An innovative graph polynomial numerical strategy for the time-fractional Kuramoto-Sivashinsky model based on the complete bipartite graph's independence polynomials

The time-fractional Kuramoto-Sivashinsky equation (TF-K-SE) models chaotic dynamics with memory effects, necessitating advanced computational techniques for stability and accuracy. Conventional methods frequently encounter instability, excessive computational costs, and dependence on transformation techniques, limiting their effectiveness in handling fractional derivatives. This study introduces the Independence Polynomial Collocation Method (ICCM), leveraging graph-theoretic independence polynomials to enhance spectral accuracy and computational efficiency. Constructing sparse operational matrices, ICCM optimizes spectral discretization while preserving nonlinear characteristics, ensuring a more stable numerical framework. Caputo fractional derivatives capture memory-dependent dynamics and provide smooth transitions between fractional and integer orders without sacrificing physical fidelity. Unlike conventional orthogonal polynomials, which require increasing degrees for refinement and dense matrix representations, ICCM introduces a graph-theoretic spectral basis, improving smoothness, maintaining linear independence, and enhancing stability. A key advantage of ICCM is its independence from controlling parameters, distinguishing it from conventional semi-analytical methods that rely on stabilization terms or artificial tuning. ICCM achieves intrinsic numerical stability without external adjustments, making it a computationally efficient alternative while accurately preserving the chaotic nature of TF-K-SE. Numerical validation across four test cases demonstrates robust spectral stability, with error norms confirming precision and smooth transitions across fractional orders. ICCM establishes a computationally efficient framework for fractional PDE modeling and nonlinear system analysis, offering a novel approach distinct from existing techniques.