Course Syllabus for

# Introduction to Partial Differential Equations Kontinuerliga system

## FMA022F, 7.5 credits

Valid from: Autumn 2013
Decided by: FN1/Anders Gustafsson
Date of establishment: 2014-01-27

## General Information

Division: Mathematics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: FMA021
Teaching language: Swedish

## Aim

Within 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 economic 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 model building - analysis - interpretation av solutions to PDE:s appearing for such systems. Another aim is the converse: 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 doctoral student should become acquainted with the use and usability of software packages for computation and simulation.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• •be able to demonstrate an ability to formulate mathematical models for phenomena in heat conduction, diffusion, wave propagation and electrostatics.
• •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 an 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 associated with 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 least squares problems.
• •have some experience and understanding of the use of mathematical and numerical software in order to solve problems related to the course.

Competences and Skills

For a passing grade the doctoral student must

• •be able to demonstrate an ability to independently choose appropriate methods to solve the three basic types of partial differential equations, and to essentially carry out the solution correctly.
• •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.

## 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 and their eigenvalues and eigenfunctions. Special functions, e.g., Bessel, Legendre, spherical harmonics. Distributions. The Fourier and Laplace transforms. Something about the numerical solution of partial differential equations.

## Course Literature

• Sparr, G. & Sparr, A.: Kontinuerliga system. 2000. ISBN 9789144013558.
• Sparr, G. & Sparr, A.: Kontinuerliga system: Övningsbok. 2000. ISBN 9789144012346.

## Instruction Details

Types of instruction: Lectures, laboratory exercises, exercises

## Examination Details

Examination formats: Written exam, miscellaneous. Computer sessions. A voluntary written test at the middle of the course provides an opportunity to collect credits for the final exam.Any credits acquired by passing the voluntary written test at the middle of the course expire after a year. Thereafter it is possible to participate in the voluntary test the following year in order to try to acquire new credits.