618-332 Applied Mathematics B

Note:

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Coordinator:

Prerequisite/s:

Timetable:

Contact:

Objectives:

On completion of this subject, students should:

Comprehend:

• how to evaluate real integrals using complex analysis;

• how to evaluate and invert Fourier, Laplace and Mellin transforms, and how these can be applied to solve differential and integral equations, to sum series and to compute asymptotic series;

• what an asymptotic expansion is and how it provides approximations;

• how to use Watson's lemma and the methods of Laplace, stationary phase and steepest descents to evaluate asymptotic expressions;

• how to find asymptotic solutions to ordinary differential equations.

Have developed:

• the necessary mathematical skills and knowledge to apply a range of mathematical techniques to correctly solve applied mathematics problems.

Appreciate:

• the power of these techniques to solve mathematical problems.

Content:

Complex analysis: Contour integration, branch cuts, evaluation of integrals, conformal mappings and applications. Integral transforms: Wave equation, Fourier series; Fourier transform, Fourier integral theorem, convolution, applications; Laplace transform, inversion, examples; application to ordinary differential equations; convolution, examples; application to partial differential equations; Mellin transform examples. Asymptotics: Asymptotic expansions, application of Mellin transform; Laplace's method for integrals, method of steepest descent, applications; method of stationary phase, examples; WKB method for ordinary differential equations.

Assessment: