TARU PUBLICATIONS
Journal of Interdisciplinary Mathematics cover
Hybrid ·Peer-reviewed·ISSN (Online): 2169-012X·ISSN (Print): 0972-0502

Monthly Journal: Publishes the methodological and theoretical role of mathematics and mathematical applications underpinning scientific research.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

New type of fractional Langevin equations of sequential derivatives

* ,

* Corresponding author · click or hover a name for details

pp. 1335–1354Vol. 28Issue 4June 2025DOI: 10.47974/JIM-1781XML
Received:
07 Jun 2023
Published Online:
05 May 2025
Article type:
Research Article
Language:
EN
Article no.:
JIM-1781
Pages:
1335–1354

Abstract

The aim of this work is in using Mittag-Leffler functions to study a new type of Langevin problems. We shall study the question of existence of unique solutions for a new type of fractional Langevin type equations of sequential derivatives. Using fixed point theory on Banach spaces, two main results are discussed. At the end, some examples are presented.

Keywords

Subject Classifications

26A3334A60

References

[1] W. M. Ahmad and R. El-Khazali, “Fractional order dynamical models of love,” Chaos, Solitons and Fractals, doi:10.1016/j.chaos.2006.01.098. 
[2] R. P. Agarwal, H. Al-Hutami, and B. Ahmad, “A Langevin-type q-variant system of nonlinear fractional integro-difference equations with nonlocal boundary conditions,” Fractal Fract., vol. 6, p. 45 (2022), doi:10.3390/fractalfract6010045. 
[3] A. Ardjouni, “Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions,” AIMS Mathematics, vol. 4, pp. 1101–1113 (2019). 
[4] Y. Bai and H. Kong, “Existence of solutions for nonlinear Caputo-Hadamard fractional differential equations via the method of upper and lower solutions,” J. Nonlinear Sci. Appl., vol. 10, pp. 5744–5752 (2017). 
[5] K. Bensassa, Z. Dahmani, Existence, uniqueness and numerical simulation for solutions of a coupled system of fractional differential equations. (Under review) 
[6] A. Bharucha-Reid, Random Integral Equations. New York, NY, USA: Academic Press (1972).
[7] C. Burgos, J. C. Cortes, A. Debbouche, L. Villafuerte, and R. J. Villanueva, “Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus,” Appl. Math. Comput., vol. 352, pp. 15–29 (2019). 
[8] A. Chen and Y. Chen, “Existence of solutions to nonlinear Langevin equation involving two fractional orders with boundary value conditions,” Bound. Value Probl., vol. 2011, Article ID 516481 (2011), doi:10.1155/2011/516481. 
[9] W. T. Coffey, Y. P. Kalmykov, and J. T. Waldron, The Langevin Equation, vol. 14. World Scientific (2004), doi:10.1142/5343. 
[10] J. M. Cooke, Y. P. Kalmykov, W. T. Coffey, and C. M. Kerskens, “Langevin equation approach to diffusion magnetic resonance imaging,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys., vol. 80 (2009). 
[11] Z. Czechowski, “Modeling of persistent time series by the nonlinear Langevin equation,” in Complexity Seismic Time Series, Elsevier,  pp. 141-160 (2018). doi: 10.1016/B978-0-12-813138-1.00005-5. 
[12] K. S. Fa, “Fractional Langevin equation and Riemann-Liouville fractional derivative,” Eur. Phys. J. E, vol. 24, pp. 139-143 (2007). 
[13] Z. Denton, P. W. Ng, and A. S. Vatsala, “Quasilinearization method via lower and upper solutions for Riemann-Liouville fractional differential equations,” Nonlinear Dyn. Syst. Theory, vol. 11, no. 3, pp. 239–251 (2011). 
[14] K. Diethelm, The Analysis of Fractional Differential Equations. Springer (2004). 
[15] A. M. A. El-Sayed and R. O. Abd El-Salam, “On the stability of a fractional-order differential equation with nonlocal initial condition,” Electron. J. Qual. Theory Differ. Equ., vol. 29, pp. 1–8 (2008). 
[16] K. S. Fa, “Fractional Langevin equation and Riemann-Liouville fractional derivative,” Eur. Phys. J. E, vol. 24, pp. 139–143 (2007). 
[17] H. Fazli and J. J. Nieto, “Fractional Langevin equation with anti-periodic boundary conditions,” Chaos Solitons Fractals, vol. 114, pp. 332–337 (2018). 
[18] H. Fazli, H. Sun, and S. Agheh, “Existence of extremal solutions of fractional Langevin equation involving nonlinear boundary conditions,” Int. J. Comput. Math., vol. 10 (2020). 
[19] H. Fazli, H. Sun, and J. J. Nieto, “Fractional Langevin equation involving two fractional orders: Existence and uniqueness revisited,” Mathematics, vol. 8, p. 743 (2020), doi:10.3390/math8050743. 
[20] H. Fazli, H. G. Sun, and S. Agheh, “Existence of extremal solutions of fractional Langevin equation involving nonlinear boundary conditions,” Int. J. Comput. Math. (2019), doi:10.1080/00207160.2020.1720662. 
[21] H. Fazli, H. G. Sun, and J. J. Nieto, “Fractional Langevin equation involving two fractional orders: Existence and uniqueness revisited,” Mathematics, vol. 8, p. 743 (2020), doi:10.3390/math8050743. 
[22] Y. Gouari, Z. Dahmani, and I. Jebril, “Application of fractional calculus on a new differential problem of Duffing type,” Adv. Math. Sci. J., vol. 12, pp. 10989–11002 (2020). 
[23] Y. Gouari, Z. Dahmani, M. M. Belhamiti, and M. Z. Sarikaya, “Uniqueness of solutions, stability, and simulations for a differential problem involving convergent series and time variable singularities,” Rocky Mountain J. Math., vol. 53, no. 4, pp. 1099-1116, Aug. (2023). 
[24] Y. Gouari, Z. Dahmani, and A. Ndiaye, “A generalized sequential problem of Lane-Emden type via fractional calculus,” Moroccan J. Pure Appl. Anal., vol. 6, pp. 168–183 (2020). 
[25] Y. Gouari, M. Rakah, and Z. Dahmani, “A sequential differential problem with Caputo and Riemann-Liouville derivatives involving convergent series,” Advances in the Theory of Nonlinear Analysis and its Applications, vol. 7, no. 2, pp. 319-225 (2023). 
[26] R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin, Mittag-Leffler Functions Related Topics and Applications, Springer, Berlin (2014). 
[27] P. Guo, C. Zeng, C. Li, and Y. Chen, “Numerics for the fractional Langevin equation driven by the fractional Brownian motion,” Fract. Calc. Appl. Anal., vol. 16, pp. 123-141 (2013). 
[28] V. Ho, “Random fractional functional differential equations,” Int. J. Nonlinear Anal. Appl., vol. 7, no. 2, pp. 253–267 (2016). 
[29] J. H. Jeon and R. Metzler, “Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys., vol. 81, no. 2, pt. 1 (2010). 
[30] A. Kaur, P. S. Takhar, D. M. Smith, J. E. Mann, and M. M. Brashears, “Fractional differential equations-based modeling of microbial survival and growth curves: model development and experimental validation,” J. Food Sci., vol. 73, no. 8, pp. 3-14 (2008). 
[31] A. A. Kilbas, H. M. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006). 
[32] A. Kilbas and S. A. Marzan, “Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions,” Differ. Equ., vol. 41, pp. 84-89 (2005). 
[33] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, vol. 204 (2006). 
[34] A. A. Kilbas, O. I. Marichev, and S. G. Samko, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Switzerland (1993). 
[35] Z. Kou and S. Kosari, “On a generalization of fractional Langevin equation with boundary conditions,” AIMS Mathematics, vol. 7, no. 1, pp. 1333-145 (2022), doi: 10.3934/math.2022079. 
[36] J. Kursawe, J. H. P. Schulz, and R. Metzler, “Transient aging in fractional Brownian and Langevin equation motion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys., vol. 88, no. 6, Dec. (2013), doi: 10.1103/PhysRevE.88.062124. 
[37] G. Ladde and V. Lakshmikantham, Random Differential Inequalities. New York, NY, USA: Academic Press (1980). 
[38] J. T. Lu, B. Z. Hu, P. Hedegard, and M. Brandbyge, “Semi-classical generalized Langevin equation for equilibrium and nonequilibrium molecular dynamics simulation,” Prog. Surf. Sci., vol. 94, pp. 21-40 (2019), doi: 10.1016/j.progsurf.2018.07.002. 
[39] E. Lutz, “Fractional Langevin equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2001). 
[40] P. Mendoza-Méndez, L. López-Flores, A. Vizcarra-Renón, L. E. Sánchez-Díaz, and Medina-Noyola, “Generalized Langevin equation for traces diffusion in atomic liquids,” Physica A, vol. 394, pp. 1-16 (2014), doi: 10.1016/j.physa.2013.09.061. 
[41] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York (1993). 
[42] M. Pineda and M. Stamatatis, “On the stochastic modelling of surface reactions through reflected chemical Langevin equations,” Comput. Chem. Eng., vol. 117, pp. 145-158 (2018), doi: 10.1016/j.compchemeng.2018.05.003. 
[43] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999). 
[44] Y. Rosikin and M. Shitikova, “Application of fractional calculus for dynamic problem of solid mechanics, novel trends and recent results,” Appl. Mech. Rev., vol. 63, pp. 1-52 (2010). 
[45] A. Salem and M. Alnegga, “Measure of non-compactness of hybrid Langevin fractional differential equations,” Axioms, vol. 9, no. 59 (2020), doi: 10.3390/axioms9020059. 
[46] A. Salem, F. Alzahrani, and L. Almaghamsi, “Fractional Langevin equations with nonlocal integral boundary conditions,” Mathematics, vol. 7, no. 5, p. 402 (2019), doi: 10.3390/math7050402. 
[47] Y. Singh and R. S. Dubey, “Fractional calculus operator with generalized k-Mittag-Leffler function,” J. Interdiscip. Math., vol. 23, no. 2, pp. 545-553 (2020), doi: 10.1080/09720502.2020.1731971. 
[48] W. Sudsutad, S. K. Ntouyas, and J. Tariboon, “Systems of fractional Langevin equations of Riemann-Liouville and Hadamard types,” Adv. Differ. Equ., vol. 2015, p. 235 (2015), doi: 10.1186/s13662-015-0566-8. 
[49] N. Tobias and R. Florian, Random Differential Equations in Scientific Computing. Poland: De Gruyter Open (2013). [Online]. Available: https://doi.org/10.2478/9788376560267.b 
[50] P. J. Torvik and R. L. Bagley, “On the appearance of fractional derivatives in the behavior of real materials,” J. Appl. Mech., vol. 51, pp. 294-298 (1984). 
[51] A. Vinodkumar, K. Malar, M. Gowrisankar, and P. Mohankumar, “Existence, uniqueness and stability of random impulsive fractional differential equations,” Acta Math. Sci., vol. 36, no. 2, pp. 428-442 (2016). 
[52] J. Wang, M. Feckan, and Y. Zhou, “Presentation of solutions of impulsive fractional Langevin equations and existence results: impulsive fractional Langevin equation,” Eur. Phys. J. Special Topics, vol. 222, pp. 1857-1874 (2013). 
[53] W. Wang, K. H. Khalid, A. Zada, S. B. Moussa, and J. Ye, “q-Fractional Langevin differential equation with q-fractional integral conditions,” Mathematics, vol. 11, p. 2132 (2023), doi: 10.3390/math11092132. 
[54] S. Zhang and X. Su, “Existence of extreme solutions for fractional-order boundary value problem using upper and lower solutions method in reverse order,” J. Fract. Calc. Appl., vol. 2, no. 6, pp. 1–14 (2012). 
[55] H. Zhou, J. Alzabut, and L. Yang, “On fractional Langevin differential equations with anti-periodic boundary conditions,” Eur. Phys. J. Spec. Topics, vol. 226, pp. 3577-3590 (2017). 
[56] R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press (2001).

Views: 283Downloads: 91Citations: 0