# MM831: Differential Equations II

## Comment

## Entry requirements

## 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

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

## Expected learning outcome

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

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

## Content

The following main topics are contained in the course:

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

## Literature

## Examination regulations

## Exam element a)

## Timing

## Tests

## Oral exam

## EKA

## Assessment

## Grading

## Identification

## Language

## Examination aids

To be announced during the course

## ECTS value

## Indicative number of lessons

## Teaching Method

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