An Efficient Off-Step Exponential Spline Technique to Solve Kuramoto-Sivashinsky and Extended Fisher-Kolmogorov Equations

In this study, we introduce a novel numerical approach for solving 1D unsteady biharmonic problems, specifically addressing the generalized Kuramoto-Sivashinsky equations. Our proposed method involves a two-level exponential spline technique with quasi-variable mesh discretization, ensuring high accuracy with a precision of orders 3 and 2 in spatial and lateral directions, respectively. The scheme utilizes three points at each time level, including two off-step and a central point. Notably, our method exhibits unconditional stability when applied to a partial differential equation of order 4. Comparative analysis with results from prior research highlights the superiority of our numerical algorithm. Furthermore, we present 2D and 3D graphs illustrating the numerical solutions of various benchmark problems, such as the Kuramoto-Sivashinsky equation and the extended Fisher-Kolmogorov equation, emphasizing the versatility and efficacy of our approach.