Swaraj Paul

Research Interest

1. Microlocal analysis and shearlets

In microlocal analysis, the singularities of a distribution f can be described through the wavefront set of f. Now a days wavefront sets play an important role in representing image processing, artifacts,  time-machine, space-times, quantum energy inequalities and cosmological models. Shearlets, whose underlying systems are discrete affine systems, provide an optimally sparse approximation of a cartoon-type image with discontinuity/singularity across a C2 smooth curve. In ongoing work we are trying to characterize microlocal behaviour of any distribution using shearlets.

2. Numerical scheme based on wavelets 

Singular integral equations arise in various problems of mathematical physics in a natural way and as such, they play an important role in varieties of fields of science and engineering. In ongoing work we are trying to develop the following using numerical scheme based on wavelets  

• Multiresolution approach of regularization of singular integral operators ˆ 

• Constructing wavelets 

• ˆ Detection of singularity of solution using wavelet coefficients of the solution

Publications

Journal Publications

• Two wavelet multipliers and Landau-Pollak-Slepian operators on locally compact abelian groups associated to right--translation-invariant functions,  Santosh Kumar Nayak and Swaraj Paul.

• Microlocal analysis and characterization of Sobolev wavefront sets using shearlets, Bin Han, Swaraj Paul and Niraj K. Shukla, Constructive Approximation, 55 (2022) pp 661-704.

• Study of the sextic and decatic anharmonic oscillators using an interpolating scale function,  K. Manimegalai, Swaraj Paul, M M Panja and Tapas Sil, Eur. Phys. J. Plus 135, 133 (2020). 

• Approximate solution of first kind singular integral equation with generalized kernel using Legendre multiwavelets, Swaraj Paul,  M M Panja & B N Mandal, Comput. Appl. Math., 38 (2019), pp 23.

• Use of Legendre multiwavelets to solve Carleman type singular integral equations, Swaraj Paul,  M M Panja & B N Mandal, Appl. Math. Model., 55 (2018), pp 522-535.

• Multiscale approximation of the solution of weakly singular second kind Fredholm integral equation in Legendre multiwavelet basis,  Swaraj Paul,  M M Panja & B N Mandal, J. Comp. Appl. Math., 300 (2016) pp 275-289.

• Wavelet based numerical solution of second kind hypersingular integral equation, Swaraj Paul,  M M Panja & B N Mandal, Appl. Math. Sci., 10 (2016) pp 2687-2707.

• Use of Legendre multiwavelets in solving second kind singular integral equations with Cauchy type kernel, Swaraj Paul,  M M Panja & B N Mandal, Investigation in Mathematical Sciences, 5 (2016), pp 10-21.

Conference proceedings 

• Comparison of Two Methods Based on Daubechies Scale Functions and Legendre Multiwavelets for Approximate Solution of Cauchy-Type Singular Integral Equation on R, Swaraj Paul & B N Mandal, Mathematics and Computing, Springer Proceedings in Mathematics & Statistics 253, Springer Nature Singapore Pte Ltd. 2018, pp 453-470.

Thesis

Some Studies on Techniques based on Multiresolution Analysis for Solution of Integral Equations, Swaraj Paul, PhD thesis, Visva-Bharati, Santiniketan (2018).