Module Code - Title:
MS4404
-
PARTIAL DIFFERENTIAL EQUATIONS
Year Last Offered:
2025/6
Hours Per Week:
Grading Type:
N
Prerequisite Modules:
MS4403
Rationale and Purpose of the Module:
To introduce the partial differential equations of applied mathematics and physics with some standard solutions and applications.
To introduce the theory and applications of first order linear and nonlinear partial differential equations of mathematical physics.
Syllabus:
[Introduction to PDEs:] Introduction to the partial differential equation of physics; classification of second order linear partial differential equations (hyperbolic, parabolic, elliptic).
[Wave equation:] Derivation of wave equation for strings and membranes; solutions by separation of variables; harmonics; d'Alembert's solution; applications to light and sound.
[Laplace's equation:] steady state heat flow; cylindrically symmetric solutions and Bessel functions; spherically symmetric solutions and Legendre functions; flow in porous media.
[Diffusion equation:] Derivation of heat/diffusion equations in one dimension; relation to Brownian motion (random walk) in two and three dimensions; application to chemical diffusion; solutions by separation of variables.
[First order PDEs:] Linear and quasilinear first order partial differential equations; characteristics; applications in chromatography, glacial flow, sedimentation; breaking waves and shocks; diffusion and dispersion (Burger's and KdV equations).
Learning Outcomes:
Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)
On successful completion of this module, students should be able to:
1. Recognize the role of partial differential equations in applied mathematics.
2. Distinguish between examples of parabolic, hyperbolic and elliptic equations.
3. Formulate mathematical problems describing applications of partial differential equations.
4. Apply the method of separation of variables to solve the heat/diffusion equation, the wave equation, and LaplaceÆs equation.
5. Recognize first-order quasilinear equations and apply the method of characteristics to construct solutions.
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:
Normal mathematics lectures.
Research Findings Incorporated in to the Syllabus (If Relevant):
Motivation from research in mathematical modelling.
Prime Texts:
Haberman, R. (2014)
Applied partial differential equations with Fourier series and boundary value problems, 5th ed.
, Pearson
Zauderer, E. (1998)
Partial Differential Equations of Applied Mathematics
, Wiley
Strauss, W. (2018)
Partial Differential Equations
, Wiley
Pinchover, Y. and Rubinstein, J. (2005)
An introduction to partial differential equations
, Cambridge University Press
Other Relevant Texts:
Levandosky, J. (2008)
Solutions manual for partial differential equations : an introduction, 2nd ed.
, Wiley
Kreyszig, E. (2010)
Advanced Engineering Mathematics, 10th ed.
, Wiley
Whitham, G. B. (1974)
Linear and Nonlinear Waves
, Wiley-Interscience
Carrier, G. F. and Pearson, C. E. (1988)
Partial Differential Equations: Theory and Techniques, 2nd ed.
, Academic Press
Lipschutz, S. (2009)
Vector analysis and an introduction to tensor analysis
, Schaum
Programme(s) in which this Module is Offered:
Semester(s) Module is Offered:
Module Leader:
Clifford.Nolan@ul.ie