This unit may only be taken by students on Mathematics and Physics programmes.
Aims & Learning Objectives:
Aims: To furnish the student with a range of analytic techniques for the solution of ODEs and PDEs with applications to advanced physical problems.
Objectives:
Students should be able to obtain the solution of certain ODEs and PDEs, and in cases interpret these in physical terms. They should also be aware of certain analytic properties associated with the solution e.g. uniqueness.
Content:
Sturm-Liouville theory: Reality of eignevalues. Orthogonality of eigenfunctions. Expansion in eigenfunctions. Approximation in mean square. Statement of completeness.
Fourier Transform: As a limit of Fourier series. Properties and applications to solution of differential equations. Frequency response of linear systems. Characteristic functions.
Linear and quasi-linear second-order PDEs in two independent variables: Cauchy-Kovalevskaya theorem (without proof). Characteristic data. Lack of continuous dependence on initial data for Caucy problem. Classification as elliptic, parabolic, and hyperbolic. Different standard forms. Constand and nonconstant coefficients.
One-dimensional wave equation: d'Alembert's solution. Uniqueness theorem for corresponding Cauchy problem (with data on a spacelike curve).
Applications to physical problems. Translate advanced physical problems into mathematical form; obtain and interpret mathematical solutions.
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