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Graduate courses

Departments' graduate courses for PhD-students.

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Syllabus for

Academic year
TMA372 - Partial differential equations, first course  
 
Syllabus adopted 2012-02-22 by Head of Programme (or corresponding)
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
Open for exchange students

Course module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0101 Examination 7,5c Grading: TH   7,5c   15 Mar 2017 pm M   08 Jun 2017 pm SB   25 Aug 2017 am SB  

In programs

TKTEM ENGINEERING MATHEMATICS, Year 3 (elective)
MPENM ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 1 (compulsory)
TKAUT AUTOMATION AND MECHATRONICS ENGINEERING, Year 3 (elective)
TKELT ELECTRICAL ENGINEERING, Year 3 (compulsory elective)
TKITE SOFTWARE ENGINEERING, Year 3 (compulsory elective)
TKITE SOFTWARE ENGINEERING, Year 2 (elective)

Examiner:

Bitr professor  Mohammad Asadzadeh



  Go to Course Homepage

Eligibility:


In order to be eligible for a second cycle course the applicant needs to fulfil the general and specific entry requirements of the programme that owns the course. (If the second cycle course is owned by a first cycle programme, second cycle entry requirements apply.)
Exemption from the eligibility requirement: Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling these requirements.

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
quadrature rules and quadrature error

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.