Advanced Math Topics

Introduction: many physical phenomena can be described by differential equations. In the context of Nanobiology you can think of processes like heat dispersion, diffusion or chemical reactions. Often these equations have to be solved numerically. First, the course will discuss how a system of ordinary differential equations (ODEs) can be solved using numerical time integration. Then we discuss how a stationary diffusion equation can be discretized with the Finite Difference Method. Finally, we combine time integration and space discretization to numerically solve the heat equation.

The theory will be explained in 4 sessions as described below:

• Lecture 1: Numerical integration and some time integration methods: Forward Euler, Backward Euler, Trapezoidal Method, Modified Euler. Implicit methods and Explicit methods. Predictor-corrector methods.

• Lecture 2: Stability of time integration methods, order of accuracy and systems of ordinary differential equations (ODEs).

• Lecture 3: Numerical differentiation and Finite Difference Method(s). Discretization of boundary conditions and properties of the discretization matrix.

• Lecture 4: Finite Difference Method (continued): (Non-constant) diffusion and the heat equation. The method of lines.
  

After every lecture, there is a practical session where students should be present in class to:

• practice with the theory discussed during lectures using pen-and-paper exercises and Python implementations;

• use Python to numerically solve given problems;

• explain their Python code, implementation choices, and pen-and-paper solutions to the present instructors and/or teaching assistants in order to receive feedback;

• prepare for the final assignment, which will be an oral exam.

Remark: whenever you choose not to be present in class, there are no possibilities to get extra explanation or feedback regarding the content of the lectures nor the practical sessions. This also includes the Python code and pen-and-paper exercises.