  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.
