Abstract

This subject covers various methods of solving systems of ordinary differential equations and partial differential equations. Special functions are studied in the framework of Sturm-Liouville theory, leading naturally into Fourier series and transforms. Laplace transforms are studied in greater depth than in previous subjects. All these methods are applied to … For more content click the Read More button below.

Syllabus

Solution of autonomous systems of differential equations;Lyapunov stability;Phase plane analysis;Series solutions to ordinary differential equations;Orthogonal functions;Legendre polynomials and series;Method of Frobenius for series solutions;Bessel's equation;Special functions;The Laplace transform and its use in solving ordinary and partial differential equations;Fourier series, Dirichlet conditions, complex form of Fourier series and Parseval's theorem;Fourier transforms;Solution … For more content click the Read More button below.

Learning outcomes

Upon successful completion of this subject, students should:
1.
be able to solve simple systems of differential equations;
2.
be able to analyse the phase plane for linear and nonlinear autonomous systems of differential equations;
3.
be able to determine the type and stability of the critical points;
4.
be able to use the gamma, beta and error functions;
5.
be able to solve ordinary differential equations using series expansions;
6.
be able to explain Sturm-Liouville theory;
7.
be able to apply orthogonality to the solution of differential equations;
8.
be able to apply Laplace transforms to the solution of differential equations;
9.
be able to calculate Fourier series representing periodic functions;
10.
be able to extend the techniques of Fourier series to nonperiodic functions by means of Fourier integrals and Fourier transforms;
11.
be able to solve partial differential equations with given initial and boundary conditions;
12.
be able to interpret mathematical models and communicate their output to non-mathematical audiences.