On solving fractional mobile/immobile equation

Abstract

In this article, a numerical efficient method for fractional mobile/immobile equation is developed. The presented numerical technique is based on the compact finite difference method. The spatial and temporal derivatives are approximated based on two difference schemes of orders O(t 2Àa ) and O(h 4 ), respectively. The proposed method is unconditionally stable and the convergence is analyzed within Fourier analysis. Furthermore, the solvability of the compact finite difference approach is proved. The obtained results show the ability of the compact finite difference.

Fourier Convergence Analysis for a Fokker-Planck Equation of Tempered Fractional Langevin-Brownian Motion

“…Next, different from the proof in [33,37], we use an alternative technique to complete the proof of Lemma 3.3 below, which removes the erroneous assumption of truncation error in the convergence analyses in the existing Fourier methods [33]. Lemma 3.3.…”

“…Many methods have been developed so far for analyzing the stability and convergence of the numerical schemes, including energy method [14], matrix property analysis method [28], and Fourier method [1,4], etc. Among them, the Fourier method is frequently used, especially when performing the numerical stability, see, e.g., [5,31,33] and the references therein. However, the existing literatures using Fourier method to do convergence analysis requires a strong assumption on the truncation error function.…”

“…. , N in [33], where the roles of ξ and η in this paper are the same, both denoting the coefficients corresponding to the Fourier series expansion of the truncation error function.…”

Numerical Solution of Two-Dimensional Time Fractional Mobile/Immobile Equation Using Explicit Group Methods

“…Liu and Li [27] presented a Crank-Nicolson (C-N) difference scheme for the one-dimensional mobile/immobile equation with time derivative of variable order. Pourbashash et al [28] proposed a compact difference scheme for solving the one-dimensional time fractional mobile/immobile equation. The stability and convergence of the numerical scheme were proved using Fourier method.…”

Application and analysis of spline approximation for time fractional mobile–immobile advection‐dispersion equation

“…A compact difference scheme for the solution of time fractional mobile-immobile equation is presented in [13]. Liu and Li [14] have used Crank-Nicolson method for the time variable fractional order mobile-immobile advection-dispersion model.…”