Module Code - Title:
MS6012
-
ADVANCED METHODS II
Year Last Offered:
2025/6
Hours Per Week:
Grading Type:
Prerequisite Modules:
Rationale and Purpose of the Module:
Syllabus:
Review: Classification of second-order linear PDEs; derivation of wave equation and heat equation; solution using separation of variables for heat, wave, and Laplace equation; use of method of characteristics for quasilinear first order pdes.
Eigenfunction expansions: Completeness and convergence of Fourier series; Gibbs phenomenon; use of Bessel, Legendre, and other special functions. Overview of numerical methods for PDEs; the role of spectral methods.
Transform methods: use of Laplace and Fourier transforms for ordinary differential equations and partial differential equations; applied spectral theory.
Greens functions: Distributions and the Dirac delta function; Green's functions for ordinary differential equations; Greens functions for Laplace equation - use of image sources; Green's functions for heat/wave equations - adjoint operators.
Advanced topics: Topics covered may include: integral equations and integrodifferential equations; systems of hyperbolic equations; gas dynamics; Riemann invariants, shock waves; Burgers and KdV equations; solitons; inverse scattering theory; water waves and various approximations; integrable systems.
Learning Outcomes:
Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)
1. Apply eigenfunction expansion methods to solve partial differential equations.
Final exam and homework for credit.
2. Select and apply suitable transform methods to solve partial differential equations.
Final exam and homework for credit.
3. Derive and solve equations for Green's functions for ordinary and partial differential equations.
Final exam and homework for credit.
4. Recognise and understand the role of certain advanced topics in the solution of partial differential equations.
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:
Haberman, R. (2003)
Applied Partial Differential Equations (4e)
, Prentice Hall
Zauderer, E., (1998)
Partial differential equations of applied mathematics (2e)
, John Wiley & Sons.
Drazin, P.G. and Johnson, R.S (2008)
Solitons: an introduction (2e)
, Cambridge University Press.
Other Relevant Texts:
Carrier, G. and Pearson, C (1990)
Partial differential equations
, Academic Press.
Kreyszig, E., (1993)
Advanced Engineering Mathematics
, Wiley
Whitham, G.B., (1974)
Linear and non-linear waves
, Wiley
Programme(s) in which this Module is Offered:
Semester(s) Module is Offered:
Module Leader:
Michael.Vynnycky@ul.ie