journal article
Open Access
Jan 07, 2026
Qualitative Analysis and Applications of Fractional Stochastic Systems with Non-Instantaneous Impulses
Abstract
Fractional stochastic differential Equations (FSDEs) with time delays and non-instantaneous impulses describe dynamical systems whose evolution relies not only on their current state but also on their historical context, random fluctuations, and impulsive effects that manifest over finite intervals rather than occurring instantaneously. This combination of features offers a more precise framework for capturing critical aspects of many real-world processes. Recent findings demonstrate the existence, uniqueness, and Ulam–Hyers stability of standard fractional stochastic systems. In this study, we extend these results to include systems characterized by FSDEs that incorporate time delays and non-instantaneous impulses. We prove the existence and uniqueness of the solution for this system using Krasnoselskii’s and Banach’s fixed-point theorems. Additionally, we present findings related to Ulam–Hyers stability. To illustrate the practical application of our results, we develop a population model that incorporates memory effects, randomness, and non-instantaneous impulses. This model is solved numerically via the Euler–Maruyama method, and graphical simulations effectively depict the dynamic behavior of the system.
Topics
No keywords indexed for this article. Browse by subject →
References
43
[1]
Djaouti, A.M., and Liaqat, M.I. (2025). Theoretical Results on the pth Moment of φ-Hilfer Stochastic Fractional Differential Equations with a Pantograph Term. Fractal Fract., 9.
10.3390/fractalfract9030134
[2]
Mannan "Dynamic analysis of ebola virus disease with non-linear incidence rate using morlet wavelet neural networks and hybrid optimization techniques" Model. Earth Syst. Environ. (2025) 10.1007/s40808-024-02256-0
[3]
Jafari "Theoretical Study of Fractional Differential Equations Under Uncertainty" Fractals (2025)
[4]
Chakrabarty "Dynamical analysis of optical soliton solutions for CGL equation with Kerr law nonlinearity in classical, truncated M-fractional derivative, beta fractional derivative, and conformable fractional derivative types" Results Phys. (2024) 10.1016/j.rinp.2024.107636
[5]
Mohammed Djaouti, A., Khan, Z.A., Liaqat, M.I., and Al-Quran, A. (2024). Existence, Uniqueness, and Averaging Principle of Fractional Neutral Stochastic Differential Equations in the Lp Space with the Framework of the Ψ-Caputo Derivative. Mathematics, 12.
10.3390/math12071037
[6]
Vu "Fuzzy fractional differential equations under Caputo–Katugampola fractional derivative approach" Fuzzy Sets Syst. (2019) 10.1016/j.fss.2018.08.001
[7]
Boucenna, D., Makhlouf, A.B., Naifar, O., Guezane-Lakoud, A., and Hammami, M.A. (2018). Linearized stability analysis of Caputo-Katugampola fractional-order nonlinear systems. arXiv.
10.23952/jnfa.2018.27
[8]
Sweilam "Numerical solutions of fractional optimal control with Caputo-Katugampola derivative" Adv. Differ. Equ. (2021) 10.1186/s13662-021-03580-w
[9]
Ma "Analysis of Caputo–Katugampola fractional differential system" Eur. Phys. J. Plus (2024) 10.1140/epjp/s13360-024-04949-y
[10]
Basti, B., Hammami, N., Berrabah, I., Nouioua, F., Djemiat, R., and Benhamidouche, N. (2021). Stability analysis and existence of solutions for a modified SIRD model of COVID-19 with fractional derivatives. Symmetry, 13.
10.3390/sym13081431
[11]
Srivastava "Fractional-order derivatives and integrals: Introductory overview and recent developments" Kyungpook Math. J. (2020)
[12]
Redhwan "Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type" Aims Math. (2020) 10.3934/math.2020240
[13]
Al-Ghafri, K.S., Alabdala, A.T., Redhwan, S.S., Bazighifan, O., Ali, A.H., and Iambor, L.F. (2023). Symmetrical solutions for non-local fractional integro-differential equations via caputo–katugampola derivatives. Symmetry, 15.
10.3390/sym15030662
[14]
Singh "Fractional dynamics and analysis of coupled Schrödinger-KdV equation with Caputo-Katugampola type memory" J. Comput. Nonlinear Dyn. (2023) 10.1115/1.4062391
[15]
Sajedi "Impulsive coupled system of fractional differential equations with Caputo–Katugampola fuzzy fractional derivative" J. Math. (2021) 10.1155/2021/7275934
[16]
Srivastava "Existence of Solution for a Katugampola Fractional Differential Equation Using Coincidence Degree Theory" Mediterr. J. Math. (2024) 10.1007/s00009-024-02658-5
[17]
Liaqat, M.I., Akgül, A., and Abu-Zinadah, H. (2023). Analytical investigation of some time-fractional Black–Scholes models by the Aboodh residual power series method. Mathematics, 11.
10.3390/math11020276
[18]
Liaqat "Significant findings Concerning a specific class of stochastic differential equations with fractional-order" Fractals (2025)
[19]
Kahouli "Stability results for neutral fractional stochastic differential equations" AIMS Math. (2024) 10.3934/math.2024158
[20]
Ali "Qualitative analysis of fractional stochastic differential equations with variable order fractional derivative" Qual. Theory Dyn. Syst. (2024) 10.1007/s12346-024-00982-5
[21]
Raheem, A., Alamrani, F.M., Akhtar, J., Alatawi, A., Alshaban, E., Khatoon, A., and Khan, F.A. (2024). Study on Controllability for ψ-Hilfer Fractional Stochastic Differential Equations. Fractal Fract., 8.
10.3390/fractalfract8120727
[22]
Ramkumar "Fractional neutral stochastic differential equations with Caputo fractional derivative: Fractional Brownian motion, Poisson jumps, and optimal control" Stoch. Anal. Appl. (2021) 10.1080/07362994.2020.1789476
[23]
Mohammed Djaouti, A., and Imran Liaqat, M. (2024). Qualitative analysis for the solutions of fractional stochastic differential equations. Axioms, 13.
10.3390/axioms13070438
[24]
Moualkia "Mittag–Leffler–Ulam stabilities for variable fractional-order differential equations driven by Lévy noise" Nonlinear Dyn. (2025) 10.1007/s11071-024-10859-6
[25]
Moualkia "An averaging result for fractional variable-order neutral differential equations with variable delays driven by Markovian switching and Lévy noise" Chaos Solitons Fractals (2024) 10.1016/j.chaos.2024.114795
[26]
Ali, Z., Abebe, M.A., and Nazir, T. (2024). Strong Convergence of Euler–Type Methods for Nonlinear Fractional Stochastic Differential Equations without Singular Kernel. Mathematics, 12.
10.3390/math12182890
[27]
Lavanya "Analysis of controllability in Caputo–Hadamard stochastic fractional differential equations with fractional Brownian motion" Int. J. Dyn. Control (2024) 10.1007/s40435-023-01244-z
[28]
Moualkia, S., and Xu, Y. (2021). On the existence and uniqueness of solutions for multidimensional fractional stochastic differential equations with variable order. Mathematics, 9.
10.3390/math9172106
[29]
Liping "A new financial chaotic model in Atangana-Baleanu stochastic fractional differential equations" Alex. Eng. J. (2021) 10.1016/j.aej.2021.04.023
[30]
Abouagwa "Impulsive stochastic fractional differential equations driven by fractional Brownian motion" Adv. Differ. Equ. (2020) 10.1186/s13662-020-2533-2
[31]
Asadzade "Finite time stability analysis for fractional stochastic neutral delay differential equations" J. Appl. Math. Comput. (2024) 10.1007/s12190-024-02174-5
[32]
Ahmed "Hilfer fractional stochastic integro-differential equations" Appl. Math. Comput. (2018) 10.1016/j.amc.2018.03.009
[33]
Abuasbeh, K., Mahmudov, N.I., and Awadalla, M. (2023). Fractional Stochastic Integro-Differential Equations with Nonintantaneous Impulses: Existence, Approximate Controllability and Stochastic Iterative Learning Control. Fractal Fract., 7.
10.3390/fractalfract7010087
[34]
Dhanalakshmi "Exponential stability of second-order fractional stochastic integro-differential equations" Filomat (2023) 10.2298/fil2309699d
[35]
Badawi "Fractional conformable stochastic integrodifferential equations: Existence, uniqueness, and numerical simulations utilizing the shifted Legendre spectral collocation algorithm" Math. Probl. Eng. (2022) 10.1155/2022/5104350
[36]
Cui "Existence result for fractional neutral stochastic integro-differential equations with infinite delay" J. Phys. A Math. Theor. (2011) 10.1088/1751-8113/44/33/335201
[37]
Garrappa "Numerical evaluation of two and three parameter Mittag–Leffler functions" Siam J. Numer. Anal. (2015) 10.1137/140971191
[38]
Singh "Dynamical analysis of fractional order biological population model with carrying capacity under Caputo-Katugampola memory" Alex. Eng. J. (2024) 10.1016/j.aej.2024.02.005
[39]
Pedjeu "Stochastic fractional differential equations: Modeling, method and analysis" Chaos Solitons Fractals (2012) 10.1016/j.chaos.2011.12.009
[40]
Wang "Ulam-Hyers stability of caputo type fuzzy fractional differential equations with time-delays" Chaos Solitons Fractals (2022) 10.1016/j.chaos.2022.111822
[41]
Fec "On the concept and existence of solution for impulsive fractional differential equations" Commun. Nonlinear Sci. Numer. Simul. (2012) 10.1016/j.cnsns.2011.11.017
[42]
Guo "The existence and Hyers–Ulam stability of solution for an impulsive Riemann–Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1 < β < 2" Bound. Value Probl. (2019) 10.1186/s13661-019-1172-6
[43]
Dai "Stability of Ulam–Hyers and Ulam–Hyers–Rassias for a class of fractional differential equations" Adv. Differ. Equ. (2020) 10.1186/s13662-020-02558-4
Metrics
1
Citations
43
References
Details
- Published
- Jan 07, 2026
- Vol/Issue
- 14(2)
- Pages
- 224
- License
- View
Authors
Funding
Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia
Award: KFU260076
Cite This Article
Muhammad Imran Liaqat, ABDELHAMID MOHAMMED DJAOUTI (2026). Qualitative Analysis and Applications of Fractional Stochastic Systems with Non-Instantaneous Impulses. Mathematics, 14(2), 224. https://doi.org/10.3390/math14020224
Related
You May Also Like
Mitigating the Multicollinearity Problem and Its Machine Learning Approach: A Review
Jireh Yi-Le Chan, Steven Mun Hong Leow · 2022
463 citations
Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial
Roberto Garrappa · 2018
418 citations
Survey on Synthetic Data Generation, Evaluation Methods and GANs
Alvaro Figueira, Bruno Vaz · 2022
344 citations
Roles and Research Trends of Artificial Intelligence in Mathematics Education: A Bibliometric Mapping Analysis and Systematic Review
Gwo-Jen Hwang, Yun-Fang Tu · 2021
271 citations
Auto-Encoders in Deep Learning—A Review with New Perspectives
Shuangshuang Chen, Wei Guo · 2023
265 citations