A numerical solution of the Korteweg-de Vries equation by pseudospectral method using Darvishi’s preconditionings
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];where ε and μ are parameters. Boundary conditions are taken asand the initial condition is taken asThe 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 formThe exact solution of Eq. (18) is taken as, [13]To solve Eq. (18) by pseudospectral method we discretize the equation in space aswhere 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) iswhere
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.
References (17)
- et al.
Galerkin methods for some model equations for nonlinear dispersive waves
J. Comput. Phys.
(1979) - et al.
On the errors incurred calculating derivatives using Chebyshev polynomials
J. Comput. Phys.
(1992) A numerical simulation and explicit solutions of KdV–Burgers’ and Lax’s seventh-order KdV equations
Chaos, Solitons & Fractals
(2006)A synergetic approach to problem of nonlinear dispersive wave propagation and interaction
- et al.
Spectral differencing with a twist
SIAM J. Sci. Comput.
(2003) - et al.
Error reduction for higher derivatives of Chebyshev collocation method using preconditioning and domain decomposition
Korean J. Comput. Appl. Math.
(1999) Preconditioning and domain decomposition schemes to solve PDEs
Int. J. Pure Appl. Math.
(2004)- et al.
Accuracy and speed in computing the Chebyshev collocation derivative
SIAM J. Sci. Comput.
(1995)
Cited by (27)
Numerical investigation for a hyperbolic annular fin with temperature dependent thermal conductivity
2016, Propulsion and Power ResearchAnalytic Study of Coupled Burgers’ Equation
2023, MathematicsSolving (3+1) D-New Hirota Bilinear Equation Using Tanh Method and New Modification of Extended Tanh Method
2023, Advances in the Theory of Nonlinear Analysis and its ApplicationsA combination of Lie group-based high order geometric integrator and delta-shaped basis functions for solving Korteweg-de Vries (KdV) equation
2021, International Journal of Geometric Methods in Modern Physics