Applied Mathematics
MP346: Mathematical Methods II (5 ECTS)
This is a mathematical methods course, and amongst the topics considered are the heat equation, Laplace's equation, Sturm-Liouville theory, the Fourier transform, and the numerical solution of partial differential equations using finite difference techniques.
Taught in Semester(s) II. Examined in Semester(s) II.
Workload: 36 hours (24 Lecture hours, 12 Tutorial hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- Solve the 1-dimensional heat equation subject to different boundary conditions and initial conditions.
- Prove orthogonality of eigensolutions and reality of eigenvalues of a Sturm-Liouville system.
- Apply Sturm-Liouville method to obtain the solution in simple examples.
- Solve the 2-dimensional Laplace equation subject to different boundary conditions in a rectangular or a rotationally symmetric region.
- Solve the 1-dimensional heat equation on an infinite region by use of the Fourier transform.
- Solve the 1-d heat equation numerically by use of the finite difference method.
Indicative Content
This is a follow on to the course Mathematical Methods MP345. Topics covered include: (i) The 1-dimensional heat equation. Introduction to Initial Value Boundary Value Problems. Solution for various boundary conditions and initial conditions. (ii) Sturm-Liouville Systems. General properties and application to simple systems. (iii) The 2-dimensional Laplace equation. Solution for various boundary conditions on a rectangular or rotationally symmetric region. (iv) The Fourier Transform. Properties, the inverse transform. Application to solving the 1-dimensional heat equation on an infinite region. (v) Finite difference methods. Application to numerically solving the 1-dimensional heat equation. Stability of numerical method.
Module Resources
Advanced engineering mathematics, Erwin Kreizig (Willey) Mathematical methods for physics and engineering, Riley, Hobson and Bence (CUP) Mathematical methods in the physical sciences, Boas (Wiley)
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