Fractional Langevin Equations with Nonlocal Integral Boundary Conditions

Salem, Ahmed; Alzahrani, Faris; Almaghamsi, Lamya

doi:10.3390/math7050402

Cited by 44 publications

(26 citation statements)

References 24 publications

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“…Inspired by the work in [4,20], in what follows we will be concerned with a more general class of Langevin equations of fractional order. The considered class will contain a nonlinearity that depends on a fractional derivative of order δ: So, let us consider the following problem:…”

Section: A Class Of Differential Equations Of Fractional Ordermentioning

confidence: 99%

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Integral Inequalities and Differential Equations via Fractional Calculus

Dahmani¹,

Belhamiti²

2020

Functional Calculus

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In this chapter, fractional calculus is used to develop some results on integral inequalities and differential equations. We develop some results related to the Hermite-Hadamard inequality. Then, we establish other integral results related to the Minkowski inequality. We continue to present our results by establishing new classes of fractional integral inequalities using a family of positive functions; these classes of inequalities can be considered as generalizations of order n for some other classical/fractional integral results published recently. As applications on inequalities, we generate new lower bounds estimating the fractional expectations and variances for the beta random variable. Some classical covariance identities, which correspond to the classical case, are generalised for any α ≥ 1, β ≥ 1. For the part of differential equations, we present a contribution that allow us to develop a class of fractional chaotic electrical circuit. We prove recent results for the existence and uniqueness of solutions for a class of Langevin-type equation. Then, by establishing some sufficient conditions, another result for the existence of at least one solution is also discussed.

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Section: A Class Of Differential Equations Of Fractional Ordermentioning

confidence: 99%

“…During the last few decades, fractional calculus has been extensively developed due to its important applications in many field of research [1][2][3][4]. On the other hand, the integral inequalities are very important in probability theory and in applied sciences.…”

Section: Introductionmentioning

confidence: 99%

Integral Inequalities and Differential Equations via Fractional Calculus

Dahmani¹,

Belhamiti²

2020

Functional Calculus

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“…ere is a clear progress on fractional Langevin equations in physics (see [21,22]). New results on Langevin equations under the variety of boundary value conditions have been published [23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Coupled System of Nonlinear Fractional Langevin Equations with Multipoint and Nonlocal Integral Boundary Conditions

Salem

Alzahrani

Alnegga

2020

Mathematical Problems in Engineering

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is research paper is about the existence and uniqueness of the coupled system of nonlinear fractional Langevin equations with multipoint and nonlocal integral boundary conditions. e Caputo fractional derivative is used to formulate the fractional differential equations, and the fractional integrals mentioned in the boundary conditions are due to Atangana-Baleanu and Katugampola. e existence of solution has been proven by two main fixed-point theorems: O'Regan's fixed-point theorem and Krasnoselskii's fixed-point theorem. By applying Banach's fixed-point theorem, we proved the uniqueness result for the concerned problem. is research paper highlights the examples related with theorems that have already been proven. HindawiMathematical Problems in Engineering Volume 2020, Article ID 7345658, 15 pages https://doi.org/10.1155/2020/7345658

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“…Recently, several contributions mindful with the uniqueness and existence results for fractional generalized Langevin equations, have been published, see [6][7][8][9][10][11][12][13][14][15] and the references given therein.…”

Section: Introductionmentioning

confidence: 99%

Multi-Point and Anti-Periodic Conditions for Generalized Langevin Equation with Two Fractional Orders

Salem

Alghamdi

2019

Fractal Fract

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With anti-periodic and a new class of multi-point boundary conditions, we investigate, in this paper, the existence and uniqueness of solutions for the Langevin equation that has Caputo fractional derivatives of two different orders. Existence of solutions is obtained by applying Krasnoselskii–Zabreiko’s and the Leray–Schauder fixed point theorems. The Banach contraction mapping principle is used to investigate the uniqueness. Illustrative examples are provided to apply of the fundamental investigations.

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Fractional Langevin Equations with Nonlocal Integral Boundary Conditions

Cited by 44 publications

References 24 publications

Integral Inequalities and Differential Equations via Fractional Calculus

Integral Inequalities and Differential Equations via Fractional Calculus

Coupled System of Nonlinear Fractional Langevin Equations with Multipoint and Nonlocal Integral Boundary Conditions

Multi-Point and Anti-Periodic Conditions for Generalized Langevin Equation with Two Fractional Orders

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