New existence results for nonlinear functional hybrid differential equations involving  the $\psi-$Caputo fractional derivative

Mfadel, Ali El; Melliani, Saïd; Elomari, M’hamed

doi:10.53006/rna.1020895

Cited by 20 publications

(9 citation statements)

References 23 publications

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“…Wahash et al examined the existence and uniqueness of solutions to nonlocal Cauchy problems for the ψ-Caputo fractional differential equations in [1]. Mfadel et al proposed novel existence results for nonlinear functional hybrid ψ-Caputo fractional differential equations of order 0 to 1 in [2]. Using variational approaches, Khaliq et al investigated the presence of weak solutions to the ψ-Caputo boundary value problems in [3].…”

Section: Introductionmentioning

confidence: 99%

Controllability of fractional dynamical systems with ψ-Caputo fractional derivative

Selvam¹,

Vellappandi²,

Govindaraj³

2023

Phys. Scr.

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The idea behind this study is to investigate the controllability of dynamical systems in terms of the ψ-Caputo fractional derivative. The Grammian matrix is used to get at necessary and sufficient controllability requirements for linear systems, which are characterized by the Mittag-Leffler functions, while the fixed point approach is used to arrive at adequate controllability criteria for nonlinear systems. The novelty of this research is to inquire into the controllability concepts by utilizing the ψ-Caputo fractional derivative. Since ψ-Caputo fractional derivatives have the advantage of capturing memory effects as well as increasing the accuracy of anticipating real-world scenarios. A few numerical examples are offered to help better understand the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Controllability of fractional dynamical systems with ψ-Caputo fractional derivative

Selvam¹,

Vellappandi²,

Govindaraj³

2023

Phys. Scr.

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“…The ϕ-Caputo derivatives have been introduced to extend the standard Caputo derivative by incorporating a variable function, enabling a more flexible definition of fractional derivatives. This generalization allows for a more accurate representation of complex systems with memory effects, enhancing modeling capabilities (see [24][25][26][27][28][29][30][31][32] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Existence and Uniqueness of Variable-Order φ-Caputo Fractional Two-Point Nonlinear Boundary Value Problem in Banach Algebra

Awad,

Fakih,

Alkhezi

2023

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Using variable-order fractional derivatives in differential equations is essential. It enables more precise modeling of complex phenomena with varying memory and long-range dependencies, improving our ability to describe real-world processes reliably. This study investigates the properties of solutions for a two-point boundary value problem associated with φ-Caputo fractional derivatives of variable order. The primary objectives are to establish the existence and uniqueness of solutions, as well as explore their stability through the Ulam-Hyers concept. To achieve these goals, Banach’s and Krasnoselskii’s fixed point theorems are employed as powerful mathematical tools. Additionally, we provide numerical examples to illustrate results and enhance comprehension of theoretical findings. This comprehensive analysis significantly advances our understanding of variable-order fractional differential equations, providing a strong foundation for future research. Future directions include exploring more complex boundary value problems, studying the effects of varying fractional differentiation orders, extending the analysis to systems of equations, and applying these findings to real-world scenarios, all of which promise to deepen our understanding of Caputo fractional differential equations with variable order, driving progress in both theoretical and applied mathematics.

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“…From a mathematical point of view, the formalization of such a problem then results in a differential equation with one or more disturbing terms. In this work, we are interested in the so-called quadratic perturbations, which recently attracted of certain researchers' interest [11,13,14,15,17,24]. We call them fractional hybrid differential equations.…”

Section: Introductionmentioning

confidence: 99%

Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations

Kajouni

Chefnaj²,

Hilal

2022

Results in Nonlinear Analysis

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In this work, we prove the existence of a solution for the initial value problem of nonlinear fractional differential equation with quadratic perturbations involving the Caputo fractional derivative ( cDα0+−ρt cDβ0+)(x(t)f(t,x(t)))=g(t,x(t)),t∈J=[0,1],1<α<2,0<β<α( cD0+α−ρt cD0+β)(x(t)f(t,x(t)))=g(t,x(t)),t∈J=[0,1],1<α<2,0<β<α with conditions x0=x(0)f(0,x(0))x0=x(0)f(0,x(0)) and \\x1=x(1)f(1,x(1))x1=x(1)f(1,x(1)). Dhage's fixed-point the theorem was used to establish this existence. As an application, we have given example to demonstrate the effectiveness of our main result.

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New existence results for nonlinear functional hybrid differential equations involving the $\psi-$Caputo fractional derivative

Cited by 20 publications

References 23 publications

Controllability of fractional dynamical systems with ψ-Caputo fractional derivative

Controllability of fractional dynamical systems with ψ-Caputo fractional derivative

Existence and Uniqueness of Variable-Order φ-Caputo Fractional Two-Point Nonlinear Boundary Value Problem in Banach Algebra

Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations

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