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2006/07 Undergraduate Module Catalogue

MATH2370 Linear Differential Equations and Transforms

10 creditsClass Size: 200

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2006/07

Pre-requisites

MATH2360 or equivalent.

This module is approved as an Elective

Module summary

Prerequisite: MATH2360 Excluded combination: may not be taken with MATH2431This module introduces a variety of techniques for the solution, subject to suitable boundary and initial conditions, of the basic PDEs of mathematical physics, which describe such ubiquitous phenomena as waves and diffusion, as well as the potential problems of gravitation, electromagnetism and fluid dynamics. The method of separation of variables leads to the solution of various linear ODEs with variable coefficients by the method of power series. The solutions can be viewed as eigenfunctions of symmetric operators, and consequently possess orthogonality properties analogous to those of the eigenvectors of a real symmetric matrix. The particular cases of the Bessel functions and the Legendre polynomials are considered in some detail and applied to the solution of various boundary value problems. The module ends with a section on Fourier and Laplace transforms. These can be used to convert a problem involving an infinite space or time domain into a new and simpler problem. One then solves the simpler problem and converts back to obtain the solution to the original problem.See the schools website or contact: a.slomson@leeds.ac.uk for more information.

Objectives

To describe the method of separation of variables for the solution of PDE?s subject to given boundary conditions, incorporating such topics as power series solution of ODE?s, orthogonality of eigenfunctions of symmetric operators, and the basic properties of Bessel and Legendre functions. To introduce Fourier and Laplace transforms and apply them to various linear boundary and initial value problems. On completion of this module, students should be able to: (a) obtain power series solutions of 2nd order homogeneous linear ODE?s; (b) test 2nd order linear differential operators for symmetry and draw appropriate conclusions from the resulting orthogonality of their eigenfunctions; (c) solve the standard PDE?s of mathematical physics in Cartesian or (2D or 3D) polar co-ordinates subject to given boundary conditions by the method of separation of variables, using Bessel and Legendre functions where necessary; (d) use Fourier and Laplace transforms to solve a range of boundary and initial value problems for linear ODE?s and PDE?s.

Syllabus

This module introduces a variety of techniques for the solution, subject to suitable boundary and initial conditions, of the basic PDE's of mathematical physics, which describe such ubiquitous phenomena as waves and diffusion, as well as the potential problems of gravitation, electromagnetism and fluid dynamics. The method of separation of variables leads to the solution of various linear ODE's with variable coefficients by the method of power series. The solutions can be viewed as eigenfunctions of symmetric operators, and consequently possess orthogonality properties analogous to those of the eigenvectors of a real symmetric matrix. The particular cases of the Bessel functions and the Legendre polynomials are considered in some detail and applied to the solution of various boundary value problems. The module ends with a section on Fourier and Laplace transforms. These can be used to convert a problem involving an infinite space or time domain into a new and simpler problem. One then solves the simpler problem and converts back to obtain the solution to the original problem. Topics covered include: Separation of variables, power series solution of ODE's, symmetric operators and orthogonality of eigen functions, Bessel and Legendre functions, their basic properties and application to boundary problems. Fourier and Laplace transforms, with applications to boundary and initial value problems.

Teaching methods

Lectures (22 hours) and examples classes (11 hours).

Methods of assessment

2 hour written examination at end of semester (85%), coursework (15%).

Reading list

The reading list is available from the Library website

Last updated: 30/03/2007

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