary conditions, such as the heat equation, the wave equation and Laplace' or Poisson's equation.

be able to show capability to identify Sturm-Liouville operators and in simple cases to find the corresponding eigenfunctions and eigenvalues.

be able to show capability to interpret functions as abstract vectors in Hilbert space.

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 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 "central" partial differential equations in simple cases.

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. Distributions. The Fourier and Laplace transforms. Somewhat about numerical solving of partial differential equations.

**Literature**

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