Course syllabus adopted 2019-02-12 by Head of Programme (or corresponding).
Overview
- Swedish nameFouriermetoder
- CodeMVE290
- Credits7.5 Credits
- OwnerTKKEF
- Education cycleFirst-cycle
- Main field of studyChemical Engineering with Engineering Physics, Mathematics, Engineering Physics
- DepartmentMATHEMATICAL SCIENCES
- GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Course round 1
- Teaching language Swedish
- Application code 54120
- Maximum participants45
- Open for exchange studentsNo
- Only students with the course round in the programme overview.
Credit distribution
Module | Sp1 | Sp2 | Sp3 | Sp4 | Summer | Not Sp | Examination dates |
|---|---|---|---|---|---|---|---|
| 0109 Examination 6 c Grading: TH | 6 c |
| |||||
| 0209 Written and oral assignments 1.5 c Grading: UG | 1.5 c |
In programmes
- TKAUT - AUTOMATION AND MECHATRONICS ENGINEERING, Year 3 (elective)
- TKKEF - CHEMICAL ENGINEERING WITH ENGINEERING PHYSICS, Year 2 (compulsory)
Examiner
Julie Rowlett- Professor, Analysis and Probability Theory, Mathematical Sciences
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)
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. Distributions and their use as solutions to differential equations.
Organisation
The course is organized in lectures and exercises (about 5h/week of each). It includes a hand-in assignment, and computer laborations may occur.
Literature
G.B. Folland: Fourier Analysis and Its Applications. Wadsworth & Brooks/Cole, Pacific Grove 1992, Chapters 1-9, plus some additional material.
Examination including compulsory elements
A written exam with about 6 problems and 2 theory questions, in 5 hours.