Exact A(α)-Stability Angles for Fractional Linear Multi-step Methods

A new criterion for evaluating exact values for the maximum angles of A(α)-stability for linear multistep methods is proposed. This criterion gives an equation to be solved for a complex point corresponding to the tangent line from the origin to the root locus curve of the stability boundary. The maximum angle for A(α)-stability sector is then obtained from the tangent point. This criterion is applied to find the A(α)-stability angles for several fractional backward difference formula type (BDF-type) numerical methods for fractional initial value problems.