# Search course

​​​​
​​

## Syllabus for

TMA372 - Partial differential equations, first course

Owner: MPENM
7,5 Credits
Grading: TH - Five, Four, Three, Not passed
Education cycle: Second-cycle
Major subject: Mathematics
Department: 11 - MATHEMATICAL SCIENCES

Teaching language: English

 Course module Credit distribution Examination dates Sp1 Sp2 Sp3 Sp4 No Sp 0101 Examination 7,5 c Grading: TH 7,5 c 09 Mar 2009 am M, 26 Aug 2009 am V, 13 Jan 2009 am V

#### In programs

MPENM ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 1 (compulsory)
TKITE SOFTWARE ENGINEERING, Year 3 (elective)
TKELT ELECTRICAL ENGINEERING, Year 3 (elective)

#### Eligibility:

For single subject courses within Chalmers programmes the same eligibility requirements apply, as to the programme(s) that the course is part of.

#### Course specific prerequisites

A solid background in modern linear algebra and calculus in one and several variables. A solid background in Fourier analysis, especially the method of separation of variables for solving PDEs.

#### Aim

This course gives an introduction to the modern theory of partial differential equations (PDEs) with applications in science and engineering. It also presents an introduction to the finite element method as a general tool for numerical solution of PDEs.

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

- Derive weak formulations of the basic initial-boundary value problems for PDEs.
- Derive stability estimates for the continuous problems and predict the influence of data.
- Formulate Galerkin finite element methods for PDEs and dynamical systems.
- Derive erroe estimates using exact solution (a priori) and numerical solution (a posteriori).
- Understand how the finite element method is implemented in computer code.
- Improve the error estimates modifying the method or employing adaptive procedure.
- Draw relevant conclusions about stability, reliability and efficiency of the methods.

#### Content

Weak solutions to elliptic, parabolic, and hyperbolic partial differential equations (PDE). Computation of approximate solutions to various PDE by the finite element method (as well as dynamical systems). Interpolation, quadrature and linear systems. A brief introduction of representation theorems and
abstract theory to justify the weak (variational) approach. A priori and a posteriori error estimates. Applications to e.g. diffusion, heat conduction, and wave propagation.

More precisely the course covers following topics:

Basic interpolations theory:
Interpolation with polynomials
Interpolations error analys

Numerical linear algebra:
Solving linear system of equation with
Jacobi's method
Gauss- Seidel and
Overrelaxation methods.
Dynamical system:
Structures in approximation with polynomials
Ill-conditioned system.

Finite element method for boundary-value problem in 1D:
Stability
Error estimates and algorithms
Finite element method för initial-value problem i 1D:
Fundamental solution
Stability
Error estimates and algorithms
The dual problem.
Lax-Milgram theorem:
Abstract formulation
Riesz representation theorem
Studies and Analysis of problems in higher space dimensions:
Finite element in higher space dimensions.
Finite element method for Poisson equation in higher space dimensions.
Finite element method for heat equation in higher space dimensions.
Stability
Error estimates ans numerical algorithmes
Finite element method for wave equation in higher space dimensions.
Fundamental solution
Stability
Error estimates ans numerical algorithmes
Finite element method for convection-diffusion equations:
Stabilitey
Error estimates ans numerical algorithmes

#### Organisation

Lectures (about 35 hours), exercises (about 21 hours) and home assignments consisting of both theoretical and computer assingments.

#### Literature

K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations, Studentlitteratur/Cambridge University press, 1996.
Asadzadeh, M. Lecture Notes in PDE (Electronic material posted in course web-site)

#### Examination

Home and computer assignments combined with written exam

Page manager Published: Thu 04 Feb 2021.