# Methods of Mathematical Modeling

Instructor: Larissa V. Panina 2016-May-25

### Course Summary

The aim of this module is to introduce practical numerical methods through application to electromagnetic problems in modern micro and nano technology. The revision of basic numerical analysis and methods along with high level programming languages (C, C++) is an important part of the module. This includes polynomial interpolation, solving non-linear equations, numerical integration and differentiation, initial value and boundary problems. The application of the numerical methods is based on three main topics: modelling the properties of heterostructures in mean-field approximation; modeling dynamics of magnetic nanoparticles and mathematical aspects of their detection based on high harmonics generation; design and modeling semiconductor heterostructures and solar cell optimization. The students will complete practical modeling projects in all thematic areas presenting their results in a graphical form (2D and 3D graphics).

### Course Format

Hours of lectureHours of discussionHours of independent studyTotal numbers of hours
3232192256

Please note that students are expected to study outside of class for three hours for every hour in class.

### Course Content

The plan is to work through the following topics

1. Basics of numerical methods
• Source of errors, stability and convergence of numerical methods;
• Modelling based on analytical solutions;
• Interpolation and inputting experimental data;
• Numerical differentiation and integration;
• Finding roots of non-linear equation.
2. Modeling magnetic properties of composites and nanoparticles (with numerical methods introduced in part 1).
• Effective permeability and permittivity in mean-field approximation;
• Understanding the size effect on static and dynamic magnetization, superparamagnetism;
• Generation of high harmonics at re-magnetization of superparamagnetic particles;
• Principles of magnetic tomography based on non-linear magnetization.
3. Initial value and boundary problems
• Differential equations with initial conditions (Euler, Predictor-Corrector , Runge-Kutta methods), stability and convergence;
• Solution of system of linear equations, finding inverse matrix;
• Method of mean-least squares for interpolation;
• Tridiagonal matrix, differential equations with boundary conditions.
4. Transport of magnetic nanoparticles and methods of their detection (with numerical methods introduced in parts 1,3).
• Equations of motion of magnetic nanoparticles in gradient magnetic fields inside viscose substance;
• Detection of magnetic nanoparticles using non-linear magnetisation;
• Magnetisation behaviour of ensemble of magnetic nanoparticles;
• Dynamics of magnetic nanoparticles in rotating magnetic field.
5. Modeling semiconductor heterostructures (with numerical methods introduced in parts 1,3).
• Properties of p-n junctions (depletion area, blocking potential, dark p-n current);
• Multilayer p-n junctions;
• Solar cells. Modeling the V-J characteristics and efficiency of realistic solar cells.

Primary textbooks:

1. F.S. Acton. Numeical methods that work. Mathematical Association of America, 1999.
2. P. Gubin. Magnetic nanoparticles. Wiley, New York, 2009.
3. I.M. Dharmadasa, Advances in thin-film solar cells, Stanford publishing Ltd, 2013