Homotopy Analysis Aboodh Transform Method for Nonlinear System of Partial Differential Equations

In this paper, a combined form of homotopy analysis method with Aboodh transform method is proposed to solve nonlinear system of partial differential equations. This method is called the homotopy analysis Aboodh transform method (HAATM). The homotopy analysis Aboodh transform method can easily be applied to many problems of nonlinear system, and is capable of reducing the size of computational work.

Keywords:

Aboodh transform method Homotopy analysis method, Nonlinear system of partial differential equationS,

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[1] N. Taghizadeh, M. Akbaria, M. Shahidia, Homotopy perturbation method and reduced differential transform method for solving (1+1)-dimensional nonlinear boussinesq equation, Int. J. Appl. Math. Comput., 5(2) (2013), 28-33.
[2] S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
[3] S. J. Liao, Beyond perturbation: Introduction to Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, 2003.
[4] S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004), 499-513.
[5] S. J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 983-997.
[6] M. Ayub, A. Rasheed and T. Hayat, Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Int. J. Eng. Sci., 41 (2003), 2091-2103.
[7] S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A, 360 (2006), 109-113.
[8] Z. Abbasa, S. Vahdatia, F. Ismaila, A. K. Dizicheha, Application of homotopy analysis method for linear integro-differential equations, Int. Math. Forum., 5(5) (2010) 237-249.
[9] V. G. Gupta, S. Gupta, Applications of homotopy analysis transform method for solving various nonlinear equations, W. Appl. Sci. J., 18(12), (2012), 1839-1846.
[10] 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), (2013), 1-5.
[11] V. G. Gupta, P. Kumar, Approximate solutions of fractional linear and nonlinear differential equations using Laplace homotopy analysis method, Int. J. Nonlinear Sci., 19(2) (2015), 113-12.
[12] V. G. Gupta, P. Kumar,Approximate solutions of fractional biological population model by homotopy analysis Sumudu transform method, Int. J. Sci. Res., 5(5) (2016), 908-917.
[13] R. K. Pandey, H. K. Mishra, Numerical simulation of time-fractional fourth order differential equations via homotopy analysis fractional Sumudu transform method, Amer. J. Num. Anal., 3(3) (2015), 52-64.
[14] S. Rathorea, D. Kumarb , J. Singh, S. Gupta, Homotopy analysis Sumudu transform method for nonlinear equations, Int. J. Ind. Math., 4(4) (2012), Article ID IJIM-00204, 13 pages.
[15] 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) (2017), 297-303.
[16] D. Ziane, M. Hamdi Cherif, Modified Homotopy analysis method for nonlinear fractional partial differential equations, Int. J. Anal. Appl., 14(1) (2017), 77-87.
[17] D. Ziane, The combined of homotopy analysis method with new transform for nonlinear partial differential equations, Malaya J. Math., 6(1) (2018), 34-40.
[18] S. Khalid, K. S. Aboodh, The new integral transform ”Aboodh transform”, Glob. J. Pure. Appl. Math., 9(1) (2013), 35-43.
[19] A. K. Hassan Sedeeg, M. M. Abdelrahim Mehgoub, Aboodh transform of homotopy perturbation method for solving system of nonlinear partial differential equations, Math. Theo. Mod., 6(8) (2016), 108-113.
[20] M. Khana, M. Asif Gondala, S. Karimi Vananib, On the coupling of homotopy perturbation and Laplace transformation for system of partial differential equations, Appl. Math. Sci., 6(10) (2012), 467-478.
[21] H. Eltayeb, A. Kılıc¸man, Application of Sumudu decomposition method to solve nonlinear system of partial differential equations, Abs. Appl. Anal., Article ID 412948 (2012), 13 pages.