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，用于逼近带无界算子系数和Caputo时间分数阶导数的非齐次分数阶微分方程柯西问题的解。该数值方法基于新获得的解公式，该公式统一了带有扇形算子系数$A$和非零初始条件的次抛物型、抛物型和次双曲型方程的温和解表示。其中涉及的积分算子采用sinc求积公式进行近似，这些公式根据$A$的谱参数、分数阶$\alpha$、初始条件的平滑性以及方程右端项$f(t)$的性质进行调整。所提出的方法对正扇形算子$A$、任意有限时间$t$（包括$t=0$）以及整个范围$\alpha\in(0,2)$均具有指数收敛性。该方法适用于实际中常见的情况，即在所考虑区间$t\in[0,T]$之外没有$f(t)$的信息。该方法的算法具有多级并行能力。我们提供了数值算例，验证了理论误差估计。