pseudo-spectral method

This paper presents a numerical scheme for solving the non-linear time fractional Klein-Gordon equation. To approximate spatial derivatives, we employ the pseudo-spectral method based on Lagrange polynomials at Chebyshev points, while using the finite difference method for time discretization. Our analysis demonstrates that this scheme is unconditionally stable, with a time convergence order of $\mathcal{O}({3 \alpha})$. Additionally, we provide numerical results in one, two, and three dimensions, highlighting the high accuracy of our approach. The significance of our proposed method lies in its ability to efficiently and accurately address the non-linear time fractional Klein-Gordon equation. Furthermore, our numerical outcomes validate the effectiveness of this scheme across different dimensions.

In this paper, a pseudo-spectral method with the Lagrange polynomial basis is proposed to solve the time-fractional advection-diffusion equation. A semi-discrete approximation scheme is used for conversion of this equation to a system of ordinary fractional differential equations. Also, to protect the high accuracy of the spectral approximation, the Mittag-Leffler function is used for the integration along the time variable. Some examples are performed to illustrate the accuracy and efficiency of the proposed method.