Integral Transforms and Integral Equations

Code: MA661 | L-T-P-C: 3-0-0-6

 

Prerequisites: MA542 Differential Equations or its equivalent

Preamble: This course aims at providing basic ideas of integral transforms and integral equations and how they can be utilized to solve initial/boundary value problems. Many problems, not easily solvable by standard methods, can be appropriately handled by means of integral transforms and integral equations. This course will benefit interested senior undergraduate, Master's and Doctoral students of a number of departments.

Syllabus: Basic integral transforms: Fourier transform, Fourier sine and cosine transforms, Laplace transform, Hankel transform, Mellin transform, Radon transform. Finite Fourier sine and cosine transforms, finite Hankel transforms. Construction of the kernels of integral transforms: kernels on a finite interval, circular region, a semi-finite interval and radially symmetric interval, kernels for discrete to continuous spectrum. Applications to ODEs, hyperbolic PDEs.

Occurrence of integral equations, Regular integral equations: Volterra integral equations, Fredholm integral equations, Volterra and Fredholm equations with regular kernels. Symmetric kernels and orthogonal systems of functions. Singular integral equations: weakly singular integral equations, Cauchy singular integral equations, hypersingular integral equations. Bernstein polynomials: properties and its use in solving integral equations. Green's function in integral equations.

Texts/References:

1. L. Debnath and D.D. Bhatta, Integral Transforms and Their Applications, Book World Enterprises, 2006.
2. M. Ya. Antimirov, A.A. Kolyshkin, R. Valliancourt, Applied Integral Transforms, CRM Monograph Series, American Mathematical Society, 2007.
3. A.D Poularikas, The Transforms and Applications Handbook, CRC Press, 1996.
4. F.G Tricomi, Integral Equations, Dover Publications Inc. New York, 1985.
5. D. Porter and D.S.G. Stirling, Integral Equations: A Practical Treatment from Spectral Theory to Applications, Cambridge University Press, 1990.
6. N.I. Muskhelishvili, Singular Integral Equations, Dover Publications Inc., New York, 2008.