The accuracy of the Cayley transform method for an evolution equation with a fractional derivative

The initial value problem for a differential equation with a fractional derivative and a positive definite operator coefficient in a Hilbert space is considered.  The exact solution involves the solving operator (expressed as an infinite series incorporating the Cayley transform of the operator coefficient, and certain polynomials of the independent variable, which is known as the Laguerre–Cayley polynomials) and the convolution integral of the solving operator with the right-hand side of the equation.  The approximate solution is expressed through the partial sum of the first $N$ terms of this series.  Then, we obtain error estimates by taking into account certain smoothness properties of the input data: at first, we prove the power rate of convergence depending on the discretization parameter $N$ in the case of a finitely differentiable right-hand side of the equation, and next, we prove the exponential rate of convergence if the right-hand side is analytic in some sense.