MM831: Differential Equations II (5 ECTS)

STADS: 13011401

Level
Master's level course

Teaching period
The course is offered in the autumn semester.

Teacher responsible
Email: debrabant@imada.sdu.dk

Timetable
Show entire timetable
Show personal time table for this course.

Comment:
Ubegrænset deltagerantal. Fælles undervisning med MM531 Differentialligninger II samt MM547 Ordinære differentialligninger: Teori, modellering og beregning

Prerequisites:
The course cannot be chosen by students, who has either followed og passed MM543, MM547 or MM531.

Academic preconditions:
Students taking the course are expected to:

• Know the concept of a function, real and complex numbers, differentiation and integration of functions of one and several variables, vector calculus, convergence of sequences, Newton’s method.
• Be familiar with: systems of linear equations, matrices, determinants,  vector spaces, scalar product and orthogonality, linear transformations, eigenvectors and eigenvalues, diagonalization, polynomials, random variables, normal distribution
• Have knowledge of how to implement algorithms as computer programs and compute numerical approximations to mathematical problems that don't allow a closed form solution.

Course introduction
The purpose of the course is to analyse and solve ordinary differential equations by computational methods.

The course builds on the knowledge acquired in the courses MM536 (Calculus for Mathematics), MM533 (Mathematical and Numerical Analysis), MM538 (Algebra and Linear Algebra), MM507 (Differential equations) / first half of MM545 (Differential equations and geometry).

The course gives an academic basis for a Master Project in several core areas of Natural Sciences.

In relation to the competence profile of the degree it is the explicit focus of the course to:

• Give skills to:
1. analyse practical and theoretical problems with the help of numerical simulation based on a suitable mathematical model
2. describe and evaluate sources of error for the modelling and calculation of a given problem
3. justify relevant models for analysis and solution and choose between them
4. analyse, model and solve given problems at a high level of abstraction, based on logical and structured reasoning
• Give knowledge and understanding of:
1. Mathematical modelling and numerical analysis in science and engineering
2. Reflection on theories, methods and practices in the field of applied mathematics.
3. Advanced models and methods in mathematics

Expected learning outcome
The learning objectives of the course are that the student demonstrates the ability to:

1. Construct, implement and analyse advanced numerical methods to compute (approximate) solutions to differential equations.
2. Give an oral presentation on an advanced topic and answer supplementary questions on the course syllabus.

Subject overview
The following main topics are contained in the course:

1. Numerical methods: (embedded) Runge-Kutta methods and adaptivity.
2. Stiffness, implicit methods, A-stability.
3. Introduction to Ito-SDEs: Ito integral, Ito process, Ito formula.
4. Numerical methods for SDEs: Euler-Maruyama and Milstein methods, weak and strong convergence.

Literature

Meddeles ved kursets start.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:
Oral exam. Internal marking, graded after Danish 7-mark scale.

Reexamination in the same exam period or immediately thereafter.

Expected working hours
The teaching method is based on three phase model.
Intro phase: 28 hours
Skills training phase: 14 hours, hereof:
 - Tutorials: 14 hours

Educational activities

Educational form
Activities during the study phase:

• preparation of exercises in study groups
• preparation of projects
• contributing to online learning activities related to the course

Language
This course is taught in Danish or English, depending on the lecturer. However, if international students participate, the teaching language will always be English.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.