Module I (15 hours)
Asymptotic expansions, Watson's lemma, method of stationary phase and saddle point method. Applications to differential equations. Behaviour of solutions near an irregular singular point, Stoke's phenomenon, Method of strained coordinates and matched asymptotic expansions. Variational principles, Lax-Milgram theorem and
applications to boundary value problem,
Module II (12 hours) Hankel transform, finite Hankel transform, Mellin transform. Solution of differential equations by integral transform methods. Construction of the kernels of integral transforms on a finite interval through Sturm-Liouville problem.
Module III (13 hours)
Calculus of variations and integral equations. Regular and singular integral equations: Volterra, Fredholm integral equations, Volterra and Fredholm equations with different types of kernels.

To introduce different transform methods.

To apply the transform techniques in solving different differential equations

After completing the course students will learn different methods and will be able to apply these methods in solving real-life problems.

C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill Book Co., 1978.

A. D. Poularikas, The Transforms and Applications Handbook, CRC Press, 1996.

J. Kevorkian and J.D. Cole, Perturbation Methods in Applied Mathematics, Springer Verlag, Berlin, 1985.

F.G Tricomi, Integral Equations, Dover Publications Inc. New York, 1985.