IDEAS home Printed from https://ideas.repec.org/a/wly/jnlaaa/v2014y2014i1n803902.html

Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations

Author

Listed:
  • Asma Ali Elbeleze
  • Adem Kılıçman
  • Bachok M. Taib

Abstract

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method.

Suggested Citation

  • Asma Ali Elbeleze & Adem Kılıçman & Bachok M. Taib, 2014. "Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:803902
    DOI: 10.1155/2014/803902
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2014/803902
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2014/803902?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. A. M. A. El-Sayed & A. Elsaid & D. Hammad, 2012. "A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein‐Gordon Equation of Arbitrary (Fractional) Orders," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    2. A. M. A. El-Sayed & A. Elsaid & D. Hammad, 2012. "A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-13, February.
    3. Momani, Shaher & Odibat, Zaid, 2007. "Numerical comparison of methods for solving linear differential equations of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1248-1255.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Hassan, Waqar Ul & Shabbir, Khurram & Zeeshan, Ahmed & Ellahi, Rahmat, 2025. "Regression analysis for thermal transport of fractional-order magnetohydrodynamic Maxwell fluid flow under the influence of chemical reaction using integrated machine learning approach," Chaos, Solitons & Fractals, Elsevier, vol. 191(C).
    2. Asma Ali Elbeleze & Adem Kılıçman & Bachok M. Taib, 2014. "Convergence of Variational Iteration Method for Solving Singular Partial Differential Equations of Fractional Order," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Mohammed Ali & Marwan Alquran & Mahmoud Mohammad, 2012. "Solitonic Solutions for Homogeneous KdV Systems by Homotopy Analysis Method," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    2. Cuicui Liao & Xiaohua Ding, 2012. "Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    3. Fukang Yin & Junqiang Song & Yongwen Wu & Lilun Zhang, 2013. "Numerical Solution of the Fractional Partial Differential Equations by the Two‐Dimensional Fractional‐Order Legendre Functions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    4. Abdon Atangana & Adem Kılıçman, 2013. "Analytical Solutions of Boundary Values Problem of 2D and 3D Poisson and Biharmonic Equations by Homotopy Decomposition Method," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    5. He, Ji-Huan, 2009. "Nonlinear science as a fluctuating research frontier," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2533-2537.
    6. Xu, Lan, 2009. "The variational iteration method for fourth order boundary value problems," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1386-1394.
    7. Roman Parovik, 2020. "Mathematical Modeling of Linear Fractional Oscillators," Mathematics, MDPI, vol. 8(11), pages 1-26, October.
    8. Deng, Hongmin & Li, Tao & Wang, Qionghua & Li, Hongbin, 2009. "A fractional-order hyperchaotic system and its synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 962-969.
    9. Veyis Turut & Nuran Güzel, 2013. "Multivariate Padé Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    10. Asma Ali Elbeleze & Adem Kılıçman & Bachok M. Taib, 2014. "Convergence of Variational Iteration Method for Solving Singular Partial Differential Equations of Fractional Order," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    11. Odibat, Zaid M., 2009. "Exact solitary solutions for variants of the KdV equations with fractional time derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1264-1270.
    12. Mohsen Alipour & Dumitru Baleanu, 2013. "Approximate Analytical Solution for Nonlinear System of Fractional Differential Equations by BPs Operational Matrices," Advances in Mathematical Physics, John Wiley & Sons, vol. 2013(1).
    13. Marwan Abukhaled, 2013. "Variational Iteration Method for Nonlinear Singular Two-Point Boundary Value Problems Arising in Human Physiology," Journal of Mathematics, Hindawi, vol. 2013, pages 1-4, February.
    14. Mossa Al-sawalha, M. & Noorani, M.S.M., 2009. "A numeric–analytic method for approximating the chaotic Chen system," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1784-1791.
    15. Soliman, A.A., 2009. "On the solution of two-dimensional coupled Burgers’ equations by variational iteration method," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1146-1155.
    16. Tuan Hoang, Manh & Nagy, A.M., 2019. "Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 24-34.
    17. Ferial Ghaemi & Robiah Yunus & Ali Ahmadian & Soheil Salahshour & Mohamed Suleiman & Shanti Faridah Saleh, 2013. "Application of Fuzzy Fractional Kinetic Equations to Modelling of the Acid Hydrolysis Reaction," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    18. Hulya Kodal Sevindir & Ali Demir, 2018. "The Solution of Initial Boundary Value Problem with Time and Space‐Fractional Diffusion Equation via a Novel Inner Product," Advances in Mathematical Physics, John Wiley & Sons, vol. 2018(1).
    19. Goh, S.M. & Noorani, M.S.M. & Hashim, I., 2009. "Efficacy of variational iteration method for chaotic Genesio system – Classical and multistage approach," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2152-2159.
    20. Tomar, Saurabh & Singh, Mehakpreet & Vajravelu, Kuppalapalle & Ramos, Higinio, 2023. "Simplifying the variational iteration method: A new approach to obtain the Lagrange multiplier," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 640-644.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:803902. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/4058 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.