Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations

In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann-Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grunwald integral (WSGI) formula in time. We introduce an auxiliary variable to transform the nonlinear fourth-order model into a low-order coupled system including two second-order equations and then discretize the resulting equations by the combined method between the BDF2 with the WSGI formula and the mixed finite element method. Further, we derive stability and error results for the fully discrete scheme. Finally, we develop two numerical examples to verify the theoretical results.

Automated summary

This article studies a class of nonlinear fourth-order partial differential equation models with the Riemann-Liouville fractional integral term in two dimensions. The mixed element method is used in space, while the second-order backward difference formula (BDF2) with the weighted and shifted Grunwald integral (WSGI) formula is used in time. Stability and error results for the fully discrete scheme are derived, and two numerical examples are provided to verify the theoretical results.