Abstract
In this paper, the one-dimensional nonlinear Sine–Gordon equation is solved using the “Exponential modified cubic B-spline differential quadrature method with the leave-one-out cross-validation (LOOCV) approach”. By employing the LOOCV approach to determine the optimal value of the parameter \(\epsilon \) involved in the basis function, the accuracy and effectiveness of the results are improved. The combination of this approach with the exponential modified cubic B-spline differential quadrature method, which is novel in the literature, is likely to attract researchers' interest. Additionally, the procedure is implemented on six examples of the Sine–Gordon equation. The results are presented in the form of tables and figures. It is demonstrated that this approach is straightforward and yields superior outcomes compared to the existing literature. This paper also presents an insightful discussion on the significant application of the Sine–Gordon equation in Josephson junctions and its crucial role in new technologies.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The authors are unable or have chosen not to specify which data have been used.
References:
Adak D, Natarajan S (2019) Virtual element method for semilinear sine-Gordon equation over polygonal mesh using product approximation technique. Math Comput Simul 172:224–243
Arora G, Bhatia GS (2020) A meshfree numerical technique based on radial basis function pseudospectral method for Fisher’s equation. Int J Nonlinear Sci Numer Simul 21(1):37–49. https://doi.org/10.1515/ijnsns-2018-0091
Arora G, Joshi V, Mittal RC (2019) Numerical simulation of nonlinear Schrodinger equation in one and two dimensions. Math Model Comput Simul 11(4):634–648. https://doi.org/10.1134/S2070048219040070
Arora G, Rani R, Emadifar H (2022a) Numerical solutions of nonlinear Schrodinger equation with applications in optical fiber communication. Optik (stuttg) 266:169661
Arora G, Rani R, Emadifar H (2022b) Soliton: a dispersion-less solution with existence and its types. Heliyon 8(June):e12122. https://doi.org/10.1016/j.heliyon.2022.e12122
Arora G, Joshi V, Mittal RC (2022c) a Spline-Based Differential Quadrature Approach To Solve Sine-Gordon Equation in One and Two Dimension. Fractals 30(7):1–14. https://doi.org/10.1142/S0218348X22501535
Bahan A, Battal S, Karakoç G, Geyikli T (2015) B-spline Differential Quadrature Method for the Modified Burgers’ Equation. Ç Ankaya Univ J Sci Eng 12(1):1–13
Bellman RE, Casti J (1971) Differential quadrature and long -term integration. J Math Anal Appl 34:235–238
Bellman RE, Kashef BG, Casti J (1972) Differential quadrature :a technique for the rapid solution of nonlinear partial differential equation. J Comput Phys 10:40–52
Bert CW, Malik M (1996) Differential quadrature in computational mechanics: a review. Appl Mech Rev 49(1):1–27
Bert CW, Jang SK, Striz AG (1988) Two new approximate methods for analyzing free vibration of structural components. AIAA J 26:612–618
Bykov VG (2014) Sine-Gordon equation and its application to tectonic stress transfer. J Seismol 18(3):497–510. https://doi.org/10.1007/s10950-014-9422-7
Dehghan M, Shokri A (2008) A Numerical method for one-dimensional nonlinear sine-Gordon equation using collocation and radial basis functions. Numer Methods Partial Differ Equ 24(2):687–698. https://doi.org/10.1002/num
Di L, Villari M, Marcucci G, Braidotti MC, Conti C (2018) Sine-Gordon soliton as a model for Hawking radiation of moving black holes and quantum soliton evaporation. J Phys Commun 2(5):055016
Jiwari R (2020) Barycentric rational interpolation and local radial basis functions based numerical algorithms for multidimensional sine-Gordon equation. Numer Methods Partial Differ Equ 37(3):1965–1992. https://doi.org/10.1002/num.22636
Kaya D (2004) An application of the modified decomposition method for two dimensional sine-Gordon equation Do. Appl Math Comput 159(1):1–9. https://doi.org/10.1016/S0096-3003(03)00820-8
Korkmaz A, Dag I (2013) Cubic B-spline differential quadrature method and stability for Burger’s equation. Eng Comput Int J Comput Aided Eng Softw 30(3):320–344
Korkmaz A, Aksoy AM, Dag I (2011) Quartic B-spline differential quadrature method. Int Nonlinear Sci 11(4):403–411
Koupaei JA, Firouznia M, Hosseini SMM (2018) Finding a good shape parameter of RBF to solve PDEs based on the particle swarm optimization algorithm. Alex Eng J 57(4):3641–3652. https://doi.org/10.1016/j.aej.2017.11.024
Lotfi M, Alipanah A (2019) Legendre spectral element method for solving sine-Gordon equation. Adv Differ Equations 2019(1):1–15. https://doi.org/10.1186/s13662-019-2059-7
Mittal RC, Bhatia R (2014) Numerical solution of nonlinear Sine-Gordon equation by modified cubic B-spline collocation method. Int J Partial Differ Equations 2014(1):1–8. https://doi.org/10.1155/2014/343497
Mittal RC, Dahiya S (2017) Numerical simulation of three-dimensional telegraphic equation using cubic B-spline differential quadrature method. Appl Math Lett 313:442–452
Msmali AH, Tamsir M, Ahmadini AAH (2021) Crank-Nicolson-DQM based on cubic exponential B-splines for the approximation of nonlinear Sine-Gordon equation. Ain Shams Eng J 12(4):4091–4097. https://doi.org/10.1016/j.asej.2021.04.004
Povich T, Xin J (2005) A Numerical Study of the Light Bullets Interaction in the (2 + 1) Sine-Gordon Equation. J Nonlinear Sci 15(1):11–25. https://doi.org/10.1007/s00332-003-0588-y
Quan JR, Chang CT (1989) New insights in solving distributed system equations by the quadrature method-I. Comput Chem Eng 13:779–788
Rani R, Arora G, Emadifar H, Khademi M (2023) Numerical simulation of one-dimensional nonlinear Schrodinger equation using PSO with exponential B-spline. Alex Eng J 79(August):644–651. https://doi.org/10.1016/j.aej.2023.08.050
Rashidinia J, Mohammadi R (2011) Tension spline solution of nonlinear sine-Gordon equation. Numer Algorithms 56(1):129–142. https://doi.org/10.1007/s11075-010-9377-x
Rippa S (2014) An algorithm for selecting a good parameter c in radial basis function interpolation. Adv Comput Math 11(November 1999):193–210
Shiralizadeh M, Alipanah A, Mohammadi M (2022) Numerical solution of one-dimensional Sine-Gordon equation using rational radial basis functions. J Math Model 10(3):387–405. https://doi.org/10.22124/jmm.2021.20458.1780
Shu C (2000) Differential quadrature and its application in engineering. London Ltd., Springer-Verlag
Shukla HS, Tamsir M (2018a) An exponential cubic B-spline algorithm for multi-dimensional convection-diffusion equations. Alex Eng J 57(3):1999–2006. https://doi.org/10.1016/j.aej.2017.04.011
Shukla HS, Tamsir M (2018b) Numerical solution of nonlinear Sine-Gordon equation by using the modified cubic B-spline differential quadrature method. Beni-Suef Univ J Basic Appl Sci 7(4):359–366. https://doi.org/10.1016/j.bjbas.2016.12.001
Singh BK, Gupta M (2021) A new efficient fourth order collocation scheme for solving sine—Gordon equation. Int J Appl Comput Math 123(7):138. https://doi.org/10.1007/s40819-021-01089-0
Singh BK, Kumar P (2018) An algorithm based on a new DQM with modified exponential cubic B-splines for solving hyperbolic telegraph equation in (2 + 1) dimension. Nonlinear Eng 7(2):113–125. https://doi.org/10.1515/nleng-2017-0106
Spiteri R, Ruuth S (2002) A new class of optimal high-order strong stability-preserving time-stepping schemes. SIAM J Numer Anal 40(2):469–491
Tamsir M, Srivastava VK, Jiwari R (2016) An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers ’ equation. Appl Math Comput 290:111–124. https://doi.org/10.1016/j.amc.2016.05.048
Tamsir M, Srivastava VK, Dhiman N, Chauhan A (2018) Numerical computation of nonlinear Fisher’s reaction-diffusion equation with exponential modified cubic B-spline differential quadrature method. Int J Appl Comput Math 4(1):1–13. https://doi.org/10.1007/s40819-017-0437-y
Yücel U (2008) Homotopy analysis method for the sine-Gordon equation with initial conditions. Appl Math Comput 203(1):387–395. https://doi.org/10.1016/j.amc.2008.04.042
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Author information
Authors and Affiliations
Contributions
All authors listed have significantly contributed to the development and the writing of this article.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflict of interest.
Additional information
Communicated by Jose Alberto Cuminato.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Rani, R., Arora, G. & Bala, K. Numerical solution of one-dimensional nonlinear Sine–Gordon equation using LOOCV with exponential B-spline. Comp. Appl. Math. 43, 188 (2024). https://doi.org/10.1007/s40314-024-02672-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-024-02672-z