IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v79y2008i3p700-715.html
   My bibliography  Save this article

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

Author

Listed:
  • Dehghan, Mehdi
  • Shokri, Ali

Abstract

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashion as finite difference methods. Numerical results are obtained for various cases involving line and ring solitons.

Suggested Citation

  • Dehghan, Mehdi & Shokri, Ali, 2008. "A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 700-715.
    • Handle: RePEc:eee:matcom:v:79:y:2008:i:3:p:700-715
    DOI: 10.1016/j.matcom.2008.04.018
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475408001778
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2008.04.018?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Dehghan, Mehdi, 2006. "Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 71(1), pages 16-30.
    2. Sheng, Q. & Khaliq, A.Q. M. & Voss, D.A., 2005. "Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(4), pages 355-373.
    3. Bratsos, A.G., 2007. "A third order numerical scheme for the two-dimensional sine-Gordon equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 76(4), pages 271-282.
    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. Li, Meng & Fei, Mingfa & Wang, Nan & Huang, Chengming, 2020. "A dissipation-preserving finite element method for nonlinear fractional wave equations on irregular convex domains," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 404-419.
    2. Saffarian, Marziyeh & Mohebbi, Akbar, 2021. "Numerical solution of two and three dimensional time fractional damped nonlinear Klein–Gordon equation using ADI spectral element method," Applied Mathematics and Computation, Elsevier, vol. 405(C).
    3. Saberi Zafarghandi, Fahimeh & Mohammadi, Maryam & Babolian, Esmail & Javadi, Shahnam, 2019. "Radial basis functions method for solving the fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 224-246.
    4. Deng, Dingwen & Wu, Qiang, 2023. "Accuracy improvement of a Predictor–Corrector compact difference scheme for the system of two-dimensional coupled nonlinear wave equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 223-249.
    5. Gupta, A.K. & Saha Ray, S., 2017. "On the solitary wave solution of fractional Kudryashov–Sinelshchikov equation describing nonlinear wave processes in a liquid containing gas bubbles," Applied Mathematics and Computation, Elsevier, vol. 298(C), pages 1-12.
    6. Adak, D. & Natarajan, S., 2020. "Virtual element method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 224-243.
    7. Jiang, Chaolong & Sun, Jianqiang & Li, Haochen & Wang, Yifan, 2017. "A fourth-order AVF method for the numerical integration of sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 144-158.
    8. Jiong Weng & Xiaojing Liu & Youhe Zhou & Jizeng Wang, 2021. "A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations," Mathematics, MDPI, vol. 9(22), pages 1-17, November.
    9. Sun, Zhengjie, 2022. "A conservative scheme for two-dimensional Schrödinger equation based on multiquadric trigonometric quasi-interpolation approach," Applied Mathematics and Computation, Elsevier, vol. 423(C).
    10. Martin-Vergara, Francisca & Rus, Francisco & Villatoro, Francisco R., 2019. "Padé numerical schemes for the sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 232-243.

    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. Jiang, Chaolong & Sun, Jianqiang & Li, Haochen & Wang, Yifan, 2017. "A fourth-order AVF method for the numerical integration of sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 144-158.
    2. Ballestra, Luca Vincenzo & Pacelli, Graziella & Radi, Davide, 2016. "A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 240-248.
    3. Adak, D. & Natarajan, S., 2020. "Virtual element method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 224-243.
    4. Ghosh, Suchismita & Deb, Anish & Sarkar, Gautam, 2016. "Taylor series approach for function approximation using ‘estimated’ higher derivatives," Applied Mathematics and Computation, Elsevier, vol. 284(C), pages 89-101.
    5. Hu, Dongdong & Cai, Wenjun & Xu, Zhuangzhi & Bo, Yonghui & Wang, Yushun, 2021. "Dissipation-preserving Fourier pseudo-spectral method for the space fractional nonlinear sine–Gordon equation with damping," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 35-59.
    6. Guo, Geyang & Lü, Shujuan & Liu, Bo, 2015. "Unconditional stability of alternating difference schemes with variable time steplengthes for dispersive equation," Applied Mathematics and Computation, Elsevier, vol. 262(C), pages 249-259.
    7. Mohammadi, Reza, 2018. "Smooth Quintic spline approximation for nonlinear Schrödinger equations with variable coefficients in one and two dimensions," Chaos, Solitons & Fractals, Elsevier, vol. 107(C), pages 204-215.
    8. Wang, Hanquan & Ma, Xiu & Lu, Junliang & Gao, Wen, 2017. "An efficient time-splitting compact finite difference method for Gross–Pitaevskii equation," Applied Mathematics and Computation, Elsevier, vol. 297(C), pages 131-144.
    9. Dehghan, Mehdi & Mohebbi, Akbar, 2008. "High-order compact boundary value method for the solution of unsteady convection–diffusion problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 683-699.
    10. Dehghan, Mehdi & Gharibi, Zeinab, 2021. "Numerical analysis of fully discrete energy stable weak Galerkin finite element Scheme for a coupled Cahn-Hilliard-Navier-Stokes phase-field model," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    11. Nehad Ali Shah & Ioannis Dassios & Essam R. El-Zahar & Jae Dong Chung & Somaye Taherifar, 2021. "The Variational Iteration Transform Method for Solving the Time-Fractional Fornberg–Whitham Equation and Comparison with Decomposition Transform Method," Mathematics, MDPI, vol. 9(2), pages 1-14, January.
    12. Martin-Vergara, Francisca & Rus, Francisco & Villatoro, Francisco R., 2019. "Padé numerical schemes for the sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 232-243.
    13. Pathak, Maheshwar & Joshi, Pratibha & Nisar, Kottakkaran Sooppy, 2022. "Numerical study of generalized 2-D nonlinear Schrödinger equation using Kansa method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 186-198.
    14. Almushaira, Mustafa, 2023. "Efficient energy-preserving eighth-order compact finite difference schemes for the sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    15. Luo, Yidong, 2020. "Galerkin method with trigonometric basis on stable numerical differentiation," Applied Mathematics and Computation, Elsevier, vol. 370(C).
    16. Dehghan, Mehdi & Saadatmandi, Abbas, 2009. "Variational iteration method for solving the wave equation subject to an integral conservation condition," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1448-1453.
    17. Kenzu Abdella & Jeet Trivedi, 2020. "Solving Multi-Point Boundary Value Problems Using Sinc-Derivative Interpolation," Mathematics, MDPI, vol. 8(12), pages 1-14, November.
    18. Shi, Dongyang & Liao, Xin & Wang, Lele, 2016. "Superconvergence analysis of conforming finite element method for nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 298-310.
    19. Gao, Yali & Mei, Liquan & Li, Rui, 2018. "Galerkin methods for the Davey–Stewartson equations," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 144-161.
    20. Wang, Huili & Shi, Baochang & Liang, Hong & Chai, Zhenhua, 2017. "Finite-difference lattice Boltzmann model for nonlinear convection-diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 334-349.

    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:eee:matcom:v:79:y:2008:i:3:p:700-715. 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: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    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.