New existence and stability results for fractional Langevin equation with three-point boundary conditions. (English) Zbl 1476.34020
Summary: The aim of the paper is to present a new technical analysis of the special case of the nonlinear Langevin equation with positive friction constant involving two fractional orders with three-point boundary conditions. Using some basic properties of the special case of the Prabhakar integral operator, we find an equivalent integral equation to the mentioned equation. We obtain a new result on existence, uniqueness and Hyers-Ulam stability by employing contraction mapping principle and Krasnoselskii’s fixed point theorem with respect to an appropriate weighted Banach space. Our result is an improvement of existing results reported in the previous literature. The consistency of the main results is demonstrated by some examples.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
Keywords:
fractional Langevin equation; Prabhakar integral operator; existence; uniqueness; Hyers-Ulam stabilityReferences:
| [1] | ZWANZIG R (2001) Nonequilibrium Statistical Mechanics. Oxford University Press, pp 3-19 · Zbl 1267.82001 |
| [2] | Coffey, WT; Kalmykov, YP; Waldron, JT, The Langevin Equation, 11-15 (2004), Singapore: World Scientific, Singapore · Zbl 1098.82001 · doi:10.1142/5343 |
| [3] | LEMONS D S, GYTHIEL A (1997) Paul Langevin’s 1908 paper “On the Theory of Brownian Motion” [“Sur la théorie du mouvement brownien,” C. R. Acad. Sci. (Paris), 146, 530-533 (1908)], Am. J. Phys. 65(11) (1997) 1079-1081. American Journal of Physics, 65, 1079 doi:10.1119/1.18725 · JFM 39.0847.03 |
| [4] | KUBO, R., The fluctuation-dissipation theorem, Reports on Progress in Physics, 29, 255-84 (1966) · Zbl 0163.23102 · doi:10.1088/0034-4885/29/1/306 |
| [5] | FAN TY (2017) Generalized Dynamics of Soft-Matter Quasicrystals. Springer Singapore, pp 40-41 · Zbl 1383.82002 |
| [6] | Zhao, WJ; Yang, SP; Wen, GL; Ren, XH, Fractional-order visco-plastic constitutive model for uniaxial ratcheting behaviors, Applied Mathematics and Mechanics (English Edition), 40, 1, 49-62 (2019) · Zbl 1416.74020 · doi:10.1007/s10483-019-2413-8 |
| [7] | Xiong, C.; Niu, Y., Fractional-order generalized thermoelastic diffusion theory, Applied Mathematics and Mechanics (English Edition), 38, 1091-1108 (2017) · Zbl 1373.74031 · doi:10.1007/s10483-017-2230-9 |
| [8] | Mainardi, F.; Pironi, P., The fractional Langevin equation: Brownian motion revisited, Extracta Mathematicae, 10, 140-154 (1996) |
| [9] | Odzijewicz, T.; Malinowska, A.; Torres, DFM, Generalized fractional calculus with applications to the calculus of variations, Computers and Mathematics with Applications, 64, 10, 3351-3366 (2012) · Zbl 1268.26012 · doi:10.1016/j.camwa.2012.01.073 |
| [10] | ORTIGUEIRAAM, MD; TENREIRO MACHADO, JA, What is a fractional derivative?, Journal of Computational Physics, 293, 4-13 (2015) · Zbl 1349.26016 · doi:10.1016/j.jcp.2014.07.019 |
| [11] | Vojta, T.; Skinner, S.; Metzler, R., Probability density of the fractional Langevin equation with reflecting walls, Physical Review E, 100, 042142 (2019) · doi:10.1103/PhysRevE.100.042142 |
| [12] | Kosinski, RA; Grabowski, A., Langevin equations for modeling evacuation processes, Acta Physica Polonica B, 3, 2, 365-377 (2010) |
| [13] | Wodkiewicz, K.; Zubairy, MS, Exact solution of a nonlinear Langevin equation with applications to photoelectron counting and noise-induced instability, Journal of Mathematical Physics, 24, 6, 1401-1404 (1983) · Zbl 0513.60057 · doi:10.1063/1.525874 |
| [14] | Bouchaud, JP; Cont, R., A Langevin approach to stock market fluctuations and crashes, European Physical Journal B, 6, 4, 543-550 (1998) · doi:10.1007/s100510050582 |
| [15] | HINCH, EJ, Application of the Langevin equation to fluid suspensions, Journal of Fluid Mechanics, 72, 3, 499-511 (1975) · Zbl 0327.76044 · doi:10.1017/S0022112075003102 |
| [16] | Schluttig, J.; Alamanova, D.; Helms, V.; Schwarz, US, Dynamics of protein-protein encounter: A Langevin equation approach with reaction patches, Journal of Chemical Physics, 129, 15, 155106 (2008) · doi:10.1063/1.2996082 |
| [17] | LUTZ, E., Fractional Langevin equation, Physical Review E, 64, 051106 (2001) · doi:10.1103/PhysRevE.64.051106 |
| [18] | Ahmad, B.; Nieto, JJ; Alsaedi, A.; El-Shahed, M., A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Analysis: Real World Applications, 13, 599-606 (2012) · Zbl 1238.34008 · doi:10.1016/j.nonrwa.2011.07.052 |
| [19] | MAHMUDOV, NI, Fractional Langevin type delay equations with two fractional derivatives, Applied Mathematics Letters, 103, 106215 (2020) · Zbl 1444.34093 · doi:10.1016/j.aml.2020.106215 |
| [20] | Fazli, H.; Nieto, JJ, Fractional Langevin equation with anti-periodic boundary conditions, Chaos, Solitons and Fractals, 114, 332-337 (2018) · Zbl 1415.34016 · doi:10.1016/j.chaos.2018.07.009 |
| [21] | TORRES, C., Existence of solution for fractional Langevin equation: variational approach, Electronic Journal of Qualitative Theory of Differential Equations, 54, 1-14 (2014) · Zbl 1324.34014 · doi:10.14232/ejqtde.2014.1.54 |
| [22] | Guo, P.; Zeng, C.; Li, C.; Chen, Y., Numerics for the fractional Langevin equation driven by the fractional Brownian motion, Fractional Calculus and Applied Analysis, 16, 123-141 (2013) · Zbl 1312.34093 |
| [23] | Salem, A.; Alzahrani, F.; Alghamdi, B., Langevin equation involving two fractional orders with three-point boundary conditions, Differential and Integral Equations, 33, 163-180 (2020) · Zbl 1488.34062 |
| [24] | Salem, A.; Alnegga, M., Fractional Langevin equations with multi-point and non-local integral boundary conditions, Cogent Mathematics and Statistics, 7, 1, 1758361 (2020) · Zbl 1486.34036 · doi:10.1080/25742558.2020.1758361 |
| [25] | Zhou, H.; Alzabut, J.; Yang, L., On fractional Langevin differential equations with anti-periodic boundary conditions, European Physical Journal Special Topics, 226, 3577-3590 (2017) · doi:10.1140/epjst/e2018-00082-0 |
| [26] | Lim, S.; Li, M.; Teo, L., Langevin equation with two fractional orders, Physics Letters A, 372, 6309-6320 (2008) · Zbl 1225.82049 · doi:10.1016/j.physleta.2008.08.045 |
| [27] | FA, KS, Fractional Langevin equation and Riemann-Liouville fractional derivative, European Physical Journal E, 24, 139-143 (2007) · doi:10.1140/epje/i2007-10224-2 |
| [28] | Darzi, R.; Agheli, B.; Nieto, JJ, Langevin Equation Involving Three Fractional Orders, Journal of Statistical Physics, 178, 986-995 (2020) · Zbl 1436.26006 · doi:10.1007/s10955-019-02476-0 |
| [29] | Fazli, H.; Sun, H.; Nieto, JJ, Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited, Mathematics, 8, 743 (2020) · doi:10.3390/math8050743 |
| [30] | Fazli H, Sun H, Agchi S (2020) Existence of extremal solutions of fractional Langevin equation involving nonlinear boundary conditions. International Journal of Computer Mathematics. doi:10.1080/00207160.2020.1720662 · Zbl 1498.34024 |
| [31] | Giusti, A.; Colombaro, I.; Garra, R.; Garrappa, R.; Polito, F.; Popolizio, M.; Mainardi, F., A practical guide to Prabhakar fractional calculus, Fractional Calculus and Applied Analysis, 23, 1, 9-54 (2020) · Zbl 1437.33019 · doi:10.1515/fca-2020-0002 |
| [32] | PRABHAKAR, TR, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Mathematical Journal, 19, 7-15 (1971) · Zbl 0221.45003 |
| [33] | DIETHELM, K., The Analysis of Fractional Differential Equations (2010), Berlin: Springer-Verlag, Berlin · Zbl 1215.34001 · doi:10.1007/978-3-642-14574-2 |
| [34] | Miller, KS; Samko, SG, Completely monotonic functions, Integral Transforms and Special Functions, 12, 389-402 (2001) · Zbl 1035.26012 · doi:10.1080/10652460108819360 |
| [35] | BERBERAN-SANTOS, MN, Properties of the Mittag-Leffler relaxation function, Journal of Mathematical Chemistry, 38, 4, 629-635 (2005) · Zbl 1101.33015 · doi:10.1007/s10910-005-6909-z |
| [36] | Hu, DL; Chen, W.; Liang, YJ, Inverse Mittag-Leffler stability of structural derivative nonlinear dynamical systems, Chaos, Solitons and Fractals, 123, 304-308 (2019) · Zbl 1448.34113 · doi:10.1016/j.chaos.2019.04.027 |
| [37] | WANG, JR; FEC̆KAN, M.; ZHOU, Y., Presentation of solutions of impulsive fractional Langevin equations and existence results, Impulsive fractional Langevin equations, European Physical Journal Special Topics, 222, 1857-1874 (2013) · doi:10.1140/epjst/e2013-01969-9 |
| [38] | SMART DR (1980) Fixed Point Theorems. Cambridge University Press, pp 31-32 · Zbl 0427.47036 |
| [39] | Kilbas, AA; Srivastava, HM; Trujillo, JJ, Theory and Applications of Fractional Differential Equations, 135-145 (2006), Amsterdam: Elsevier, Amsterdam · Zbl 1092.45003 |
| [40] | Zeng, CB; Chen, YQ, Global Padé approximations of the generalized Mittag-Leffler function and its inverse, Fractional Calculus and Applied Analysis, 18, 6, 1492-1506 (2015) · Zbl 1333.26007 · doi:10.1515/fca-2015-0086 |
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