We consider a Cauchy problem for the inhomogeneous differential equation given in terms of an unbounded linear operator $A$ and the Caputo fractional derivative of order $\alpha \in (0, 2)$ in time. The previously known representation of the mild solution to such a problem does not have a conventional variation-of-constants like form, with the propagator derived from the associated homogeneous problem. Instead, it relies on the existence of two propagators with different analytical properties. This fact limits theoretical and especially numerical applicability of the existing solution representation. Here, we propose an alternative representation of the mild solution to the given problem that consolidates the solution formulas for sub-parabolic, parabolic and sub-hyperbolic equations with a positive sectorial operator $A$ and non-zero initial data. The new representation is solely based on the propagator of the homogeneous problem and, therefore, can be considered as a more natural fractional extension of the solution to the classical parabolic Cauchy problem. By exploiting a trade-off between the regularity assumptions on the initial data in terms of the fractional powers of $A$ and the regularity assumptions on the right-hand side in time, we show that the proposed solution formula is strongly convergent for $t \geq 0$ under considerably weaker assumptions compared to the standard results from the literature. Crucially, the achieved relaxation of space regularity assumptions ensures that the new solution representation is practically feasible for any $\alpha \in (0, 2)$ and is amenable to the numerical evaluation using uniformly accurate quadrature-based algorithms.

翻译：我们考虑由无界线性算子 $A$ 和时间阶 $\alpha \in (0, 2)$ 的卡普托分数阶导数所描述的非齐次微分方程的柯西问题。此类问题的温和解的已知表示不具有常规的常数变易形式，即不包含由相关齐次问题导出的传播子。相反，它依赖于两个具有不同解析性质的传播子的存在性。这一事实限制了现有解表示的理论，尤其是数值适用性。本文针对给定问题提出了一种温和解的替代表示，该表示将具有正扇形算子 $A$ 和非零初始数据的亚抛物型、抛物型和亚双曲型方程的解公式统一起来。新表示仅基于齐次问题的传播子，因此可视为经典抛物型柯西问题解的一种更自然的分数阶推广。通过利用以 $A$ 的分数幂表述的初始数据正则性假设与右端项在时间上的正则性假设之间的权衡，我们证明了在相比文献中标准结果显著更弱的假设条件下，所提出的解公式对 $t \geq 0$ 强收敛。关键的是，空间正则性假设的放宽确保了新解表示对任意 $\alpha \in (0, 2)$ 均具有实际可行性，并且适用于基于一致精度求积算法的数值评估。