IDEAS home Printed from https://ideas.repec.org/a/gam/jeners/v14y2021i23p7831-d685428.html
   My bibliography  Save this article

Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schr ö dinger Equation with Energy and Mass Conversion

Author

Listed:
  • Xuan Liu

    (Department of Mathematics, Hanshan Normal University, Chaozhou 515041, China)

  • Muhammad Ahsan

    (Department of Mathematics, University of Swabi, Swabi 23200, Pakistan)

  • Masood Ahmad

    (Department of Basic Sciences, University of Engineering and Technology Peshawar, Peshawar 25000, Pakistan)

  • Muhammad Nisar

    (Department of Mathematics and Statistics, Macquarie University, Sydney, NSW 2109, Australia
    Department of Mathematics, FATA University, Darra Adam Khel 26100, Pakistan)

  • Xiaoling Liu

    (Department of Mathematics, Hanshan Normal University, Chaozhou 515041, China)

  • Imtiaz Ahmad

    (Department of Mathematics, University of Swabi, Swabi 23200, Pakistan)

  • Hijaz Ahmad

    (Department of Computer Engineering, Biruni University, Istanbul 34025, Turkey
    Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy)

Abstract

This article is concerned with the numerical solution of nonlinear hyperbolic Schr o ¨ dinger equations (NHSEs) via an efficient Haar wavelet collocation method (HWCM). The time derivative is approximated in the governing equations by the central difference scheme, while the space derivatives are replaced by finite Haar series, which transform it to full algebraic form. The experimental rate of convergence follows the theoretical statements of convergence and the conservation laws of energy and mass are also presented, which strengthens the proposed method to be convergent and conservative. The Haar wavelets based on numerical results for solitary wave shape of | φ | are discussed in detail. The proposed approach provides a fast convergent approximation to the NHSEs. The reliability and efficiency of the method are illustrated by computing the maximum error norm and the experimental rate of convergence for different problems. Comparisons are performed with various existing methods in recent literature and better performance of the proposed method is shown in various tables and figures.

Suggested Citation

  • Xuan Liu & Muhammad Ahsan & Masood Ahmad & Muhammad Nisar & Xiaoling Liu & Imtiaz Ahmad & Hijaz Ahmad, 2021. "Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schr ö dinger Equation with Energy and Mass Conversion," Energies, MDPI, vol. 14(23), pages 1-17, November.
    • Handle: RePEc:gam:jeners:v:14:y:2021:i:23:p:7831-:d:685428
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/1996-1073/14/23/7831/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/1996-1073/14/23/7831/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ahsan, Muhammad & Ahmad, Imtiaz & Ahmad, Masood & Hussian, Iltaf, 2019. "A numerical Haar wavelet-finite difference hybrid method for linear and non-linear Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 13-25.
    2. Hsiao, Chun-Hui & Wang, Wen-June, 2001. "Haar wavelet approach to nonlinear stiff systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 57(6), pages 347-353.
    3. Saeed, Umer & ur Rehman, Mujeeb, 2015. "Haar wavelet Picard method for fractional nonlinear partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 264(C), pages 310-322.
    4. Nazir, Shah & Shahzad, Sara & Wirza, Rahmita & Amin, Rohul & Ahsan, Muhammad & Mukhtar, Neelam & García-Magariño, Iván & Lloret, Jaime, 2019. "Birthmark based identification of software piracy using Haar wavelet," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 144-154.
    5. Pervaiz, Nosheen & Aziz, Imran, 2020. "Haar wavelet approximation for the solution of cubic nonlinear Schrodinger equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    6. Hsiao, C.H., 2004. "Haar wavelet approach to linear stiff systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 64(5), pages 561-567.
    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. Ahsan, Muhammad & Bohner, Martin & Ullah, Aizaz & Khan, Amir Ali & Ahmad, Sheraz, 2023. "A Haar wavelet multi-resolution collocation method for singularly perturbed differential equations with integral boundary conditions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 166-180.
    2. Ahsan, Muhammad & Lei, Weidong & Bohner, Martin & Khan, Amir Ali, 2024. "A high-order multi-resolution wavelet method for nonlinear systems of differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 215(C), pages 543-559.
    3. Xiaofei Qin & Youhua Fan & Hongjun Li & Weidong Lei, 2022. "A Direct Method for Solving Singular Integrals in Three-Dimensional Time-Domain Boundary Element Method for Elastodynamics," Mathematics, MDPI, vol. 10(2), pages 1-17, January.

    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. Ahsan, Muhammad & Lei, Weidong & Bohner, Martin & Khan, Amir Ali, 2024. "A high-order multi-resolution wavelet method for nonlinear systems of differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 215(C), pages 543-559.
    2. Pervaiz, Nosheen & Aziz, Imran, 2020. "Haar wavelet approximation for the solution of cubic nonlinear Schrodinger equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    3. Mart Ratas & Jüri Majak & Andrus Salupere, 2021. "Solving Nonlinear Boundary Value Problems Using the Higher Order Haar Wavelet Method," Mathematics, MDPI, vol. 9(21), pages 1-12, November.
    4. Ahsan, Muhammad & Bohner, Martin & Ullah, Aizaz & Khan, Amir Ali & Ahmad, Sheraz, 2023. "A Haar wavelet multi-resolution collocation method for singularly perturbed differential equations with integral boundary conditions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 166-180.
    5. Siraj-ul-Islam, & Haider, Nadeem & Aziz, Imran, 2018. "Meshless and multi-resolution collocation techniques for parabolic interface models," Applied Mathematics and Computation, Elsevier, vol. 335(C), pages 313-332.
    6. Tian, Yongge & Herzberg, Agnes M., 2006. "A-minimax and D-minimax robust optimal designs for approximately linear Haar-wavelet models," Computational Statistics & Data Analysis, Elsevier, vol. 50(10), pages 2942-2951, June.
    7. Singh, Randhir & Guleria, Vandana & Singh, Mehakpreet, 2020. "Haar wavelet quasilinearization method for numerical solution of Emden–Fowler type equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 174(C), pages 123-133.
    8. H. Çerdik Yaslan, 2021. "Numerical solution of the nonlinear conformable space–time fractional partial differential equations," Indian Journal of Pure and Applied Mathematics, Springer, vol. 52(2), pages 407-419, June.
    9. Igor Sinitsyn & Vladimir Sinitsyn & Eduard Korepanov & Tatyana Konashenkova, 2022. "Bayes Synthesis of Linear Nonstationary Stochastic Systems by Wavelet Canonical Expansions," Mathematics, MDPI, vol. 10(9), pages 1-14, May.
    10. Usman, Muhammad & Hamid, Muhammad & Liu, Moubin, 2021. "Novel operational matrices-based finite difference/spectral algorithm for a class of time-fractional Burger equation in multidimensions," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    11. Xie, Jiaquan & Wang, Tao & Ren, Zhongkai & Zhang, Jun & Quan, Long, 2019. "Haar wavelet method for approximating the solution of a coupled system of fractional-order integral–differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 163(C), pages 80-89.
    12. Karabulut, Gokhan & Bilgin, Mehmet Huseyin & Doker, Asli Cansin, 2020. "The relationship between commodity prices and world trade uncertainty," Economic Analysis and Policy, Elsevier, vol. 66(C), pages 276-281.
    13. Dehestani, H. & Ordokhani, Y. & Razzaghi, M., 2018. "Fractional-order Legendre–Laguerre functions and their applications in fractional partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 433-453.
    14. Rawani, Mukesh Kumar & Verma, Amit Kumar & Verma, Lajja, 2024. "Numerical treatment of Burgers' equation based on weakly L-stable generalized time integration formula with the NSFD scheme," Applied Mathematics and Computation, Elsevier, vol. 467(C).
    15. Hsiao, Chun-Hui, 2015. "A Haar wavelets method of solving differential equations characterizing the dynamics of a current collection system for an electric locomotive," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 928-935.
    16. Lepik, Ü., 2005. "Numerical solution of differential equations using Haar wavelets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(2), pages 127-143.
    17. Ahsan, Muhammad & Ahmad, Imtiaz & Ahmad, Masood & Hussian, Iltaf, 2019. "A numerical Haar wavelet-finite difference hybrid method for linear and non-linear Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 13-25.
    18. Sowa, Marcin, 2018. "Application of SubIval in solving initial value problems with fractional derivatives," Applied Mathematics and Computation, Elsevier, vol. 319(C), pages 86-103.
    19. Amin, Rohul & Shah, Kamal & Asif, Muhammad & Khan, Imran, 2021. "A computational algorithm for the numerical solution of fractional order delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 402(C).
    20. Saeed, Umer, 2017. "CAS Picard method for fractional nonlinear differential equation," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 102-112.

    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:jeners:v:14:y:2021:i:23:p:7831-:d:685428. 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 (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.