Extended Legendre Wavelet Methods for Advanced Differential Equations

Legendre wavelet–based approximation techniques have shown promising performance in solving differential equations and function approximation problems. However, existing studies are largely restricted to low-dimensional, deterministic models with fixed-resolution schemes and limited theoretical analysis. This paper addresses these gaps by proposing an extended Legendre wavelet framework that incorporates rigorous error analysis, adaptive multiresolution strategies, and applicability to nonlinear and fractional-order differential equations. The proposed approach enhances accuracy, stability, and computational efficiency while broadening the scope of Legendre wavelet methods to more realistic and complex mathematical models. Numerical experiments demonstrate the superiority of the extended framework over classical fixed-scale Legendre wavelet approximations.