This paper presents a numerical scheme for solving the non-linear time fractional Klein-Gordon equation. To approximate spatial derivatives, we employ the pseudo-spectral method based on Lagrange polynomials at Chebyshev points, while using the finite difference method for time discretization. Our analysis demonstrates that this scheme is unconditionally stable, with a time convergence order of $\mathcal{O}({3 \alpha})$. Additionally, we provide numerical results in one, two, and three dimensions, highlighting the high accuracy of our approach. The significance of our proposed method lies in its ability to efficiently and accurately address the non-linear time fractional Klein-Gordon equation. Furthermore, our numerical outcomes validate the effectiveness of this scheme across different dimensions.
[1] D. Baleanu, J. A. T. Machado, and A. C. J. Luo, Fractional dynamics and control, Springer Science & Business Media, 2011.
[2] A. Barone, F. Esposito, C. J. Magee, and A. C. Scott, Theory and applications of the sine-Gordon equation, La Rivista del Nuovo Cimento (1971-1977), 1(2) (1971), 227–267.
[3] B. Batiha, M. S. M. Noorani, and I. Hashim, Numerical solution of sine-Gordon equation by variational iteration method, Phys. Lett. A, 370(5-6) (2007), 437–440.
[4] J. P. Boyd, Chebyshev and Fourier spectral methods, Dover Publications, 2001.
[5] C. Canuto and A. Quarteroni, Spectral methods, Wiley Online Library, 2006.
[6] H. Chen, S. Lu¨, W. Chen, and others, A fully discrete spectral method for the nonlinear time fractional KleinGordon equation, Taiwanese J. Math., 21(1) (2017), 231–251.
[7] K. Devendra, J. Singh, and D. Baleanu, A hybrid computational approach for Klein-Gordon equations on Cantor sets, Nonlinear Dyn., 87(1) (2017), 511–517.
[8] K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, Springer, 2010.
[9] R. A. El-Nabulsi, Generalized Klein-Gordon and Dirac equations from nonlocal kinetic approach, Zeitschrift fu¨r Naturforschung A, 71(9) (2016), 817–821.
[10] S. M. El-Sayed, The decomposition method for studying the Klein–Gordon equation, Chaos Solitons Fractals, 18(5) (2003), 1025–1030.
[11] S. Esmaeili, Solving 2D time-fractional diffusion equations by a pseudospectral method and Mittag-Leffler function evaluation, Math. Methods Appl. Sci., 40(6) (2017), 1838–1850.
[12] H. R. Ghazizadeh, M. Maerefat, and A. Azimi, Explicit and implicit finite difference schemes for fractional Cattaneo equation, J. Comput. Phys., 229(19) (2010), 7042–7057.
[13] A. K. Golmankhaneh, A. K. Golmankhaneh, and D. Baleanu, On nonlinear fractional Klein-Gordon equation, Signal Processing, 91(3) (2011), 446–451.
[14] H. Gómez, I. Colominas, F. Navarrina, and M. Casteleiro, A mathematical model and a numerical model for hyperbolic mass transport in compressible flows, Heat and Mass Transfer, 45(2) (2008), 219–226.
[15] R. M. Hafez, Y.H. Youssri, and A.G. Atta, Jacobi rational operational approach for time-fractional sub-diffusion equation on a semi-infinite domain, Contemp. Math., 4(4) (2023), 853–876.
[16] R. Hilfer, Applications of fractional calculus in physics, World Scientific, 2000.
[17] R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge university press, 1990.
[18] K. Hosseini, P. Mayeli, and R. Ansari, Modified Kudryashov method for solving the conformable time-fractional Klein-Gordon equations with quadratic and cubic nonlinearities, Optik (Stuttg.), 130 (2017), 737–742.
[19] J. Klafter, S. C. Lim, and R. Metzler, Fractional dynamics: recent advances, World Scientific, 2012.
[20] P. Lyu and S. Vong, A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations, Numer. Algorithms, 78 (2018), 485–511.
[21] F. Mainardi, Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, World Scientific, 2010.
[22] A. Mohebbi, M. Abbaszadeh, and M. Dehghan, High-order difference scheme for the solution of linear time fractional klein-gordon equations, Numer. Methods Partial Differential Equations, 30(4) (2014), 1234–1253.
[23] M. Moustafa, Y.H. Youssri, and A.G. Atta, Explicit Chebyshev-Galerkin scheme for the time-fractional diffusion equation, Internat. J. Modern Phys. C, 35(01) (2024), 2450002.
[24] A. M. Nagy, Numerical solution of time fractional nonlinear Klein-Gordon equation using Sinc-Chebyshev collocation method, Appl. Math. Comput., 310 (2017), 139–148.
[25] B. Nemati Saray, M. Lakestani, and C. Cattani, Evaluation of mixed Crank-Nicolson scheme and Tau method for the solution of Klein-Gordon equation, Appl. Math. Comput., 331 (2018), 169–181.
[26] A. Ozkan and E. M. Ozkan, Exact solutions of the space time-fractional Klein-Gordon equation with cubic nonlinearities using some methods, Computational Methods for Differential Equations, 10(3) (2022), 674–685.
[27] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic press, Vol. 198, 1998.
[28] Z. Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56(2) (2006), 193–209.
[29] S. Taherkhani, I. Najafi Khalilsaraye, and B. Ghayebi, A pseudospectral Sinc method for numerical investigation of the nonlinear time-fractional Klein-Gordon and sine-Gordon equations, Computational Methods for Differential Equations, 11(2) (2023), 357–368.
[30] L. N. Trefethen, Spectral methods in MATLAB, SIAM, 2000.
[31] D. L. Turcotte, Fractals and chaos in geology and geophysics, Cambridge university press, 1997.
[32] F. J. Valdes-Parada, J. A. Ochoa-Tapia, and J. Alvarez-Ramirez, Effective medium equation for fractional Cattaneo’s diffusion and heterogeneous reaction in disordered porous media, Phys. A, 369(2) (2006), 318–328.
[33] S. Vong and Z. Wang, A compact difference scheme for a two dimensional fractional Klein-Gordon equation with Neumann boundary conditions, J. Comput. Phys., 274 (2014), 268–282.
[34] S. Vong and Z. Wang, A high-order compact scheme for the nonlinear fractional Klein–Gordon equation, Numer. Methods Partial Differential Equations, 31(3) (2015), 706–722.
[35] A. M. Wazwaz, The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations, Appl. Math. Comput., 167(2) (2005), 1196–1210.
[36] B. D. Welfert, Generation of pseudospectral differentiation matrices I, SIAM J. Numer. Anal., 34(4) (1997), 1640–1657.
[37] F. Yin, J. Song, and F. Lu, A coupled method of Laplace transform and Legendre wavelets for nonlinear KleinGordon equations, Math. Methods Appl. Sci., 37(6) (2014), 781–792.
[38] Y. H. Youssri, Two Fibonacci operational matrix pseudo-spectral schemes for nonlinear fractional Klein-Gordon equation, Internat. J. Modern Phys. C, 33(04) (2022), 2250049.
[39] Y. H. Youssri and A.G. Atta, Modal spectral Tchebyshev Petrov-Galerkin stratagem for the time-fractional nonlinear Burgers’ equation, Iran. j. numer. anal. optim., 14(1) (2024), 172–199.
[40] Y. H. Youssri, M. I. Ismail, and A.G. Atta, Chebyshev Petrov-Galerkin procedure for the time-fractional heat equation with nonlocal conditions, Phys. Scr., 99(1) (2024), 015251.
[41] E. Yusufoğlu, The variational iteration method for studying the Klein-Gordon equation, Appl. Math. Lett., 21(7) (2008), 669–674.
Mirzaei, S. and Shokri, A. (2025). Numerical study of the non-linear time fractional Klein-Gordon equation using the Pseudo-spectral method. Computational Methods for Differential Equations, 13(2), 479-493. doi: 10.22034/cmde.2024.58979.2500
MLA
Mirzaei, S. , and Shokri, A. . "Numerical study of the non-linear time fractional Klein-Gordon equation using the Pseudo-spectral method", Computational Methods for Differential Equations, 13, 2, 2025, 479-493. doi: 10.22034/cmde.2024.58979.2500
HARVARD
Mirzaei, S., Shokri, A. (2025). 'Numerical study of the non-linear time fractional Klein-Gordon equation using the Pseudo-spectral method', Computational Methods for Differential Equations, 13(2), pp. 479-493. doi: 10.22034/cmde.2024.58979.2500
CHICAGO
S. Mirzaei and A. Shokri, "Numerical study of the non-linear time fractional Klein-Gordon equation using the Pseudo-spectral method," Computational Methods for Differential Equations, 13 2 (2025): 479-493, doi: 10.22034/cmde.2024.58979.2500
VANCOUVER
Mirzaei, S., Shokri, A. Numerical study of the non-linear time fractional Klein-Gordon equation using the Pseudo-spectral method. Computational Methods for Differential Equations, 2025; 13(2): 479-493. doi: 10.22034/cmde.2024.58979.2500
)