High-order adaptive solutions for coupled fractional Riccati equations via Daubechies redundant frames

Source abstract

Coupled fractional Riccati equations play a fundamental role in modeling complex systems with memory effects and anomalous diffusion, frequently arising in engineering, control theory, finance, and quantum mechanics. Their analytical and numerical treatment remains highly challenging due to the nonlocal nature of fractional-order derivatives and the presence of nonlinear coupling terms. This study introduces an adaptive numerical framework that combines the Caputo fractional derivative with redundant Daubechies wavelet frames. The method leverages multi-resolution analysis, compact support, and controlled redundancy to achieve accurate approximation of both localized and global solution features, particularly in scenarios characterized by singular behavior and long-range memory effects. Comprehensive numerical experiments show that the proposed scheme provides superior accuracy, stability, and computational efficiency compared to conventional approaches. These results underscore the effectiveness of wavelet-based techniques in solving fractional differential systems and highlight their promise for advancing real-world modeling and simulation.