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 the accuracy of results from algorithms using numerical error analysis;
6. choose or modify algorithms for new numerical problems.
Understanding scientific models and solving complicated engineering problems require 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.|
In their bachelor's programme, an engineering student learns several mathematical methods through several calculus and linear algebra courses. Starting from this bachelor's level, this course has three aims:
1. Refresh and enrich the students' knowledge of calculus, linear algebra, and Fourier analysis,
2. Expose the student to other areas such as numerical analysis and mathematical algorithms, and ordinary, stochastic and partial differential equations,
3. Practising programming using Matlab with an emphasis on efficiency and correctness. After this course, the student can 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 intuition for the mathematical methods. A few tutorials focus on Matlab programming, but students may use Python if they prefer. In the homework problems, the student deals with neuroscience and biomedical engineering applications.