Analysis of a new type of fractional linear multistep method of order two with improved stability

We present and investigate a new type of implicit fractional linear multistep method of order two for fractional initial value problems. The method is obtained from the second order super convergence of the Gr\"unwald-Letnikov approximation of the fractional derivative at a non-integer shift point. The proposed method is of order two consistency and coincides with the backward difference method of order two for classical initial value problems when the order of the derivative is one. The weight coefficients of the proposed method are obtained from the Gr\"unwald weights and hence computationally efficient compared with that of the fractional backward difference formula of order two. The stability properties are analyzed and shown that the stability region of the method is larger than that of the fractional Adams-Moulton method of order two and the fractional trapezoidal method. Numerical result and illustrations are presented to justify the analytical theories.