# American Institute of Mathematical Sciences

• Previous Article
Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain
• EECT Home
• This Issue
• Next Article
The influence of the physical coefficients of a Bresse system with one singular local viscous damping in the longitudinal displacement on its stabilization
doi: 10.3934/eect.2022012
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Impulsive conformable fractional stochastic differential equations with Poisson jumps

 Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt

* Corresponding author: Hamdy M. Ahmed

Received  August 2021 Revised  January 2022 Early access March 2022

In this article, periodic averaging method for impulsive conformable fractional stochastic differential equations with Poisson jumps are discussed. By using stochastic analysis, fractional calculus, Doob's martingale inequality and Cauchy-Schwarz inequality, we show that the solution of the conformable fractional impulsive stochastic differential equations with Poisson jumps converges to the corresponding averaged conformable fractional stochastic differential equations with Poisson jumps and without impulses.

Citation: Hamdy M. Ahmed. Impulsive conformable fractional stochastic differential equations with Poisson jumps. Evolution Equations and Control Theory, doi: 10.3934/eect.2022012
##### References:
 [1] R. Almeida, N. R. Bastos, M. Teresa and T. Monteiro, Modeling some real phenomena by fractional differential equations, Mathematical Methods in the Applied Sciences, 39 (2016), 4846-4855.  doi: 10.1002/mma.3818. [2] S. Qureshi, A. Yusuf, A. A. Shaikh, M. Inc and D. Baleanu, Fractional modeling of blood ethanol concentration system with real data application, An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013143.  doi: 10.1063/1.5082907. [3] D. Baleanu, H. Mohammadi and S. Rezapour, A fractional differential equation model for the COVID-19 transmission by using the Caputo-Fabrizio derivative, Advances in difference equations, 2020 (2020), 1-27. [4] E. Bas and R. Ozarslan, Real world applications of fractional models by Atangana-Baleanu fractional derivative, IMA Chaos, Solitons and Fractals, 116 (2018), 121-125.  doi: 10.1016/j.chaos.2018.09.019. [5] R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002. [6] M. Han, Y. Xu and B. Pei, A mixed stochastic differential equations: Averaging principle result, Applied Mathematics Letters, 112 (2021), 106705.  doi: 10.1016/j.aml.2020.106705. [7] B. Pei, Y. Xu and J. L. Wu, Stochastic averaging for stochastic differential equations driven by fractional Brownian motion and standard Brownian motion, Applied Mathematics Letters, 100 (2020), 106006.  doi: 10.1016/j.aml.2019.106006. [8] J. Cui and N. Bi, Averaging principle for neutral stochastic functional differential equations with impulses and non-Lipschitz coefficients, Statistics and Probability Letters, 163 (2020), 108775.  doi: 10.1016/j.spl.2020.108775. [9] J. K. Liu, W. Xu and Q. Guo, Averaging principle for impulsive stochastic partial differential equations, Stoch. Dynam., 21 (2021), 2150014.  doi: 10.1142/S0219493721500143. [10] W. Mao and X. Mao, An averaging principle for neutral stochastic functional differential equations driven by Poisson random measure, Adv. Difference Equ., 2016 (2016), 77 pp. doi: 10.1088/1751-8113/44/33/335201. [11] H. M. Ahmed and Q. Zhu, The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jumps, Applied Mathematics Letters, 112 (2021), 106755.  doi: 10.1016/j.aml.2020.106755. [12] J. Liu and W. Xu, An averaging result for impulsive fractional neutral stochastic differential equations, Applied Mathematics Letters, 114 (2021), 106892.  doi: 10.1016/j.aml.2020.106892. [13] W. J. Xu, W. Xu and S. Zhang, The averaging principle for stochastic differential equations with Caputo fractional derivative, Applied Mathematics Letters, 93 (2019), 79-84.  doi: 10.1016/j.aml.2019.02.005. [14] D. F. Luo, Q. X. Zhu and Z. G. Luo, An averaging principle for stochastic fractional differential equations with time delays, Applied Mathematics Letters, 105 (2020), 106290.  doi: 10.1016/j.aml.2020.106290. [15] M. Hannabou, K. Hilal and A. Kajouni, Existence and uniqueness of mild solutions to impulsive nonlocal Cauchy problems, Journal of Mathematics, 2020 (2020), 5729128. [16] L. Shen and J. T. Sun, Global existence of solutions for stochastic impulsive differential equations, Acta Mathematica Sinica, 27 (2011), 773-780.  doi: 10.1007/s10114-011-8650-9.

show all references

##### References:
 [1] R. Almeida, N. R. Bastos, M. Teresa and T. Monteiro, Modeling some real phenomena by fractional differential equations, Mathematical Methods in the Applied Sciences, 39 (2016), 4846-4855.  doi: 10.1002/mma.3818. [2] S. Qureshi, A. Yusuf, A. A. Shaikh, M. Inc and D. Baleanu, Fractional modeling of blood ethanol concentration system with real data application, An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013143.  doi: 10.1063/1.5082907. [3] D. Baleanu, H. Mohammadi and S. Rezapour, A fractional differential equation model for the COVID-19 transmission by using the Caputo-Fabrizio derivative, Advances in difference equations, 2020 (2020), 1-27. [4] E. Bas and R. Ozarslan, Real world applications of fractional models by Atangana-Baleanu fractional derivative, IMA Chaos, Solitons and Fractals, 116 (2018), 121-125.  doi: 10.1016/j.chaos.2018.09.019. [5] R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002. [6] M. Han, Y. Xu and B. Pei, A mixed stochastic differential equations: Averaging principle result, Applied Mathematics Letters, 112 (2021), 106705.  doi: 10.1016/j.aml.2020.106705. [7] B. Pei, Y. Xu and J. L. Wu, Stochastic averaging for stochastic differential equations driven by fractional Brownian motion and standard Brownian motion, Applied Mathematics Letters, 100 (2020), 106006.  doi: 10.1016/j.aml.2019.106006. [8] J. Cui and N. Bi, Averaging principle for neutral stochastic functional differential equations with impulses and non-Lipschitz coefficients, Statistics and Probability Letters, 163 (2020), 108775.  doi: 10.1016/j.spl.2020.108775. [9] J. K. Liu, W. Xu and Q. Guo, Averaging principle for impulsive stochastic partial differential equations, Stoch. Dynam., 21 (2021), 2150014.  doi: 10.1142/S0219493721500143. [10] W. Mao and X. Mao, An averaging principle for neutral stochastic functional differential equations driven by Poisson random measure, Adv. Difference Equ., 2016 (2016), 77 pp. doi: 10.1088/1751-8113/44/33/335201. [11] H. M. Ahmed and Q. Zhu, The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jumps, Applied Mathematics Letters, 112 (2021), 106755.  doi: 10.1016/j.aml.2020.106755. [12] J. Liu and W. Xu, An averaging result for impulsive fractional neutral stochastic differential equations, Applied Mathematics Letters, 114 (2021), 106892.  doi: 10.1016/j.aml.2020.106892. [13] W. J. Xu, W. Xu and S. Zhang, The averaging principle for stochastic differential equations with Caputo fractional derivative, Applied Mathematics Letters, 93 (2019), 79-84.  doi: 10.1016/j.aml.2019.02.005. [14] D. F. Luo, Q. X. Zhu and Z. G. Luo, An averaging principle for stochastic fractional differential equations with time delays, Applied Mathematics Letters, 105 (2020), 106290.  doi: 10.1016/j.aml.2020.106290. [15] M. Hannabou, K. Hilal and A. Kajouni, Existence and uniqueness of mild solutions to impulsive nonlocal Cauchy problems, Journal of Mathematics, 2020 (2020), 5729128. [16] L. Shen and J. T. Sun, Global existence of solutions for stochastic impulsive differential equations, Acta Mathematica Sinica, 27 (2011), 773-780.  doi: 10.1007/s10114-011-8650-9.
 [1] Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272 [2] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [3] David Cheban, Zhenxin Liu. Averaging principle on infinite intervals for stochastic ordinary differential equations. Electronic Research Archive, 2021, 29 (4) : 2791-2817. doi: 10.3934/era.2021014 [4] Teresa Faria, Rubén Figueroa. Positive periodic solutions for systems of impulsive delay differential equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022070 [5] Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems and Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025 [6] Xinping Zhou, Yong Li, Xiaomeng Jiang. Periodic solutions in distribution of stochastic lattice differential equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022123 [7] Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667 [8] Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031 [9] Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025 [10] Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 723-739. doi: 10.3934/dcdss.2020040 [11] Yinuo Wang, Chuandong Li, Hongjuan Wu, Hao Deng. Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary value. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1767-1776. doi: 10.3934/dcdss.2022005 [12] Jehad O. Alzabut. A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay. Conference Publications, 2007, 2007 (Special) : 35-43. doi: 10.3934/proc.2007.2007.35 [13] Teresa Faria, José J. Oliveira. On stability for impulsive delay differential equations and application to a periodic Lasota-Wazewska model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2451-2472. doi: 10.3934/dcdsb.2016055 [14] Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 [15] Monica Motta, Caterina Sartori. Generalized solutions to nonlinear stochastic differential equations with vector--valued impulsive controls. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 595-613. doi: 10.3934/dcds.2011.29.595 [16] Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353 [17] Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037 [18] Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6483-6510. doi: 10.3934/dcdsb.2021030 [19] Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197 [20] Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257

2021 Impact Factor: 1.169

Article outline