A novel adaptive multi-scale wavelet Galerkin method for solving fuzzy hybrid differential equations

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Abstract
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.

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