Module Code - Title:
MS6011
-
ADVANCED METHODS 1
Year Last Offered:
2025/6
Hours Per Week:
Grading Type:
Prerequisite Modules:
Rationale and Purpose of the Module:
Syllabus:
Ordinary differential equations. Phase plane analysis. Fixed points, stability, bifurcations. Hopf bifurcation, multiple scales. Relaxation oscillations. Applications to the Van der Pol oscillator, Belousov-Zhabotinskii reaction, activator-inhibitor systems.
Review of complex analysis, particularly Taylor/Laurent series, contour integration, branch cuts, the complex Fourier and Laplace transforms and inversion contours.
Applications of complex analysis, including topics from: representation of solutions of Laplace and biharmonic equations via analytic functions Plemelj formulae, Hilbert problem on the real line, Hilbert transform.
Advanced topics from: solution of mixed boundary value problems motivated by thin airfoil theory and the theory of cracks in elastic solids, Riemann-Hilbert problems, Wiener-Hopf method, singular integral equations.
Laplaces method. Method of steepest descent. The Airy integral. Stokes phenomenon.
Learning Outcomes:
Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)
On successful completion of this module students will be able to:
1. Use phase plane analysis to visualise the solutions of nonlinear systems of differential equations, and analyse their long-term behaviour.
Final exam and homework for credit.
2. Determine whether a complex function is differentiable, define and evaluate path integrals, find Taylor and Laurent expansions of complex functions, calculate residues and use the residue theorem to evaluate real integrals and sums of real series.
Final exam and homework for credit.
3. Solve mixed boundary value problems motivated by applications
Final exam and homework for credit.
4. Use asymptotic methods to solve problems with large or small parameters.
Final exam and homework for credit.
Affective (Attitudes and Values)
N/A
Psychomotor (Physical Skills)
N/A
How the Module will be Taught and what will be the Learning Experiences of the Students:
Research Findings Incorporated in to the Syllabus (If Relevant):
Prime Texts:
Priestley, H. A (2003)
Introduction to Complex Analysis (2e)
, Oxford University Press
Ablowitz, M.J. and Fokas A.S. (2003)
Complex Variables: Introduction and Applications (2e)
, Cambridge University Press
Keener, J. (2000)
Principles of Applied Mathematics: Transformation and Approximation (2e)
, Perseus Books Group
Other Relevant Texts:
Drazin, P.G (2008)
Nonlinear Systems
, Cambridge University Press
Ockendon, J., Howison, S., Lacey, A. and Movichan, A (2003)
Applied Partial Differential Equations
, Oxford University Press
Baker, G.L. and Gollub, J.P (1996)
Chaotic Dynamics: An Introduction
, Cambridge University Press
Golubitsky, M. and Stewart I. (2002)
The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space
, Springer
Programme(s) in which this Module is Offered:
Semester(s) Module is Offered:
Module Leader:
Stephen.OBrien@ul.ie