A Newton Linearized Crank-Nicolson Method for the Nonlinear Space Fractional Sobolev Equation

Abstract

In this paper, one class of finite difference scheme is proposed to solve nonlinear space fractional Sobolev equation based on the Crank-Nicolson (CN) method. Firstly, a fractional centered finite difference method in space and the CN method in time are utilized to discretize the original equation. Next, the existence, uniqueness, stability, and convergence of the numerical method are analyzed at length, and the convergence orders are proved to be O τ 2 + h 2 in the sense of l 2 -norm, H α / 2 -norm, and l ∞ -norm. Finally, the extensive numerical examples are carried out to verify our theoretical results and show the effectiveness of our algorithm in simulating spatial fractional Sobolev equation.

Numerical analysis of time-fractional Sobolev equation for fluid-driven processes in impermeable rocks

“…Haq and Hussain [26,27] developed a meshless scheme based on radial basis functions (RBFs). Beshtokov [28] proposed a finite-difference algorithm, while Qin et al [29] employed a Newton linearized scheme based on the Crank-Nicolson technique and Zhao et al [30] used a finite-volume element (FVE) approach.…”