A Coupling Method of Homotopy Perturbation and Aboodh Transform for Solving Nonlinear Fractional Heat - Like Equations
International Journal of Systems Science and Applied Mathematics
Volume 1, Issue 4, November 2016, Pages: 63-68
Received: Oct. 8, 2016; Accepted: Oct. 18, 2016; Published: Nov. 15, 2016
Views 2535      Downloads 89
Author
Mohand M. Abdelrahim Mahgoub, Department of Mathematics, Faculty of Science & Technology, Omdurman Islamic University, Khartoum, Sudan; Mathematics Department Faculty of Sciences and Arts, Almikwah-Albaha University, Albaha, Saudi Arabia
Article Tools
Follow on us
Abstract
In this paper, we present the solution of nonlinear fractional Heat - Like equations by using Aboodh transform homotopy perturbation method (ATHPM). The proposed method was derived by combining Aboodh transform and homotopy perturbation method. This method is seen as a better alternative method to some existing techniques for such realistic problems. The results showed the efficiency and accuracy of the combined Aboodh transform and homotopy perturbation method.
Keywords
Homotopy Decomposition Method, Nonlinear Fractional Heat - Like Equation, Aboodh Transform
To cite this article
Mohand M. Abdelrahim Mahgoub, A Coupling Method of Homotopy Perturbation and Aboodh Transform for Solving Nonlinear Fractional Heat - Like Equations, International Journal of Systems Science and Applied Mathematics. Vol. 1, No. 4, 2016, pp. 63-68. doi: 10.11648/j.ijssam.20160104.15
Copyright
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
K. B. Oldham and J. Spanier, “The Fractional Calculus”, Academic Press, New York, NY, USA, (1974).
[2]
I. Podlubny. “Fractional Differential Equations”, Academic Press, NewYork, NY, USA, (1999).
[3]
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations”, Elsevier, Amsterdam, The Netherlands, (2006).
[4]
J. F. Cheng, Y. M. Chu, Solution to the linear fractional differential equation using Adomian decomposition method, Mathematical Problems in Engineering, 2011, doi: 10.1155 /2011/587068
[5]
J. H. He, A coupling method of a homotopy technique and a perturbation technique for nonlinear problems, International Journal of Non- Linear Mechanics, vol. 35, 2000, pp. 37-43.
[6]
J. H. He, New interpretation of homotopy perturbation method, International Journal of Modern Physics B, vol. 20, 2006b, pp. 2561- 2668
[7]
] J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, (1999), pp. 257–262.
[8]
J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, vol. 167 (1-2), 1998, pp. 57– 68.
[9]
Esmail Hesameddini, Mohsen Riahi, Habibolla Latifizadeh, A coupling method of Homotopy technique and Laplace transform for nonlinear fractional differential equations, International Journal of Advances in Applied Sciences (IJAAS) Vol. 1, No. 4, December 2012, pp. 159~170.
[10]
Abdon Atangana and Adem Kihcman, The Use of Sumudu Transform for Solving Certain Nonlinear Fractional Heat-Like Equations, Hindawi Publishing Corporation, Abstract and Applied Analysis, Volume 2013
[11]
Rodrigue Batogna Gnitchogna, Abdon Atangana, Comparison of Homotopy Perturbation Sumudu Transform method and Homotopy Decomposition method for solving nonlinear Fractional Partial Differential Equations, Advances in Applied and Pure Mathematics.
[12]
Abdolamir Karbalaie, Mohammad Mehdi Montazer, Hamed Hamid Muhammed, New Approach to Find the Exact Solution of Fractional Partial Differential Equation, WSEAS TRANSACTIONS on MATHEMATICS, Issue 10, Volume 11, October 2012.
[13]
M. Khalid, Mariam Sultana, Faheem Zaidi and Uroosa Arshad, Application of Elzaki Transform Method on Some Fractional Differential Equations, Mathematical Theory and Modeling, Vol. 5, No. 1, 2015.
[14]
Abdon Atangana and Adem Kihcman, The Use of Sumudu Transform for Solving Certain Nonlinear Fractional Heat-Like Equations, Hindawi Publishing Corporation, Abstract and Applied Analysis, Volume 2013.
[15]
Eltayeb A. Yousif, Solution of Nonlinear Fractional Differential Equations Using the Homotopy Perturbation Sumudu Transform Method, Applied Mathematical Sciences, Vol. 8, 2014, no. 44, 2195 - 2210
[16]
G. Adomian, Solving frontier problems of physics: The decomposition method, Kluwer Academic Publishers, Boston and London, 1994.
[17]
J. S. Duan, R. Rach, D. Buleanu, and A. M. Wazwaz, “A review of the Adomian decomposition method and its applications to fractional differential equations,” Communications in FractionalCalculus, vol. 3, no. 2, (2012). pp. 73–99.
[18]
K. S. Aboodh, The New Integral Transform “Aboodh Transform” Global Journal of pure and Applied Mathematics, 9 (1), 35-43 (2013).
[19]
K. S. Aboodh, Application of New Transform “Aboodh transform” to Partial Differential Equations, Global Journal of pure and Applied Math, 10 (2), 249-254 (2014).
[20]
Mohand M. Abdelrahim Mahgob “ Homotopy Perturbation Method And Aboodh Transform For Solving Sine –Gorden And Klein – Gorden Equations” International Journal of Engineering Sciences & Research Technology, 5 (10): October, 2016
[21]
Mohand M. Abdelrahim Mahgob and Abdelilah K. Hassan Sedeeg “The Solution of Porous Medium Equation by Aboodh Homotopy Perturbation Method ” American Journal of Applied Mathematics 2016; 4 (5): 217-221.
[22]
Abdelilah K. Hassan Sedeeg and Mohand M. Abdelrahim Mahgoub, “ Aboodh Transform Homotopy Perturbation Method For Solving System Of Nonlinear Partial Differential Equations,” Mathematical Theory and Modeling Vol. 6, No. 8, 2016,
[23]
Abdelilah K. Hassan Sedeeg and Mohand M. Abdelrahim Mahgoub, “Combine Aboodh Transform And Homotopy Perturbation Method For Solving Linear And Nonlinear Schrodinger Equations,” International Journal of Development Research Vol. 06, Issue, 08, pp. 9085-9089, August, 2016.
[24]
Abdelbagy A. Alshikh and Mohand M. Abdelrahim Mahgoub, “A Comparative Study Between Laplace Transform and Two New Integrals “ELzaki” Transform and “AboodhTransform,” Pure and Applied Mathematics Journal 2016; 5 (5): 145-150.
[25]
T. M. Elzaki and E. M. A. Hilal, Solution of linear and nonlinear partial differential equations using mixture of Elzaki transform and the projected differential transform method, Math. Theo. & Model., 2 (2012), 50-59.
[26]
Sh. Chang, Il Chang, A new algorithm for calculating one-dimensional differential transform of nonlinear functions, Appl. Math. & Compu. 195 (2008), 799-808.
[27]
G. C. Wu, “New trends in the variational iteration method,” Communications in Fractional Calculus, vol. 2, pp. 59–75, 2011.
[28]
G. C. Wu and D. Baleanu, “Variational iteration method for fractional calculus—a universal approach by Laplace transform,” Advances in Difference Equations, vol. 2013, article 18, 2013.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186