Analysis of New Type of Second-order Fractional Linear Multi-step Method with Improved Stability: ANALYSIS OF A NEW TYPE OF SECOND-ORDER FRACTIONAL LINEAR MULTI-STEP METHOD

We present and investigate a new type of implicit fractional linear multi-step method of order two
for fractional initial value problems. The method is obtained from the second-order super-convergence of the
Grünwald-Letnikov approximation of the fractional derivative at a non-integer shift point. The method coincides
with the backward difference method of order two for the classical initial value problem when the order of the
derivative is one. The weight coefficients of the proposed method are obtained from the Grünwald weights and are
hence computationally efficient compared with that of the fractional backward difference formula of order two. The
stability properties are analyzed and it is 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 results and
illustrations are presented to justify the analytical theories.