A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes

ABSTRACT

We consider finite difference methods for solving nonlinear fractional differential equations in the Caputo fractional derivative sense with non-uniform meshes. Under the assumption that the Caputo derivative of the solution of the fractional differential equation is suitably smooth [C. Li, Q. Yi, and A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J. Comput. Phys. 316 (2016), pp. 614–631] obtained the error estimates of finite difference methods with non-uniform meshes. However, the Caputo derivative of the solution of the fractional differential equation in general has a weak singularity near the initial time. In this paper, we obtain the error estimates of finite difference methods with non-uniform meshes when the Caputo fractional derivative of the solution of the fractional differential equation has lower smoothness. The convergence result shows clearly how the regularity of the Caputo fractional derivative of the solution affect the order of convergence of the finite difference methods. Numerical results are presented that confirm the sharpness of the error analysis.

KEYWORDS:

• Nonlinear fractional differential equation
• predictor–corrector method
• error estimates
• non-uniform meshes
• trapezoid formula

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M12
• 65M06
• 65M70
• 35S10

Acknowledgments

The work of the first author was carried out during her stay at the University of Chester, which is supported financially by Shanxi province government, P. R. China. She thanks the Department of Mathematics, University of Chester for its warm hospitality and providing a very good working condition for her during her stay in Chester.

Disclosure statement

No potential conflict of interest was reported by the authors.