Abstract
A spline is a piecewise defined special function that is usually comprised of polynomials of a certain degree. These polynomials are supposed to generate a smooth curve by connecting at given data points. In this work, an application of fifth degree basis spline functions is presented for a numerical investigation of the Kuramoto–Sivashinsky equation. The finite forward difference formula is used for temporal integration, whereas the basis splines, together with a new approximation for fourth order spatial derivative, are brought into play for discretization in space direction. In order to corroborate the presented numerical algorithm, some test problems are considered and the computational results are compared with existing methods.
Highlights
The Kuramoto–Sivashinsky (KS) equation, a canonical nonlinear evolution equation, crops up in mathematical modeling of several physical phenomena indicating reaction– diffusion systems, unstable drift waves in plasmas, pattern formation on thin hydrodynamic films, flame front instability, long waves on the interface between two viscous fluids, fluid flow on a vertical plate and spatially uniform oscillating chemical reaction in some homogeneous medium [1, 2]
Mittal and Arora [27] explored the numerical solution to KS equation by means of the Crank–Nicolson scheme and quintic B-spline (QnBS) functions
8 Conclusion In this work, an application of a new quintic polynomial B-spline approximation approach has been presented for a numerical investigation of the Kuramoto–Sivashinsky equation
Summary
The Kuramoto–Sivashinsky (KS) equation, a canonical nonlinear evolution equation, crops up in mathematical modeling of several physical phenomena indicating reaction– diffusion systems, unstable drift waves in plasmas, pattern formation on thin hydrodynamic films, flame front instability, long waves on the interface between two viscous fluids, fluid flow on a vertical plate and spatially uniform oscillating chemical reaction in some homogeneous medium [1, 2]. Lai and Ma [25] proposed a lattice Boltzmann model for solving the nonlinear KS equation, A mesh free approach based on radial basis functions was used in [26] for an approximate solution of the generalized KS equation. Mittal and Arora [27] explored the numerical solution to KS equation by means of the Crank–Nicolson scheme and quintic B-spline (QnBS) functions. The authors in [29] presented a numerical approach based on basis spline functions for an approximate solution of the KS equation. Ersoy and Dag [31] proposed an exponential cubic B-spline method for numerical solution of KS equation. Mittal and Dahiya [6] proposed a differential quadrature method based on QnBS functions for solving the generalized KS equation.
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