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. Finally, a short introduction to the Laplace transform along with its applications will be given.

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 and phase lines.
Week 3: Malthusian and logistic models.
Week 4: Alle effects.
Week 5: Delayed logistic equation.
Week 6: Equations with analytic coefficients and Cauchy-Euler equidimensional equations.
Week 7: Method of Frobenius.
Week 8: Heating and cooling of buildings.
Week 9: The mass-spring oscillator.
Week 10: Free mechanical vibrations.
Week 11: The Laplace transform and its basic properties.
Week 12: Inverse of the Laplace transform.
Week 13: Further properties of the Laplace transform.
Week 14: Applications of the Laplace transform to differential equations.
Week 15: Applications of the Laplace transform to differential equations – continuation.

Reading list

Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.
Nagle, Saff, Snider, Fundamentals of Differential Equations, Pearson, 2012.
Schiff, The Laplace Transform, Theory and Applications, Springer, 1999.