Stability properties of fractional second linear multistep methods in the implicit form: Theory and applications

Abstract

The main purpose of this paper is to numerically solve the fractional differential equations (FDE)s with the fractional order in (1, 2) using the implicit forms of the special case of fractional second linear multistep methods (FSLMM)s. The studies are focused on the stability properties and proving that the proposed methods are A(α)−stable. For this purpose, after introducing the FSLMMs, the implicit family of FSLMMs based on fractional backward difference formula 1 (FBDF1) are constructed which have the first, and second order of convergence.The stability regions of the proposed methods are thoroughly studied. Furthermore, in order to show the validity of the proposed theories, some numerical examples are reported. Finally, the application of proposed method for solving the Bagley-Torvik (B-T) equation is also presented.