New fractional linear multi-step methods of order four with improved stabilities

Abstract

We present two implicit fractional linear multi-step methods (FLMM) of order four for fractional initial value problems. These FLMMs are of a new type that has not appeared before in the literature. The methods are obtained from the second order super-convergence of the Grünwald-Letnikov approximation of the fractional derivative at a non-integer shift point, taking advantage of the fact that the error coefficients of this superconvergence vanish not only at first order, but also at the third order terms. The weight coefficients of the methods are obtained from the Grünwald weights and hence computationally efficient compared with that of the fractional backward difference formula method of order four. The stability regions of the proposed methods are larger than that of the fractional Adams-Moulton method and the fractional backward difference formula method. Numerical results and illustrations are presented to justify results.