International Journal of Theoretical and Applied Mathematics

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A Comparative Study of Homotopy Perturbation Aboodh Transform Method and Homotopy Decomposition Method for Solving Nonlinear Fractional Partial Differential Equations

Received: 03 October 2016    Accepted: 05 November 2016    Published: 30 November 2016
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Abstract

In this paper, we present the solution of nonlinear fractional partial differential equations by using the Homotopy Perturbation Aboodh Transform Method (HPATM) and Homotopy Decomposition Method (HDM). The Two methods introduced an efficient tool for solving a wide class of linear and nonlinear fractional differential equations. The results shown that the (HDM) has an advantage over the (HPATM) that it takes less time and using only the inverse operator to solve the nonlinear problems and there is no need to use any other inverse transform as in the case of (HPATM).

DOI 10.11648/j.ijtam.20160202.14
Published in International Journal of Theoretical and Applied Mathematics (Volume 2, Issue 2, December 2016)
Page(s) 45-51
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Homotopy Decomposition Method, Integral Transforms, Nonlinear Fractional Differential Equation, Aboodh Transform

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, New York, 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] A. A. Kilbas, M. Saigo, R. K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms and Special Functions, vol.15, 2004, pp. 31–49.
[10] A. Yildirim, Analytical approach to Fokker- Planck equation with space- and time-fractional derivatives by means of the homotopy perturbation method. Journal of King Saud University- Science, vol.22, No.4, 2010, pp. 257-264.
[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] G. Adomian, Solving frontier problems of physics: The decomposition method, Kluwer Academic Publishers, Boston and London, 1994.
[13] 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 Fractional Calculus, vol. 3, no. 2, (2012). pp. 73–99.
[14] A. Atangana and Aydin Secer. “Time-fractional Coupled- the Korteweg-de Vries Equations” Abstract Applied Analysis, In press (2013).
[15] D. D. Ganji, “The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer,” PhysicsLetters A, vol. 355, no. 4-5, (2006). pp. 337–341.
[16] J. Singh, D. Kumar, and Sushila, “Homotopy perturbation Sumudu transform method for nonlinear equations,” Advancesin Applied Mathematics and Mechanics, vol. 4, (2011), pp. pp 165–175.
[17] D. D. Ganji and M. Rafei, “Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method,” Physics Letters A, vol. 356, no. 2, (2006)., pp. 131–137.
[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 and Abdelbagy A. Alshikh “On The Relationship Between Aboodh Transform and New Integral Transform " ZZ Transform”, Mathematical Theory and Modeling, Vol.6, No.9, 2016.
[21] Mohand M. Abdelrahim Mahgoub “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.
[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] 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.
[25] 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.
[26] 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.
Author Information
  • 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

  • Mathematics Department Faculty of Education- Holy Quran and Islamic Sciences University, Khartoum, Sudan; Mathematics Department Faculty of Sciences and Arts-Almikwah-Albaha University, Albaha, Saudi Arabia

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    Mohand M. Abdelrahim Mahgoub, Abdelilah K. Hassan Sedeeg. (2016). A Comparative Study of Homotopy Perturbation Aboodh Transform Method and Homotopy Decomposition Method for Solving Nonlinear Fractional Partial Differential Equations. International Journal of Theoretical and Applied Mathematics, 2(2), 45-51. https://doi.org/10.11648/j.ijtam.20160202.14

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    Mohand M. Abdelrahim Mahgoub; Abdelilah K. Hassan Sedeeg. A Comparative Study of Homotopy Perturbation Aboodh Transform Method and Homotopy Decomposition Method for Solving Nonlinear Fractional Partial Differential Equations. Int. J. Theor. Appl. Math. 2016, 2(2), 45-51. doi: 10.11648/j.ijtam.20160202.14

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    AMA Style

    Mohand M. Abdelrahim Mahgoub, Abdelilah K. Hassan Sedeeg. A Comparative Study of Homotopy Perturbation Aboodh Transform Method and Homotopy Decomposition Method for Solving Nonlinear Fractional Partial Differential Equations. Int J Theor Appl Math. 2016;2(2):45-51. doi: 10.11648/j.ijtam.20160202.14

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  • @article{10.11648/j.ijtam.20160202.14,
      author = {Mohand M. Abdelrahim Mahgoub and Abdelilah K. Hassan Sedeeg},
      title = {A Comparative Study of Homotopy Perturbation Aboodh Transform Method and Homotopy Decomposition Method for Solving Nonlinear Fractional Partial Differential Equations},
      journal = {International Journal of Theoretical and Applied Mathematics},
      volume = {2},
      number = {2},
      pages = {45-51},
      doi = {10.11648/j.ijtam.20160202.14},
      url = {https://doi.org/10.11648/j.ijtam.20160202.14},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ijtam.20160202.14},
      abstract = {In this paper, we present the solution of nonlinear fractional partial differential equations by using the Homotopy Perturbation Aboodh Transform Method (HPATM) and Homotopy Decomposition Method (HDM). The Two methods introduced an efficient tool for solving a wide class of linear and nonlinear fractional differential equations. The results shown that the (HDM) has an advantage over the (HPATM) that it takes less time and using only the inverse operator to solve the nonlinear problems and there is no need to use any other inverse transform as in the case of (HPATM).},
     year = {2016}
    }
    

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    T1  - A Comparative Study of Homotopy Perturbation Aboodh Transform Method and Homotopy Decomposition Method for Solving Nonlinear Fractional Partial Differential Equations
    AU  - Mohand M. Abdelrahim Mahgoub
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    JO  - International Journal of Theoretical and Applied Mathematics
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    EP  - 51
    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.ijtam.20160202.14
    AB  - In this paper, we present the solution of nonlinear fractional partial differential equations by using the Homotopy Perturbation Aboodh Transform Method (HPATM) and Homotopy Decomposition Method (HDM). The Two methods introduced an efficient tool for solving a wide class of linear and nonlinear fractional differential equations. The results shown that the (HDM) has an advantage over the (HPATM) that it takes less time and using only the inverse operator to solve the nonlinear problems and there is no need to use any other inverse transform as in the case of (HPATM).
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