An implicit difference scheme for the fourth-order nonlinear partial integro-differential equations

Abstract

In this paper, an implicit difference scheme is constructed and analyzed for the fourth-order nonlinear partial integro-differential equations. The backward Euler scheme and convolution quadrature are combined for time derivative and Riemann–Liouville (R–L) fractional integral terms. A fully discrete difference scheme is established with the space discretization by the standard central difference approximation. For nonlinear convection term $u{u}_x$, by the Galerkin method based on piecewise linear test functions, we can handle nonlinear term implicitly and attain a system of nonlinear algebraic equations. The stability and convergence are derived by the energy method. Especially, the existence and uniqueness of the numerical solutions are strictly proved for the nonlinear system. Also, we introduce three methods for solving the nonlinear systems, where two existing methods and a new linearized iterative algorithm are given to show the efficiency of our scheme. Numerical results are consistent with the theoretical analysis.

Keywords:

• Fourth-order nonlinear equations
• multi-term kernels
• finite difference method
• existence and uniqueness
• stability and convergence

AMS subject classification (2020):

• 45K05
• 65M06
• 65M12
• 65M15

Acknowledgments

The authors are grateful for helpful suggestions from reviewers.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [grant numbers 12126321] and Scientific Research Fund of Hunan Provincial Education Department [grant number 21B0550].