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

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