Course syllabus for Applied mathematics

Course syllabus adopted 2021-02-26 by Head of Programme (or corresponding).

Overview

  • Swedish nameTillämpad matematik
  • CodeTMA683
  • Credits7.5 Credits
  • OwnerTKKMT
  • Education cycleFirst-cycle
  • Main field of studyMathematics
  • DepartmentMATHEMATICAL SCIENCES
  • GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail

Course round 1

  • Teaching language Swedish
  • Application code 53115
  • Maximum participants135
  • Open for exchange studentsNo
  • Only students with the course round in the programme overview.

Credit distribution

0115 Project 1.5 c
Grading: UG		0 c1.5 c0 c0 c0 c0 c
0215 Examination 6 c
Grading: TH		0 c6 c0 c0 c0 c0 c
  • 15 Jan 2022 am J
  • 11 Apr 2022 am J
  • 18 Aug 2022 pm J

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

Analysis
  • Calculus (single and several variables): Complex numbers, series, trigonometry, Green's formula, Stokes's and Gauss theorems.
  • Integral calculus: Partial integration, partial devision of rational functions, numerical integration and multiple integrals.
  • Differential equations: Linear ordinary differential equation of first (scalar and system) and second order.
Linear Algebra
  • Linear system of equations
  • Matrix algebra
  • Linear spaces and eigenvalue problem.
Programming in MATLAB

Aim

The aim of this course is to study numerical, as well as analytical solutions for partial (and ordinary) differential equations (PDEs). To solve PDEs is one of the most modern mathematical tools applied in science and engineering.

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

Passing this course  the student should be able to:
  • solve partial and ordinary differential equations,  as (time dependent or stationary) heat equation, wave equation, convection-diffusion and reaction-diffusion equation, approximatively using finite element method.
  • construct numerical algorithms and implement them (in MatLab) and illustracte their results graphically.  
  • Laplace transform and inverse Laplace transform
  • solve ordinary differential equations and integro-differential equations using Laplace transforms.
  • determine Fourier series for periodic functions, sine and cosine series for functions defined in an interval.
  • solve problems of heat- and wave equations using the method of separation of variables.
  • explain and in some cases prove the theorems concerning the subject above that are covered in the lectures.

Content

This course covers mathematical models in 1D (and 2D) for the processes in science and engineering. These are phenomena described by (partial) differential equations derived from, the conservation laws for, e.g. heat (energy) and mass.  Typical examples are reaction, production, diffusion and convection.

The course is in two parts. The first part treats approximate solutions for differential equations using piecewise polynomials: the finite element method (FEM). This is due to the fact that, in general,  the problems mentioned above lack in having closed form analytic solutions and the finite element method is more flexible than the other numerical approaches. Implementing the method plays a central role in this part of the course. This part contains also examples of typical finite difference approximations for time-dependent problems as well as ordinary differential equations.

The second part of the course is about the Fourier methods: Laplace transforms, Fourier series and separation of variables technique. These methods yield analytical solutions, when they exist, and also give indications how to construct the numerical methods. The Laplace transforms are also crucial tools in some other courses in chemical engineering.

Organisation

The course consists of lectures, exercises and a project (home and computer assignments).

Some (minor) parts  that will not be covered in the lectures are left to self-study. These moments are equally important and integrated part of the whole course.

Working with exercises, computer labs and the project (home assignments) play an important role during the course and clarify the theoretical content from practical point views. The course consists of two parts: one part of 6 hp and another one of 1.5 hp. The 6 hp part is examined through a written examination. To pass the 1.5 hp part it is required passing both the project and compuer-lab assignments (specified on the web-site of the course). The project part is performed cooperating with the course in transport theory.

Literature

M. Asadzadeh, An introduction to the finite element method (FEM). Part I. Problems in 1-D. (electronic , posted on course web-site).

M. Asadzadeh and F. Bengzon, Lecture notes in Fourier analysis (electronic , posted on course web-site).

References and supplemtary course material (including exercised for each chapter) are posted on the course web-site.

Examination including compulsory elements

Written exam of problem solving nature with theoretical aspects (corresponds to 6 hp) and home and computer assignment: project (which are corresponding to 1,5 hp)

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.