MATH4008 Advanced Ordinary Differential Equations This is former 463/MA4008/MT4882 SEMESTER: Second CONTACT: CREDIT RATING: 15
 Aims: To introduce students to some current research problems of importance in the solution of differential equations. Intended Learning Outcomes: On successful completion of this module, students will be able to: Develop numerical methods for solving differential-algebraic equations, delay differential equations, Hamiltonian problems and high order differential equations. Identify numerical methods that preserve the qualitative behaviour of the solution of the problem. Recognise some of the numerical difficulties that can occur when solving problems arising in scientific and industrial applications. Pre-requisites: 157, 211, 362 (ex-UMIST) Dependent Courses: None Course Description: This module introduces some topics from the field of ordinary differential equations, in which there has been much research activity recently. The module begins by discussing the numerical solution of differential-algebraic equations (DAEs), consisting of coupled systems of ordinary differential equations (ODEs) and algebraic equations. Next, differential equations with delay terms are introduced. Features such as the propagation of discontinuities are discussed and numerical methods for solving such problems are described. Then, the numerical solution of Hamiltonian problems is discussed. The property of symplecticness characterises Hamiltonian problems and we investigate numerical methods that preserve this property. Finally, for MMath students only, we consider the numerical solution of higher order differential equations. Teaching Mode: 27 Lectures 6 Tutorials Private Study: 117 hours Recommended Texts: E Hairer, S P Norsett and G Wanner, Ordinary Differential Equations I: Nonstiff Problems, (2nd edition), 1993, Springer-Verlag. E Hairer and G Wanner, Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems, (2nd edition), 1996, Springer-Verlag. Assessment Methods: Coursework: 20% (for MMath); 25% (for MSc) Coursework Mode: Project, set in Week 2, deadline in Week 9. Examination: 80% (for MMath); 75% (for MSc) For MMath students, the examination is of 2 hours duration at the end of the Second Semester. For MSc students, the examination is of one and a half hours duration in April.
 No. of Lectures Syllabus 6 Differential-algebraic equations (DAEs): Basic types of DAEs, including fully-implicit and semi-explicit DAEs. Solvability and index. Order of convergence of the BDF and Runge-Kutta methods. 1 Case Study: A mathematical model of an industrial problem will be studied. The problem may vary from year to year but will give rise to a system of differential-algebraic equations. This model will be investigated further in the coursework assignment. 6 Delay-differential equations: Method of steps. Propagation of discontinuities. Dense output. 5 Hamiltonian problems: The property of symplecticness. Symplectic integrators. 2 Dynamical systems theory: Long-term dynamics, fixed points, bifurcations. 7 (For MMath students only) Higher order differential equations: direct and indirect methods, convergence and stability properties, Runge-Kutta-Nyström methods and hybrid methods.

Last revised August, 2006