# American Institute of Mathematical Sciences

September 2018, 8(3): 337-350. doi: 10.3934/naco.2018022

## Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations

 1 Faculty of Science and Technology, Universiti Sains Islam Malaysia, Malaysia (USIM), Negeri Sembilan, Malaysia 2 Institute for Mathematical Research, Universiti Putra Malaysia (UPM), Malaysia

* Corresponding author: Zainidin Eshkuvatov

Received  April 2017 Revised  March 2018 Published  June 2018

TIn this note, we review homotopy perturbation method (HPM), Discrete HPM, Chebyshev polynomials and its properties. Moreover, the convergences of HPM and error term of Chebyshev polynomials were discussed. Then, linear singular integral equations (SIEs) and hyper-singular integral equations (HSIEs) are solved by combining modified HPM together with Chebyshev polynomials. Convergences of the mixed method for the linear HSIEs are also obtained. Finally, illustrative examples and comparisons with different methods are presented.

Citation: Zainidin Eshkuvatov. Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 337-350. doi: 10.3934/naco.2018022
##### References:
 [1] M. Abdulkawi, N. M. A. Nik Long and Z. K. Eshkuvatov, Numerical solution of hypersingular integral equations, International Journal of Pure and Applied Mathematics, 69 (2011), 265-274. [2] M. M. Abdulkawi, N. M. A. Nik Long and Z. K. Eshkuvatov, Numerical solution of hypersingular integral equations, Int. J. Pure Appl. Math., 69 (2011), 265-274. [3] T. Allahviranloo and M. Ghanbari, Discrete homotopy analysis method for the nonlinear Fredholm integral equations, Ain Shams Engineering Journal, 2 (2011), 133-140. doi: 10.1016/j.asej.2011.06.002. [4] J. Biazar and H. Aminikhah, Study of convergence of homotopy perturbation method for systems of partial differential equations, Computers and Mathematics with Applications, 58 (2009), 2221-2230. doi: 10.1016/j.camwa.2009.03.030. [5] S. H. Behiry, R. A. Abd-Elmonem and A. M. Gomaa, Discrete adomian decomposition solution of nonlinear Fredholm integral equation, Ain Shams Engineering Journal, 1 (2010), 97-101. doi: 10.1016/j.asej.2010.09.009. [6] Y. Sha Chan, A. C. Fannjiang and G. H. Paulino, Integral equations with hypersingular kernels, theory and applications to fracture mechanics, International Journal of Engineering Science, 41 (2003), 683-720. doi: 10.1016/S0020-7225(02)00134-9. [7] J. D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell Publishing Company, Waltham, Massachusetts, 1968. [8] G. Davydov, E. V. Zakharov and Yu. V. Pimenov, Some computational aspects of the hypersingular integral equation method in electrodynamics, Computational Mathematics and Modeling, 15 (2004), 105-109. doi: 10.1023/A:1022072215887. [9] V. M. Dyke, Perturbation Methods in Fluid Mechanics, The Parabolic Press, Stanford, California, 1975. [10] M. Dehghan and F. Shakeri, Solution of an integro-differential equation arising in oscillating magnetic fields using He's homotopy perturbation method, Prog. Electromagn. Res., 78 (2008), 361-376. doi: 10.2528/PIER07090403. [11] A. A. Daschioglu, A Chebyshev polynomial approach for linear Fredholm-Volterra integro-differential equations in the most general form, Applied Mathematics and Computation, 181 (2006), 103-112. doi: 10.1016/j.amc.2006.01.018. [12] A. A. Daschioglu and M. Sezer, Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations, Journal of the Franklin Institute, 342 (2005), 688-701. doi: 10.1016/j.jfranklin.2005.04.001. [13] Z. K. Eshkuvatov, F. S. Zulkarnain, N. M. A. Nik Long and Z. Muminov, Modified homotopy perturbation method for solving hypersingular integral equations of the first kind, Springer Plus, 5 (2016), 1-21. doi: 10.1186/s40064-016-3070-z. [14] Z. K. Eshkuvatov, N. M. A. Nik Long and M. Abdulkawi, Approximate solution of singular integral equations of the first kind with Cauchy kernel, Journal of Applied Mathematics Letter., 22 (2009), 651-657. doi: 10.1016/j.aml.2008.08.001. [15] A. Golbabai and M. Javidi, Application of He's homotopy perturbation method for nth-order integro-differential equations, Appl. Math. Comput., 190 (2007), 1409-1416. doi: 10.1016/j.amc.2007.02.018. [16] M. Ghasemi, M. T. Kajani and A. Davari, Numerical solution of the nonlinear Volterra-Fredholm integral equations by using homotopy perturbation method, Appl. Math. Comput., 188 (2007), 446-449. doi: 10.1016/j.amc.2006.10.015. [17] A. Golbabai and B. Keramati, Solution of nonlinear Fredholm integral equations of the first kind using modified homotopy perturbation method, Chaos, Solitons, and Fractals, 39 (2009), 2316-2321. doi: 10.1016/j.chaos.2007.06.120. [18] A. Ghorbani and J. Saberi-Nadjafi, Exact solutions for nonlinear integral equations by a modified homotopy perturbation method, Comput. Math. Appl., 28 (2006), 1032-1039. doi: 10.1016/j.camwa.2008.01.030. [19] J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178 (1999), 257-262. doi: 10.1016/S0045-7825(99)00018-3. [20] J. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Inter. J. Non-linear Mech., 35 (2000), 37-43. doi: 10.1016/S0020-7462(98)00085-7. [21] J. H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput., 135 (2003), 73-79. doi: 10.1016/S0096-3003(01)00312-5. [22] J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation, 151 (2004), 287-292. doi: 10.1016/S0096-3003(03)00341-2. [23] J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26 (2005), 695-700. doi: 10.1016/j.chaos.2005.03.006. [24] J. H. He, New interpretation of homotopy perturbation method, Internat. J. Modern Phys. B, 20 (2006), 2561-2568. doi: 10.1142/S0217979206034819. [25] W. Al-Hayani, Solving nth-order integro-differential equations using the combined Laplace transform-adomian decomposition method, Applied Mathematics, 4 (2013), 882-886. [26] H. Jafari, M. Alipour and H. Tajadodi, Convergence of homotopy perturbation method for solving integral equations, Thai J. Math., 8 (2010), 511-520. [27] M. Javidi and A. Golbabai, Modified homotopy perturbation method for solving non-linear Fredholm integral equations, Chaos Solitons Fractals, 40 (2009), 1408-1412. doi: 10.1016/j.chaos.2007.09.026. [28] M. Kanoria and B. N. Mandal, Water wave scattering by a submerged circular-arc-shaped plate, Fluid Dynamics, 31 (2002), 317-331. doi: 10.1016/S0169-5983(02)00136-3. [29] I. K. Lifanov, Singular Integral Equations and Discrete Vortices, VSP, Utrecht, The Netherlands, 1996. [30] I. K. Lifanov, L. N. Poltavskii and G. M. Vainikko, Hypersingular Integral Equations and Their Applications, Chapman Hall/CRC, CRC Press, Boca Raton, London, 2004. [31] Y. Mahmoudi, Modified homotopy perturbation method for solving a class of hyper-singular Integral equations of second kind, Journal of Statistics and Mathematics Studies, 1 (2015), 8-18. [32] K. Maleknejad, S. Sohrabi and Y. Rostami, Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, Applied Mathematics and Computation, 188 (2007), 123-128. doi: 10.1016/j.amc.2006.09.099. [33] B. N. Mandal and Subhra Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials, Appl. Math. Comput., 190 (2007), 1707-1716. doi: 10.1016/j.amc.2007.02.058. [34] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, A CRC Press Company, 2000. [35] N. M. A. Nik Long and Z. K. Eshkuvatov, Hypersingular integral equation for multiple curved cracks problem in plane elasticity, International J of Solid Structure, 46 (2009), 2611-2617. [36] A. H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981. [37] A. H. Nayfeh, Problems in Perturbation, John Wiley & Sons, New York, 1985. [38] N. F. Parsons and P. A. Martin, Scattering of water waves by submerged curved plates and by surface-piercing flat plates, Appl. Ocean Res., 16 (1994), 129-139. doi: 10.1016/0141-1187(94)90024-8. [39] J. I. Ramos, Piecewise homotopy methods for nonlinear ordinary differential equations, Appl. Math. Comput., 198 (2008), 92-116. doi: 10.1016/j.amc.2007.08.030. [40] S. Shahmorad and S. Ahdiaghdam, Approximate solution of a system of singular integral equations of the first kind by using Chebyshev polynomials, arXiv: 1508.01873v1. [41] M. Shaban, S. Kazem and J. A. Rad, A modification of the homotopy analysis method based on Chebyshev operational matrices, Mathematical and Computer Modelling, 57 (2013), 1227-1239. doi: 10.1016/j.mcm.2012.09.024. [42] R. Vahidi and M. Isfahani, On the homotopy perturbation method and the Adomian decomposition method for solving Abel integral equations of the second kind, Applied Mathematical Sciences, 5 (2011), 799-804. [43] E. Yusufo'glu, A homotopy perturbation algorithm to solve a system of Fredholm-Volterra type integral equations, Mathematical and Computer Modelling, 47 (2008), 1099-1107. doi: 10.1016/j.mcm.2007.06.022. [44] M. Odibat Zaid, A new modification of the homotopy perturbation method for linear and nonlinear operators, Applied Mathematics and Computation, 189 (2007), 746-753. doi: 10.1016/j.amc.2006.11.188. [45] F. S. Zulkarnain, Z. K. Eshkuvatov, N. M. A. Nik Long and F. Ismail, Modified homotopy perturbation method for solving hypersingular integral equations of the second kind, AIP Conference Proceedings, 1739 (2016). doi: 10.1063/1.4952507.

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##### References:
 [1] M. Abdulkawi, N. M. A. Nik Long and Z. K. Eshkuvatov, Numerical solution of hypersingular integral equations, International Journal of Pure and Applied Mathematics, 69 (2011), 265-274. [2] M. M. Abdulkawi, N. M. A. Nik Long and Z. K. Eshkuvatov, Numerical solution of hypersingular integral equations, Int. J. Pure Appl. Math., 69 (2011), 265-274. [3] T. Allahviranloo and M. Ghanbari, Discrete homotopy analysis method for the nonlinear Fredholm integral equations, Ain Shams Engineering Journal, 2 (2011), 133-140. doi: 10.1016/j.asej.2011.06.002. [4] J. Biazar and H. Aminikhah, Study of convergence of homotopy perturbation method for systems of partial differential equations, Computers and Mathematics with Applications, 58 (2009), 2221-2230. doi: 10.1016/j.camwa.2009.03.030. [5] S. H. Behiry, R. A. Abd-Elmonem and A. M. Gomaa, Discrete adomian decomposition solution of nonlinear Fredholm integral equation, Ain Shams Engineering Journal, 1 (2010), 97-101. doi: 10.1016/j.asej.2010.09.009. [6] Y. Sha Chan, A. C. Fannjiang and G. H. Paulino, Integral equations with hypersingular kernels, theory and applications to fracture mechanics, International Journal of Engineering Science, 41 (2003), 683-720. doi: 10.1016/S0020-7225(02)00134-9. [7] J. D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell Publishing Company, Waltham, Massachusetts, 1968. [8] G. Davydov, E. V. Zakharov and Yu. V. Pimenov, Some computational aspects of the hypersingular integral equation method in electrodynamics, Computational Mathematics and Modeling, 15 (2004), 105-109. doi: 10.1023/A:1022072215887. [9] V. M. Dyke, Perturbation Methods in Fluid Mechanics, The Parabolic Press, Stanford, California, 1975. [10] M. Dehghan and F. Shakeri, Solution of an integro-differential equation arising in oscillating magnetic fields using He's homotopy perturbation method, Prog. Electromagn. Res., 78 (2008), 361-376. doi: 10.2528/PIER07090403. [11] A. A. Daschioglu, A Chebyshev polynomial approach for linear Fredholm-Volterra integro-differential equations in the most general form, Applied Mathematics and Computation, 181 (2006), 103-112. doi: 10.1016/j.amc.2006.01.018. [12] A. A. Daschioglu and M. Sezer, Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations, Journal of the Franklin Institute, 342 (2005), 688-701. doi: 10.1016/j.jfranklin.2005.04.001. [13] Z. K. Eshkuvatov, F. S. Zulkarnain, N. M. A. Nik Long and Z. Muminov, Modified homotopy perturbation method for solving hypersingular integral equations of the first kind, Springer Plus, 5 (2016), 1-21. doi: 10.1186/s40064-016-3070-z. [14] Z. K. Eshkuvatov, N. M. A. Nik Long and M. Abdulkawi, Approximate solution of singular integral equations of the first kind with Cauchy kernel, Journal of Applied Mathematics Letter., 22 (2009), 651-657. doi: 10.1016/j.aml.2008.08.001. [15] A. Golbabai and M. Javidi, Application of He's homotopy perturbation method for nth-order integro-differential equations, Appl. Math. Comput., 190 (2007), 1409-1416. doi: 10.1016/j.amc.2007.02.018. [16] M. Ghasemi, M. T. Kajani and A. Davari, Numerical solution of the nonlinear Volterra-Fredholm integral equations by using homotopy perturbation method, Appl. Math. Comput., 188 (2007), 446-449. doi: 10.1016/j.amc.2006.10.015. [17] A. Golbabai and B. Keramati, Solution of nonlinear Fredholm integral equations of the first kind using modified homotopy perturbation method, Chaos, Solitons, and Fractals, 39 (2009), 2316-2321. doi: 10.1016/j.chaos.2007.06.120. [18] A. Ghorbani and J. Saberi-Nadjafi, Exact solutions for nonlinear integral equations by a modified homotopy perturbation method, Comput. Math. Appl., 28 (2006), 1032-1039. doi: 10.1016/j.camwa.2008.01.030. [19] J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178 (1999), 257-262. doi: 10.1016/S0045-7825(99)00018-3. [20] J. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Inter. J. Non-linear Mech., 35 (2000), 37-43. doi: 10.1016/S0020-7462(98)00085-7. [21] J. H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput., 135 (2003), 73-79. doi: 10.1016/S0096-3003(01)00312-5. [22] J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation, 151 (2004), 287-292. doi: 10.1016/S0096-3003(03)00341-2. [23] J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26 (2005), 695-700. doi: 10.1016/j.chaos.2005.03.006. [24] J. H. He, New interpretation of homotopy perturbation method, Internat. J. Modern Phys. B, 20 (2006), 2561-2568. doi: 10.1142/S0217979206034819. [25] W. Al-Hayani, Solving nth-order integro-differential equations using the combined Laplace transform-adomian decomposition method, Applied Mathematics, 4 (2013), 882-886. [26] H. Jafari, M. Alipour and H. Tajadodi, Convergence of homotopy perturbation method for solving integral equations, Thai J. Math., 8 (2010), 511-520. [27] M. Javidi and A. Golbabai, Modified homotopy perturbation method for solving non-linear Fredholm integral equations, Chaos Solitons Fractals, 40 (2009), 1408-1412. doi: 10.1016/j.chaos.2007.09.026. [28] M. Kanoria and B. N. Mandal, Water wave scattering by a submerged circular-arc-shaped plate, Fluid Dynamics, 31 (2002), 317-331. doi: 10.1016/S0169-5983(02)00136-3. [29] I. K. Lifanov, Singular Integral Equations and Discrete Vortices, VSP, Utrecht, The Netherlands, 1996. [30] I. K. Lifanov, L. N. Poltavskii and G. M. Vainikko, Hypersingular Integral Equations and Their Applications, Chapman Hall/CRC, CRC Press, Boca Raton, London, 2004. [31] Y. Mahmoudi, Modified homotopy perturbation method for solving a class of hyper-singular Integral equations of second kind, Journal of Statistics and Mathematics Studies, 1 (2015), 8-18. [32] K. Maleknejad, S. Sohrabi and Y. Rostami, Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, Applied Mathematics and Computation, 188 (2007), 123-128. doi: 10.1016/j.amc.2006.09.099. [33] B. N. Mandal and Subhra Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials, Appl. Math. Comput., 190 (2007), 1707-1716. doi: 10.1016/j.amc.2007.02.058. [34] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, A CRC Press Company, 2000. [35] N. M. A. Nik Long and Z. K. Eshkuvatov, Hypersingular integral equation for multiple curved cracks problem in plane elasticity, International J of Solid Structure, 46 (2009), 2611-2617. [36] A. H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981. [37] A. H. Nayfeh, Problems in Perturbation, John Wiley & Sons, New York, 1985. [38] N. F. Parsons and P. A. Martin, Scattering of water waves by submerged curved plates and by surface-piercing flat plates, Appl. Ocean Res., 16 (1994), 129-139. doi: 10.1016/0141-1187(94)90024-8. [39] J. I. Ramos, Piecewise homotopy methods for nonlinear ordinary differential equations, Appl. Math. Comput., 198 (2008), 92-116. doi: 10.1016/j.amc.2007.08.030. [40] S. Shahmorad and S. Ahdiaghdam, Approximate solution of a system of singular integral equations of the first kind by using Chebyshev polynomials, arXiv: 1508.01873v1. [41] M. Shaban, S. Kazem and J. A. Rad, A modification of the homotopy analysis method based on Chebyshev operational matrices, Mathematical and Computer Modelling, 57 (2013), 1227-1239. doi: 10.1016/j.mcm.2012.09.024. [42] R. Vahidi and M. Isfahani, On the homotopy perturbation method and the Adomian decomposition method for solving Abel integral equations of the second kind, Applied Mathematical Sciences, 5 (2011), 799-804. [43] E. Yusufo'glu, A homotopy perturbation algorithm to solve a system of Fredholm-Volterra type integral equations, Mathematical and Computer Modelling, 47 (2008), 1099-1107. doi: 10.1016/j.mcm.2007.06.022. [44] M. Odibat Zaid, A new modification of the homotopy perturbation method for linear and nonlinear operators, Applied Mathematics and Computation, 189 (2007), 746-753. doi: 10.1016/j.amc.2006.11.188. [45] F. S. Zulkarnain, Z. K. Eshkuvatov, N. M. A. Nik Long and F. Ismail, Modified homotopy perturbation method for solving hypersingular integral equations of the second kind, AIP Conference Proceedings, 1739 (2016). doi: 10.1063/1.4952507.
Comparisons with other methods
 $x$ Error term in [45] Error of HPM Error of MHPM in [28] -1 0 0 0 -0.5 $13*10^{-17}$ 0 0 0. 0 0 0 0.5 $7.8*10^{-18}$ 0 0 1 0 0 0
 $x$ Error term in [45] Error of HPM Error of MHPM in [28] -1 0 0 0 -0.5 $13*10^{-17}$ 0 0 0. 0 0 0 0.5 $7.8*10^{-18}$ 0 0 1 0 0 0
Comparisons with other methods
 $x$ Error term in [45] Error of MHPM [28] -1.0 0.0 0 -0.8 $2.1\cdot 10^{-10}$ 0 -0.4 $3.4\cdot 10^{-9}$ 0 0.0 $2.0\cdot 10^{-9}$ 0 0.4 $3.3\cdot 10^{-9}$ 0 0.8 $2.7\cdot 10^{-9}$ 0 1.0 0.0 0
 $x$ Error term in [45] Error of MHPM [28] -1.0 0.0 0 -0.8 $2.1\cdot 10^{-10}$ 0 -0.4 $3.4\cdot 10^{-9}$ 0 0.0 $2.0\cdot 10^{-9}$ 0 0.4 $3.3\cdot 10^{-9}$ 0 0.8 $2.7\cdot 10^{-9}$ 0 1.0 0.0 0
Error terms for different value of $n$
 $x$ Exact Solution Error MHPM for m=n=6 Error MHPM for m=n=10 -0.9999 0.14140368029 1.0851729 10?4 1.4040446 10?10 -0.901 3.94739842327 3.5501594 10?4 1.8203460 10?9 -0.436 5.75413468725 3.0648319 10?4 2.6221447 10?8 -0.015 5.03721659280 8.6784464 10?5 1.7385004 10?8 0.015 4.96222081226 1.9992451 10?5 1.8524990 10?8 0.436 3.69436233615 3.3916634 10?1 1.9700974 10?8 0.901 1.49541222584 1.2745747 10?4 1.3403888 10?8 0.9999 0.04714084491 3.3327100 10?5 3.5400390 10?10
 $x$ Exact Solution Error MHPM for m=n=6 Error MHPM for m=n=10 -0.9999 0.14140368029 1.0851729 10?4 1.4040446 10?10 -0.901 3.94739842327 3.5501594 10?4 1.8203460 10?9 -0.436 5.75413468725 3.0648319 10?4 2.6221447 10?8 -0.015 5.03721659280 8.6784464 10?5 1.7385004 10?8 0.015 4.96222081226 1.9992451 10?5 1.8524990 10?8 0.436 3.69436233615 3.3916634 10?1 1.9700974 10?8 0.901 1.49541222584 1.2745747 10?4 1.3403888 10?8 0.9999 0.04714084491 3.3327100 10?5 3.5400390 10?10
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