Analysis of the 2D Wavelets Technique for the Coupled Nonlinear Variable-Order Time-Fractional Sine-Gordon Equations in Complex Dynamical Systems

This paper presents a novel approach for solving coupled nonlinear variable-order time-fractional sine-Gordon equations (CNVOTF-SGEs) by employing two-dimensional wavelets. When the solu tion of a fractional differential equation is non-smooth, weakly regular, or singular, polynomialbased methods encounter significant challenges, as polynomials are not well-suited for approximating non smooth functions. In contrast, wavelet-based methods can approximate such functions with both accuracy and efficiency. Wavelet-based techniques often produce near-diagonal matrices, such as integral operational matrices, which significantly accelerate computations by reducing the overall computational time. Wavelets can be easily extended to 2D and 3D, which makes them highly ef fective for solving multidimensional fractional partial differential equations. The original equations are reduced to an algebraic system of equations (ASEs) by approximating the unknown functions with Jacobi and Chebyshev cardinal wavelets and by applying operational matrices together with collocation points. After solving the ASEs, we observed that the numerical results were better when using JWs compared to CCWs. Theoretical results on convergence and error estimation of the proposed method are also established. Several illustrative examples confirm the accuracy, efficiency, and applicability of the two-dimensional wavelet technique.