Search course

Use the search function to find more information about the study programmes and courses available at Chalmers. When there is a course homepage, a house symbol is shown that leads to this page.

Graduate courses

Departments' graduate courses for PhD-students.

​​​​
​​

Syllabus for

Academic year
MVE030 - Fourier analysis  
Fourieranalys
 
Syllabus adopted 2022-05-02 by Head of Programme (or corresponding)
Owner: TKTFY
6,0 Credits
Grading: TH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Education cycle: First-cycle
Main field of study: Chemical Engineering with Engineering Physics, Mathematics, Engineering Physics
Department: 11 - MATHEMATICAL SCIENCES


Teaching language: Swedish
Application code: 57142
Open for exchange students: No
Maximum participants: 180
Only students with the course round in the programme plan

Module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0105 Examination 6,0 c Grading: TH   6,0 c   18 Mar 2022 am J   08 Jun 2022 am J   23 Aug 2022 am J DIG

In programs

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

Examiner:

Julie Rowlett

  Go to Course Homepage


Eligibility

General entry requirements for bachelor's level (first cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Specific entry requirements

The same as for the programme that owns the course.
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

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)

After completing the course, the student will be able to solve partial differential equations using separation of variables, eigenfunction and Fourier series expansions, eigenfunction expansions using Sturm-Liouville problems, as well as Fourier and Laplace transforms.  In addition, the student will be able to apply the theoretical concepts of Hilbert spaces to solving physical problems.  In this regard, the student will be able to determine, based on the geometry of the physical domain and the character of the equation, which orthogonal system, such as trigonometric functions, Bessel functions, or orthogonal polynomials, is best suited to solve the physical problem.  The student will also be able to use Fourier series to compute sums, like the sum of 1/n^2.  The student will learn how to use Fourier transforms to compute tricky integrals.

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.

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 TM.

Literature

Custom course book plus some additional material.

Examination including compulsory elements

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



The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers on educational support due to disability.


Page manager Published: Thu 04 Feb 2021.