After this course, the student is able to|
1. find roots of nonlinear functions;
2. linearize near equilibria of an ODE and classify equilibria using eigenvalues;
3. simulate deterministic and stochastic ODEs as well as perform quadrature;
4. to construct solutions of second order PDE’s equation using Fourier series or finite differences;
5. assess accuracy of results from algorithms using numerical error analysis;
6. choose or modify algorithms for new numerical problems.
The understanding of scientific models and solving complicated engineering problems requires the correct use of programming and mathematical algorithms. Typical problems involve numerical solutions of nonlinear equations, simulations of dynamical systems, stability via eigenvalues, finite-dimensional approximations of spatial systems by discretisation or Fourier analysis. The art is not just to be able to solve such problems but also to have an intuition for the accuracy of the solution. We will develop some analytical insight, derive several algorithms and discuss error analysis.
During the bachelor study, an engineering student learns several mathematical methods through a number of calculus and linear algebra courses. Starting from this bachelor level, this course has three aims:
1. Refresh and enrich the students' knowledge on calculus, linear algebra, and Fourier analysis,
2. Expose the student to other areas such as numerical analysis and mathematical algorithms, and stochastic and partial differential equations,
3. Practising programming using Matlab with emphasis on efficiency and correctness. After this course the student has the ability to efficiently simulate and analyse models given by differential equations. In a more complicated setting, the student will be able to choose and adopt a method for an application.
The course consists of lectures providing an intuition to the mathematical methods. A few tutorials focus on Matlab programming. The homework problems deal with applications in neuroscience and biomedical engineering.