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An efficient finite difference scheme for fractional partial differential equation arising in electromagnetic waves model | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 16، دوره 14، شماره 10، دی 2023، صفحه 163-178 اصل مقاله (612.24 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2023.28651.3962 | ||
| نویسنده | ||
| Vijay Kumar Patel* | ||
| School of Advanced Sciences and Languages, VIT Bhopal University, Bhopal, India | ||
| تاریخ دریافت: 20 مهر 1401، تاریخ بازنگری: 03 بهمن 1401، تاریخ پذیرش: 16 بهمن 1401 | ||
| چکیده | ||
| We present an unconditionally stable finite difference scheme (FDS) for the fractional partial differential equation (PDE) arising in the electromagnetic waves, which contains both initial and Dirichlet boundary conditions. The Riemann-Liouville fractional derivatives in time are discretized by a finite difference scheme of order $\mathcal{O}\left( \Delta t^{3-\alpha}\right)$ and $\mathcal{O}\left( \Delta t^{3-\beta}\right)$, $1<\beta < \alpha < 2$ and the Laplacian operator is discretized by central difference approximation. The proposed stable FDS schemes transform the fractional PDE into a tridiagonal system. Theoretically, uniqueness, unconditionally stability, error bound, and convergence of FDS are investigated. Moreover, the accuracy of the order of convergence $\mathcal{O}\left( \Delta t^{3-\alpha}+ \Delta t^{3-\beta}+\Delta x^2 \right)$ of the scheme is investigated. Finally, numerical results are reported to illustrate our optimal error bound, order of convergence, and efficiency of proposed schemes. | ||
| کلیدواژهها | ||
| Fractional PDE؛ Finite difference scheme؛ Reimann-Liouville fractional derivative؛ Convergence analysis | ||
| مراجع | ||
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[1] B. Beumer, M. Kovcs and M.M. Meerschaert, Numerical solutions for fractional reaction diffusion equations, Comput. Math. Appl. 55 (2008), 2212–2226. [2] L. Chen, R.H. Nochetto, E. Otsarola and A.J. Salgado, Multilevel methods for nonuniformly elliptic operators and fractional diffusion, Math. Comput. 85 (2016), 2583–2607. [3] W. Deng, Finite element method for the space and time fractional Fokker–Planck equation, SIAM J. Numer. Anal. 47 (2008), 204–226. [4] N. Engheta, Fractional curl operator in electromagnetics, Microw. Opt. Technol. Lett. 17 (1998), no. 2, 86–91. [5] J.F. Kelly, R.J. McGough and M.M. Meerschaert, Analytical time-domain Green’s functions for power-law media, J. Acous. Soc. Amer. 124 (2008), 2861–2872. [6] S. Kumar, S. Kumar and S. Sumit, A posteriori error estimation for quasilinear singularly perturbed problems with integral boundary condition, Numer. Algor. 89 (2022), 791-–809. [7] S. Kumar, S. Kumar and Sumit, High-order convergent methods for singularly perturbed quasilinear problems with integral boundary conditions, Math. Method Appl. Sci, (2020), https://doi.org/10.1002/mma.6854. [8] K. Kumar, R.K. Pandey, S. Sharma and Y. Xu, Numerical scheme with convergence for a generalized time-fractional Telegraph-type equation, Numer. Meth. Partial Differ. Equ. 35 (2019), no. 3, 1164–1183. [9] K. Kumar, R.K. Pandey and F. Sultana, Numerical schemes with convergence for generalized fractional integrodifferential equations, J. Comput. Appl. Math. 388 (2021), no. 1, 113318. [10] S. Kumar, S. Sumit and J.V. Aguiar, A high order convergent numerical method for singularly perturbed time-dependent problems using mesh equidistribution, Math. Comput. Simul. 199 (2022), 287–306. [11] S. Kumar, S. Sumit and J.V. Aguiar, A parameter-uniform grid equidistribution method for singularly perturbed degenerate parabolic convection–diffusion problems, J. Comput Appl. Math. 404 (2022), 113273. [12] X. Li and C. Xu, The existence and uniqueness of the weak solution of the space–time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys. 8 (2010), 1016–1051. [13] C. Li and F. Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, 2015. [14] F.R. Lin, S.W. Yang and X.Q. Jin, Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys. 256 (2014), 109–117. [15] F. Liu, V. Anh and I. Turner, Numerical solution of the space fractional Fokker–Planck equation, J. Comput. Appl. Math. 166 (2004), 209–219. | ||
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