Kuramoto-Sivashinsky equation: Numerical solution using two quintic B-splines and differential quadrature method

Abstract

This paper aims to obtain the numerical solution of a one-dimension fourth-order Kuramoto-Sivashinsky (KS) equation, which has application in the context of flame propagation. This study combines the implementation of two new mechanisms quintic Uniform Algebraic Hyperbolic (QUAH) tension B-spline and quintic Uniform Algebraic Trigonometric (QUAT) tension B-spline with a renowned method known as Differential Quadrature Method (DQM). DQM is used to approximate the derivatives. Then, to obtain the weighting coefficients both quintic splines QUAH and QUAT are implemented with DQM and the original boundary value problem was then transformed into an ordinary differential equation. The Runge-Kutta 43 technique was used to solve the resultant ODE. With this approach, the outcomes are precise and close to the exact solution. The differential quadrature technique has a significant advantage over the previous approaches because it prevents the perturbation in order to find the better results for the given nonlinear equations. The accuracy of the suggested method is then illustrated numerically resolving four test problems and calculating the error norms L2 and L∞. Calculated results are displayed graphically and in tabular format for quick and simple access. This paper introduces an original work, to the best of the author’s knowledge. The obtained solution and graphical observations show that QUAH and QUAT tension B-splines with DQM is a strong and dependable methods for estimating nonlinear problem solutions. Quintic B-splines are higher-order interpolation method compared to lower-degree B-splines, which results in more accurate approximations, and provides smooth and continuous representations of functions. Moreover, DQM when combined with quintic B-splines, can help reduce numerical dispersion, and can be applied to a wide range of problems.