Applied Mathematics
MP237: Mechanics II (5 ECTS)
This course consists principally of an introduction to the theory and applications of partial differential equations. Topics covered include the heat equation, the wave equation, Laplace's equation, and a brief introduction to the special theory of relativity.
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:
- formulate a well-posed problem for the heat equation;
- solve some initial boundary value problems for the heat equation using the method of separation of variables;
- find the general solution to the one-dimensional wave equation using characteristic variables;
- construct solutions to the one-dimensional wave equation on an infinite and semi-infinite domain using characteristic variables;
- construct separable variable solutions to the wave equation;
- construct separable variable solutions to Laplace's equation;
- state Einstein's two postulates of special relativity;
- perform simple calculations in special relativity involving time dilation, length contraction, velocity transformations, energy and momentum.
Indicative Content
- An introduction to partial differential equations, the heat equation as a model for heat flow, boundary conditions, initial conditions, well-posed problems, separable variable solutions of the heat equation.
- The wave equation as a model for the vibrations of a string, characteristic variables for the wave equation, the general solution of the one-dimensional wave equation, D'Alembert's solution, solution of the wave equation on a semi-infinite domain using characteristic variables, solution of the wave equation on a finite domain using the method of separation of variables.
- Laplace's equation in two dimensions, solutions to Laplace's equation in rectangular domains using the method of separation of variables.
- Introduction to the theory of special relativity, Einstein's two postulates of special relativity, the Lorentz transformation, length contraction, time dilation, the velocity transformation, relativistic mass, momentum and energy, the transformation law for momentum and energy.
Module Resources
Advanced Engineering Mathematics, 10th Edition, E. Kreyszig, John Wiley & Sons.
An Introduction to Mechanics, D. Kleppner & R. Kolenkow, Cambridge University Press.
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