Course syllabus for Partial differential equations

Course syllabus adopted 2024-02-05 by Head of Programme (or corresponding).

Overview

  • Swedish namePartiella differentialekvationer
  • CodeMVE695
  • Credits7.5 Credits
  • OwnerTKTEM
  • Education cycleFirst-cycle
  • Main field of studyMathematics, Engineering Physics
  • DepartmentMATHEMATICAL SCIENCES
  • GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail

Course round 1

  • Teaching language Swedish
  • Application code 59128
  • Block schedule
  • Open for exchange studentsNo
  • Only students with the course round in the programme overview.

Credit distribution

0123 Examination 6 c
Grading: TH		 6 c
  • 13 Jan 2025 am J
  • 15 Apr 2025 pm J
  • 18 Aug 2025 pm J
0223 Laboratory, part B 1.5 c
Grading: UG		 1.5 c

In programmes

Examiner

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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

Single variable calculus, multi-variable analysis and Fourier analysis

Aim

Partial differential equations are powerful computational tools in science and technology with a myriad of applications. For this reason a course in partial differential equations is essential to the engineering mathematics program and would be beneficial for students at several other engineering disciplines for which this course is an elective.

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

The student should be able to show a basic understanding of
the fundamental types of partial differential equations,
and the basics of distribution theory.
The student should be able to give variational formulations
of boundary value problems and to use the finite element method to
solve these numerically.
The student should be able to solve initial value problems using
Fourier analysis and eigenfunctions.

The student should be able to formulate boundary value problems as integral equations and to use Nyström discretization to solve these.
The student should be able to use fundamental solutions, Green's
formulas and Green's functions.

Content

Systems av ordinary differential equations.
Change of variables for first order partial differential equations.
Heat, wave and Laplace's equations.
Linear PDE problems, classification and well posedness.
Introduction to distribution theory.
Variational methods for boundary value problems. The Sobolev space and Lax-Milgram's theorem.
Introduction to the finite element method.
Fourier transformation of tempered distributions, for solving inital value problems.
Laplace eigenfunctions, solving initial/boundary value problems, and Rayleigh-Ritz approximation.
Solving boundary value problems with single- and double layer potentials.
Nyström discretization of integral equations.
Harmonic functions, Green's functions, mean value theorems and maximum principles.
The initial value problem for Maxwell's equations and Huygens principle.

Organisation

Lectures, problems solving sessions and compulsary computer projects about FEM and integral equations.

Literature

Rosén, Andreas: Partial differential equations, from theory to coding.

Examination including compulsory elements

Compulsory computer projects and a written exam. Bonus points on the exam are obtained based on the quality of the lab report.

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.