Well-posedness and cold start criteria for differential equations with a modified Riemann–Liouville fractional operator

This paper investigates the start-up dynamics of a fractional system subjected to a weakly singular power-law input. For $ 1 < \alpha < 2 $, a modified Riemann–Liouville operator is introduced to prescribe the initial displacement while avoiding the singular contribution generated by a constant state. The associated fractional trace is explicitly identified, and the initial-value problem is reformulated as an equivalent Volterra integral equation. We establish the existence, uniqueness, and continuous dependence of solutions under locally integrable forcing for the fractional system. For the power-law excitation $ At^{-\mu} $ for $ 0 < \mu < 1 $, an explicit response representation in terms of Mittag–Leffler functions is obtained. The short-time behavior reveals that a nonzero fractional trace produces a singular ordinary velocity, thereby selecting the zero-trace branch when finite velocity is required. On this branch, a sharp cold-start criterion is established:$ f'(0) = 0 $ if and only if $ \alpha-\mu > 1 $. The critical and subcritical regimes are further classified, showing that zero initial velocity may coexist with an unbounded but locally integrable acceleration. The analysis is extended to more general excitations of the form $ t^{-\mu}h(t) $, demonstrating that the leading local behavior of the modulation function determines the effective start-up exponent. Numerical experiments with a piecewise-linear product-integration scheme reproduce the zero-trace cold start and the singular ordinary velocity associated with a nonzero trace. Mesh grading near $ t = 0 $ yields essentially second-order convergence in the latter case. These results identify the compatibility between weakly singular forcing and zero ordinary initial velocity and show how the associated start-up singularity affects both the solution behavior and numerical convergence.