Mathematical methods in natural and social sciences 1000-135MMN
The aim of the lecture is to present some basic methods of dynamical systemsand partial differential equations that are essential in the modern description of natural and social processes. In the past Mathematical Methods of Physics were a basis for description of physical processes. Nowadays it is important to know modern mathematical methods used in description of processes in Natural and Social Sciences. The methods refer to typical nonlinear equations that are used in description. The plan of the lecture is as follows: Poincar´ e–Bendixson Theorem; Grobman–Hartman Theorem; Methods of Small Parameter, Singular Perturbations; Conservation Laws, Methods of Characteristics; Diffusion Processes; Reaction–Diffusion Equations; Semigroups theory; Deterministic chaos. The theory is illustrated by numerous examples including those in Economy, Biology, Medicine, Social Sciences and Technology.
Main fields of studies for MISMaP
physics
computer science
Type of course
Course coordinators
Learning outcomes
1. The knowledge of basic mathematical structures corresponding to processes in biology, medicine and social sciences
2) The knowledge of mathematical technics in analysis of models
(a) The Poincare'-Bendixson theorem,
(b) The Grobman-Hartman theorem,
(c) Methods of small parameter, singular perturbation,
(d) Initial layer, Boundary layer, shock waves,
(e) Tikhonov-Vasil'eva theory,
(f) The characteristic methods,
(g) The similiaryty methods,
(h) Travelling waves,
(i) Existence, uniqueness, Maximum Principle,
(j) Energy estimates and asymptotic behaviour,
(k) Patterns
(l) Semigroup theory
(m) Deterministic Chaos
Assessment criteria
score system and a written exam
Bibliography
1. J. Banasiak, M. Lachowicz, Methods of small parameter in mathematical biology, Birkhüser 2014.
2. M.W. Hirsch, S. Smale, R.L. Devaney, Differential equations, dynamical systems, and an introduction to chaos, Academic Press 2004.
3. J.D. Logan, An Introduction to Nonlinear Partial Differential Equations, Wiley Interscience 2008.
4. L. Perko, Differential Equations and Dynamical Systems, Springer 2001.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
- Bachelor's degree, first cycle programme, Mathematics
- Master's degree, second cycle programme, Mathematics
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: