On a Nonlinear Fractional Langevin Equation of Two Fractional Orders with a Multiplicative Noise
Abstract
:1. Introduction
- 1.
- The above stochastic Langevin equation of two fractional orders in (2) can find its application in modeling dynamical processes that exhibit both fractal (phenomena in hierarchical or porous media) and fractional (systems with long-term memory and long-range interaction) properties.
- 2.
- The addition of randomness (noise term) to the particles’ dynamics takes care of the challenges due to the complexity of a medium environment that particles may encounter.
- 3.
- 4.
- Moreso, the main difficulty when considering the Wiener process is the use of Itô isometry in computing the second moment (energy growth) of the mild solution.
- 5.
Formulation of Solution
- and
- For ,
2. Preliminaries
3. Main Results
Upper Moment Bound
- Case 1: For . Given that , then is continuous and increasing (continuously increasing), hence,Further, for and for all , is continuously increasing, andThus,
- Case 2: For , we follow similar steps in case 1, which shows that is continuously increasing andOn the other hand, is continuously increasing, andTherefore,
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Czechowski, Z. Modeling of Persistent time series by the nonlinear Langevin equation. In Complexity of Seismic Time Series; Elsevier: Amsterdam, The Netherlands, 2018; pp. 141–160. [Google Scholar]
- Coffey, W.T.; Kalmykov, Y.P.; Waldron, J.T. The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, 2nd ed.; World Scientific Publishing: Singapore, 2004. [Google Scholar]
- Yu, T.; Deng, K.; Luo, M. Existence and uniqueness of solutions of initial value problems for nonlinear equation involving two fractional orders. Commun. Nonlinear Sci. Numer Simulat. 2014, 19, 1661–1668. [Google Scholar] [CrossRef]
- Yang, S.; Deng, M.; Ren, R. Stochastic resonance of fractional-order Langevin equation driven by periodic modulated noise with mass fluctuation. Adv. Differ. Equ. 2020, 2020, 81. [Google Scholar] [CrossRef] [Green Version]
- Kubo, R. The fluctuation-dissipation theorem. Rep. Prog. Phys. 1966, 29, 255–284. [Google Scholar] [CrossRef] [Green Version]
- Fazli, H.; Sun, H.-G.; Nieto, J.J. Fractional Langevin Equations Involving Two Fractional Orders: Existence and Uniqueness Revisted. Mathematics 2020, 8, 745. [Google Scholar] [CrossRef]
- Lisý, V.; Tóthová, J. Fractional Langevin Equation Model for Characterization of Anomalous Brownian Motion from NMR Signals. Math. Model. Comput. Phys. 2018, 173, 02013. [Google Scholar] [CrossRef] [Green Version]
- Mainardi, F.; Pironi, P. The fractional langevin equation: Brownian motion revisited. Extr. Math. 1996, 10, 140–154. [Google Scholar]
- Ahmad, B.; Nieto, J.J. Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions. Int. J. Differ. Equ. 2010, 2010, 649486. [Google Scholar] [CrossRef] [Green Version]
- Chen, A.; Chen, Y. Existence of solutions to nonlinear Langevin equation involving two fractional orders with boundary value problems. Bound. Value Probl. 2011, 2011, 516481. [Google Scholar] [CrossRef] [Green Version]
- Ahmad, B.; Nieto, J.J.; Alsaedi, A.; El-Shahred, M. A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal. Real World Appl. 2012, 13, 599–606. [Google Scholar] [CrossRef]
- Kou, Z.; Kosari, S. On a generalization of fractional Langevin equation with boundary conditions. AIMS Math. 2022, 7, 1333–1345. [Google Scholar] [CrossRef]
- Salem, A.; Alzahrani, F.; Almaghamsi, L. Langevin equation involving one fractional order with three-point boundary conditions. J. Nonlinear Sci. Appl. 2019, 12, 791–798. [Google Scholar] [CrossRef]
- Salem, A.; Alzahrani, F.; Almaghamsi, L. Fractional Langevin Equations with Nonlocal Integral Boundary Conditions. Mathematics 2019, 7, 402. [Google Scholar] [CrossRef] [Green Version]
- Zhou, Z.; Qiao, Y. Solutions for a class of fractional Langevin equations with integral and anti-periodic boundary condition. Bound. Value Probl. 2018, 2018, 152. [Google Scholar] [CrossRef]
- Baghani, O. On fractional Langevin equation involving two fractional orders. Commun. Nonlinear Sci. Numer. Simul. 2017, 42, 675–681. [Google Scholar] [CrossRef]
- Karatzas, I.; Shreve, S.E. Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics), 2nd ed.; Springer: New York, NY, USA, 1991. [Google Scholar]
- Guo, P.; Zeng, C.; Li, C.; Chen, Y.-Q. Numerics for the fractional Langevin Equation driven by the fractional Brownian motion. Fract. Calc. Appl. Anal. 2013, 16, 124–141. [Google Scholar] [CrossRef]
- Kobelev, V.; Romanov, E. Fractional Langevin quation to Describe Anomalous Diffusion. Prog. Theor. Phys. Suppl. 2000, 139, 470–476. [Google Scholar] [CrossRef] [Green Version]
- Lipovan, O. A retarded Gronwall-Like Inequality and Its Applications. J. Math. Anal. Appl. 2000, 252, 389–401. [Google Scholar] [CrossRef] [Green Version]
- Agarwal, R.P.; Deng, S.; Zhang, W. Generalization of a retarded Gronwall-like inequality and its applications. Appl. Math. Comput. 2005, 165, 599–612. [Google Scholar] [CrossRef]
- Pineda, M.; Stamatakis, M. On the stochastic modeling of surface reactions through reflected chemical Langevin equations. Comput. Chem. Eng. 2018, 117, 145–158. [Google Scholar] [CrossRef] [Green Version]
- Lu, J.T.; Hu, B.Z.; Hedegard, P.; Brandbyge, M. Semi-classical Langevin equation for equilibrium and nonequilibrium molecular dynamical simulation. Prog. Surf. Sci. 2019, 94, 21–40. [Google Scholar] [CrossRef]
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Omaba, M.E.; Nwaeze, E.R. On a Nonlinear Fractional Langevin Equation of Two Fractional Orders with a Multiplicative Noise. Fractal Fract. 2022, 6, 290. https://doi.org/10.3390/fractalfract6060290
Omaba ME, Nwaeze ER. On a Nonlinear Fractional Langevin Equation of Two Fractional Orders with a Multiplicative Noise. Fractal and Fractional. 2022; 6(6):290. https://doi.org/10.3390/fractalfract6060290
Chicago/Turabian StyleOmaba, McSylvester Ejighikeme, and Eze R. Nwaeze. 2022. "On a Nonlinear Fractional Langevin Equation of Two Fractional Orders with a Multiplicative Noise" Fractal and Fractional 6, no. 6: 290. https://doi.org/10.3390/fractalfract6060290