Computational approaches for analyzing fractional impulsive systems in differential equations

Document Type : Research Paper

Authors

1 Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.

2 Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.

Abstract

This research introduces an algorithmically efficient framework for analyzing the fractional impulsive system, which can be seen as specific instances of the broader fractional Lorenz impulsive system. Notably, these systems find pertinent applications within the financial domain. To this end, the utilization of cubic splines is embraced to effectively approximate the fractional integral within the context of the system. The outcomes derived from this method are subsequently compared with those yielded by alternative techniques documented in existing literature, all pertaining to the integration of functions.
Furthermore, the proposed methodology is not only applied to the resolution of the fractional impulsive system, but also extended to encompass scenarios involving the fractional Lorenz system with impulsive characteristics. The discernible effects stemming from the selection of disparate impulse patterns are meticulously demonstrated. In synthesis, this paper endeavors to present a pragmatic and proficient resolution to the intricate challenges posed by impulsive systems.

Keywords

Main Subjects

References

  • [1] S. Aly, A. Al-Qahtani, H. B. Khenous, and G. M. Mahmoud, Impulsive Control and Synchronization of Complex Lorenz Systems, Abstract and Applied Analysis, 2014 (2014), 1–9.
  • [2] A. Babaei and S. Banihashemi, Reconstructing unknown nonlinear boundary conditions in a time-fractional inverse reaction-diffusion-convection problem, Numerical Methods for Partial Differential Equations, 35(3) (2019), 976–992.
  • [3] S. Banihashemi, H. Jafari, and A. Babaei, A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay, Discrete and Continuous Dynamical Systems - Series S, 10 (2020).
  • [4] A. H. Bukhari, M. A. Z. Raja, M. Shoaib, and A. K. Kiani, Fractional order Lorenz based physics informed SARFIMA-NARX model to monitor and mitigate megacities air pollution, Chaos, Solitons and Fractals, 161 (2022), 112375.
  • [5] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophysical Journal International, 13(5) (1967), 529–539.
  • [6] M. Caputo, Elasticit e dissipazione, Zanichelli, Bologna, Italy, 1969.
  • [7] J. Chen and M. Jiang, Stability of Memristor-based Fractional-order Neural Networks with Mixed Time-delay and Impulsive, Neural Processing Letters, 2022 (2022).
  • [8] C. Djuikem, F. Grognard, and S. Touzeau, Impulsive modelling of rust dynamics and predator releases for biocontrol, Mathematical Biosciences, 356 (2023), 108968.
  • [9] A. Faghih and P. Mokhtary, A new fractional collocation method for a system of multi-order fractional differential equations with variable coefficients, Journal of Computational and Applied Mathematics, 383 (2021), 113139.
  • [10] T. A. Faree and S. K. Panchal, Existence of solution for impulsive fractional differential equations with nonlocal conditions by topological degree theory, Results in Applied Mathematics, 18 (2023), 100377.
  • [11] C. Feng, L. Li, Y. Liu, and Z. Wei, Global Dynamics of the Chaotic Disk Dynamo System Driven by Noise, Complexity, 2020 (2020), 1–9.
  • [12] J. Fu, M. Yu, and T.-D. Ma, Modified impulsive synchronization of fractional order hyperchaotic systems, Chinese Physics B, 20(12) (2011).
  • [13] F. Ghanbari, K. Ghanbari, and P. Mokhtary, Generalized JacobiGalerkin method for nonlinear fractional differential algebraic equations, Computational and Applied Mathematics, 37(4) (2018), 5456–5475.
  • [14] F. Gholami Bahador, P. Mokhtary, and M. Lakestani, Mixed Poisson-Gaussian noise reduction using a time-space fractional differential equations, Information Sciences, 647 (2023), 119417.
  • [15] C. Ionescu, A. Lopes, D. Copot, J. A. T. Machado, and J. H. T. Bates, The role of fractional calculus in modeling biological phenomena: A review, Communications in Nonlinear Science and Numerical Simulation, 51 (2017), 141–159.
  • [16] B. Jalili, P. Jalili, A. Shateri, and D. Domiri Ganji, Rigid plate submerged in a Newtonian fluid and fractional differential equation problems via Caputo fractional derivative, Partial Differential Equations in Applied Mathematics, 6 (2022), 100452.
  • [17] W. Jung and J. H. Lee, Quantile Impulse Response Analysis with Applications in Macroeconomics and Finance, in Essays in Honor of Joon Y. Park: Econometric Methodology in Empirical Applications, Emerald Publishing Limited, (2023), 99–131.
  • [18] R. Karimi, A. Dabiri, J. Cheng, and E. A. Butcher, Probabilistic-Robust Optimal Control for Uncertain Linear Time-delay Systems by State Feedback Controllers with Memory, 2018 Annual American Control Conference (ACC), 2018.
  • [19] A. Kumar, R. K. Vats, A. Kumar, and D. N. Chalishajar, Numerical approach to the controllability of fractional order impulsive differential equations, Demonstratio Mathematica, 53(1) (2020), 193–207.
  • [20] X. Li and S. Song, Impulsive Systems with Delays, Springer Singapore, 2022.
  • [21] Z. Li, L. Chen, and Z. Liu, Periodic solution of a chemostat model with variable yield and impulsive state feedback control, Applied Mathematical Modelling, 36(3) (2012), 1255–1266.
  • [22] E. N. Lorenz, Deterministic Nonperiodic Flow, Journal of the Atmospheric Sciences, 20(2) (1963), 130–141.
  • [23] Y. Guan, Z. Zhao, and X. Lin, On the existence of solutions for impulsive fractional differential equations, Advances in Mathematical Physics, 2017 (2017).
  • [24] R. Ma, J. Wu, K. Wu, and X. Pan, Adaptive fixed-time synchronization of Lorenz systems with application in chaotic finance systems, Nonlinear Dynamics, 109(4) (2022), 3145–3156.
  • [25] A. Martynyuk, G. Stamov, I. Stamova, and E. Gospodinova, Formulation of Impulsive Ecological Systems Using the Conformable Calculus Approach: Qualitative Analysis, Mathematics, 11(10) (2023), 2221.
  • [26] L. Mei, H. Sun, and Y. Lin, Numerical method and convergence order for second-order impulsive differential equations, Advances in Difference Equations, 2019(1) (2019).
  • [27] B. P. Moghaddam, Z. S. Mostaghim, A. A. Pantelous, and J. A. Tenreiro Machado, An integro quadratic spline-based scheme for solving nonlinear fractional stochastic differential equations with constant time delay, Communications in Nonlinear Science and Numerical Simulation, 92 (2021), 105475.
  • [28] B. P. Moghaddam, M. Pishbin, Z. S. Mostaghim, O. S. Iyiola, A. Galhano, and A. M. Lopes, A Numerical Algorithm for Solving Nonlocal Nonlinear Stochastic Delayed Systems with Variable-Order Fractional Brownian Noise, Fractal and Fractional, 7(4) (2023), 293.
  • [29] Z. Moniri, B. P. Moghaddam, and M. Zamani Roudbaraki, An Efficient and Robust Numerical Solver for Impulsive Control of Fractional Chaotic Systems, Journal of Function Spaces, 2023 (2023), 1–10.
  • [30] Z. S. Mostaghim, B. P. Moghaddam, and H. S. Haghgozar, Computational technique for simulating variable-order fractional Heston model with application in US stock market, Mathematical Sciences, 12(4) (2018), 277–283.
  • [31] Z. S. Mostaghim, B. P. Moghaddam, and H. S. Haghgozar, Numerical simulation of fractional-order dynamical systems in noisy environments, Computational and Applied Mathematics, 37(5) (2018), 6433–6447.
  • [32] G. Narayanan, M. S. Ali, R. Karthikeyan, G. Rajchakit, and A. Jirawattanapanit, Impulsive control strategies of mRNA and protein dynamics on fractional-order genetic regulatory networks with actuator saturation and its oscillations in repressilator model, Biomedical Signal Processing and Control, 82 (2023), 104576.
  • [33] N. Nyamoradi, Existence of solutions for a class of second-order differential equations with impulsive effects, Mathematical Methods in the Applied Sciences, 38(18) (2015), 5023–5033.
  • [34] Z. Odibat, V. S. Erturk, P. Kumar, A. B. Makhlouf, and V. Govindaraj, An Implementation of the Generalized Differential Transform Scheme for Simulating Impulsive Fractional Differential Equations, Mathematical Problems in Engineering, 2022 (2022), 1–11.
  • [35] B. P. Moghaddam, A. Dabiri, Z. S. Mostaghim, and Z. Moniri, Numerical solution of fractional dynamical systems with impulsive effects, International Journal of Modern Physics C, 34(01) (2023).
  • [36] B. P. Moghaddam, A. Dabiri, and J. A. T. Machado, Application of variable-order fractional calculus in solid mechanics, in D. Bleanu and A. Mendes Lopes (eds.), Applications in Engineering, Life and Social Sciences, Part A, De Gruyter, 2019, 207–224.
  • [37] X. J. Ran, M. Z. Liu, and Q. Y. Zhu, Numerical methods for impulsive differential equation, Mathematical and Computer Modelling, 48(1-2) (2008), 46–55.
  • [38] D. Raghavan and S. Nagarajan, Convergence on iterative learning control of Hilfer fractional impulsive evolution equations, Asian Journal of Control, 25(3) (2022), 2183–2193.
  • [39] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, 1993.
  • [40] A. Shahnazi-Pour, B. P. Moghaddam, and A. Babaei, Numerical simulation of the Hurst index of solutions of fractional stochastic dynamical systems driven by fractional Brownian motion, Journal of Computational and Applied Mathematics, 386 (2021), 113210.
  • [41] I. Stamova and G. Stamov, Applied Impulsive Mathematical Models, Springer International Publishing, 2016.
  • [42] G. Stamov and I. Stamova, Extended Stability and Control Strategies for Impulsive and Fractional Neural Networks: A Review of the Recent Results, Fractal and Fractional, 7(4) (2023), 289.
  • [43] I. M. Stamova and A. G. Stamov, Impulsive control on the asymptotic stability of the solutions of a Solow model with endogenous labor growth, Journal of the Franklin Institute, 349(8) (2012), 2704–2716.
  • [44] I. Stamova and J. Henderson, Practical stability analysis of fractional-order impulsive control systems, ISA Transactions, 64 (2016), 77–85.
  • [45] Y. Talaei, S. Shahmorad, and P. Mokhtary, A new recursive formulation of the Tau method for solving linear Abel-Volterra integral equations and its application to fractional differential equations, Calcolo, 56(4) (2019), 1–20.
  • [46] U. S. Thounaojam, Stochastic chaos in chemical Lorenz system: Interplay of intrinsic noise and nonlinearity, Chaos, Solitons and Fractals, 165 (2022), 112763.
  • [47] Q. Wang and M. Wang, Existence of solution for impulsive differential equations with indefinite linear part, Applied Mathematics Letters, 51 (2016), 41–47.
  • [48] G.-C. Wu, D. Q. Zeng, and D. Baleanu, Fractional Impulsive Differential Equations: Exact Solutions, Integral Equations and Short Memory Case, Fractional Calculus and Applied Analysis, 22(1) (2019), 180–192.
  • [49] X. Wu, J. A. Lu, C. K. Tse, J. Wang, and J. Liu, Impulsive control and synchronization of the Lorenz systems family, Chaos, Solitons and Fractals, 31(3) (2007), 631–638.
  • [50] X. Yang, Existence and multiplicity of weak solutions for a nonlinear impulsive (q, p)-Laplacian dynamical system, Advances in Difference Equations, 2017(1) (2017).
  • [51] X. Zhang, D. Li, and X. Zhang, Adaptive impulsive synchronization for a class of fractional order complex chaotic systems, Journal of Vibration and Control, 25(10) (2019), 1614–1628.
  • [52] Y. Zhao, C. Luo, and H. Chen, Existence Results for Non-instantaneous Impulsive Nonlinear Fractional Differential Equation Via Variational Methods, Bulletin of the Malaysian Mathematical Sciences Society, 43(3) (2019), 2151–2169.

 

Computational Methods for Differential Equations
Volume 13, Issue 2
March 2025
Pages 384-394

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History

  • Receive Date: 27 April 2024
  • Revise Date: 09 June 2024
  • Accept Date: 25 June 2024

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