Multistep schemes for one and two dimensional electromagnetic wave models based on fractional derivative approximation

Abstract

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 O(τ3−α)&O(τ3−β), 1<β<α<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 O(τ3−α+τ3−β+h2) and for 2D FDMEWs, accuracy of order O(τ3−α+τ3−β+h12+h22) are investigated, where 1<β<α<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.