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 $\alpha$ and the smoothness of the first initial condition, as well as to the properties of the equation's right-hand side $f(t)$. The resulting method possesses exponential convergence for positive sectorial $A$, any finite $t$, including $t = 0$ and the whole range $\alpha \in (0,2)$. It is suitable for a practically important case, when no knowledge of $f(t)$ is available outside the considered interval $t \in [0, T]$. The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates.

翻译：本文提出了一种指数收敛的数值方法，用于逼近含无界算子系数和Caputo时间分数阶导数的非齐次分数阶微分方程的Cauchy问题解。该方法基于新建立的解公式，该公式整合了具有扇形算子系数A和非零初值的次抛物型、抛物型和次双曲型方程的温和解表示。利用sinc求积公式逼近涉及积分算子，这些公式针对A的谱参数、分数阶α、第一初值条件的光滑性以及方程右端项f(t)的性质进行定制。所得方法对正扇形算子A、任意有限时间t（包括t=0）和整个区间α∈(0,2)均具有指数收敛性。该方法适用于实际重要情形，即无需了解所考虑区间t∈[0,T]之外的f(t)信息。算法具备多级并行能力。我们提供数值算例验证理论误差估计。