### A Comparison between Solving of Two-Dimensional Nonlinear Fredholm Integral Equations of the Second Kind by the Optimal Homotopy Asymptotic Method and Homotopy Perturbation Method

#### Abstract

In this article, we present the optimal homotopy asymptotic method (OHAM) and homotopy perturbation method (HPM) for solving 2-dimensional nonlinear Fredholm integral equations of the second kind. A comparison is made between these methods to solve 2-dimensional nonlinear Fredholm integral equations of the second kind. The results show that the presented methods are very powerful and simple techniques in solving 2-dimensional nonlinear Fredholm integral equations of the second kind.

doi:10.14456/WJST.2015.35

#### Keywords

#### Full Text:

PDF#### References

M Abdou, L El-Kalla and A Al-Bugami. New approach for convergence of the series solution to a class of Hammerstein integral equations. Int. J. Appl. Math. Comput. 2011; 3, 261-9.

Z Avazzadeh and M Heydari. Chebyshev polynomials for solving two dimensional linear and nonlinear integral equations of the second kind. Comput. Appl. Math. 2012; 31, 127-42.

E Babolian, S Bazm and P Lima. Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions. Comm. Nonlinear Sci. Numer. Simulat. 2011; 16, 1164-75.

H Guoqiang and W Jiong. Extrapolation of nystrom solution for two-dimensional nonlinear Fredholm integral equations. J. Comput. Appl. Math. 2001; 134, 259-68.

M Heydari, Z Avazzadeh, H Navabpour and G Loghmani. Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem II. High dimensional problems. Appl. Math. Model. 2013; 37, 432-42.

W Xie and F Lin. A fast numerical solution method for two dimensional Fredholm integral equations of the second kind. Appl. Math. Comput. 2009; 59, 1709-19.

J He. Homotopy perturbation technique. Comput. Meth. Appl. Mech. Eng. 1999; 178, 257-62.

J He. A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int. J. Nonlinear Mech. 2000; 35, 37-43.

J He. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl. Math. Comput. 2004; 151, 287-92.

J He. Comparison of homotopy perturbation method and homotopy analysis method. Appl. Math. Comput. 2004; 156, 527-39.

J He. Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 2006; 20, 1141-99.

J He. Homotopy perturbation method for solving boundary value problems. Phys. Lett. A 2006; 50, 87-8.

M El-Shahed. Application of He’s Homotopy perturbation method to Volterra’s integro-differential equation. Int. J. Nonlinear Sci. Numer. Simulat. 2005; 6,163-8.

AA Hemeda. Homotopy perturbation method for solving systems of nonlinear coupled equations. Appl. Math. Sci. 2012; 6, 4787-800.

S Aminsadrabad. Solution for inverse space-dependent heat source problems by homotopy perturbation method. Appl. Math. Sci. 2012; 6, 575 - 8.

N Herişanu, V Marinca, T Dordea and G Madescu. A new analytical approach to nonlinear vibration of an electrical machine. Proc. Rom. Acad. Ser. A 2008; 9, 229-36.

V Marinca and N Herisanu. Application of optimal homotopy asymptotic method for solving non-linear equations arising in heat transfer. Int. Comm. Heat Mass Tran. 2008; 35, 710-5.

V Marinca, N Herisanu and I Nemes. Optimal homotopy asymptotic method with application to thin film flow. Cent. Eur. J. Phys. 2008; 6, 648-53.

V Marinca, N Herisanu, C Bota and B Marinca. An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate. Appl. Math. Lett. 2009; 22, 245-51.

MS Hashmi, N Khan and S Iqbal. Numerical solutions of weakly singular Volterra integral equations using the optimal homotopy asymptotic method. Comput. Math. Appl. 2012; 64, 1567-74.

NR Anakira, AK Alomari and I Hashim. Optimal homotopy asymptotic method for solving delay differential equations. Math. Prob. Eng. 2013; 2013, 498902.

M Almousa and A Ismail. Optimal homotopy asymptotic method for solving the linear Fredholm integral equations of the first kind. Abstr. Appl. Anal. 2013; 2013, 278097.

F Mabood, WA Khan and A Ismail. Analytical solution for radiation effects on heat transfer in Blasius flow. Int. J. Mod. Eng. Sci. 2013; 2, 63-72.

F Mabood, WA Khan and A Ismail. Series solution for steady heat transfer in a heat-generating fin with convection and radiation. Math. Prob. Eng. 2013; 2013, 806873.

### Refbacks

- There are currently no refbacks.

**Online ISSN: 2228-835X****http://wjst.wu.ac.th **

**Last updated:**13 February 2019