Abstract

This paper investigates a class of Hadamard variable-order fractional differential equations in which the fractional order is determined by the cumulative distribution function (CDF) of a continuous random variable. The proposed framework establishes a novel connection between probability theory and variable-order fractional calculus by allowing the memory index of the fractional operator to evolve according to a prescribed distribution law. To facilitate the analysis, the CDF-based variable order is considered through a piecewise-constant representation on a finite partition of the interval, which transforms the original problem into a family of Hadamard fractional differential equations of constant order on successive subintervals. Existence and uniqueness results are established by converting the differential problem into an equivalent fractional integral equation and applying the Banach contraction principle in suitable Banach spaces. Sufficient conditions ensuring the well-posedness of the problem are derived in terms of explicit bounds involving the fractional order and the nonlinear term. In addition, the Ulam–Hyers stability of the proposed model is investigated, and stability criteria are obtained under the same analytical framework. To illustrate the applicability of the theoretical results, a numerical example involving a CDF-generated variable-order function is presented. The example verifies the assumptions of the existence, uniqueness, and stability theorems and demonstrates the effect of piecewise-constant approximations of the cumulative distribution function on the resulting numerical solutions. The obtained results show that the proposed CDF-based Hadamard variable-order framework provides a mathematically consistent setting for studying fractional differential equations whose memory characteristics depend on probabilistic distributions.

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