On Hyers–Ulam stability of fractional differential equations with Prabhakar derivatives

Abstract

In this article, we study the Hyers–Ulam stability of the linear and nonlinear fractional differential equations with the Prabhakar derivative. By using the Laplace transform, we show that the introduced fractional differential equations with the Prabhakar fractional derivative is Hyers–Ulam stable. The results generalize the stability of ordinary and fractional differential equations in the Riemann–Liouville sense.

Numerical Solution via Operational Matrix for Solving Prabhakar Fractional Differential Equations

“…is kind of integral had been applied in anomalous dielectrics [12], viscoelasticity [13], kinetic equation [14], and diffusion equation [15]. Besides that, some new concepts or theories were derived to suit this Prabhakar operator, for example, in [16], Hyers-Ulam stability of fractional differential equations with Prabhakar derivatives was investigated, and stability analysis of fractional differential equations with Prabhakar derivative was studied in [17]. Furthermore, since this Prabhakar operator involves more parameters, existing numerical methods may not be applicable for solving the fractional differential equation defined in the Prabhakar sense.…”

Analytical Solutions for the Equal Width Equations Containing Generalized Fractional Derivative Using the Efficient Combined Method

“…e main reason for choosing the three-parameter Mittag-Leffler function in this paper is related to its application in many models such as disordered materials and heterogeneous models [34], Havriliak-Negami models [35,36], viscoelasticity models [37], stochastic models [38], probability models [39], spherical stellar models [40], Poisson models [41], and fractional models or integral models [42][43][44][45]. Many research studies have been conducted in numerical fields to obtain numerical solutions of fractional differential equations such as the Adomian decomposition method [46], the q-homotopy analysis transform method (q-HATM) [47], the Laplace transform method [48], the method based on the implementation of an iterative perturbation method [49], the numerical method based on the Petrov-Galerkin method [50], the Petrov-Galerkin finite element scheme [51], the local discontinuous Galerkin method [52], and other methods [24,27,30,53].…”

Asymptotic Stability of Distributed-Order Nonlinear Time-Varying Systems with the Prabhakar Fractional Derivatives

“…This type of fractional derivative was introduced by Garra et al [20] in that it is considered in terms of the generalized Mittag-Leffler function and can be considered as a generalization of the most popular definitions of fractional derivatives. In the field of stability and asymptotic stability, several papers have been published as follows: in [21], the Hyers-Ulam stability of the linear and nonlinear differential equations of fractional order with Prabhakar derivative by using the Laplace transform method is studied and the authors show that the fractional equation introduced is Hyers-Ulam stable, and in [22], the authors obtained the stability regions of differential systems of fractional order with the Prabhakar fractional derivatives. For this purpose, in Section 2, we recall some definitions and lemmas in generalized fractional calculus.…”