Fractional second linear multistep methods: the explicit forms for solving fractional differential equations and stability analysis

Abstract

In this paper, the explicit forms of the fractional second linear multistep methods (FSLMMs) are introduced for solving fractional differential equations (FDEs) of the fractional-order in $(1,2)$. These explicit FSLMMs are constructed based on fractional backward difference formulas 1, 2, and 3 (FBDF1, FBDF2, and FBDF3) with the first, second, third, and fourth orders of convergence. Also, the monotonicity of these FBDFs is considered when the order of fractional derivatives lies into $(1,2)$. The order of consistency, linear stability, and the order of convergence of these explicit methods are analysed. Moreover, the stability regions of the proposed methods are completely studied in the stability topic. Finally, four experimental examples are presented to confirm the proposed theories.

Keywords:

• Fractional second linear multistep methods
• linear stability
• Caputo fractional derivative
• consistency

Mathematics Subject Classification (MSC):

• 34XX
• 34Axx

Acknowledgments

The authors wish to thank Professor Changpin Li and for helpful discussions interest in this work.

Disclosure statement

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

Additional information

Funding

The work of second author was supported by the University of Tabriz [grant number 3902].