Course Syllabus for

# Mathematics - Systems and transforms Matematik - System och transformer

## FMAF05F, 7 credits

Valid from: Autumn 2018
Decided by: Professor Thomas Johansson
Date of establishment: 2018-10-08

## 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: FMAF05
Teaching language: Swedish

## Aim

To make the postgraduate student familiar with several concepts and techniques that are important in modern engineering mathematics.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be familiar with the significance of eigenvalues in the context of stability and resonance, in linear systems, with continuous as well as discrete time.
• be able to describe and use the concepts of linearity, time invariance, stability, causality, impulse response and transfer function.
• be able to describe the structure of an exponential matrix, and be able to compute exponential matrices in simple cases.
• be able to define the concept of convolution, continuous and discrete, and to use convolutions both in the context of linear, time invariant systems and in the description of certain types of integral equations.

Competences and Skills

For a passing grade the doctoral student must

• be able to demonstrate an ability to independently choose appropriate methods to solve systems of linear differential and difference equations, and to carry out the solution essentially correctly.
• be able to demonstrate an ability to use eigenvalue techniques, elementary distribution theory, function theory, Fourier and Laplace transforms and convolutions in problem solving within the theory of linear systems.
• 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 manner and with clear logic be able to explain the solution to mathematical problems within the framework of the course.

## Course Contents

● Systems of linear differential equations: Equations in state form. Solution via diagonalization. Stability. Stationary solutions and transients. Solution via exponential matrix. ● Input/output relations: Linearity, time and space invariance, stability, causality. Convolutions. Elementary distribution theory. Transfer and frequency functions. ● Fourier analysis: The Laplace and Fourier transforms. Inversion formulae, the convolution theorem and Plancherel's theorem. Transform theory and analytic functions. Applications to differential equations and systems of differential equations.

## Course Literature

• math-ssp: Lineära system.. KF-Sigma, 1997.
• math-ssp: Övningar i Lineära system.. KF-Sigma, 2009.

## Instruction Details

Types of instruction: Lectures, exercises

## Examination Details

Examination formats: Written exam, written assignments