MM547: Ordinary Differential Equations: Theory, Modelling and Simulation (10 ECTS)

STADS: 13013201

Level
Bachelor course

Teaching period
The course is offered in the autumn semester.

Teacher responsible
Email: zimmermann@imada.sdu.dk

Timetable
Show entire timetable
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Comment:
Ubegrænset deltagerantal. Fælles undervisning med MM531/MM831 Differentialligninger II samt MM507 Ordinære differentialligninger

Prerequisites:
None.

The course cannot be chosen by students, who followed MM507, MM531, MM545, or MM831.

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, Banach’s fixed point theorem, 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 introduce modelling of problems from science and engineering by ordinary differential equations and to analyse and solve these equations both by analytic tools (when appropriate) and by computational methods.

The course builds on the knowledge acquired in the courses MM536 (Calculus for Mathematics), MM533 (Mathematical and Numerical Analysis), and one of MM505 (Linear Algebra) or MM538 (Algebra and Linear Algebra). The course is of high multidisciplinary value and gives an academic basis for a Bachelor Project in several core areas of Natural Sciences, as well as the courses MM546 (Partial differential equations: theory, modelling and simulation) and MM5CC (Computational physics).

 

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

• Give the competence to :
1. handle complex and development-oriented situations in study and work contexts.
• Give skills to:
1. analyse practical and theoretical problems with the help of numerical simulation based on a suitable mathematical model
2. analyse the qualitative and quantitative properties of a given problem
3. describe and evaluate sources of error for the modelling and calculation of a given problem
4. justify relevant models for analysis and solution and choose between them
• 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.

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

1. Formulate a differential equation as a model for a simple problem
2. Solve differential equations by methods taught in the course
3. Find steady states and analyse the asymptotic 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
The following main topics are contained in the course:

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. Gronwall's 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-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:

1. Mandatory assignments. Pass/fail, internal marking by teacher. (5 ECTS)
2. Oral exam. Danish 7-mark scale, internal marking. (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: 28 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.