Asymptotic Analysis (20cr)
| Code | School | Level | Credits | Semesters |
| MATH4086 | Mathematical Sciences | 4 | 20 | Autumn UK |
- Code
- MATH4086
- School
- Mathematical Sciences
- Level
- 4
- Credits
- 20
- Semesters
- Autumn UK
Summary
Mathematical models based on systems of ordinary or partial differential equations are used in a vast range of disciplines, ranging from classical fields such as fluid and solid mechanics to more recent applications in mathematical biology and finance. The complexity of these models is often so great that numerical methods are the only ones available to construct solutions. However, in this course we will learn how to make analytical progress in the presence of a small parameter using asymptotic methods, to determine similarity solutions and to obtain qualitative information using the techniques of dynamical systems theory.
Topics will include:
• Asymptotic expansions and order symbols.
• Asymptotic solutions of algebraic equations.
• Laplace’s method and the method of stationary phase.
• The method of matched asymptotic expansions.
• The method of multiple scales.
• The Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) expansion.
• The centre manifold theorem.
• Lyapunov’s theorems.
• Bifurcation theory, including local and global bifurcations.
Target Students
Single and Joint Honours students from the School of Mathematical Sciences. Also available to Natural Sciences and Mathematical Physics students. Available to Postgraduate Taught students and Postgraduate Research students.
Classes
Classes timetabled centrally, some of which may be used for lectures or workshops.
Assessment
- 100% Exam 1 (3-hour): Written Exam In Person
Assessed by end of autumn semester
Educational Aims
This course introduces various analytical methods for the solution of ordinary and partial differential equations, focussing on asymptotic techniques and dynamical systems theory. Students taking this course will build on their understanding of differential equations covered in the prerequisite modules.Learning Outcomes
L1 - apply a variety of asymptotic methods to obtain systematic approximations to integrals and to
solutions of algebraic and differential equations;
L2 - interpret mathematical results derived by asymptotic methods, qualitatively and quantitatively;
L3 - define and use appropriate notation and terminology related to asymptotic methods;
L4 - determine the bifurcation structure of first and second order nonlinear ordinary differential equations;
L5 - perform bifurcation analysis of dynamical systems exhibiting a range of local and global bifurcations.