A Fast High-Order Compact Difference Scheme for Time-Fractional KS Equation with the Generalized Burgers’ Type Nonlinearity

Abstract

This work integrates the fast Alikhanov method with a compact scheme to solve the time-fractional Kuramoto–Sivashinsky (KS) equation with the generalized Burgers’ type nonlinearity. Initially, the Alikhanov algorithm, designed to handle the Caputo fractional derivative on non-uniform time grids, effectively avoids the initial singularity. Additionally, the combination of the Alikhanov method with the sum-of-exponentials (SOE) technique significantly reduces both computational cost and memory requirements. By discretizing the spatial direction using a compact finite difference method, a fully discrete scheme is developed, achieving fourth-order convergence in the spatial domain. Stability and convergence are analyzed through energy methods. Several numerical examples are provided to validate the theoretical framework, demonstrating that the proposed algorithm is accurate, stable, and efficient.

Keywords

• time-fractional Kuramoto–Sivashinsky equation
• fast Alikhanov method
• sum-of-exponentials technique
• non-uniform time steps
• compact difference method