Document Type : Research Paper

Author

• Vijay Kumar Patel

School of Advanced Sciences and Languages, VIT Bhopal University, Bhopal, India

Abstract

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.

Keywords

• Fractional PDE
• Finite difference scheme
• Reimann-Liouville fractional derivative
• Convergence analysis