Langevin differential equation in frame of ordinary and Hadamard fractional derivatives under three point boundary conditions

Yassine Adjabi; Mohammad Esmael Samei; Mohammed M. Matar; Jehad Alzabut; Yassine Adjabi; Mohammad Esmael Samei; Mohammed M. Matar; Jehad Alzabut

doi:10.3934/math.2021171

AIMS Mathematics

2021, Volume 6, Issue 3: 2796-2843. doi: 10.3934/math.2021171

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Research article

Langevin differential equation in frame of ordinary and Hadamard fractional derivatives under three point boundary conditions

1.
Department of Mathematics, Faculty of Sciences, University of M'hamed Bougara, Boumerdès, Dynamic Systems Laboratory, Faculty of Mathematics, U.S.T.H.B., Algeria
2.
Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan, Iran
3.
Department of Mathematics, Al-Azhar University-Gaza, State of Palestine
4.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia

Received: 26 July 2020 Accepted: 29 November 2020 Published: 06 January 2021
MSC : 26A33, 34A08, 34A12, 34B15

In this paper, we study a type of Langevin differential equations within ordinary and Hadamard fractional derivatives and associated with three point local boundary conditions

$ \mathcal{D}_{1}^{\alpha} \left( \mathrm{D}^{2} + \lambda^{2}\right) x(t) = f\left( t, x(t), \mathcal{D}_1^{\alpha} \left[ x\right] (t) \right), $

$ \mathrm{D}^{2} x\left(1 \right) = x(1) = 0 $, $ x(e) = \beta x(\xi) $, for $ t\in \left(1, e\right) $ and $ \xi \in (1, e] $, where $ 0 < \alpha < 1 $, $ \lambda, \beta > 0 $, $ \mathcal{D}_1^\alpha $ denotes the Hadamard fractional derivative of order $ \alpha $, $ \mathrm{D} $ is the ordinary derivative and $ f:[1, e]\times C([1, e], \mathbb{R})\times C([1, e], \mathbb{R})\rightarrow C([1, e], \mathbb{R}) $ is a continuous function. Systematical analysis of existence, stability and solution's dependence of the addressed problem is conducted throughout the paper. The existence results are proven via the Banach contraction principle and Schaefer fixed point theorem. We apply Ulam's approach to prove the Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stability of solutions for the problem. Furthermore, we investigate the dependence of the solution on the parameters. Some illustrative examples along with graphical representations are presented to demonstrate consistency with our theoretical findings.
- Hadamard fractional derivative,
- fractional Langevin equation,
- discretization method,
- continuous dependence of solution
Citation: Yassine Adjabi, Mohammad Esmael Samei, Mohammed M. Matar, Jehad Alzabut. Langevin differential equation in frame of ordinary and Hadamard fractional derivatives under three point boundary conditions[J]. AIMS Mathematics, 2021, 6(3): 2796-2843. doi: 10.3934/math.2021171

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Abstract

In this paper, we study a type of Langevin differential equations within ordinary and Hadamard fractional derivatives and associated with three point local boundary conditions

$ \mathcal{D}_{1}^{\alpha} \left( \mathrm{D}^{2} + \lambda^{2}\right) x(t) = f\left( t, x(t), \mathcal{D}_1^{\alpha} \left[ x\right] (t) \right), $

$ \mathrm{D}^{2} x\left(1 \right) = x(1) = 0 $, $ x(e) = \beta x(\xi) $, for $ t\in \left(1, e\right) $ and $ \xi \in (1, e] $, where $ 0 < \alpha < 1 $, $ \lambda, \beta > 0 $, $ \mathcal{D}_1^\alpha $ denotes the Hadamard fractional derivative of order $ \alpha $, $ \mathrm{D} $ is the ordinary derivative and $ f:[1, e]\times C([1, e], \mathbb{R})\times C([1, e], \mathbb{R})\rightarrow C([1, e], \mathbb{R}) $ is a continuous function. Systematical analysis of existence, stability and solution's dependence of the addressed problem is conducted throughout the paper. The existence results are proven via the Banach contraction principle and Schaefer fixed point theorem. We apply Ulam's approach to prove the Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stability of solutions for the problem. Furthermore, we investigate the dependence of the solution on the parameters. Some illustrative examples along with graphical representations are presented to demonstrate consistency with our theoretical findings.

References

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