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

作者：Liu Yanzhi; Roberts Jason; Yan Yubin

来源：International Journal of Computer Mathematics, 2018, 95(6-7): 1151-1169.

DOI：10.1080/00207160.2017.1381691

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

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.

• 出版日期2018