Spectral collocation approach for solving the time-fractional Kuramoto-Sivashinsky equation using the Fibonacci coefficient polynomials

This work presents an algorithm that applies the typical collocation method to solve the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Our suggested new basis functions are the Fibonacci coefficient polynomials. We develop new theoretical results of these polynomials, such as their integer and fractional derivatives. The proposed approach efficiently estimates spatial and temporal derivatives using the operational matrices (OMs) of derivatives of the introduced polynomials. The approach converts the TFKSE controlled by their conditions into a non-linear system of equations that may be handled numerically. A thorough error and convergence analysis study for the proposed Fibonacci coefficient expansion is presented. Several illustrative numerical examples, including those with known exact solutions, are presented to confirm the efficiency and applicability of the suggested approach, even with fewer terms of basis functions. In addition, comparisons with some numerical methods are presented to verify our numerical approach's high accuracy.