MM534: Differential Equations, Computing and Modeling (10 ECTS)

STADS: 13008521

Level
Bachelor course

Teaching period
The course is offered in the autumn semester.

Teacher responsible
Email: achim@imada.sdu.dk

Timetable
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Comment:
Ubegrænset deltagerantal.

Prerequisites:
None

Academic preconditions:
Linear Algebra, Mathematical and Numerical Analysis should be known.

Course introduction
To introduce modeling of problems from science and engineering by ordinary differential equations. To analyse and solve these equations both by analytic tools (when appropriate) and by computational methods.

Expected learning outcome

1. Formulate a differential equation as a model for a simple problem.
2. Solve differential equations by analytical and numerical methods taught in the course.
3. Find steady states and analyze the assymptotic behaviour of simple systems of differential equations.
4. Construct, implement and analyse numerical methods to compute (approximate) solutions to differential equations.
5. Give an oral presentation and answer supplementary questions on the course syllabus and the problems solved in mandatory assignments.

Subject overview
1.1. First order differential equations and mathematical models.
1.2. Slope fields and initial value problems.
1.3. Euler's approximation.
1.4. Existence and uniqueness, Picard-Lindelöf theorem (as application of fixed point theorem).
1.5. Gronwalls Lemma and the convergence of Euler's method.
1.6. Analytic tools: integrating factors, separation of variables, and exact equations.

2.1. Systems of first order linear differential equations, and linear higher order differential equations: fundamental solutions, the solution space.
2.2. The Wronskian, Abel's theorem.
2.3. Analytic tools: undetermined coefficients and the variation of parameters

3. Numerical methods: (embedded) Runge-Kutta methods and adaptivity.

4. Stiffness, implicit methods, A-stability.

5.1. Introduction to Ito-SDEs: Ito integral, Ito process, Ito formula.
5.2 Numerical methods for SDEs: Euler-Mayurama and Milstein methods, weak and strong convergence. 

Literature
There isn't any litterature for the course at the moment.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:

1. Mandatory assignments. Pass/fail, internal evaluation by teacher. (5 ECTS)
2. Oral exam. Danish 7 mark scale, internal examiner. (5 ECTS)

Re-exam in the same exam period or immediately thereafter. The re-exam may be a different type than the ordinary exam.

Expected working hours
The teaching method is based on three phase model.
Intro phase: 56 hours
Skills training phase: 28 hours, hereof:
 - Tutorials: 14 hours
 - Laboratory exercises: 14 hours

Educational activities

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.