  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.
