A(0) and A(π/2) stabilities of fractional LMMs for fractional IVPs

This paper investigates the stability properties of fractional linear multistep methods (FLMMs) for solving fractional initial value problems (FIVPs). Lower order FLMMs are widely used to solve FIVP due to their simple stability properties. Higher order methods have not been used because their stability behaviors are highly restricted. We analyze the stability properties of implicit FLMMs and demonstrate that certain FLMMs fail to be unconditionally or A (π/2)-stable. We determine the maximum threshold values for the fractional order γ up to which the FLMM methods are A (0) or A (π/2)-stable. Theoretical stability conditions are derived, and numerical experiments are presented to validate the findings. We use these properties to modify the FIVP into a system of FIVP and use higher order FLMMs giving A (0) and A (π/2)-stabilities. The results provide crucial insights into the limitations of FLMMs and opens doors to improve the applicability in solving FIVPs with higher order FLMMs.