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Author: Grafiati
Published: 6 September 2023

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1

Lovejoy, Shaun. "Fractional relaxation noises, motions and the fractional energy balance equation." Nonlinear Processes in Geophysics 29, no. 1 (February 25, 2022): 93–121. http://dx.doi.org/10.5194/npg-29-93-2022.

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Abstract. We consider the statistical properties of solutions of the stochastic fractional relaxation equation and its fractionally integrated extensions that are models for the Earth's energy balance. In these equations, the highest-order derivative term is fractional, and it models the energy storage processes that are scaling over a wide range. When driven stochastically, the system is a fractional Langevin equation (FLE) that has been considered in the context of random walks where it yields highly nonstationary behaviour. An important difference with the usual applications is that we instead consider the stationary solutions of the Weyl fractional relaxation equations whose domain is −∞ to t rather than 0 to t. An additional key difference is that, unlike the (usual) FLEs – where the highest-order term is of integer order and the fractional term represents a scaling damping – in the fractional relaxation equation, the fractional term is of the highest order. When its order is less than 1/2 (this is the main empirically relevant range), the solutions are noises (generalized functions) whose high-frequency limits are fractional Gaussian noises (fGn). In order to yield physical processes, they must be smoothed, and this is conveniently done by considering their integrals. Whereas the basic processes are (stationary) fractional relaxation noises (fRn), their integrals are (nonstationary) fractional relaxation motions (fRm) that generalize both fractional Brownian motion (fBm) as well as Ornstein–Uhlenbeck processes. Since these processes are Gaussian, their properties are determined by their second-order statistics; using Fourier and Laplace techniques, we analytically develop corresponding power series expansions for fRn and fRm and their fractionally integrated extensions needed to model energy storage processes. We show extensive analytic and numerical results on the autocorrelation functions, Haar fluctuations and spectra. We display sample realizations. Finally, we discuss the predictability of these processes which – due to long memories – is a past value problem, not an initial value problem (that is used for example in highly skillful monthly and seasonal temperature forecasts). We develop an analytic formula for the fRn forecast skills and compare it to fGn skill. The large-scale white noise and fGn limits are attained in a slow power law manner so that when the temporal resolution of the series is small compared to the relaxation time (of the order of a few years on the Earth), fRn and its extensions can mimic a long memory process with a range of exponents wider than possible with fGn or fBm. We discuss the implications for monthly, seasonal, and annual forecasts of the Earth's temperature as well as for projecting the temperature to 2050 and 2100.
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2

Eab, C. H., and S. C. Lim. "Fractional generalized Langevin equation approach to single-file diffusion." Physica A: Statistical Mechanics and its Applications 389, no. 13 (July 2010): 2510–21. http://dx.doi.org/10.1016/j.physa.2010.02.041.

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3

Taloni, Alessandro, and Fabio Marchesoni. "Interacting Single-File System: Fractional Langevin Formulation Versus Diffusion-Noise Approach." Biophysical Reviews and Letters 09, no. 04 (December 2014): 381–96. http://dx.doi.org/10.1142/s1793048014400050.

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We review the latest advances in the analytical modelling of single file diffusion. We focus first on the derivation of the fractional Langevin equation that describes the motion of a tagged file particle. We then propose an alternative derivation of the very same stochastic equation by starting from the diffusion-noise formalism for the time evolution of the file density. [Formula: see text] Special Issue Comments: This article presents mathematical formulations and results on the dynamics in files with applied potential, yet also general files. This article is connected to the Special Issue articles about the zig zag phenomenon,72 advanced statistical properties in single file dynamics,73 and expanding files.74
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4

Lim, S. C., and L. P. Teo. "Modeling single-file diffusion with step fractional Brownian motion and a generalized fractional Langevin equation." Journal of Statistical Mechanics: Theory and Experiment 2009, no. 08 (August 13, 2009): P08015. http://dx.doi.org/10.1088/1742-5468/2009/08/p08015.

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5

Taloni, Alessandro. "Kubo Fluctuation Relations in the Generalized Elastic Model." Advances in Mathematical Physics 2016 (2016): 1–16. http://dx.doi.org/10.1155/2016/7502472.

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The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. In this paper we show that the Fractional Langevin Equation (FLE) is a suitable framework for the study of the tracer (probe) particle dynamics, when an external force actsonlyon a single pointx→⋆(tagged probe) belonging to the system. With the help of the Fox function formalism we study the scaling behaviour of the noise- and force-propagators for large and short times (distances). We show that the Kubo fluctuation relations are exactly fulfilled when a time periodic force is exerted on the tagged probe. Most importantly, by studying the large and low frequency behaviour of the complex mobility we illustrate surprising nontrivial physical scenarios. Our analysis shows that the system splits into two distinct regions whose size depends on the applied frequency, characterized by very different response to the periodic perturbation exerted, both in the phase shift and in the amplitude.
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6

Qiu, Lini, Guitian He, Yun Peng, Hui Cheng, and Yujie Tang. "Noise Spectral of GML Noise and GSR Behaviors for FGLE with Random Mass and Random Frequency." Fractal and Fractional 7, no. 2 (February 10, 2023): 177. http://dx.doi.org/10.3390/fractalfract7020177.

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Due to the interest of anomalous diffusion phenomena and their application, our work has widely studied a fractional-order generalized Langevin Equation (FGLE) with a generalized Mittag–Leffler (GML) noise. Significantly, the spectral of GML noise involving three parameters is well addressed. Furthermore, the spectral amplification (SPA) of an FGLE has also been investigated. The generalized stochastic resonance (GSR) phenomenon for FGLE only influenced by GML noise has been found. Furthermore, material GSR for FGLE influenced by two types of noise has been studied. Moreover, it is found that the GSR behaviors of the FGLE could also be induced by the fractional orders of the FGLE.
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7

DEVI, AMITA, ANOOP KUMAR, THABET ABDELJAWAD, and AZIZ KHAN. "EXISTENCE AND STABILITY ANALYSIS OF SOLUTIONS FOR FRACTIONAL LANGEVIN EQUATION WITH NONLOCAL INTEGRAL AND ANTI-PERIODIC-TYPE BOUNDARY CONDITIONS." Fractals 28, no. 08 (June 25, 2020): 2040006. http://dx.doi.org/10.1142/s0218348x2040006x.

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In this paper, we deal with the existence and uniqueness (EU) of solutions for nonlinear Langevin fractional differential equations (FDE) having fractional derivative of different orders with nonlocal integral and anti-periodic-type boundary conditions. Also, we investigate the Hyres–Ulam (HU) stability of solutions. The existence result is derived by applying Krasnoselskii’s fixed point theorem and the uniqueness of result is established by applying Banach contraction mapping principle. An example is offered to ensure the validity of our obtained results.
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8

Barakat, Mohamed A., Abd-Allah Hyder, and Doaa Rizk. "New fractional results for Langevin equations through extensive fractional operators." AIMS Mathematics 8, no. 3 (2022): 6119–35. http://dx.doi.org/10.3934/math.2023309.

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<abstract><p>Fractional Langevin equations play an important role in describing a wide range of physical processes. For instance, they have been used to describe single-file predominance and the behavior of unshackled particles propelled by internal sounds. This article investigates fractional Langevin equations incorporating recent extensive fractional operators of different orders. Nonperiodic and nonlocal integral boundary conditions are assumed for the model. The Hyres-Ulam stability, existence, and uniqueness of the solution are defined and analyzed for the suggested equations. Also, we utilize Banach contraction principle and Krasnoselskii fixed point theorem to accomplish our results. Moreover, it will be apparent that the findings of this study include various previously obtained results as exceptional cases.</p></abstract>
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9

Zhang, Binlin, Rafia Majeed, and Mehboob Alam. "On Fractional Langevin Equations with Stieltjes Integral Conditions." Mathematics 10, no. 20 (October 19, 2022): 3877. http://dx.doi.org/10.3390/math10203877.

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In this paper, we focus on the study of the implicit FDE involving Stieltjes integral boundary conditions. We first exploit some sufficient conditions to guarantee the existence and uniqueness of solutions for the above problems based on the Banach contraction principle and Schaefer’s fixed point theorem. Then, we present different kinds of stability such as UHS, GUHS, UHRS, and GUHRS by employing the classical techniques. In the end, the main results are demonstrated by two examples.
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10

Ahmad, Bashir, and Juan J. Nieto. "Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions." International Journal of Differential Equations 2010 (2010): 1–10. http://dx.doi.org/10.1155/2010/649486.

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We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.
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11

Sau Fa, Kwok. "Fractional Langevin equation and Riemann-Liouville fractional derivative." European Physical Journal E 24, no. 2 (October 2007): 139–43. http://dx.doi.org/10.1140/epje/i2007-10224-2.

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12

Baghani, Omid. "On fractional Langevin equation involving two fractional orders." Communications in Nonlinear Science and Numerical Simulation 42 (January 2017): 675–81. http://dx.doi.org/10.1016/j.cnsns.2016.05.023.

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13

Xu, Mengrui, Shurong Sun, and Zhenlai Han. "SOLVABILITY FOR IMPULSIVE FRACTIONAL LANGEVIN EQUATION." Journal of Applied Analysis & Computation 10, no. 2 (2020): 486–94. http://dx.doi.org/10.11948/20180170.

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14

Lim, S. C., Ming Li, and L. P. Teo. "Langevin equation with two fractional orders." Physics Letters A 372, no. 42 (October 2008): 6309–20. http://dx.doi.org/10.1016/j.physleta.2008.08.045.

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15

Darzi, Rahmat, Bahram Agheli, and Juan J. Nieto. "Langevin Equation Involving Three Fractional Orders." Journal of Statistical Physics 178, no. 4 (January 2, 2020): 986–95. http://dx.doi.org/10.1007/s10955-019-02476-0.

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16

Wang, Wuyang, Khansa Hina Khalid, Akbar Zada, Sana Ben Moussa, and Jun Ye. "q-Fractional Langevin Differential Equation with q-Fractional Integral Conditions." Mathematics 11, no. 9 (May 2, 2023): 2132. http://dx.doi.org/10.3390/math11092132.

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The major goal of this manuscript is to investigate the existence, uniqueness, and stability of a q-fractional Langevin differential equation with q-fractional integral conditions. We demonstrate the existence and uniqueness of the solution to the proposed q-fractional Langevin differential equation using the Banach contraction principle and Schaefer’s fixed-point theorem. We also elaborate on different kinds of Ulam stability. The theoretical outcomes are verified by examples.
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17

Zhao, Kaihong. "Stability of a Nonlinear ML-Nonsingular Kernel Fractional Langevin System with Distributed Lags and Integral Control." Axioms 11, no. 7 (July 21, 2022): 350. http://dx.doi.org/10.3390/axioms11070350.

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The fractional Langevin equation has more advantages than its classical equation in representing the random motion of Brownian particles in complex viscoelastic fluid. The Mittag–Leffler (ML) fractional equation without singularity is more accurate and effective than Riemann–Caputo (RC) and Riemann–Liouville (RL) fractional equation in portraying Brownian motion. This paper focuses on a nonlinear ML-fractional Langevin system with distributed lag and integral control. Employing the fixed-point theorem of generalised metric space established by Diaz and Margolis, we built the Hyers–Ulam–Rassias (HUR) stability along with Hyers–Ulam (HU) stability of this ML-fractional Langevin system. Applying our main results and MATLAB software, we have carried out theoretical analysis and numerical simulation on an example. By comparing with the numerical simulation of the corresponding classical Langevin system, it can be seen that the ML-fractional Langevin system can better reflect the stationarity of random particles in the statistical sense.
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18

Guo, Peng, Changpin Li, and Fanhai Zeng. "Numerical simulation of the fractional Langevin equation." Thermal Science 16, no. 2 (2012): 357–63. http://dx.doi.org/10.2298/tsci110407073g.

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In this paper, we study the fractional Langevin equation, whose derivative is in Caputo sense. By using the derived numerical algorithm, we obtain the displacement and the mean square displacement which describe the dynamic behaviors of the fractional Langevin equation.
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19

Zhao, Kaihong. "Existence, Stability and Simulation of a Class of Nonlinear Fractional Langevin Equations Involving Nonsingular Mittag–Leffler Kernel." Fractal and Fractional 6, no. 9 (August 26, 2022): 469. http://dx.doi.org/10.3390/fractalfract6090469.

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The fractional Langevin equation is a very effective mathematical model for depicting the random motion of particles in complex viscous elastic liquids. This manuscript is mainly concerned with a class of nonlinear fractional Langevin equations involving nonsingular Mittag–Leffler (ML) kernel. We first investigate the existence and uniqueness of the solution by employing some fixed-point theorems. Then, we apply direct analysis to obtain the Ulam–Hyers (UH) type stability. Finally, the theoretical analysis and numerical simulation of some interesting examples show that there is a great difference between the fractional Langevin equation and integer Langevin equation in describing the random motion of free particles.
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20

Fazli, Hossein, HongGuang Sun, and Juan J. Nieto. "Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited." Mathematics 8, no. 5 (May 8, 2020): 743. http://dx.doi.org/10.3390/math8050743.

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We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s fixed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to find the fixed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefficient. Two examples are discussed to illustrate the feasibility of the main theorems.
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21

Kobelev, V., and E. Romanov. "Fractional Langevin Equation to Describe Anomalous Diffusion." Progress of Theoretical Physics Supplement 139 (2000): 470–76. http://dx.doi.org/10.1143/ptps.139.470.

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22

Darzi, Rahmat. "New Existence Results for Fractional Langevin Equation." Iranian Journal of Science and Technology, Transactions A: Science 43, no. 5 (August 20, 2019): 2193–203. http://dx.doi.org/10.1007/s40995-019-00748-8.

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23

Zhao, Kaihong. "Existence and uh-stability of integral boundary problem for a class of nonlinear higher-order Hadamard fractional Langevin equation via Mittag-Leffler functions." Filomat 37, no. 4 (2023): 1053–63. http://dx.doi.org/10.2298/fil2304053z.

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The Langevin equation is a very important mathematical model in describing the random motion of particles. The fractional Langevin equation is a powerful tool in complex viscoelasticity. Therefore, this paper focuses on a class of nonlinear higher-order Hadamard fractional Langevin equation with integral boundary value conditions. Firstly, we employ successive approximation and Mittag-Leffler function to transform the differential equation into an equivalent integral equation. Then the existence and uniqueness of the solution are obtained by using the fixed point theory. Meanwhile, the Ulam-Hyers (UH) stability is proved by inequality technique and direct analysis.
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24

Kou, Zheng, and Saeed Kosari. "On a generalization of fractional Langevin equation with boundary conditions." AIMS Mathematics 7, no. 1 (2021): 1333–45. http://dx.doi.org/10.3934/math.2022079.

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<abstract><p>In this work, we consider a generalization of the nonlinear Langevin equation of fractional orders with boundary value conditions. The existence and uniqueness of solutions are studied by using the results of the fixed point theory. Moreover, the previous results of fractional Langevin equations are a special case of our problem.</p></abstract>
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25

Zeng, Caibin, Qigui Yang, and YangQuan Chen. "Bifurcation dynamics of the tempered fractional Langevin equation." Chaos: An Interdisciplinary Journal of Nonlinear Science 26, no. 8 (August 2016): 084310. http://dx.doi.org/10.1063/1.4959533.

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26

Zhong, Suchuan, Kun Wei, Shilong Gao, and Hong Ma. "Stochastic Resonance in a Linear Fractional Langevin Equation." Journal of Statistical Physics 150, no. 5 (December 21, 2012): 867–80. http://dx.doi.org/10.1007/s10955-012-0670-z.

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27

Gao Shi-Long, Zhong Su-Chuan, Wei Kun, and Ma Hong. "Overdamped fractional Langevin equation and its stochastic resonance." Acta Physica Sinica 61, no. 10 (2012): 100502. http://dx.doi.org/10.7498/aps.61.100502.

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28

Koyama, Junji, and Hiroaki Hara. "Fractional Brownian motions described by scaled Langevin equation." Chaos, Solitons & Fractals 3, no. 4 (July 1993): 467–80. http://dx.doi.org/10.1016/0960-0779(93)90031-u.

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29

Fazli, Hossein, and Juan J. Nieto. "Fractional Langevin equation with anti-periodic boundary conditions." Chaos, Solitons & Fractals 114 (September 2018): 332–37. http://dx.doi.org/10.1016/j.chaos.2018.07.009.

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30

Omaba, McSylvester Ejighikeme, and Eze R. Nwaeze. "On a Nonlinear Fractional Langevin Equation of Two Fractional Orders with a Multiplicative Noise." Fractal and Fractional 6, no. 6 (May 26, 2022): 290. http://dx.doi.org/10.3390/fractalfract6060290.

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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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31

Salem, Ahmed, and Noorah Mshary. "On the Existence and Uniqueness of Solution to Fractional-Order Langevin Equation." Advances in Mathematical Physics 2020 (October 30, 2020): 1–11. http://dx.doi.org/10.1155/2020/8890575.

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In this work, we give sufficient conditions to investigate the existence and uniqueness of solution to fractional-order Langevin equation involving two distinct fractional orders with unprecedented conditions (three-point boundary conditions including two nonlocal integrals). The problem is introduced to keep track of the progress made on exploring the existence and uniqueness of solution to the fractional-order Langevin equation. As a result of employing the so-called Krasnoselskii and Leray-Schauder alternative fixed point theorems and Banach contraction mapping principle, some novel results are presented in regarding to our main concern. These results are illustrated through providing three examples for completeness.
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32

Rizwan, Rizwan. "Existence Theory and Stability Analysis of Fractional Langevin Equation." International Journal of Nonlinear Sciences and Numerical Simulation 20, no. 7-8 (November 18, 2019): 833–48. http://dx.doi.org/10.1515/ijnsns-2019-0053.

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AbstractIn this paper, we consider a non local boundary value problem of nonlinear fractional Langevin equation with non-instantaneous impulses. Initially, we form a standard framework to originate a formula of solutions to our proposed model and then implement the concept of generalized Ulam–Hyers–Rassias using Diaz–Margolis’s fixed point theorem over a generalized complete metric space.
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33

Zheng, Yongjun, He Li, Bo Deng, and Min Lin. "Control of Stochastic Resonance in Overdamped Fractional Langevin Equation." International Journal of Signal Processing, Image Processing and Pattern Recognition 6, no. 6 (December 31, 2013): 275–84. http://dx.doi.org/10.14257/ijsip.2013.6.6.25.

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34

Torres, Cesar. "Existence of solution for fractional Langevin equation: Variational approach." Electronic Journal of Qualitative Theory of Differential Equations, no. 54 (2014): 1–14. http://dx.doi.org/10.14232/ejqtde.2014.1.54.

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35

Wu, Jing-Nuo, Hsin-Chien Huang, Szu-Cheng Cheng, and Wen-Feng Hsieh. "Fractional Langevin Equation in Quantum Systems with Memory Effect." Applied Mathematics 05, no. 12 (2014): 1741–49. http://dx.doi.org/10.4236/am.2014.512167.

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36

Yukunthorn, Weera, Sotiris K. Ntouyas, and Jessada Tariboon. "Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions." Advances in Difference Equations 2014, no. 1 (2014): 315. http://dx.doi.org/10.1186/1687-1847-2014-315.

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37

Vitali, Silvia, Vittoria Sposini, Oleksii Sliusarenko, Paolo Paradisi, Gastone Castellani, and Gianni Pagnini. "Langevin equation in complex media and anomalous diffusion." Journal of The Royal Society Interface 15, no. 145 (August 2018): 20180282. http://dx.doi.org/10.1098/rsif.2018.0282.

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The problem of biological motion is a very intriguing and topical issue. Many efforts are being focused on the development of novel modelling approaches for the description of anomalous diffusion in biological systems, such as the very complex and heterogeneous cell environment. Nevertheless, many questions are still open, such as the joint manifestation of statistical features in agreement with different models that can also be somewhat alternative to each other, e.g. continuous time random walk and fractional Brownian motion. To overcome these limitations, we propose a stochastic diffusion model with additive noise and linear friction force (linear Langevin equation), thus involving the explicit modelling of velocity dynamics. The complexity of the medium is parametrized via a population of intensity parameters (relaxation time and diffusivity of velocity), thus introducing an additional randomness, in addition to white noise, in the particle's dynamics. We prove that, for proper distributions of these parameters, we can get both Gaussian anomalous diffusion, fractional diffusion and its generalizations.
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38

Tariboon, Jessada, Sotiris K. Ntouyas, and Chatthai Thaiprayoon. "Nonlinear Langevin Equation of Hadamard-Caputo Type Fractional Derivatives with Nonlocal Fractional Integral Conditions." Advances in Mathematical Physics 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/372749.

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We study existence and uniqueness of solutions for a problem consisting of nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions. A variety of fixed point theorems are used, such as Banach’s fixed point theorem, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative, and Leray-Schauder’s degree theory. Enlightening examples illustrating the obtained results are also presented.
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39

Zhao, Jing, Peifen Lu, and Yiliang Liu. "Existence and Numerical Simulation of Solutions for Fractional Equations Involving Two Fractional Orders with Nonlocal Boundary Conditions." Journal of Applied Mathematics 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/268347.

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We study a boundary value problem for fractional equations involving two fractional orders. By means of a fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. In addition, we describe the dynamic behaviors of the fractional Langevin equation by using theG2algorithm.
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40

Salem, Ahmed. "EXISTENCE RESULTS OF SOLUTIONS FOR ANTI-PERIODIC FRACTIONAL LANGEVIN EQUATION." Journal of Applied Analysis & Computation 10, no. 6 (2020): 2557–74. http://dx.doi.org/10.11948/20190419.

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41

Baleanu, Dumitru, Rahmat Darzi, and Bahram Agheli. "Existence Results for Langevin Equation Involving Atangana-Baleanu Fractional Operators." Mathematics 8, no. 3 (March 12, 2020): 408. http://dx.doi.org/10.3390/math8030408.

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A new form of nonlinear Langevin equation (NLE), featuring two derivatives of non-integer orders, is studied in this research. An existence conclusion due to the nonlinear alternative of Leray-Schauder type (LSN) for the solution is offered first and, following that, the uniqueness of solution using Banach contraction principle (BCP) is demonstrated. Eventually, the derivatives of non-integer orders are elaborated in Atangana-Baleanu sense.
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42

Lingyun, Han, Li Yuehua, Chen Jianfei, and Wang Jianqiao. "Stochastic Resonance Induced by over Damped Fractional-order Langevin Equation." Information Technology Journal 12, no. 8 (April 1, 2013): 1650–54. http://dx.doi.org/10.3923/itj.2013.1650.1654.

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43

Anh, V. V., and N. N. Leonenko. "Fractional Stokes–Boussinesq–Langevin equation and Mittag-Leffler correlation decay." Theory of Probability and Mathematical Statistics 98 (August 19, 2019): 5–26. http://dx.doi.org/10.1090/tpms/1060.

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44

Dhaniya, Sombir, Anoop Kumar, Aziz Khan, Thabet Abdeljawad, and Manar A. Alqudah. "Existence Results of Langevin Equations with Caputo–Hadamard Fractional Operator." Journal of Mathematics 2023 (June 23, 2023): 1–12. http://dx.doi.org/10.1155/2023/2288477.

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In this manuscript, we deal with a nonlinear Langevin fractional differential equation that involves the Caputo–Hadamard and Caputo fractional operators, with nonperiodic and nonlocal integral boundary conditions. The results presented in this study establish the existence, uniqueness, and Hyers–Ulam (HU) stability of the solution to the proposed equation. We achieved our main result by using the Banach contraction mapping principle and Krasonoselskii’s fixed point theorem. Furthermore, we introduce an application to demonstrate the validity of the results of our findings.
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45

Bian, Yukun, Xiuli Cao, Peng Li, and Nanrong Zhao. "Understanding chain looping kinetics in polymer solutions: crowding effects of microviscosity and collapse." Soft Matter 14, no. 39 (2018): 8060–72. http://dx.doi.org/10.1039/c8sm01499j.

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46

Hamid, Beddani. "Upper-solution or lower-solution method for Langevin equations with n fractional orders." Filomat 35, no. 14 (2021): 4743–54. http://dx.doi.org/10.2298/fil2114743h.

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Abstract:
In this paper, we study a nonlinear Langevin equation involving n-parameter singular fractional orders ?i(i=1,2), and ? with initial conditions. By means of an interesting fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations.
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47

Fang, Di, and Lei Li. "Numerical approximation and fast evaluation of the overdamped generalized Langevin equation with fractional noise." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 2 (February 18, 2020): 431–63. http://dx.doi.org/10.1051/m2an/2019067.

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The generalized Langevin equation (GLE) is a stochastic integro-differential equation that has been used to describe the movement of microparticles with sub-diffusion phenomenon. It has been proved that with fractional Gaussian noise (fGn) mostly considered by biologists, the overdamped Generalized Langevin equation satisfying fluctuation dissipation theorem can be written as a fractional stochastic differential equation (FSDE). In this work, we present both a direct and a fast algorithm respectively for this FSDE model in order to numerically study ergodicity. The strong orders of convergence are proven for both schemes, where the role of the memory effects can be clearly observed. We verify the convergence theorems using linear forces, and then verify the convergence to Gibbs measure algebraically for the double well potentials in both 1D and 2D setups. Our work is new in numerical analysis of FSDEs and provides a useful tool for studying ergodicity. The idea can also be used for other stochastic models involving memory.
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Vojta, Thomas, and Alex Warhover. "Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls." Journal of Statistical Mechanics: Theory and Experiment 2021, no. 3 (March 1, 2021): 033215. http://dx.doi.org/10.1088/1742-5468/abe700.

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49

Magdziarz, M., and A. Weron. "Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series." Studia Mathematica 181, no. 1 (2007): 47–60. http://dx.doi.org/10.4064/sm181-1-4.

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50

Sudsutad, Weerawat, Bashir Ahmad, Sotiris K. Ntouyas, and Jessada Tariboon. "Impulsively hybrid fractional quantum Langevin equation with boundary conditions involving Caputo q-fractional derivatives." Chaos, Solitons & Fractals 91 (October 2016): 47–62. http://dx.doi.org/10.1016/j.chaos.2016.05.002.

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