Analysis of Implicit Solutions for a Coupled System of Hybrid Fractional Order Differential Equations with Hybrid Integral Boundary Conditions in Banach Algebras

Awad, Yahia; Alkhezi, Yousuf

doi:10.3390/sym15091758

Cited by 4 publications

(3 citation statements)

References 33 publications

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“…The VOFBVP (8) can be written as Equation (27). Now, by Lemma 5, Equation ( 27) can be written in the interval [0, T 1 ] as Equation (29), which can also be written as:…”

Section: Existence and Uniqueness Of Solutionsmentioning

confidence: 99%

“…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%

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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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“…The VOFBVP (8) can be written as Equation (27). Now, by Lemma 5, Equation ( 27) can be written in the interval [0, T 1 ] as Equation (29), which can also be written as:…”

Section: Existence and Uniqueness Of Solutionsmentioning

confidence: 99%

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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“…In [40], the authors investigated the existence and uniqueness of implicit solutions in a coupled symmetry system of hybrid fractional order differential equations, along with hybrid integral boundary conditions in Banach algebra. The methodology centers on a hybrid fixed-point theorem that involves mixed Lipschitz and Carathéodory conditions, serving to establish the existence of solutions.…”

mentioning

confidence: 99%

New Trends on the Mathematical Models and Solitons Arising in Real-World Problems

Baskonus

2023

Symmetry

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Stability Results for Nonlinear Implicit $ϑ$ -Caputo Fractional Differential Equations with Fractional Integral Boundary Conditions

Kaddoura,

Awad

2023

International Journal of Differential Equations

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This article examines the necessary conditions for the unique existence of solutions to nonlinear implicit ϑ-Caputo fractional differential equations accompanied by fractional order integral boundary conditions. The analysis draws upon Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Furthermore, the circumstances leading to the attainment of Ulam–Hyers–Rassias forms of stability are established. An illustrative example is provided to demonstrate the derived findings.

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Analysis of Implicit Solutions for a Coupled System of Hybrid Fractional Order Differential Equations with Hybrid Integral Boundary Conditions in Banach Algebras

Cited by 4 publications

References 33 publications

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

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

New Trends on the Mathematical Models and Solitons Arising in Real-World Problems

Stability Results for Nonlinear Implicit $ϑ$ -Caputo Fractional Differential Equations with Fractional Integral Boundary Conditions

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Analysis of Implicit Solutions for a Coupled System of Hybrid Fractional Order Differential Equations with Hybrid Integral Boundary Conditions in Banach Algebras

Cited by 4 publications

References 33 publications

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

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

New Trends on the Mathematical Models and Solitons Arising in Real-World Problems

Stability Results for Nonlinear Implicit ϑ-Caputo Fractional Differential Equations with Fractional Integral Boundary Conditions

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Stability Results for Nonlinear Implicit $ϑ$ -Caputo Fractional Differential Equations with Fractional Integral Boundary Conditions