# Numerical Modelling of Physical Phenomena in Engineering, Biology and Medicine

## Subject description

Lectures:

A brief overview of the basic procedures of modelling in engineering and biology; determination of the system and its surroundings, the selection and mathematical description of variables that describe the system and the observation time.

Numerical methods for solving systems of linear algebraic equations and nonlinear algebraic equations.

The optimization procedures.

Numerical methods for solving ordinary differential equations.

Formulation of partial differential equations with appropriate initial and boundary conditions.

Numerical solution of partial differential equations; basics of finite difference method and finite element method.

The basics of cellular automata and Monte Carlo methods.

Laboratory work:

Solving of various problems in biology and medicine using Matlab, its toolboxes (Partial Differential Equation Toolbox) and Comsol Multiphysics program.

## Objectives and competences

During this course students will gain knowledge about the modeling and the use of numerical methods for solving problems in engineering, biology and medicine. They will learn the basic procedures of mathematical model construction on the basis of typical examples from technique, medicine and biology. They will learn the basics of cellular automata modeling,  Monte Carlo methods and optimization procedures.

Mostly, the course deals with numerical methods for solving partial differential equations. Students learn the theoretical basis of the finite difference and finite element method. On the basis of solving simple problems at the beginning and more complex problems during the course the students familiarize with the advantages and limitations of numerical methods. An important part of the learning process is an analysis of the calculated values and their comparison with the experimentally obtained values in cases where the results of the corresponding measurements are available.

The diversity of cases offers students an useful knowledge in the wider field of engineering and science.

## Teaching and learning methods

Lectures; solving typical problems in the context of laboratory work; solving complex tasks in the context of laboratory work and independent work at home.

## Expected study results

After successful completion of the course, students should be able to:

• describe the concepts of modelling in technology, biology in medicine
• choose the appropriate method for numerical solving of mathematical formulations
• state the theory of finite element method as the main method for numerical solving of partial differential equations
• use numerical methods for solving simple and more complex cases