Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

We present an exponentially convergent numerical method to approximate the solution of Cauchy problem for inhomogeneous fractional differential equation with unbounded operator coefficient and Caputo derivative in time. The is based on newly obtained formula that consolidates mild representations sub-parabolic, parabolic sub-hyperbolic equations sectorial $A$ non-zero initial data. involved integral operators are approximated using sinc-quadrature formulas tailored spectral parameters $A$, order $\alpha$ smoothness first condition, as well properties equation's right-hand side $f(t)$. resulting possesses exponential convergence positive any finite $t$, including $t = 0$ whole range $\alpha \in (0,2)$. It suitable a practically important case, when no knowledge $f(t)$ available outside considered interval [0, T]$. algorithm capable multi-level parallelism. provide examples confirm theoretical error estimates.