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

Yifan Qin,Wahidullah Niazi,Xiaocheng Yang,Yunzhu Ren,Yinghong Xu,Maria Alessandra Ragusa

doi:10.1155/2021/9979791

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.

Highlights

The main propose of this paper is to construct one class of the Newton linearized finite difference method based on CN discretization in temporal direction to efficiently solve the following spatial fractional Sobolev equation:
Chen et al [5] proposed a Newton linearized compact finite difference scheme to numerically solve a class of Sobolev equations based on the CN method and proved the unique solvability, convergence, and stability of the proposed scheme
The main work in this paper is to develop an efficient Newton linearized CN method to solve the nonlinear space fractional Sobolev problem (1)

Summary

Introduction

The main propose of this paper is to construct one class of the Newton linearized finite difference method based on CN discretization in temporal direction to efficiently solve the following spatial fractional Sobolev equation:. Çelik and Duman [7] investigated the CN method to approximate the fractional diffusion equation with the Riesz fractional derivative in a finite domain. Chen et al [5] proposed a Newton linearized compact finite difference scheme to numerically solve a class of Sobolev equations based on the CN method and proved the unique solvability, convergence, and stability of the proposed scheme. The theoretical results are verified by several numerical examples

Preliminaries

Second-Order CN Method and Theoretical Analysis

CCCCCCCCCCCCA: ð31Þ

Numerical Examples

Conclusion