Langevin equation involving one fractional order with three-point boundary conditions

Salem, Ahmed; Alzahrani, Faris; Almaghamsi, Lamya

doi:10.22436/jnsa.012.12.02

Cited by 4 publications

(6 citation statements)

References 23 publications

(24 reference statements)

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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%

“…Also, boundary value problems of fractional differential equations have occupied an important area in the fractional calculus domain, since these problems appear in several applications of sciences and engineering, like mechanics, chemistry, electricity, chemistry, biology, finance, and control theory. For more details, we refer the reader to [3,[18][19][20].…”

Section: Introductionmentioning

confidence: 99%

“…Then, by establishing some sufficient conditions on the data of the problem, another result for the existence of at least one solution is also discussed. The considered class has some relationship with the good paper in [20].…”

Section: Introductionmentioning

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%

Section: Introductionmentioning

confidence: 99%

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Integral Inequalities and Differential Equations via Fractional Calculus

Dahmani¹,

Belhamiti²

2020

Functional Calculus

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“…See [11,[13][14][15] and their references for further studies on the fractional Langevin equation of two fractional orders with different boundary conditions. Motivated by Baghani [16] and the authors of the papers [6,13], we consider the initial value problem of the stochastic nonlinear fractional Langevin equation of two fractional orders:…”

Section: Introductionmentioning

confidence: 99%

On a Nonlinear Fractional Langevin Equation of Two Fractional Orders with a Multiplicative Noise

Omaba

Nwaeze

2022

Fractal Fract

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We consider a stochastic nonlinear fractional Langevin equation of two fractional orders Dβ(Dα+γ)ψ(t)=λϑ(t,ψ(t))w˙(t),0<t≤1. Given some suitable conditions on the above parameters, we prove the existence and uniqueness of the mild solution to the initial value problem for the stochastic nonlinear fractional Langevin equation using Banach fixed-point theorem (Contraction mapping theorem). The upper bound estimate for the second moment of the mild solution is given, which shows exponential growth in time t at a precise rate of 3c1expc3t2(α+β)−1+c4t2α−1 on the parameters α>1 and α+β>1 for some positive constants c1,c3 and c4.

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“…The generalized Langevin equation which was concerned with describing the fractal and memory properties, was proposed by Kubo [13], in 1966. Ever after, Langevin equations have exhausted the attention of many authors [2,3,7,8,16,20,21,23,32].…”

Section: Introductionmentioning

confidence: 99%

On multidimensional fractional Langevin equations in terms of Caputo derivatives

Taieb¹,

Boumessaoud²,

Salmi³

2020

Malaya J. Mat

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In this paper, we consider a more general and multidimensional fractional Langevin equations with nonlinear terms that involve some unknown functions and their Caputo derivatives. Using some fixed point theorems, we obtain new results on the existence and uniqueness of solutions in addition to the existence of at least one solution. We also define and prove the generalized Ulam-Heyers stability of solutions for the considered equations. Some examples are provided to illustrate the applications of our results.

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Langevin equation involving one fractional order with three-point boundary conditions

Cited by 4 publications

References 23 publications

Integral Inequalities and Differential Equations via Fractional Calculus

Integral Inequalities and Differential Equations via Fractional Calculus

On a Nonlinear Fractional Langevin Equation of Two Fractional Orders with a Multiplicative Noise

On multidimensional fractional Langevin equations in terms of Caputo derivatives

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