Linear algebra, equilibrium and minimization problems. Applications on networks. Duality and calculus of variation, essential and natural boundary conditions. Systems of ordinary differential equations, linear and nonlinear. Phase plane, stability, bifurcation. Numerical methods for the solution of nonlinear systems and differential equations. Applications on mechanical and ecological systems.
Solution of systems of linear equations. Symmetrical positive definite matrices. Minimization problems. Eigenvalues and dynamical systems.
Assignments: One assignment every second week, from paper and pencil work to parameter studies of dynamical models in ecology and mechanics.
The overall goal of the course is to give basic knowledge of applied and numerical mathematics useful for scientific and engineering modeling. Especially, the close connection between the properties of mathematical models and their successful numerical treatment is emphasized.
This understanding means that after the course you should be able to
- identify and describe discrete equilibrium models using a network approach;
- relate equilibrium problems to minimum principles and solve simple constraint minimization problems via the Lagrange multiplier approach;
- formulate variational problems starting from simple physical principles and derive the corresponding Euler-Lagrange equation such that you can derive basic equations for continuous equilibrium problems in one, two, and three space dimensions;
- model (spatially discrete) time-dependent systems by ordinary differential equations;
- investigate the stability of autonomous systems and explore geometrically the phase space of 2D dynamical systems using analytical and numerical tools;
- derive asymptotic expansions for certain simple singular perturbation problems;
- understand the relation between convergence, consistency, and stability of numerical methods;
- understand essential properties of, and proof error estimations for, numerical methods for solving stationary and instationary problems such that you can compare different methods and select suitable algorithms for given problems;
- analyze and select iterative methods for large linear systems.