Module I (10 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 (10 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 (10 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
and apply the transform techniques in solving various differential equations.

Upon completing the course, students will gain proficiency in various methods and will be able to apply them effectively to solve many real-world 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,

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