MM831: Differential Equations II

See Danish version.

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

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.

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).

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

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.

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.

At the faculty of science, teaching is organized after the three-phase model ie. intro, training and study phase.
In order to enable students to achieve the learning objectives for the course, the teaching is organised in such a way that there are 28 lectures, class lessons, etc. on a semester. These teaching activities are reflected in an estimated allocation of the workload of an average student as follows:

• Intro phase (lectures, class lessons) - 28 hours
• Training phase: 14 hours

Activities during the study phase:

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