A numerical method without accuracy saturation for a fractional-order evolution equation in a Banach space

We study an initial-value problem for a fractional-order evolution equation involving a strongly positive operator in a Banach space. A numerical method free from accuracy saturation is developed and justified for both the homogeneous and inhomogeneous cases. The approach is based on applying the Cayley transform to the operator coefficient and approximating the solution operator by a partial sum of its defining series. A detailed analysis reveals a power-type convergence rate that depends automatically on the smoothness of the input data.