MM547: Ordinary Differential Equations: Theory, Modelling and Simulation

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

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.

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).

• 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.

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.

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.

Activities during the study phase:

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