Document Type : Research Paper

**Authors**

Department of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, Iran

**Abstract**

In this paper, a pseudospectral method is proposed for solving the nonlinear time-fractional Klein-Gordon and sine-Gordon equations. The method is based on the Sinc operational matrices. A finite difference scheme is used to discretize the Caputo time-fractional derivative, while the spatial derivatives are approximated by the Sinc method. The convergence of the full discretization of the problem is studied. Some numerical examples are presented to confirm the accuracy and efficiency of the proposed method. The numerical results are compared with the analytical solution and the reported results in the literature.

**Keywords**

- [1] L. Adibmanesha, and J. Rashidinia, Sinc and B-Spline scaling functions for time-fractional convection-diffusion equations, J. King Saud Univ. Sci., 33 (2021), 101343.
- [2] M. R. Azizi and A. Khani, Sinc operational matrix method for solving the Bagley-Torvik equation, Comput. Methods Differ. Equ., 5 (2017), 56-66.
- [3] P. J. Caudrey, I. C. Eilbeck, and J. D. Gibbon, The Sine-Gordon as a model classical field theory, Il Nuovo Cimento B, 25 (1975) 497-512.
- [4] M. S. H. Chowdhury and I. Hashim, Application of homotopy-perturbation method to Klein-Gordon and sine- Gordon equations, Chaos Solit. Fractals., 39 (2009), 1928-1935.
- [5] M. Dehghan, M. Abbaszadeh, and A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations, Eng. Anal. Bound. Elem., 50 (2015), 412-434.
- [6] R. K. Dodd, H. C. Morris, J. Eilbeck, and J. Gibbon, Soliton and nonlinear wave equations, London and New York: Academic Press, 1982.
- [7] W. Greiner, Relativistic quantum mechanics, springer, Berlin, 2000.
- [8] G. Hariharan, Haar wavelet method for solving the Klein-Gordon and the Sine-Gordon equations, Int. J. Nonlinear Sci., 11 (2011), 180-189.
- [9] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
- [10] M. M. Khader and M. H. Adel, Numerical solutions of fractional wave equations using an efficient class of FDM based on the Hermite formula, Adv. Differ. Equ., 2016 (2016), 1-10.
- [11] N. Laskin and G. Zaslavsky, Nonlinear fractional dynamics on a lattice with long range interaction, Physica A, 368 (2006), 38-54.
- [12] M. Lotfi, and A. Alipanah, Legendre spectral element method for solving sine-Gordon equation, Adv. Differ. Equ., 2019 (2019), 1-15.
- [13] J. Lund and K. L. Bowers, Sinc methods for quadrature and differential equations, Society for Industrial and Applied Mathematics, 1992.
- [14] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9 (1996), 23-8.
- [15] K. Maleknejad, J. Rashidinia, and T. Eftekhari, Existence, uniqueness, and numerical solutions for two- dimensional nonlinear fractional Volterra and Fredholm integral equations in a Banach space, Comput. Appl. Math., 39 (2020), 1-22.
- [16] K. Maleknejad, J. Rashidinia, and T. Eftekhari, Operational matrices based on hybrid functions for solving general nonlinear two-dimensional fractional integro-differential equations, Comput. Appl. Math., 39 (2020), 1-34.
- [17] A. Mohebbi and M. Dehghan, High-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods, Math. Comput. Model., 51 (2010), 537-549.
- [18] N. Moshtaghi and A. Saadatmandi, Polynomial-Sinc collocation method combined with the Legendre-Gauss quad- rature rule for numerical solution of distributed order fractional differential equations, Rev. Real Acad. Cienc. Exactas Fis. Nat. Serie A: Mat., 115 (2021), 1-23.
- [19] N. Moshtaghi and A. Saadatmandi, Numerical solution of time fractional cable equation via the Sinc-bernoulli collocation method, J. Appl. Comput. Mech., 7 (2021), 1916-1924.
- [20] M. Nabati, S. Taherifar, and M. Jalalvand, Sinc-Galerkin approach for thermal analysis of moving porous fin subject to nanoliquid flow with different shaped nanoparticles, Math. Sci., (2021), 1-16.
- [21] M. Nabati, M. Jalalvand, and S. Taherifar, Sinc collocation approach through thermal analysis of porous fin with magnetic field, J. Therm. Anal. Calorim., 144 (2021), 2145-2158.
- [22] A. M. Nagy, Numerical solution of time fractional nonlinear Klein-Gordon equation using Sinc-Chebyshev collo- cation method, Appl. Math. Comput., 310 (2017), 139-148.
- [23] O. Nikan, Z. Avazzadeh, and J. T. Machado, Numerical investigation of fractional nonlinear sine-Gordon and Klein-Gordon models arising in relativistic quantum mechanics, Eng. Anal. Bound. Elem., 120 (2020), 223-237.
- [24] N. Noghrei, A. Kerayechian, and A. R. Soheili, Gaussian radial basis function and quadrature Sinc method for two-dimensional space-fractional diffusion equations, Math. Sci., (2021), 1-10.
- [25] K. Parand, M. Dehghan, and A. Pirkhedri, The Sinc-collocation method for solving the Thomas-Fermi equation, J. Comput. Appl. Math., 237 (2013), 244-252 .
- [26] W. Qiu, D. Xu, and J. Guo, Numerical solution of the fourth-order partial integro-differential equation with multi-term kernels by the Sinc-collocation method based on the double exponential transformation, Appl. Math. Comput., 392 (2021), 125693.
- [27] A. Saadatmandi, A. Khani, and M. R. Azizi, Numerical calculation of fractional derivatives for the sinc functions via Legendre polynomials, Interdiscip. Math. Sci., 5 (2020), 71-86.
- [28] A. Saadatmandi, M. Dehghan, and M. R. Azizi, The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4125- 4136.
- [29] R. Sassaman and A. Biswas, 1-soliton solution of the perturbed Klein-Gordon equation, Phys. Express., 1 (2011), 9-14.
- [30] A. Secer, S. Alkan, M. A. Akinlar, and M. Bayram, Sinc-Galerkin method for approximate solutions of fractional order boundary value problems, Bound. Value Probl., 2013 (2013), 1-14.
- [31] I. M. Sokolov, J. Klafter, and A. Blumen, Fractional kinetics, Phys. Today, 55 (2002), 48-54.
- [32] F. Stenger, Handbook of Sinc Numerical Methods, CRC Press, New York, NY, USA, 2011.
- [33] F. Stenger, Handbook of Sinc numerical methods. CRC Press, 2016.
- [34] H. G. Sun, W. Chen, C. Li, and Y. Q. Chen, Fractional differential models for anomalous diffusion, Phys. A: Stat. Mech. Appl., 389 (2010), 2719-2724.
- [35] M. Yaseen, M. Abbas, and B. Ahmad, Numerical simulation of the nonlinear generalized time-fractional Klein- Gordon equation using cubic trigonometric B-spline functions, Math. Methods Appl. Sci., 44 (2021), 901-916.
- [36] F. Yin, T. Tian, J. Song, and M. Zhu, Spectral methods using Legendre wavelets for nonlinear Klein Sine-Gordon equations, J. Comput. Appl. Math., 275 (2015), 321-334.

April 2023

Pages 357-368

**Receive Date:**24 January 2022**Revise Date:**14 August 2022**Accept Date:**24 October 2022**First Publish Date:**26 October 2022