The Finite Difference Methods for Fractional Ordinary Differential Equations

Abstract

Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis.

Keywords:

• Caputo derivative
• Convergence
• Fractional Adams method
• Fractional differential equations
• Fractional Euler method
• High order methods
• Riemman–Liouville derivative
• Stability

Mathematics Subject Classification:

• 34A08
• 65L05
• 65L12

ACKNOWLEDGMENTS

The present work was partially supported by Key Program of Shanghai Municipal Education Commission under grant no. 12ZZ084 and Shanghai Leading Academic Discipline Project under grant no. S30104.