Introduction to Ordinary Differential Equations

COURSE OBJECTIVES

Understanding Ordinary Differential Equations and their basic theory. Knowing and understanding numerical solution techniques. Connecting this knowledge to simulation and machine learning.

COURSE DESCRIPTION

Ordinary Differential Equations (ODEs) are the most mathematical tool for modelling and quantifying time dependent processes. Chemical reactions, population growth, mechanical systems are examples for this. Newton's second law is in fact an ODE. In this course, we give an introduction into the basic concepts underlying ODEs from a modeling point of view as well as from a mathematical point of view. We then consider numerical methods for the numerical solution of ODEs and investigate properties such as approximation error and stability. This will include Runge-Kutta Methods and so called BDF methods. We will also shortly investigate modern approaches such as parallel-in-time integration. Numerical examples, programming, and mathematical analysis will be the tools for getting towards an understanding of dynamical systems and their properties.

EXAMINATION INFORMATION
During the semester, the student will work on assignment sheets for practicing. A written exam will be held (closed book) at the end of the semester.

REFERENCES

• Iserles, Arieh. A first course in the numerical analysis of differential equations. No. 44. Cambridge university press, 2009.
• Deuflhard, Peter, and Folkmar Bornemann. Scientific computing with ordinary differential equations. Vol. 42. Springer Science & Business Media, 2012.
• E. Hairer, S. P. Nørsett, and G. Wanner. Solving ordinary differential equations. I, volume 8 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 1993. Nonstiff problems.
• E. Hairer and G. Wanner. Solving ordinary differential equations. II, volume 14 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 1996. Stiff and differential-algebraic problems.