A boundary value problem for a random-order fractional differential equation

Lopez-Cresencio, Omar U.; Ariza-Hernandez, Francisco J.; Sanchez-Ortiz, Jorge; Arciga-Alejandre, Martin P.

doi:10.1016/j.rinam.2022.100328

Cited by 1 publication

(1 citation statement)

References 11 publications

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“…Recently, a comprehensive review of variable fractional calculus was given in [13]. Also, fractional derivatives where the order is a random variable have been defined, studied, and applied to differential equations ( [14]). One of the most studied and applied cases of variable order is when there is a partition of the interval of consideration, and the order is a constant on each subinterval, i.e., the order is a piecewise constant function.…”

Section: Introductionmentioning

confidence: 99%

Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative

O’Regan,

Agarwal,

Hristova

et al. 2024

Mathematics

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An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues.

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Section: Introductionmentioning

confidence: 99%