Murugesh V, Priyadharshini M, Sharma Yogesh Kumar, Aldossary Sultan Mesfer, Kundu Shakti, Malik Saksham, Adem Kedir Botamo, Hashmi Arshad
Department of CSE, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur, AP, India.
Department of Computer Science & Engineering, Faculty of Science and Technology (IcfaiTech), ICFAI Foundation for Higher Education, Hyderabad, 501203, India.
Sci Rep. 2025 Jul 28;15(1):27510. doi: 10.1038/s41598-025-11915-w.
Complex dynamical systems are represented by fuzzy hybrid differential equations (FHDEs), which describe systems that have mixed discrete and continuous behaviours with uncertainty. These equations are indispensable for control engineering, biology, and economic forecasting, as they model real-world phenomena. Nevertheless, it is intrinsically challenging to solve FHDEs because the dynamics, discontinuities, and uncertainties in parameters and conditions are all nonlinear and fuzzy. Traditional numerical methods, such as the Runge-Kutta-Fehlberg (RKF5) method, Finite Difference Methods (FDM), and spectral approaches, generally fail to provide accurate, stable, and efficient solutions, especially in problems with discontinuities, sharp transitions, or the propagation of uncertainty. The primary objective of this paper is to introduce a novel scheme called the AMWWG, which effectively solves FHDEs. AMWG method combines the advantages of wavelet-based multi-resolution analysis with those of the Galerkin projection technique. The method utilises local error estimates to refine the solution domain, adapting to the solution characteristics: fine refinement is used in areas of steep gradients, discontinuities, and fuzzy transitions, while coarse refinement is used everywhere else. The selective refinement approach enables the method to utilise minimal computational efforts where they are most needed, resulting in a significant order of magnitude reduction in computational cost with no loss in solution accuracy. Numerical experiments are performed, yielding extensive results reported on several benchmark FHDEs, which are corroborated with results from known analytical solutions and other complex examples with high nonlinearity and discrete switching behaviours. It is demonstrated that the AMWG method is more accurate, has lower memory requirements, and is faster than traditional methods. Additionally, it demonstrates a superior ability to handle both fuzzy uncertainty and sharp transitions. Thus, the AMWG method is demonstrated to be a powerful, flexible, and scalable numerical tool for solving FHDEs, offering a high degree of flexibility and significant potential for application in large-scale scientific or engineering problems.
复杂动力系统由模糊混合微分方程（FHDEs）表示，这些方程描述了具有混合离散和连续行为且带有不确定性的系统。由于这些方程能够对现实世界的现象进行建模，所以它们在控制工程、生物学和经济预测中不可或缺。然而，求解FHDEs本质上具有挑战性，因为参数和条件中的动力学、不连续性以及不确定性都是非线性且模糊的。传统数值方法，如龙格 - 库塔 - 费尔贝格（RKF5）方法、有限差分法（FDM）和谱方法，通常无法提供准确、稳定且高效的解，特别是在存在不连续性、急剧转变或不确定性传播的问题中。本文的主要目标是引入一种名为AMWWG的新颖方案，该方案能有效求解FHDEs。AMWG方法将基于小波的多分辨率分析的优点与伽辽金投影技术的优点相结合。该方法利用局部误差估计来细化求解域，以适应解的特征：在梯度陡峭、不连续和模糊转变的区域使用精细细化，而在其他地方使用粗细化。这种选择性细化方法使该方法能够在最需要的地方使用最少的计算量，从而在不损失解精度的情况下显著降低计算成本。进行了数值实验，得到了关于几个基准FHDEs的大量结果，这些结果与已知解析解以及其他具有高非线性和离散切换行为的复杂示例的结果相佐证。结果表明，AMWG方法比传统方法更准确、内存需求更低且速度更快。此外，它在处理模糊不确定性和急剧转变方面表现出卓越的能力。因此，AMWG方法被证明是一种用于求解FHDEs的强大、灵活且可扩展的数值工具，在大规模科学或工程问题中具有高度的灵活性和巨大的应用潜力。
