Stability of the mixed Caputo fractional integro-differential equation by means of weighted space method

Dai, Qun; Liu, Shidong

doi:10.3934/math.2022140

MATH

2022

DOI: 10.3934/math.2022140

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Stability of the mixed Caputo fractional integro-differential equation by means of weighted space method

Qun Dai¹,

Shidong Liu

Abstract: <abstract>In this research work, we consider a class of nonlinear fractional integro-differential equations containing Caputo fractional derivative and integral derivative. We discuss the stabilities of Ulam-Hyers, Ulam-Hyers-Rassias, semi-Ulam-Hyers-Rassias for the nonlinear fractional integro-differential equations in terms of weighted space method and Banach fixed-point theorem. After the demonstration of our results, an example is given to illustrate the results we obtained.</ab… Show more

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Cited by 3 publications

References 29 publications

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On coupled Gronwall inequalities involving a $ \psi $-fractional integral operator with its applications

Jiang¹,

Bai

2022

MATH

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<abstract>In this paper, we obtain a new generalized coupled Gronwall inequality through the Caputo fractional integral with respect to another function $ \psi $. Based on this result, we prove the existence and uniqueness of solutions for nonlinear delay coupled $ \psi $-Caputo fractional differential system. Moreover, the Ulam-Hyers stability of solutions for $ \psi $-Caputo fractional differential system is discussed. An example is also presented to demonstrate the application of main results.</abstract>

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On coupled Gronwall inequalities involving a $ \psi $-fractional integral operator with its applications

Jiang¹,

Bai

2022

MATH

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An approximate approach for fractional singular delay integro-differential equations

Peykrayegan

Ghovatmand

Skandari

et al. 2022

MATH

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<abstract>In this article, we present Jacobi-Gauss collocation method to numerically solve the fractional singular delay integro-differential equations, because such methods have better superiority, capability and applicability than other methods. We first apply a technique to replace the delay function in the considered equation and suggest an equivalent system. We then propose a Jacobi-Gauss collocation approach to discretize the obtained system and to achieve an algebraic system. Having solved the algebraic system, an approximate solution is gained for the original equation. Three numerical examples are solved to show the applicability of presented approximate approach. Obtaining the approximations of the solution and its fractional derivative simultaneously and an acceptable approximation by selecting a small number of collocation points are advantages of the suggested method.</abstract>

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Caputo-Hadamard fractional boundary-value problems in $ {\mathfrak{L}}^\mathfrak{p} $-spaces

Murad,

Rafeeq,

Abdeljawad

2024

MATH

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<abstract>The focal point of this investigation is the exploration of solutions for Caputo-Hadamard fractional differential equations with boundary conditions, and it follows the initial formulation of a model that is intended to address practical problems. The research emphasizes resolving the challenges associated with determining precise solutions across diverse scenarios. The application of the Burton-Kirk fixed-point theorem and the Kolmogorov compactness criterion in $ {\mathfrak{L}}^\mathfrak{p} $-spaces ensures the existence of the solution to our problem. Banach's theory is crucial for the establishment of solution uniqueness, and it is complemented by utilizing the Hölder inequality in integral analysis. Stability analyses from the Ulam-Hyers perspective provide key insights into the system's reliability. We have included practical examples, tables, and figures, thereby furnishing a comprehensive and multifaceted examination of the outcomes.</abstract>

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Stability of the mixed Caputo fractional integro-differential equation by means of weighted space method

Cited by 3 publications

References 29 publications

On coupled Gronwall inequalities involving a $ \psi $-fractional integral operator with its applications

On coupled Gronwall inequalities involving a $ \psi $-fractional integral operator with its applications

An approximate approach for fractional singular delay integro-differential equations

Caputo-Hadamard fractional boundary-value problems in $ {\mathfrak{L}}^\mathfrak{p} $-spaces

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