1. Introduction

The Kuramoto-Sivashinsky equation (KSe) is a non-linear fourth order partial differential equation (PDE) discovered separately by Kuramoto and Sivashinsky in the study of nonlinear stability of travelling waves. Sivashinsky [1] came up with the equation while modelling small thermal diffusive instabilities in laminar flame fronts. Kuramoto [2–5] derived the equation in the study of the Belousov-Zhabotinsky reaction as a model of diffusion induced chaos. The KSe is of interest to many researchers because of its ability to describe several physical contexts such as long waves on thin films or on the interface between two viscous fluids [6] and unstable drift waves in plasmas. The equation is also used as a model to describe spatially uniform oscillating chemical reaction in a homogeneous medium and fluctuations in fluid films on inclines [7]. In one dimension, consider the KSe of the form

$$
\frac{
\partial u
}{
\partial t
} + \mu \frac{
\partial u
}{
\partial \mathbf{x}
} + \frac{
\partial^2 u
}{
\partial \mathbf{x}^2
} + \frac{
\partial^4 u
}{
\partial \mathbf{x}^4
} = 0, \qquad \qquad t > 0. \tag{1}
$$

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

The second derivative term is an energy source and thus has a distributing effect. The nonlinear term is a correction to the phase speed and responsible for transferring energy. The fourth derivative term is the dominating term and is responsible for stabilising the equation. Several methods have been used to solve the KSe numerically and these include Chebyshev spectral collocation method [8], Quintic B-spline collocation method [9], Lattice Boltzmann method [10], meshless method of lines [11], Fourier spectral method [12] and septic B-spline collocation method [13].
