A Mixed Element Algorithm Based on the Modified L1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model

Jinfeng Wang,Hong Li,Baoli Yin,Zhichao Fang,Yang Liu

doi:10.3390/fractalfract5040274

Abstract

In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified L1-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal L2 error estimates are performed and the feasibility is validated by the calculated data.

Highlights

Fourth-order fractional partial differential equations (PDEs) including fourth-order fractional subdiffusion models [1,2,3] and fourth-order fractional diffusion-wave models [2,4,5] can be founded in many fields of science and engineering
By introducing two auxiliary functions and using some techniques, we propose a new mixed element algorithm
The structure of this article is as follows: in Section 2, we provide some numerical approximation formulas, propose a new mixed element scheme, and prove the stability of the derived scheme; in Section 3, we derive optimal error estimates for three variables; in Section 4, some numerical data are computed and discussed; in Section 5, we give some concluding remarks

Summary

Introduction

Fourth-order fractional partial differential equations (PDEs) including fourth-order fractional subdiffusion models [1,2,3] and fourth-order fractional diffusion-wave models [2,4,5] can be founded in many fields of science and engineering. There have been many efficient numerical algorithms for solving linear or nonlinear fourth-order fractional subdiffusion and diffusion-wave models. Considered different mixed element methods to solve fourth-order nonlinear fractional subdiffusion models with the first-order time derivative and developed numerical theories including stability and convergence. We propose a new mixed element algorithm to solve the following nonlinear fourth-order time-fractional diffusion-wave model: β. (1) by the introduction of two auxiliary functions, we reduce the nonlinear fourth-order time-fractional diffusion-wave model to a low-order coupled system; (2) we turn order β ∈ (1, 2) into order α ∈ (0, 1) for the Riemann–Liouville fractional derivative; (3) we approximate the derived coupled system with a fractional derivative with order α ∈ (0, 1). The structure of this article is as follows: in Section 2, we provide some numerical approximation formulas, propose a new mixed element scheme, and prove the stability of the derived scheme; in Section 3, we derive optimal error estimates for three variables; in Section 4, some numerical data are computed and discussed; in Section 5, we give some concluding remarks

Numerical Approximation and Stability

A Priori Error Estimate

Numerical Tests

Concluding Remarks