DN2266 Mathematical Models, Analysis and Simulation Part 1 7.5 credits

• Educational level

  Second cycle

• Academic level (A-D)

  C

• Subject area

  Mathematics
  

• Grade scale

  A, B, C, D, E, FX, F

• Course web page

• Course page on KTH Social

Learning outcomes

The overall goal of the course is to give basic knowledge of applied and numerical mathematics useful for scientific and engineering modeling. Especially, the close connection between the properties of mathematical models and their successful numerical treatment is emphasized.

This understanding means that after the course you should be able to

• identify and describe discrete equilibrium models using a network approach;
• relate equilibrium problems to minimum principles and solve simple constraint minimization problems via the Lagrange multiplier approach;
• formulate variational problems starting from simple physical principles and derive the corresponding Euler-Lagrange equation such that you can derive basic equations for continuous equilibrium problems in one, two, and three space dimensions;
• model (spatially discrete) time-dependent systems by ordinary differential equations;
• investigate the stability of autonomous systems and explore geometrically the phase space of 2D dynamical systems using analytical and numerical tools;
• derive asymptotic expansions for certain simple singular perturbation problems;
• understand the relation between convergence, consistency, and stability of numerical methods;
• understand essential properties of, and proof error estimations for, numerical methods for solving stationary and instationary problems such that you can compare different methods and select suitable algorithms for given problems;
• analyze and select iterative methods for large linear systems.

Course main content

Linear algebra, equilibrium and minimization problems. Applications on networks. Duality and calculus of variation, essential and natural boundary conditions. Systems of ordinary differential equations, linear and nonlinear. Phase plane, stability, bifurcation. Numerical methods for the solution of nonlinear systems and differential equations. Applications on mechanical and ecological systems.

Solution of systems of linear equations. Symmetrical positive definite matrices. Minimization problems. Eigenvalues and dynamical systems.

Assignments: One assignment every second week, from paper and pencil work to parameter studies of dynamical models in ecology and mechanics.

Eligibility

Single course students: 90 university credits including 45 university credits in Mathematics or Information Technology. English B, or equivalent.

Prerequisites

DN1212 or DN1240 or corresponding and experiences in programming in Matlab.

Literature

Meddelas senast 4 veckor före kursstart på kursens hemsida. Tidigare läsår användes G. Strang: Computational Science and Engineering, Wellesley-Cambridge.

Examination

• LAB1 - Laboratory Task, 3.7 credits, grade scale: P, F
• TEN1 - Examination, 3.8 credits, grade scale: A, B, C, D, E, FX, F

In this course all the regulations of the code of honor at the School of Computer science and Communication apply, see: http://www.kth.se/csc/student/hederskodex/1.17237?l=en_UK.

Requirements for final grade

Examination (TEN1; 3,8 university credits).
Assignments (LAB1; 3,7 university credits).

Offered by

SCI/Mathematics

Contact

Anna-Karin Tornberg, akto@kth.se

Examiner

Anna-Karin Tornberg <akto@kth.se>

Version

Course plan valid from: Autumn 09.
Examination information valid from: Autumn 07.