3172.22 - Mathematics 2
Mathematics 1 (3040.21) or similar.
The purpose of the course is provide participants with tools, e.g. infinite series, to analyse and solve differential equations and systems of differential equations.
Solution of homogeneous/inhomogeneous differential equations and systems of differential equations. Transfer functions. Infinite series, power series, Fourier series. Applications of infinite series for solving differential equations, the exponential matrix. Stability. Introduction to wavelets. Nonlinear differential equations and chaos. Using MAPLE to analyse and solve differential equations and systems of differential equations.
Learning and teaching approaches
Classroom teaching (80 hours), self studies and written assignments (around 126 hours). The teaching will be in Faroese.
On successful completion of the course, the student should be able to: • Determine the solutions to n’th order homogeneous differential equations. • Determine the solutions to linear homogeneous systems of differential equations. • Master the transfer function and apply it to solutions of inhomogeneous differential equations. • Apply existence and uniqueness theorems for systems of differential equations. • Distinguish between linear/nonlinear systems and test the behavior of simple nonlinear systems. • Apply phase plane analysis on systems of differential equations. • Find equilibrium points of systems of differential equations. • Determine and argue for the stability of linear systems of differential equations. • Know the differences between various types of convergence (absolute, conditional, point-wise, uniform) of infinite series and identify them. • Estimate the number of terms which is needed in order to obtain a desired approximation of an infinite series. • Determine the Fourier series for simple periodic functions, clarify their convergence properties, and approximation-theoretic properties • Apply Maple to calculations and control of the results. • Apply Fourier series and various other types of infinite series to solution of differential equations. • Understand the principles behind wavelets.
4-hours written exam, and assessment of written assignments. All usual examination aids (included computers) are allowed. The exam and assignments account for 75% and 25% of the final course grade respectively.
Ole Christensen:Differentialligninger og uendelige rækker, DTU Matematik, August 2012, ISBN: 9788788764833.
Mortan Janusarson Thomsen