A Convergent and Stable Framework for the Fractional Kuramoto–Sivashinsky Equation

Abstract

This work presents an efficient analytical framework based on the Natural Residual Power Series Method (NRPSM) for solving several forms of the time-fractional Kuramoto–Sivashinsky equation with the Caputo derivative. The proposed method avoids discretization and linearization while producing rapidly convergent analytical series solutions. Earlier residual power series treatments assert convergence under a contractivity assumption without verifying it for the equation at hand. We close this gap by deriving an explicit formula for the contraction constant directly from the problem data, so the convergence criterion is checkable before any computation begins. A rigorous theoretical analysis is established through explicit contraction conditions, convergence proofs in the Sobolev space H4(R), and an explicit geometric-type error estimate that quantifies how the fractional order governs the convergence rate through two competing effects, without presuming a uniform direction of influence. Stability with respect to perturbations in the initial data is also proven using a fractional Gronwall inequality. Numerical results demonstrate excellent agreement with exact and previously published solutions, achieving very small absolute errors using only a few series terms. The obtained results confirm that the NRPSM is an accurate, stable, and computationally efficient approach for nonlinear fractional evolution equations.