Dissipativity and contractivity of the second-order averaged L1 method for fractional Volterra functional differential equations

Yin Yang; Aiguo Xiao; Yin Yang; Aiguo Xiao

doi:10.3934/nhm.2023032

Networks and Heterogeneous Media

2023, Volume 18, Issue 2: 753-774. doi: 10.3934/nhm.2023032

Previous Article Next Article

Research article Special Issues

Dissipativity and contractivity of the second-order averaged L1 method for fractional Volterra functional differential equations

Yin Yang ,
Aiguo Xiao ^,

Hunan Key Laboratory for Computation and Simulation in Science and Engineering & National Center for Applied Mathematics in Hunan & Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan, Hunan 411105, China

Received: 16 December 2022 Revised: 10 February 2023 Accepted: 16 February 2023 Published: 03 March 2023

This paper focuses on the dissipativity and contractivity of a second-order numerical method for fractional Volterra functional differential equations (F-VFDEs). Firstly, an averaged L1 method for the initial value problem of F-VFDEs is presented based on the averaged L1 approximation for Caputo fractional derivative together with an appropriate piecewise interpolation operator for the functional term. Then the averaged L1 method is proved to be dissipative with an absorbing set and contractive with an algebraic decay rate. Finally, the numerical experiments further confirm the theoretical results.
- fractional Volterra functional differential equations,
- averaged L1 method,
- dissipativity,
- contractivity,
- algebraic decay rate
Citation: Yin Yang, Aiguo Xiao. Dissipativity and contractivity of the second-order averaged L1 method for fractional Volterra functional differential equations[J]. Networks and Heterogeneous Media, 2023, 18(2): 753-774. doi: 10.3934/nhm.2023032

Related Papers:

Abstract

This paper focuses on the dissipativity and contractivity of a second-order numerical method for fractional Volterra functional differential equations (F-VFDEs). Firstly, an averaged L1 method for the initial value problem of F-VFDEs is presented based on the averaged L1 approximation for Caputo fractional derivative together with an appropriate piecewise interpolation operator for the functional term. Then the averaged L1 method is proved to be dissipative with an absorbing set and contractive with an algebraic decay rate. Finally, the numerical experiments further confirm the theoretical results.

References

[1]	S. Abbas, Existence of solutions to fractional order ordinary and delay differential equations and applications, Electron. J. Differ. Equ., 2011 (2011), 1–11.
[2]	J. Alzabut, A. G. M. Selvam, R. A. El-Nabulsi, V. Dhakshinamoorthy, M. E. Samei, Asymptotic stability of nonlinear discrete fractional pantograph equations with non-local initial conditions, Symmetry, 13 (2021), 473. https://doi.org/10.3390/sym13030473 doi: 10.3390/sym13030473
[3]	J. A. D. Applelby, I. Győri, D. W. Reynolds, On exact convergence rates for solutions of linear systems of Volterra difference equations, J. Difference Equ. Appl., 12 (2006), 1257–1275. https://doi.org/10.1080/10236190600986594 doi: 10.1080/10236190600986594
[4]	K. Balachandran, S. Kiruthika, J. J. Trujillo, Existence of solutions of nonlinear fractional pantograph equations, Acta Math. Sci., 33 (2013), 712–720. https://doi.org/10.1016/S0252-9602(13)60032-6 doi: 10.1016/S0252-9602(13)60032-6
[5]	M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338 (2008), 1340–1350. https://doi.org/10.1016/j.jmaa.2007.06.021 doi: 10.1016/j.jmaa.2007.06.021
[6]	N. D. Cong, H. T. Tuan, Existence, uniqueness, and exponential boundedness of global solutions to delay fractional differential equations, Mediterr. J. Math., 14 (2017), 1–12. https://doi.org/10.1007/s00009-017-0997-4 doi: 10.1007/s00009-017-0997-4
[7]	Y. Jalilian, R. Jalilian, Existence of solution for delay fractional differential equations, Mediterr. J. Math., 10 (2013), 1731–1747. https://doi.org/10.1007/s00009-013-0281-1 doi: 10.1007/s00009-013-0281-1
[8]	B. Ji, H. Liao, Y. Gong, L. Zhang, Adaptive second-order Crank-Nicolson time-stepping schemes for time-fractional molecular beam epitaxial growth models, SIAM J. Sci. Comput., 42 (2020), B738–B760. https://doi.org/10.1137/19M1259675 doi: 10.1137/19M1259675
[9]	E. Kaslik, S. Sivasundaram, Analytical and numerical methods for the stability analysis of linear fractional delay differential equations, J. Comput. Appl. Math., 236 (2012), 4027–4041. https://doi.org/10.1016/j.cam.2012.03.010 doi: 10.1016/j.cam.2012.03.010
[10]	K. Krol, Asymptotic properties of fractional delay differential equations, Appl. Math. Comput., 218 (2011), 1515–1532. https://doi.org/10.1016/j.amc.2011.04.059 doi: 10.1016/j.amc.2011.04.059
[11]	V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal., 69 (2008), 3337–3343. https://doi.org/10.1016/j.na.2007.09.025 doi: 10.1016/j.na.2007.09.025
[12]	D. Li, C. Zhang, Long time numerical behaviors of fractional pantograph equations, Math. Comput. Simul., 172 (2020), 244–257. https://doi.org/10.1016/j.matcom.2019.12.004 doi: 10.1016/j.matcom.2019.12.004
[13]	S. Li, A review of theoretical and numerical analysis for nonlinear stiff Volterra functional differential equations, Front. Math. China, 4 (2009), 23–48. https://doi.org/10.1007/s11464-009-0003-y doi: 10.1007/s11464-009-0003-y
[14]	S. Li, High order contractive Runge-Kutta methods for Volterra functional differential equations, SIAM J. Numer. Anal., 47 (2010), 4290–4325. https://doi.org/10.1137/080741148 doi: 10.1137/080741148
[15]	S. Li, Numerical Analysis for Stiff Ordinary and Functional Differential Equations, Xiangtan: Xiangtan University Press, 2018.
[16]	C. Lubich, On the stability of linear multistep methods for Volterra convolution equations, IMA J. Numer. Anal., 3 (1983), 439–465. https://doi.org/10.1093/imanum/3.4.439 doi: 10.1093/imanum/3.4.439
[17]	K. Mustapha, An L1 approximation for a fractional reaction-diffusion equation, a second-order error analysis over time-graded meshes, SIAM J. Numer. Anal., 58 (2020), 1319–1338. https://doi.org/10.1137/19M1260475 doi: 10.1137/19M1260475
[18]	J. Robinson, Infinite-dimensional Dynamical Systems, Cambridge: Cambridge University Press, 2001.
[19]	J. Shen, F. Zeng, M. Stynes, Second-order error analysis of the averaged L1 scheme $\overline{ L1}$ for time-fractional initial-value and subdiffusion problems, Sci. China Math., 2022. https://doi.org/10.1007/s11425-022-2078-4
[20]	A. Stuart, A. Humphries, Dynamical Systems and Numerical Analysis, Cambridge: Cambridge Unversity Press, 1996.
[21]	M. Stynes, A survey of the L1 scheme in the discretisation of time-fractional problems, Numer. Math. Theor. Meth. Appl., 15 (2022), 1173–1192. https://doi.org/10.4208/nmtma.OA-2022-0009s doi: 10.4208/nmtma.OA-2022-0009s
[22]	R. Temam, Infinite-dimensional Dynamical System in Mechanics and Physics, Berlin: Springer, 1997.
[23]	D. Wang, A. Xiao, Dissipativity and contractivity for fractional-order systems, Nonlinear Dyn., 80 (2015), 287–294. https://doi.org/10.1007/s11071-014-1868-1 doi: 10.1007/s11071-014-1868-1
[24]	D. Wang, A. Xiao, H. Liu, Dissipativity and stability analysis for fractional functional differential equations, Fract. Calc. Appl. Anal., 18 (2015), 1399–1422. https://doi.org/10.1515/fca-2015-0081 doi: 10.1515/fca-2015-0081
[25]	D. Wang, A. Xiao, S. Sun, Asymptotic behavior of solutions to time fractional neutral functional differential equations, J. Comput. Appl. Math., 382 (2021), 113086. https://doi.org/10.1016/j.cam.2020.113086 doi: 10.1016/j.cam.2020.113086
[26]	D. Wang, A. Xiao, J. Zou, Long-time behavior of numerical solutions to nonlinear fractional ODEs, ESAIM Math. Model. Numer. Anal., 54 (2020), 335–358. https://doi.org/10.1051/m2an/2019055 doi: 10.1051/m2an/2019055
[27]	D. Wang, J. Zou, Dissipativity and contractivity analysis for fractional functional differential equations and their numerical approximations, SIAM J. Numer. Anal., 57 (2019), 1445–1470. https://doi.org/10.1137/17M1121354 doi: 10.1137/17M1121354
[28]	F. Wang, D. Chen, X. Zhang, Y. Wu, The existence and uniqueness theorem of the solution to a class of nonlinear fractional order system with time delay, Appl. Math. Lett., 53 (2016), 45–51. https://doi.org/10.1016/j.aml.2015.10.001 doi: 10.1016/j.aml.2015.10.001
[29]	W. Wang, C. Zhang, Dissipativity of variable-stepsize Runge- Kutta methods for nonlinear functional differential equations with application to Nicholson's blowflies models, Commun. Nonlinear Sci. Numer. Simulat., 97 (2021), 105723. https://doi.org/10.1016/j.cnsns.2021.105723 doi: 10.1016/j.cnsns.2021.105723
[30]	Z. Wang, X. Huang, G. Shi, Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay, Comput. Math. Appl., 62 (2011), 1531–1539. http://dx.doi.org/10.1016/j.camwa.2011.04.057 doi: 10.1016/j.camwa.2011.04.057
[31]	L. Wen, S. Li, Dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 324 (2006), 696–706. https://doi.org/10.1016/j.jmaa.2005.12.031 doi: 10.1016/j.jmaa.2005.12.031
[32]	L. Wen, Y. Yu, S. Li, Dissipativity of Runge- Kutta methods for Volterra functional differential equations, Appl. Numer. Math., 61 (2011), 368–381. https://doi.org/10.1016/j.apnum.2010.11.002 doi: 10.1016/j.apnum.2010.11.002
[33]	L. Wen, Y. Yu, W. Wang, Generalized Halanay inequalities for dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 347 (2008), 169–178. https://doi.org/10.1016/j.jmaa.2008.05.007 doi: 10.1016/j.jmaa.2008.05.007
[34]	Y. Yan, C. Kou, Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay, Math. Comput. Simul., 82 (2012), 1572–1585. http://dx.doi.org/10.1016/j.matcom.2012.01.004 doi: 10.1016/j.matcom.2012.01.004
[35]	Y. Zhou, M. Stynes, Optimal convergence rates in time-fractional discretisations: the L1, $\overline{L1}$ and Alikhanov schemes, East Asian J. Appl. Math., 12 (2022), 503–520. https://doi.org/10.4208/eajam.290621.220921 doi: 10.4208/eajam.290621.220921

Reader Comments

Your name:*

Email:*
© 2023 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)