A Tight Wavelet Frames‐Based Method for Numerically Solving Fractional Riccati Differential Equations

ABSTRACT This paper introduces an innovative numerical framework for solving fractal‐type fractional Riccati differential equations, utilizing tight wavelet frames constructed from Coiflet wavelet scaling functions. Central to this approach is a novel fractal‐type fractional derivative, meticulously designed using integral operators to encapsulate the intricate, self‐similar properties of fractal structures. By bridging the gap between traditional fractional calculus and systems with fractal dynamics, this derivative represents a significant advancement in mathematical modeling. Integrated into a sophisticated numerical scheme, the proposed method demonstrates exceptional accuracy and computational efficiency, surpassing existing techniques such as Legendre‐Galerkin and spline‐based methods. The advantages of using tight wavelet frames include their robustness in handling complex solution behaviors and their ability to capture subtle features in fractional systems, making the method particularly well‐suited for real‐world applications in science and engineering. This work not only provides a powerful tool for addressing complex fractional dynamics but also paves the way for broader applications of fractional calculus in a variety of domains.