Modified homotopy perturbation method for solving hypersingular integral equations of the first kind.

Z K Eshkuvatov,Z Muminov,N M A Nik Long,F S Zulkarnain

doi:10.1186/s40064-016-3070-z

Z K Eshkuvatov, Z Muminov + Show 2 more

Open Access

https://doi.org/10.1186/s40064-016-3070-z

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Abstract

Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [−1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190:1707−1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3):265–274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24:636–641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points.

Highlights

Hypersingular integral equations (HSIEs) arise a variety of mixed boundary value problems in mathematical physics such as water wave scattering (Kanoria and Mandal 2002), radiation problems involving thin submerged plates (Parsons and Martin 1994) and fracture mechanics (Chan et al 2003; Nik Long and Eshkuvatov 2009)
We present the application of standard Homotopy perturbation method (HPM) for solving hypersingular integral equations of the first kind (5)
In this work, the standard and modified HPM are used to find the approximate solution of the first kind HSIE

Summary

Introduction

Hypersingular integral equations (HSIEs) arise a variety of mixed boundary value problems in mathematical physics such as water wave scattering (Kanoria and Mandal 2002), radiation problems involving thin submerged plates (Parsons and Martin 1994) and fracture mechanics (Chan et al 2003; Nik Long and Eshkuvatov 2009). In Panda et al (2015), a modified Lagrange approach is presented to obtain approximate numerical solutions of Fredholm integral equations of the second kind. Ghorbani and Saberi-Nadjafi (2006) added a series of parameter and selective functions to HPM to find the semi-analytical solutions of nonlinear Fredholm and Volterra integral equations.

Results

Conclusion