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Multistep schemes for one and two dimensional electromagnetic wave models based on fractional derivative approximation. (English) Zbl 1441.78029
Summary: In this article, multistep finite difference schemes are developed for solving one dimensional (1D) and two dimensional (2D) fractional differential models of electromagnetic waves (FDMEWs) arising from dielectric media which contain both initial and Dirichlet boundary conditions. The Caputo’s fractional derivatives in time are discretized by a difference scheme of order \(\mathcal{O} ( \tau^{3 - \alpha} )\) & \(\mathcal{O} ( \tau^{3 - \beta} )\), \(1 < \beta < \alpha < 2\), and the Laplacian operator is approximated by central difference discretization. The proposed multistep schemes transform the FDMEWs into the tridiagonal system for 1D case and pentagonal system for 2D case. Theoretical unconditional stability, convergence analysis and error bounds are investigated. For 1D FDMEWs, accuracy of order \(\mathcal{O} ( \tau^{3 - \alpha} + \tau^{3 - \beta} + h^2 )\) and for 2D FDMEWs, accuracy of order \(\mathcal{O} ( \tau^{3 - \alpha} + \tau^{3 - \beta} + h_1^2 + h_2^2 )\) are investigated, where \(1 < \beta < \alpha < 2\). Several test examples are included to verify the reliability and computational efficiency of the proposed schemes which support our theoretical findings for both 1D and 2D cases.
MSC:
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
35Q60 PDEs in connection with optics and electromagnetic theory
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