Generation of the Trigonometric Cubic B-Spline Collocation Solutions for the Kuramoto-Sivashinsky(KS) Equation

TL;DR

This paper introduces a novel numerical method using trigonometric cubic B-splines combined with Crank-Nicolson for solving the Kuramoto-Sivashinsky equation, demonstrating its effectiveness through initial boundary value problems.

Contribution

It adapts trigonometric cubic B-splines to collocation for KS equation and reduces higher order derivatives to a coupled system for numerical solution.

Findings

Accurate solutions for KS equation demonstrated

Method effectively handles higher order derivatives

Numerical results validate the proposed approach

Abstract

A recent type of B-spline functions, namely trigonometric cubic B-splines, are adapted to the collocation method for the numerical solutions of the Kuramoto-Sivashinsky equation. Having only first and second order derivatives of the trigonometric cubic B-splines at the nodes forces us to convert the Kuramoto-Sivashinsky equation to a coupled system of equations by reducing the order of the higher order terms. Crank-Nicolson method is applied for the time integration of the space discretized system resulted by trigonometric cubic B-spline approach. Some initial boundary value problems are solved to show the validity of the proposed method.