MAGIC022: Mathematical Methods

This is a core applied module, whose aim is to explain a range of advanced mathematical methods for students doing research in applied mathematics. The style of the course is not theorems and proofs, though the ideas in it can be made fully rigorous with more effort and methods from real and complex analysis. Instead we will gain a working knowledge of powerful tools and approximations at the heart of much theory in applied mathematics and physics. 

Students will be expected to undertake reading from selected texts, as well as try out example problems to reinforce the material covered in lectures. A number of topics are suggested below and depending on the time available, most will be covered.  
Assessment: The assessment for this module will be in the form of a take-home exam at the end of the course. 


 Recommended books: 

•  Bender and Orsag, Advanced mathematical methods for scientists and engineers. McGraw Hill.
•  Bleistan and Handlesman, Asymptotic expansions of integrals
•  Hinch, Perturbation methods. C.U.P.
•  Ablowitz & Fokas Complex Variables, C.U.P. 
• Lighthill Generalised Functions, Dover. 

It is assumed that students have done some real and complex analysis.

•  Advanced differential equations, series solution, classification of singularities. Properties near ordinary and regular singular points. Approximate behaviour near irregular singular points. Method of dominant balance. Airy, Gamma and Bessel functions. 
•  Asymptotic methods. Boundary layer theory. Regular and singular perturbation problems. Uniform approximations. Interior layes. LG approximation, WKBJ method. 
•  Generalised functions. Basic definitions and properties. 
•  Revision of basic complex analysis. Laurent expansions. Singularities. Cauchy's Theorem. Residue calculus. Plemelj formula. 
•  Transform methods. Fourier transform. FT of generalised functions. Laplace Transform. Properties of Gamma function. Mellin Transform. Analytic continuation of Mellin transforms. 
•  Asymptotic expansion of integrals. Laplace's method. Watson's Lemma. Method of stationary phase. Method of steepest descent. Estimation using Mellin transform technique. 
•  Conformal mapping. Riemann-Hilbert problems.