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