IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i8p722-d255928.html

On New Solutions of Time-Fractional Wave Equations Arising in Shallow Water Wave Propagation

Author

Listed:
  • Rajarama Mohan Jena

    (Department of Mathematics, National Institute of Technology, Rourkela 769008, India)

  • Snehashish Chakraverty

    (Department of Mathematics, National Institute of Technology, Rourkela 769008, India)

  • Dumitru Baleanu

    (Department of Mathematics, Faculty of Art and Sciences, Cankaya University, Balgat, 06530 Ankara, Turkey
    Institute of Space Sciences, 077125 Magurele-Bucharest, Romania)

Abstract

The primary objective of this manuscript is to obtain the approximate analytical solution of Camassa–Holm (CH), modified Camassa–Holm (mCH), and Degasperis–Procesi (DP) equations with time-fractional derivatives labeled in the Caputo sense with the help of an iterative approach called fractional reduced differential transform method (FRDTM). The main benefits of using this technique are that linearization is not required for this method and therefore it reduces complex numerical computations significantly compared to the other existing methods such as the perturbation technique, differential transform method (DTM), and Adomian decomposition method (ADM). Small size computations over other techniques are the main advantages of the proposed method. Obtained results are compared with the solutions carried out by other technique which demonstrates that the proposed method is easy to implement and takes small size computation compared to other numerical techniques while dealing with complex physical problems of fractional order arising in science and engineering.

Suggested Citation

  • Rajarama Mohan Jena & Snehashish Chakraverty & Dumitru Baleanu, 2019. "On New Solutions of Time-Fractional Wave Equations Arising in Shallow Water Wave Propagation," Mathematics, MDPI, vol. 7(8), pages 1-13, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:8:p:722-:d:255928
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/8/722/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/8/722/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Odibat, Zaid & Momani, Shaher, 2008. "Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 167-174.
    Full references (including those not matched with items on IDEAS)

    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. F. Z. Geng & X. M. Li, 2012. "A New Method for Riccati Differential Equations Based on Reproducing Kernel and Quasilinearization Methods," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    2. Jim Gatheral & Radoš Radoičić, 2019. "Rational Approximation Of The Rough Heston Solution," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-19, May.
    3. Fathy, Mohamed & Abdelgaber, K.M., 2022. "Approximate solutions for the fractional order quadratic Riccati and Bagley-Torvik differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    4. S. Balaji, 2014. "Legendre Wavelet Operational Matrix Method for Solution of Riccati Differential Equation," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2014, pages 1-10, June.
    5. Shloof, A.M. & Senu, N. & Ahmadian, A. & Salahshour, Soheil, 2021. "An efficient operation matrix method for solving fractal–fractional differential equations with generalized Caputo-type fractional–fractal derivative," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 415-435.
    6. Yanqin Liu & Fengsheng Xu & Xiuling Yin, 2013. "Variational Approximate Solutions of Fractional Nonlinear Nonhomogeneous Equations with Laplace Transform," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    7. Abolvafaei, Mahnaz & Ganjefar, Soheil, 2020. "Maximum power extraction from wind energy system using homotopy singular perturbation and fast terminal sliding mode method," Renewable Energy, Elsevier, vol. 148(C), pages 611-626.
    8. M. Motawi Khashan & Rohul Amin & Muhammed I. Syam, 2019. "A New Algorithm for Fractional Riccati Type Differential Equations by Using Haar Wavelet," Mathematics, MDPI, vol. 7(6), pages 1-12, June.
    9. H. X. Mamatova & Z. K. Eshkuvatov & Sh. Ismail, 2023. "A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind," Mathematics, MDPI, vol. 11(20), pages 1-30, October.
    10. S M, Sivalingam & Kumar, Pushpendra & Govindaraj, V., 2023. "A novel numerical scheme for fractional differential equations using extreme learning machine," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 622(C).
    11. Antony Vijesh, V. & Roy, Rupsha & Chandhini, G., 2015. "A modified quasilinearization method for fractional differential equations and its applications," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 687-697.
    12. Yalçın Öztürk & Ayşe Anapalı & Mustafa Gülsu & Mehmet Sezer, 2013. "A Collocation Method for Solving Fractional Riccati Differential Equation," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    13. Xiaomin Wang, 2014. "A Coiflets‐Based Wavelet Laplace Method for Solving the Riccati Differential Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    14. Cveticanin, L., 2009. "Application of homotopy-perturbation to non-linear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 221-228.
    15. 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.
    16. W. M. Abd-Elhameed & Y. H. Youssri, 2014. "New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    17. 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).
    18. Waleed Mohamed Abd-Elhameed & Badah Mohamed Badah, 2021. "New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas," Mathematics, MDPI, vol. 9(13), pages 1-28, July.
    19. Nur Amirah Zabidi & Zanariah Abdul Majid & Adem Kilicman & Faranak Rabiei, 2020. "Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method," Mathematics, MDPI, vol. 8(10), pages 1-23, October.
    20. Mahmoud Abd El-Hady & Mohamed El-Gamel & Homan Emadifar & Atallah El-shenawy, 2025. "Analysis of RL electric circuits modeled by fractional Riccati IVP via Jacobi-Broyden Newton algorithm," PLOS ONE, Public Library of Science, vol. 20(1), pages 1-28, January.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    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:gam:jmathe:v:7:y:2019:i:8:p:722-:d:255928. 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: MDPI Indexing Manager The email address of this maintainer does not seem to be valid anymore. Please ask MDPI Indexing Manager to update the entry or send us the correct address (email available below). General contact details of provider: https://www.mdpi.com .

    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.