Elements Of Applied Mathematics

Timetable

Module aim (aims)

The main goal of Elements of Applied Mathematics is to present some applications of nonlinear analysis, especially of ordinary differential equations. In particular, mathematical models involving first order differential equations as well as some applications of linear differential equations and systems of linear differential equations will be discussed. Moreover, the Frobenius method of solving higher order differential equations will be presented along with its applications.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

One has to have basic knowledge in mathematical analysis and ordinary differential equations.

Syllabus

Week 1: Basic existence and uniqueness theorems in differential equations theory.
Week 2: Direction fields, phase lines, phase plane analysis.
Week 3: Malthusian and logistic models.
Week 4: Alle effects.
Week 5: Delayed logistic equation.                                                                                                                        Week 6: Volterra-Lotka model.                                                                                                                              Week 7: Continuous epidemic model.                                                                                                                   Week 8: Difference equations, growth-decay model.                                                                                            Week 9: Discrete-time logistic model.                                                                                                                   Week 10: Discrete-time host-parasitoid system.
Week 11: Equations with analytic coefficients and Cauchy-Euler equidimensional equations.
Week 12: Method of Frobenius.                                                                                                                            Week 13: Applications of Frobenius method to Bessel's and Legendre's equations.                                           Week 14: Applications of Frobenius method to description of hydrogen-like atom.
Week 15: Heating and cooling of buildings.

Reading list

 Logan, Wolensky, Mathematical Methods in Biology, Wiley&Sons, 2009.                                                        Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.                                                                            Martin, Elementray Differential Equations with Boundary Value Problems, McGraw-Hill, 1984.
Nagle, Saff, Snider, Fundamentals of Differential Equations, Pearson, 2012.