Search Advanced

AIMS Mathematics

2021, Volume 6, Issue 7: 6749-6780. doi: 10.3934/math.2021397
Previous Article Next Article
Research article Special Issues

Langevin equation with nonlocal boundary conditions involving a $ \psi $-Caputo fractional operators of different orders

  • 1.
    Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Islamabad, Pakistan
  • 2.
    Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
  • 3.
    Group of Mathematics, Faculty of Engineering, Ostim Technical University, Ankara 06374, Turkey
  • 4.
    Department of Mathematics, Faculty of Sciences, University of M'hamed Bougara, U.M.B.B., Algeria
  • 5.
    Dynamic Systems Laboratory, Faculty of Mathematics, U.S.T.H.B., Algeria
  • 6.
    Department of Mathematics, Hodeidah University, Al-Hodeidah, Yemen
  • Received: 13 December 2020 Accepted: 07 April 2021 Published: 21 April 2021

  • MSC : 34A08, 26A33, 34A12, 34D20

  • This paper studies Langevin equation with nonlocal boundary conditions involving a $ \psi $-Caputo fractional operators of different orders. By the aid of fixed point techniques of Krasnoselskii and Banach, we derive new results on existence and uniqueness of the problem at hand. Further, a new $ \psi $-fractional Gronwall inequality and $ \psi $-fractional integration by parts are employed to prove Ulam-Hyers and Ulam-Hyers-Rassias stability for the solutions. Examples are provided to demonstrate the advantage of our major results. The proposed results here are more general than the existing results in the literature which can be obtained as particular cases.

    Citation: Arjumand Seemab, Mujeeb ur Rehman, Jehad Alzabut, Yassine Adjabi, Mohammed S. Abdo. Langevin equation with nonlocal boundary conditions involving a $ \psi $-Caputo fractional operators of different orders[J]. AIMS Mathematics, 2021, 6(7): 6749-6780. doi: 10.3934/math.2021397

    Related Papers:

  • This paper studies Langevin equation with nonlocal boundary conditions involving a $ \psi $-Caputo fractional operators of different orders. By the aid of fixed point techniques of Krasnoselskii and Banach, we derive new results on existence and uniqueness of the problem at hand. Further, a new $ \psi $-fractional Gronwall inequality and $ \psi $-fractional integration by parts are employed to prove Ulam-Hyers and Ulam-Hyers-Rassias stability for the solutions. Examples are provided to demonstrate the advantage of our major results. The proposed results here are more general than the existing results in the literature which can be obtained as particular cases.


    加载中


    [1] A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006.
    [2] R. Herrmann, Fractional Calculus for Physicist, World Scientific Public, 2014.
    [3] M. Yavuz, European option pricing models described by fractional operators with classical and generalized Mittag‐Leffler kernels, Numer. Methods Partial Differ. Equations, 2020. Available from: https://doi.org/10.1002/num.22645.
    [4] M. R. Jena, S. Chakraverty, M. Yavuz, Two-hybrid techniques coupled with an integral transform for caputo time-fractional Navier-Stokes equations, Prog. Fract. Differ. Appl., 6 (2020), 201–213. doi: 10.18576/pfda/060304
    [5] I. Podlubny, Fractional Differential Equations, San Diego: Academic Press, 1999.
    [6] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, Berlin: Springer-Verlag, 2010.
    [7] N. Sene, Mathematical views of the fractional Chua's electricalcircuit described by the Caputo-Liouville derivative, Rev. Mex. Fis., 67 (2020), 91–99.
    [8] N. Sene, Cascade of fractional differential equations and eeneralized Mittag-Leffler stability, Int. J. Math. Modell. Comput., 10 (2020), 25–35.
    [9] H. M. Fahad, M. Rehman, Generalized substantial fractional operators and well-posedness of Cauchy problem, Bull. Malays. Math. Sci. Soc., 2020. Available from: https://doi.org/10.1007/s40840-020-01008-4.
    [10] A. Keten, M. Yavuz, D. Baleanu, Nonlocal Cauchy problem via a fractional operator involving power kernel in Banach spaces, Fractal Fractional, 3 (2019), 27. doi: 10.3390/fractalfract3020027
    [11] M. Yavuz, Novel solution methods for initial boundary value problems of fractional order with conformable differentiation, Int. J. Optim. Control: Theor. Appl., 8 (2018), 1–7.
    [12] S. G. Samko, A. A. Kilbas, O. I. Mariche, Fractional Integrals and Derivatives, translated from the 1987 Russian original, Yverdon: Gordon and Breach, 1993.
    [13] T. J. Osler, Fractional derivatives of a composite function, SIAM J. Math. Anal., 1 (1970), 288–293.
    [14] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460–481. doi: 10.1016/j.cnsns.2016.09.006
    [15] V. S. Kiryakova, Generalized Fractional Calculus and Applications, New York: John Wiley & Sons Inc., 1994.
    [16] O. P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fractional Calculus Anal. Appl., 15 (2012), 700–711.
    [17] M. D. Kassim, N. E. Tatar, Stability of logarithmic type for a Hadamard fractional differential problem, J. Pseudo-Differ. Oper. Appl., 11 (2020), 447–466. doi: 10.1007/s11868-019-00285-3
    [18] Y. Luchko, J. J. Trujillo, Caputo-type modification of the Erdélyi-Kober fractional derivative, Fractional Calculus Anal. Appl., 10 (2007), 249–276.
    [19] B. Ahmad, M. M. Matar, O. M. EL-Salmy, Existence of solutions and ulam stability for Caputo type sequential fractional differential equations of order α ∈ (2, 3), Int. J. Anal. Appl., 15 (2017), 86–101.
    [20] M. A. Darwich, S. K. Ntouyas, Existence results for a fractional functional differential equation of mixed type, Commun. Appl. Nonlinear Anal., 15 (2008), 47–55.
    [21] V. Obukhovskii, P. Zecca, M. Afanasova, On some boundary value problems for fractional feedback control systems, Differ. Equations Dyn. Sys., 2018. Available from: https://doi.org/10.1007/s12591-018-0435-5.
    [22] M. El-Shahed, Positive solutions for boundary value problem of nonlinear fractional differential equation, Abstr. Appl. Anal., 2007 (2007), 010368.
    [23] E. M. Elsayed, On the existence and stability of solution of boundary value problem for fractional integro-differential equations with complex order, Filomat, 32 (2018).
    [24] R. A. Yan, S. R. Sun, Z. L. Han, Existence of solutions of boundary value problems for Caputo fractional differential equations on time scales, Bull. Iran. Math. Soc., 42 (2016), 247–262.
    [25] Y. Zhao, S. Sun, Z. Han, Q. Li, The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simu., 16 (2011), 2086–2097.
    [26] H. Qin, X. Zuo, J. Liu, Existence and controllability results for fractional impulsive integrodifferential systems in Banach spaces, Abstr. Appl. Anal., 2013 (2013), 295837.
    [27] R. Sakthivel, Y. Ren, N. I. Mahmudov, On the approximate controllability of semilinear fractional differential systems, Comput. Math. Appl., 62 (2011), 1451–1459.
    [28] G. Rajchakit, A. Pratap, R. Raja, J. Cao, J. Alzabut, C. Huang, Hybrid control scheme for projective lag synchronization of Riemann-Liouville sense fractional order memristive BAM neural networks with mixed delays, Mathematics, 7 (2019), 759. Available from: https://doi.org/10.3390/math7080759.
    [29] M. Ismail, U. Saeed, J. Alzabut, M. ur Rehman, Approximate solutions for fractional boundary value problems via Green-CAS method, Mathematics, 7 (2019), 1164. Available from: https://doi.org/10.3390/math7121164.
    [30] F. Mainradi, P. Pironi, The fractional Langevin equation: Brownian motion revisted, Extracta Math., 10 (1996), 140–154.
    [31] B. Ahmad, A. Alsaedi, S. Salem, On a nonlocal integral boundary value problem of nonlinear Langevin equation with different fractional orders, Adv. Differ. Equations, 2019 (2019), 1–44. doi: 10.1186/s13662-018-1939-6
    [32] H. Baghani, J. Alzabut, Juan Nieto, A coupled system of Langevin differential equations of fractional order and associated to anti–periodic boundary conditions, Math. Meth. Appl. Sci., 2020. Available from: https://doi.org/10.1002/mma.6639.
    [33] C. Kiataramkul, K. N. Sotiris, J. Tariboon, A. Kijjathanakorn, Generalized Sturm-Liouville and Langevin equations via Hadamard fractional derivatives with anti-periodic boundary conditions, Boundary Value Prob., 2016 (2016), 217. doi: 10.1186/s13661-016-0725-1
    [34] W. Sudsutad, S. K. Ntouyas, J. Tariboon, Systems of fractional Langevin equations of Riemann-Liouville and Hadamard types, Adv. Differ. Equations, 2015 (2015), 1–24.
    [35] A. Berhail, N. Bouache, M. M. Matar, J. Alzabut, On nonlocal integral and derivative boundary value problem of nonlinear Hadamard Langevin equation with three different fractional orders, Bol. Soc. Mat. Mex., 26 (2020), 303–318. doi: 10.1007/s40590-019-00257-z
    [36] X. Li, S. Sun, Y. Sun, Existence of solutions for fractional Langevin equation with infinite-point boundary conditions, Appl. Math. Comput., 53 (2016), 1–10.
    [37] W. Yukunthorn, S. K. Ntouyas, J. Tariboon, Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions, Adv. Differ. Equations, 2014 (2014), 315. doi: 10.1186/1687-1847-2014-315
    [38] Z. Zhou, Y. Qiao, Solutions for a class of fractional Langevin equations with integral and anti-periodic boundary conditions, Boundary Value Prob., 2018 (2018), 152. doi: 10.1186/s13661-018-1070-3
    [39] H. Zhou, J. Alzabut, L. Yang, On fractional Langevin differential equations with anti-periodic boundary conditions, Eur. Phys. J. Spec. Top., 226 (2017), 3577–3590. doi: 10.1140/epjst/e2018-00082-0
    [40] B. Ahmad, J. J. Nieto, A. Alsaedi, M. El-Shahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal., 13 (2012), 599–606. doi: 10.1016/j.nonrwa.2011.07.052
    [41] S. M. Ulam, A collection of mathematical problems, Bull. Am. Math. Soc., 60 (1960), 361–363.
    [42] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27 (1941), 222–224. doi: 10.1073/pnas.27.4.222
    [43] T. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. doi: 10.1090/S0002-9939-1978-0507327-1
    [44] J. R. Wang, Z. Lin, Ulam's type stability of Hadamard type fractional integral equations, Filomat, 28 (2014), 1323–1331. doi: 10.2298/FIL1407323W
    [45] S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Palm Harbor: Hadronic Press, 2001.
    [46] M. Ahmad, A. Zada, J. Alzabut, Hyres-Ulam stability of coupled system of fractional differential equations of Hilfer-Hadamard type, Demonstr. Math., 52 (2019), 283–295. doi: 10.1515/dema-2019-0024
    [47] M. Ahmad, A. Zada, J. Alzabut, Stability analysis for a nonlinear coupled implicit switched singular fractional differential system with $p$-Laplacian, Adv. Differ. Equations, 2019 (2019), 436. doi: 10.1186/s13662-019-2367-y
    [48] D. R. Smart, Fixed Point Theorems, Cambridge: University Press, 1980.
    [49] J. Alzabut, Y. Adjabi, W. Sudsutad, M. ur Rehman, New generalizations for Gronwall type inequalities involving a $\psi$-fractional operator and their applications, AIMS Math., 6 (2021), 5053–5077. doi: 10.3934/math.2021299
    [50] R. Gorenflo, A. A. Kilbas, F. Mainardi, S. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Berlin: Springer, 2014.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(3025) PDF downloads(205) Cited by(7)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(5)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog