A numerical solution of the Korteweg-de Vries equation by pseudospectral method using Darvishi’s preconditionings

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Abstract

In this paper, the Korteweg-de Vries equation is solved by pseudospectral method. In pseudospectral method we have to use matrix vector multiplication to compute derivative(s) of the unknown function. To reduce roundoff error in this multiplication we use Darvishi’s preconditionings. The numerical results have been compared with the exact solution. They are in good agreement with each other because graphs of exact solution and numerical solutions are very similar.

Introduction

The Korteweg-de Vries (KdV) equation is an important nonlinear equation used in the study of nonlinear dispersive waves. This equation has the following from [16];ut+εuux+μ3ux3=0,axb,t0,where ε and μ are parameters. Boundary conditions are taken asu(a,t)=-2sech2(a-4t),u(b,t)=-2sech2(b-4t)and the initial condition is taken asu(x,0)=-2sech2(x).The KdV equation is used to describe many important physical phenomena. Some of these studies are shallow water and ion acoustic plasma waves (see [17]). Zabusky [17] and Fornber et al. [8] have presented an analytical solution to particular KdV problems. Goda [11] and Vliengenthart [15] have used finite difference scheme to obtain the numerical solution of Eq. (1).

Alexander and Morris [1] have described Galerkin techniques using cubic splines and interior interpolation functions with quintic polynomial boundary functions to solve the problem. Soliman [12] have used the collocation solution with septic spline to set up the solution of the problem.

In this article, we use the pseudospectral method using Darvishi’s preconditionings to set up the numerical solution of the KdV equation.

Section snippets

Pseudospectral method

If a function f(x) is not periodic, we can approximate it by polynomials in x. However, it is well known that the Lagrange interpolation polynomial based on equally spaced points does not give a satisfactory approximation to general smooth f. In fact, as the number of collocation points increases, interpolation polynomial diverges [10]. This poor behavior of polynomial interpolation can be avoided for smoothly differentiable functions by removing the restriction to equally spaced collocation

Governing equation

The KdV equation takes the formut+εuux+μ3ux3=0,axb,t0.The exact solution of Eq. (18) is taken as, [13]u(x,t)=-2sech2(x-4t).To solve Eq. (18) by pseudospectral method we discretize the equation in space asut=-εuDu-μD3uwhere D is the differentiation matrix, and as we use the Tchebychev–Gauss–Lobatto collocation points hence if Dr be the matrix mapping u  u(r), then we have Dr = Dr, [10]. The kth component of Eq. (20) isut(xk,t)=-εu(xk,t)j=0Ndkju(xj,t)-μj=0Ndkj(3)u(xj,t)where D3=(dkj(3)

Numerical results

In this section we solve Eq. (18) for different values of a, b and N, number of collocation points. The obtained results for different values of parameters, time and number of collocation points compared with the exact solution. To show the efficiency of our methods we plot the graphs of exact solution and numerical solutions. As Tchebychev–Gauss–Lobatto collocation points are in interval [−1, 1], therefore we have to map interval [a, b] to [−1, 1] by a linear mapping. And so we have to change the

Conclusion

Mathematical modeling of physical systems often leads to nonlinear evolution equations. Many of the models are based on simple integrable models such as the Korteweg-de Vries equation. The last few decades have seen an enormous growth of the application of nonlinear models and the development nonlinear concepts. In this study, we solve the Korteweg-de Vries equation by pseudospectral method using Darvishi’s preconditionings. As plotted graphs show, the numerical results are very well.

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