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## Syllabus for

MVE030 - Fourier analysis

Owner: TKTFY
6,0 Credits
Grading: TH - Five, Four, Three, Not passed
Education cycle: First-cycle
Major subject: Chemical Engineering with Engineering Physics, Mathematics, Engineering Physics
Department: 11 - MATHEMATICAL SCIENCES

Teaching language: Swedish

 Course module Credit distribution Examination dates Sp1 Sp2 Sp3 Sp4 Summer course No Sp 0105 Examination 6,0c Grading: TH 6,0c 17 Mar 2017 am M, 07 Jun 2017 am SB, 22 Aug 2017 am SB

#### In programs

TKTFY ENGINEERING PHYSICS, Year 2 (compulsory)
TKTEM ENGINEERING MATHEMATICS, Year 2 (compulsory)

#### Examiner:

Docent  Julie Rowlett

#### 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

Real analysis, Multivariable analysis, Linear algebra, Complex mathematical analysis.

#### Aim

The course introduces Fourier methods in the program. These methods are powerful mathematical tools in technology and science.

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

The goal is to give the student a firm background in solution techniques for partial differential equations using separation of variables, eigenfunction and Fourier series expansions, as well as Fourier and Laplace transforms. Fourier methods usually lead to solutions in the form of trigonometric series or integrals like the Poisson integral. Based on the geometry of the physical domain and the character of the equation, the trigonometric functions in the series expansion can be replaced by other orthogonal systems, e.g. Bessel functions or Legendre, Hermite or Laguerre polynomials. These solution techniques are closely related to the theory of partial differntial equations, and to distribution theory. The course gives an introductory understanding of these fields; in particular the distribution derivative is treated.

#### Content

The method of separation of variables. Trigonometric Fourier series and their convergence. Examples of initial and boundary value problems for partial differential equations: the heat equation, the wave equation, Laplace/Poisson's equation. Orthogonal systems of functions, completeness, Sturm-Liouville eigenvalue problems. Various solution techniques like homogenization, superposition and eigenfunction expansion. Bessel functions and orthogonal polynomials (Legendre, Hermite and Laguerre polynomials). Solution methods in spherical or cylindrical coordinates. Fourier transforms and their applications to partial differential equations. Signal analysis, discrete Fourier transforms and Fast Fourier transforms. Laplace transforms with applications. Short introduction to distributions and their derivatives.

#### Organisation

The course is organized in lectures and exercises (about 5h/week of each). Computerized hand-in assignments may occur. The course and the examination coincide for the programs F and Kf.

#### Literature

G.B. Folland: Fourier Analysis and Its Applications. Wadsworth & Brooks/Cole, Pacific Grove 1992, essentially Chapters 1-8, plus some additional material.

#### Examination

Written exam with about 6 problems and 2 theory questions (5 hours).

Page manager Published: Thu 04 Feb 2021.