A Fast H2N2 Finite Difference Scheme for the Fractional Sine-Gordon Equation

doi:10.1007/s42967-024-00456-0

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (2): 490-506.doi: 10.1007/s42967-024-00456-0

A Fast H2N2 Finite Difference Scheme for the Fractional Sine-Gordon Equation

Baoting Wang¹, Guoyu Zhang²

1. Department of Mathematics, Shanghai University, Shanghai, 200444, China;
2. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, Inner Mongolia, China

收稿日期:2024-05-31 修回日期:2024-07-18 出版日期:2026-04-07 发布日期:2026-04-07
通讯作者: Guoyu Zhang,E-mail:guoyu_zhang@imu.edu.cn E-mail:guoyu_zhang@imu.edu.cn
基金资助:
This work is supported by the Natural Science Foundation of Inner Mongolia Autonomous Region of China No. 2021BS01003.

A Fast H2N2 Finite Difference Scheme for the Fractional Sine-Gordon Equation

Baoting Wang¹, Guoyu Zhang²

1. Department of Mathematics, Shanghai University, Shanghai, 200444, China;
2. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, Inner Mongolia, China

Received:2024-05-31 Revised:2024-07-18 Online:2026-04-07 Published:2026-04-07
Contact: Guoyu Zhang,E-mail:guoyu_zhang@imu.edu.cn E-mail:guoyu_zhang@imu.edu.cn
Supported by:
This work is supported by the Natural Science Foundation of Inner Mongolia Autonomous Region of China No. 2021BS01003.

摘要/Abstract

摘要： In this paper, based on the H2N2 method (a method for approximating the Caputo fractional derivative of order \begin{document}$ \alpha \in (1\text {, } 2) $\end{document} derived by the quadratic Hermite and Newton interpolation polynomials), a direct finite difference scheme with second-order accuracy in space and \begin{document}$ (3-\alpha ) $\end{document}th order accuracy in time is constructed for the fractional sine-Gordon equation. The stability and convergence of the difference scheme are analyzed theoretically. Further, to improve the computational efficiency, a fast algorithm based on the sum-of-exponentials method is adopted to construct a fast difference scheme. In addition, the initial singularity of the solution is also discussed. Numerical outcomes verify the validity of the direct and fast difference schemes and the correctness of theoretical results.

关键词: Fractional sine-Gordon equation, H2N2 method, Finite difference scheme, Fast algorithm

Abstract: In this paper, based on the H2N2 method (a method for approximating the Caputo fractional derivative of order \begin{document}$ \alpha \in (1\text {, } 2) $\end{document} derived by the quadratic Hermite and Newton interpolation polynomials), a direct finite difference scheme with second-order accuracy in space and \begin{document}$ (3-\alpha ) $\end{document}th order accuracy in time is constructed for the fractional sine-Gordon equation. The stability and convergence of the difference scheme are analyzed theoretically. Further, to improve the computational efficiency, a fast algorithm based on the sum-of-exponentials method is adopted to construct a fast difference scheme. In addition, the initial singularity of the solution is also discussed. Numerical outcomes verify the validity of the direct and fast difference schemes and the correctness of theoretical results.

Key words: Fractional sine-Gordon equation, H2N2 method, Finite difference scheme, Fast algorithm

中图分类号:

Baoting Wang, Guoyu Zhang. A Fast H2N2 Finite Difference Scheme for the Fractional Sine-Gordon Equation[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 490-506.

参考文献

[1] Dehghan, M., Abbaszadeh, M., Mohebbi, A.: An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations. Eng. Anal. Bound. Elem. 50, 412–434 (2015)
[2] Deng, D., Chen, J., Wang, Q.: Energy-preserving Du Fort-Frankel difference schemes for solving sine-Gordon equation and coupled sine-Gordon equations. Numer. Algorithms 93(3), 1045–1081 (2023)
[3] Deng, D., Wang, Q.: A class of weighted energy-preserving Du Fort-Frankel difference schemes for solving sine-Gordon-type equations. Commun. Nonlinear Sci. Numer. Simul. 117, 106916 (2023)
[4] Faridi, W.A., Asjad, M.I., Jhangeer, A.: The fractional analysis of fusion and fission process in plasma physics. Phys. Scr. 96(10), 104008 (2021)
[5] Gul, Z., Ali, A., Ahmad, I.: Dynamics of AC-driven sine-Gordon equation for long Josephson junctions with fast varying perturbation. Chaos Solitons Fractals 107, 103–110 (2018)
[6] Hasanpour, M., Behroozifar, M., Tafakhori, N.: Numerical solution of fractional sine-Gordon equation using spectral method and homogenization. Int. J. Nonlinear Sci. Numer. Simul. 20(7/8), 811–822 (2019)
[7] Hu, W., Deng, Z.: Interaction effects of DNA, RNA-polymerase, and cellular fluid on the local dynamic behaviors of DNA. Appl. Math. Mech. 41, 623–636 (2020)
[8] Jiang, S., Zhang, J., Zhang, Q., Zhang, Z.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21(3), 650–678 (2017)
[9] Kang, X., Feng, W., Cheng, K., Guo, C.: An efficient finite difference scheme for the 2D sine-Gordon equation (2017). arXiv:1706.08632
[10] Li, C., Zeng, F.: Numerical Methods for Fractional Calculus, vol. 24. CRC Press, Boca Raton (2015)
[11] Liu, Z., Lyu, S., Liu, F.: Fully discrete spectral methods for solving time fractional nonlinear sine-Gordon equation with smooth and non-smooth solutions. Appl. Math. Comput. 333, 213–224 (2018)
[12] Ma, T., Zheng, Q., Fu, Y.: Optimal error estimation of two fast structure-preserving algorithms for the Riesz fractional sine-Gordon equation. Commun. Nonlinear Sci. Numer. Simul. 119, 107067 (2023)
[13] Macías-Díaz, J.: Nonlinear wave transmission in harmonically driven Hamiltonian sine-Gordon regimes with memory effects. Chaos Solitons Fractals 142, 110362 (2021)
[14] Martin-Vergara, F., Rus, F., Villatoro, F.R.: Padé schemes with Richardson extrapolation for the sine-Gordon equation. Commun. Nonlinear Sci. Numer. Simul. 85, 105243 (2020)
[15] Ray, S.S.: A new analytical modelling for nonlocal generalized Riesz fractional sine-Gordon equation. J. King Saud Univ. Sci. 28(1), 48–54 (2016)
[16] Sadiya, U., Inc, M., Arefin, M.A., Uddin, M.H.: Consistent travelling waves solutions to the non-linear time fractional Klein-Gordon and sine-Gordon equations through extended tanh-function approach. J. Taibah Univ. Sci. 16(1), 594–607 (2022)
[17] Shen, J., Li, C., Sun, Z.Z.: An H2N2 interpolation for Caputo derivative with order in (1, 2) and its application to time-fractional wave equations in more than one space dimension. J. Sci. Comput. 83(2), 38 (2020)
[18] Sher, M., Khan, A., Shah, K., Abdeljawad, T.: Fractional-order sine-Gordon equation involving nonsingular derivative. Fractals 31(10), 2340007 (2023)
[19] Sun, Z.Z.: Numerical Solution of Partial Differential Equations, 2nd edn. Science Press, Beijing (2003)
[20] Taherkhani, S., Najafi Khalilsaraye, I., Ghayebi, B.: A pseudospectral Sinc method for numerical investigation of the nonlinear time-fractional Klein-Gordon and sine-Gordon equations. Comput. Methods Differ. Equ. 11(2), 357–368 (2023)
[21] Yan, Y., Sun, Z.Z., Zhang, J.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme. Commun. Comput. Phys. 22(4), 1028–1048 (2017)
[22] Zhou, Y., Luo, Z.: A Crank-Nicolson finite difference scheme for the Riesz space fractional-order parabolic-type sine-Gordon equation. Adv. Differ. Equ. 2018(1), 1–7 (2018)
[23] Zhou, Y., Luo, Z.: An optimized Crank-Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation. Adv. Differ. Equ. 2019(1), 1–15 (2019)

A Fast H2N2 Finite Difference Scheme for the Fractional Sine-Gordon Equation

A Fast H2N2 Finite Difference Scheme for the Fractional Sine-Gordon Equation

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