A Mixed Element Algorithm Based on the Modified <i>L</i>1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model

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

Format: Article
Diterbitkan: MDPI AG 2021-12-01

Deskripsi

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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>1</mn></mrow></semantics></math></inline-formula>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> error estimates are performed and the feasibility is validated by the calculated data.