Asymptotic behavior of solutions to time fractional neutral functional differential equations. (English) Zbl 1450.34051
Summary: In this paper, we derive a new fractional Halanay-like inequality, which is used to characterize the long-term behavior of time fractional neutral functional differential equations (F-NFDEs) of Hale type with order \(\alpha\in(0,1)\). The contractivity and dissipativity of F-NFDEs are established under almost the same assumptions as those for classical integer-order NFDEs. In contrast to the exponential decay rate for NFDEs, the F-NFDEs are proved to have a polynomial decay rate. A numerical scheme based on the \(\mathcal{L}1\) method together with linear interpolation is constructed and applied to several examples to illustrate the theoretical results and to reveal the quite different long-term decay rate in the solutions between F-NFDEs and NFDEs.
MSC:
| 34K25 | Asymptotic theory of functional-differential equations |
| 34K40 | Neutral functional-differential equations |
| 34K37 | Functional-differential equations with fractional derivatives |
| 34K38 | Functional-differential inequalities |
Keywords:
fractional Halanay inequality; fractional neutral functional differential equations; contractivity and dissipativityReferences:
| [1] | Brunner, H., Volterra Integral Equations: An Introduction to Theory and Applications (2017), Cambridge Univ. Press · Zbl 1376.45002 |
| [2] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier Sci. Ltd · Zbl 1092.45003 |
| [3] | Podlubny, Igor, Fractional Differential Equations (1999), Academic Press: Academic Press San Diego, CA · Zbl 0924.34008 |
| [4] | Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1, 1-77 (2000) · Zbl 0984.82032 |
| [5] | Bellen, A.; Zennaro, M., Numerical Methods for Delay Differential Equations (2013), Oxford Univ. Press · Zbl 0749.65042 |
| [6] | Yan, Y.; Kou, C., Stability analysis for a fractional differential model of HIV infection of CD \(4 +\) T-cells with time delay, Math. Comput. Simulation, 82, 1572-1585 (2012) · Zbl 1253.92037 |
| [7] | Wang, D.; Xiao, A., Dissipativity and contractivity for fractional-order systems, Nonlinear Dynam., 80, 1-2, 287-294 (2015) · Zbl 1345.37086 |
| [8] | Wang, D.; Xiao, A.; Liu, H., Dissipativity and stability analysis for fractional functional differential equations, Fract. Calc. Appl. Anal., 18, 6, 1399-1422 (2015) · Zbl 1348.34136 |
| [9] | Wang, D.; Xiao, A.; Zou, J., Long-time behavior of numerical solutions to nonlinear fractional ODEs, ESAIM Math. Model. Numer., 54, 335-358 (2020) · Zbl 1441.65059 |
| [10] | Wang, D.; Zou, J., Dissipativity and contractivity analysis for fractional functional differential equations and their numerical approximations, SIAM J. Numer. Anal., 57, 3, 1445-1470 (2019) · Zbl 1423.34093 |
| [11] | Hale, J. K., Asymptotic Behaviour of Dissipative Systems (1998), Am. Math. Soc: Am. Math. Soc New York |
| [12] | Agarwal, R. P.; Zhou, Y.; He, Y., Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59, 3, 1095-1100 (2010) · Zbl 1189.34152 |
| [13] | Cong, N. D.; Tuan, H. T., Existence, uniqueness, and exponential boundedness of global solutions to delay fractional differential equations, Mediterr. J. Math., 14, 193 (2017) · Zbl 1379.34071 |
| [14] | Lakshmikantham, V., Theory of fractional functional differential equations, Nonlinear Anal., 69, 10, 3337-3343 (2008) · Zbl 1162.34344 |
| [15] | Ravichandran, C.; Jothimani, K.; Baskonus, H. M.; Valliammal, N., New results on nondensely characterized integro-differential equations with fractional order, Eur. Phys. J. Plus, 133, 109 (2018) |
| [16] | Alqudah, Manar A.; Ravichandran, C.; Abdeljawad, Thabet; Valliammal, N., New results on Caputo fractional-order neutral differential inclusions without compactness, Adv. Differential Equations, 2019, 528 (2019) · Zbl 1487.34149 |
| [17] | Jothimani, K.; Kaliraj, K.; Hammouch, Zakia; Ravichandran, C., New results on controllability in the framework of fractional integro-differential equations with nondense domain, Eur. Phys. J. Plus, 134, 441 (2019) |
| [18] | Rakkiyappan, R.; Velmurugan, G.; Cao, J., Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays, Nonlinear Dynam., 78, 4, 2823-2836 (2014) · Zbl 1331.34154 |
| [19] | Velmurugan, G.; Rakkiyappan, R.; Vembarasan, V.; Cao, J.; Alsaedi, A., Dissipativity and stability analysis of fractional-order complex-valued neural networks with time delay, Neural Netw., 86, 42-53 (2017) · Zbl 1432.34101 |
| [20] | Yuan, J.; Zhao, L.; Huang, C.; Xiao, M., Novel results on bifurcation for a fractional-order complex-valued neural network with leakage delay, Phys. A, 514, 868-883 (2019) · Zbl 07562421 |
| [21] | Liu, S.; Wu, X.; Zhang, Y. J.; Yang, R., Asymptotical stability of Riemann-Liouville fractional neutral systems, Appl. Math. Lett., 69, 168-173 (2017) · Zbl 1375.34116 |
| [22] | Tuan, H. T.; Trinh, H., A qualitative theory of time delay nonlinear fractional-order systems, SIAM J. Control Optim., 58, 3, 1491-1518 (2020) · Zbl 1450.34059 |
| [23] | Ravichandran, C.; Valliammal, N.; Nieto, J. J., New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J. Franklin Inst., 356, 1535-1565 (2019) · Zbl 1451.93032 |
| [24] | Valliammal, N.; Ravichandran, C.; Park, Ju H., On the controllability of fractional neutral integro-differential delay equations with nonlocal conditions, Math. Methods Appl. Sci., 40, 5044-5055 (2017) · Zbl 1385.34054 |
| [25] | Wen, L.; Wang, W.; Yu, Y., Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations, Nonlinear Anal., 72, 3-4, 1746-1754 (2010) · Zbl 1220.45008 |
| [26] | Wang, W.; Zhang, C., Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space, Numer. Math., 115, 3, 451-474 (2010) · Zbl 1193.65140 |
| [27] | Li, S., A review of theoretical and numerical analysis for nonlinear stiff Volterra functional differential equations, Front. Math. China, 4, 1, 23-48 (2009) · Zbl 1396.65120 |
| [28] | Hu, G. D.; Liu, M., Stability criteria of linear neutral systems with multiple delays, IEEE Trans. Autom. Control, 52, 4, 720-724 (2007) · Zbl 1366.34098 |
| [29] | Hu, P.; Huang, C., Analytical and numerical stability of nonlinear neutral delay integro-differential equations, J. Franklin Inst., 348, 6, 1082-1100 (2011) · Zbl 1226.65111 |
| [30] | Wang, W.; Zhang, C.; Li, D., Asymptotic stability of exact and discrete solutions for neutral multidelay-integro-differential equations, Appl. Math. Model., 35, 9, 4490-4506 (2011) · Zbl 1225.65125 |
| [31] | Wu, S.; Gan, S., Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations, Comput. Math. Appl., 55, 2426-2443 (2008) · Zbl 1142.45306 |
| [32] | Zhang, C.; Vandewalle, S., Stability criteria for exact and discrete solutions of neutral multidelay-integro-differential equations, Adv. Comput. Math., 28, 4, 383-399 (2008) · Zbl 1154.65101 |
| [33] | Zhao, J.; Fan, Y.; Xu, Y., Stability of symmetric Runge-Kutta methods for neutral delay integro-differential equations, SIAM J. Numer. Anal., 55, 1, 328-348 (2017) · Zbl 1359.65312 |
| [34] | Garrappa, R.; Kasli, E., On initial conditions for fractional delay differential equations, Commun. Nonlinear Sci. Numer. Simul., 90, Article 105359 pp. (2020) · Zbl 1480.34103 |
| [35] | Jin, B.; Lazarov, R.; Thomée, V.; Zhou, Z., On nonnegativity preservation in finite element methods for subdiffusion equations, Math. Comp., 86, 307, 2239-2260 (2017) · Zbl 1364.65197 |
| [36] | Alikhanov, A. A., Boundary value problems for the diffusion equation of the variable order in differential and difference settings, Comput. Math. Appl., 219, 8, 3938-3946 (2012) · Zbl 1311.35332 |
| [37] | Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 2, 1533-1552 (2007) · Zbl 1126.65121 |
| [38] | Sun, Z.; Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56, 2, 193-209 (2006) · Zbl 1094.65083 |
| [39] | Wang, S.; Wen, L., Numerical dissipativity of neutral integro-differential equations with delay, Int. J. Comput. Math., 94, 3, 536-553 (2017) · Zbl 1365.65290 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
