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## Syllabus for

MVE100 - Transforms and differential equations
Transformer- och differentialekvationer

Owner: TKMAS
7,5 Credits
Grading: TH - Five, Four, Three, Fail
Education cycle: First-cycle
Major subject: Mathematics
Department: 11 - MATHEMATICAL SCIENCES

Teaching language: Swedish
Application code: 55135
Open for exchange students: No
Block schedule: A+
Only students with the course round in the programme plan

 Module Credit distribution Examination dates Sp1 Sp2 Sp3 Sp4 Summer course No Sp 0106 Project 7,5c Grading: TH 7,5c

#### In programs

TKITE SOFTWARE ENGINEERING, Year 3 (elective)
TKMAS MECHANICAL ENGINEERING, Year 3 (elective)

Erik Broman

#### Eligibility:

In order to be eligible for a first cycle course the applicant needs to fulfil the general and specific entry requirements of the programme(s) that has the course included in the study programme.

#### Course specific prerequisites

Calculus, linear algebra and mathematical software
comparable to TMV225, TMV 151, TMV166, TMV255 and TME135.

#### Aim

The aim of the course is to give basic knowledge of transform methods
(Fourier transform, Laplace transform, discrete Fourier transform and
Fourier series), which are important tools for solving differential
equations, as well as for system analysis and signal processing. Moreover,
the course treats eigenvalue and boundary value problems for differential
operators. The course gives a theoretical foundation for various
mathematical methods, and ability to apply them to concrete situations in
physics and engineering, such as coupled oscillations, oscillating beams
and analogue filters.

#### Learning outcomes (after completion of the course the student should be able to)

- Expand functions as Fourier series.
- Compute Laplace and Fourier transforms, as well as the inverse transforms, of certain functions.
- Solve differential equations using the Laplace transform.
- Determine normal modes and eigenfrequencies of linear dynamical systems.
- Determine the motion of such systems under external forces.
- Determine eigenvalues and eigenfunctions to boundary value problems.
- Determine the transfer function, frequency response and impulse response to a dynamical system.
- Solve partial differential equations using separation of variables.
- Describe and motivate methods of solutions and conditions for their applicability.

#### Content

The course treats step and impulse functions, Laplace and Fourier transform, Fourier series, discrete Fourier transform, and eigenvalue problems and boundary value problems for differential operators.

These mathematical tools are used to analyze various problems in physics and engineering leading to differential equations. Examples include oscillating strings and beams, loaded beams and coupled oscillating systems. Dynamical systems are also studied from the viewpoint of control engineering, leading to concepts such as transfer functions, frequency response and impulse response. Examples include analogue filters.

There is a possibility for adapting the course according to the interests and needs of the students.

An important part of the course is to use mathematical software for solving and visualizing problems mentioned above.

#### Organisation

The teaching consists of lectures and tutoring in connection with hand-in problems. Detailed information will be given at the course homepage before the start of the course.

#### Literature

Glyn James: Advanced modern engineering mathematics, (Pearson) Chapters 2, 4 and 5.
These chapters are also included in: Series and Transforms, Compiled by B Behrens, J Madjarova (Pearson).

F Eriksson, C-H Fant and K Holmåker: Differentialekvationer och egenvärdesproblem (Department of Mathematical Sciences, Chalmers University and University of Gothenburg).

#### Examination including compulsory elements

Examination for grades 3 and 4 is based on hand-in problems and discussion of their solution. For grade 5, it is in addition based on a theory exam after agreement with the examinor.

Published: Wed 26 Feb 2020.