High-order orthogonal spline collocation method with graded meshes for two-dimensional fractional evolution integro-differential equation

Abstract

We consider the initial boundary value problem for a fractional evolution integro-differential equation with a weakly singular kernel arising from viscoelastic rods and membranes in a bounded smooth domain. In our numerical scheme, an arbitrary-order orthogonal spline collocation (OSC) method is used for spatial approximation and combined with the Crank-Nicolson (CN) alternating direction implicit (ADI) method for time approximation, and the integral term is approximated via the product integration (PI) rule. The detailed stability and convergence of the CN ADI OSC method in $L^2$ norm are proved. Finally, some numerical simulations are provided to illustrate our theoretical error estimates. In addition, the comparison between the non-uniform mesh and the uniform mesh shows the effectiveness of the non-uniform mesh when the solution is not sufficiently smooth.

Keywords:

• Two-dimensional evolution integro-differential equations
• weakly singular kernel
• graded meshes
• orthogonal spline collocation method
• alternating direction implicit scheme

AMS subject classifications (2020)::

• 35R11
• 45E10
• 65M70
• 65M15

Acknowledgments

This research was partly supported by the National Natural Science Foundation of China (Nos. $12101080,12026263$). Hubei Superior and Distinctive Discipline Group of ‘Mechatronics and Automobiles’ (XKQ2021018).

Disclosure statement

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

Additional information

Funding

This work was supported by National Natural Science Foundation of China [Grant Numbers 12026263,12101080], Hubei Superior and Distinctive Discipline Group of ‘Mechatronics and Automobiles’ [Grant Number XKQ2021018].