The combined of Homotopy analysis method with new transform for nonlinear partial differential equations

Djelloul Ziane

doi:10.26637/MJM0601/0005

Abstract

The idea proposed in this work is to extend the Aboodh transform method to resolve the nonlinear partial differential equations by combining them with the so-called homotopy analysis method (HAM). This method can be called homotopy analysis aboodh transform method (HAATM). The results obtained by the application to the proposed examples show that this method is easy to apply and can therefore be used to solve other nonlinear partial differential equations.

Keywords:

Homotopy analysis method, Aboodh transform method, nonlinear partial differential equations

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Djelloul Ziane 1Laboratory of mathematics and its applications (LAMAP), University of Oran1 Ahmed Ben Bella, Oran, 31000, Algeria.

Pages: 34-40
Date Published: 01-01-2018
Vol. 6 No. 01 (2018): Malaya Journal of Matematik (MJM)

V.G. Gupta, S. Gupta, Applications of Homotopy Analysis Transform Method for Solving Various Nonlinear Equations, W. Appl. Sci. J., 18(12), 1839-1846, 2012.

V.G. Gupta, P. Kumar, Approximate Solutions of Fractional Linear and Nonlinear Differential Equations Using Laplace Homotopy Analysis Method, Int. J. of nonlinear Sci., 19(2), 113-12, 2015.

V.G. Gupta, P. Kumar, Approximate Solutions of Fractional Biological Population Model by Homotopy Analysis Sumudu Transform Method, Int. J. of Sci. and Res., 5(5), 908-917,2016.

A.K. Hassan Sedeeg, M.M. Abdelrahim Mehgoub, Aboodh Transform of Homotopy Perturbation Method for Solving System of Nonlinear Partial Differential Equations, Math. Theo. and Mod., 6(8), 108-113, 2016.

E.M.A. Hilal, T.M. Elzaki, Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method, J. of Fun. Spa., Vol. 2014, A. ID 790714, 5 pp, 2014.

M. Hussain, M. Khan, A Variational Iterative Method for Solving the Linear and Nonlinear Klein-Gordon Equations, Appl. Math. Sci., 4(39), 1931-1940, 2010.

S. Khalid, K.S. Aboodh, The New Integral Transform"Aboodh Transform", Glo. J. of pure and appl. Math., 9(1), 35-43, 2013.

M.M. Khader, S. Kumar, S. Abbasbandy, New homotopy analysis transform method for solving the discontinued problems arising in nanotechnology, Chin. Phys. B., 22(11), $110201(1-5), 2013$.

A. Khan, M. Junaid, I. Khan, F. Ali, K. Shah, D. Khan,Application of Homotopy Natural Transform Method to the Solution of Nonlinear partial differential equations,Sci. Int. (Lahore), 29(1), 297-303, 2017.

S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.

S.J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, 2003.

S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147, 499-513, 2004.

S.J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Com. Nonlinear Sci. Num. Sim., 14, 983-997, 2009.

M.S. Mohamed, F. Al-Malki, M. Al-humyani, Homotopy Analysis Transform Method for Time-Space Fractional Gas Dynamics Equation, Gen. Math. Notes, 24(1), 1-16, 2014

M. Matinfar, M. Saeidy, M. Mahdavi, M. Rezaei, Variational Iteration Method for Exact Solution of Gas Dynamic Equation Using He's Polynomials, Bulletin of Math. Anal. and Appl., 3(3), 50-55, 2011.

R.K. Pandey, H.K. Mishra, Numerical Simulation of Timefractional Fourth Order Differential Equations via Homotopy Analysis Fractional Sumudu Transform Method, Ame. J. of Num. Anal., 3(3), 52-64, 2015.

S. Rathorea, D. Kumarb , J. Singh, S. Gupta, Homotopy Analysis Sumudu Transform Method for Nonlinear Equations, Int. J. Ind. Math., 4(4), A., ID IJIM-00204, 13 p, 2012.

K.M. Saad, A.A. AL-Shomrani, An Application of Homotopy Analysis Transform Method for Riccati Differential Equation of Fractional Order, J. of Fra. Cal. and Appl., $7(1), 61-72,2016$

K. Wang, S. Liu, Application of new iterative transform method and modified fractional homotopy analysis transform method for fractional Fornberg-Whitham equation, J. Nonlinear Sci. Appl., 9, 2419-2433, 2016. method and modified fractional homotopy analysis transform method for fractional Fornberg-Whitham equation, J. Nonlinear Sci. Appl., 9, 2419-2433, 2016.

D. Ziane, M. Hamdi Cherif, Modified Homotopy Analysis Method for Nonlinear fractional partial differential equations, Int. J. of Anal. and Appl., 14(1), 77-87, 2017.

D. Ziane, Application of Homotopy Analysis Method Combined with Elzaki Transform for Fractional Porous Medium Equation, J. of Appr. Theo. and Appl. Math., 6, $1-19,2016$.