Numerical simulation of the generalized regularized long wave equation by He’s variational iteration method

doi:10.1016/j.matcom.2005.06.002

Mathematics and Computers in Simulation

Volume 70, Issue 2, 8 September 2005, Pages 119-124

https://doi.org/10.1016/j.matcom.2005.06.002 Get rights and content

Abstract

The solution for the generalized regularized long wave (for short, GRLW) equation based on variational iteratiom method, is exactly obtained. In this method, the solution is calculated in the form of convergent power series with easily computable componentes. This approach does need linearization, weak nonlinearity assumptions or perturbation theory. The results reveal that the method is very effective and convenient.

Introduction

The regularized long wave (RLW) equation is an important nonlinear wave equation. Solitary waves are wave packet or pulses, which propagate in nonlinear dispersive media. Due to dynamical balance between the nonlinear and dispersive effects, these waves retain a stable wave form. The regularized long wave (RLW) equation is an alternative description of nonlinear dispersive waves to the more usual Kortewege–de vries (KdV) equation [1]. Numerical solutions are based on finite difference techniques [2], [3]. Rung–Kutta method [4] and Galerkin’s method [5] have been given. Alexander and Morris [5] constructed a global trial function mainly from cubic splines. Gardner and Gardner [6] uses the Galerkin’s method and cubic B-spline as element shape function to construct an implicit finite element solution. The least squares method using linear space–time finite elements, are used to solve the RLW equation [7]. Soliman and Raslan [8] solved the RLW equation by using collocation method using quadratic B-spline at the mid point.

Soliman and Hussien used the collocation method with septic spline to solve the RLW equation [9], in the end Soliman [10] using the finite difference method with the similarity solution of the partial differential equations to obtain the numerical scheme for RLW equation. This approach eliminates the difficult associated to the boundary.

The variational iteration method was first proposed by He [11], [12], [13], [14], and was successfully applied to autonomous ordinary differential equations by He [15], to nonlinear polycrystalline solids [16], nonlinear partial differential equations [17], and other fields. The combination of a perturbation method, variational iteration method, method of variation of constants, and averaging method are used to establish an approximate solution of one degree of freedom weakly nonlinear system in [18].

The purpose of this paper is to apply the variational iterations method proposed by He [11], [12], [13], [14], [19], [20] to solve the generalized long wave equation, and the analytical solutions are obtained. Furthermore, we will show that considerably better approximations related to the accuracy level would be obtained, if numerical solution is needed.

Section snippets

Variational iteration method

To illustrate its basic concepts of variational iteration method, we consider the following differential equations: $L u + N u = g (x),$ where L is a linear operator, N a nonlinear operator, and $g (x)$ is an inhomogeneous term.

According to the variational iteration method, we can construct a correct functional as follows: $u_{n + 1} (x) = u_{n} (x) + \int_{0}^{x} λ {L u_{n} (τ) + N ũ_{n} (τ) - g (τ)} d τ,$ where $λ$ is a general Lagrangian multiplier [11], [12], [13], [14], which can be identify optimally via the variational theory [11], [12], [13], [14]

Analysis of the method on GRLW equation

To illustrate the variational iteration method in this study, we consider the standard form of the GRLW equation, such as: $\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} + ν \frac{\partial (u^{p})}{\partial x} - μ \frac{\partial^{3} u}{\partial x^{2} \partial t} = 0,$ and the initial condition given in the form: $u (x, 0) = A {[{sech}^{2} (K (x + x_{0}))]}^{1 / (p - 1)},$ where $K = \sqrt{\frac{c - 1}{c}} (\frac{p - 1}{2 μ}), A = (\frac{(p + 1) (c - 1)}{2 ν})$ , c and $x_{0}$ are arbitrary constants, and $p \geq 2 .$

To solve Eq. (3) by means of variational iteration method, we consider the correct functional of Eq. (3):

$u_{n + 1} (x, t) = u_{n} (x, t) + \int_{0}^{t} λ {{(u_{t})}_{n} + u_{n x} + ν {(ũ^{p})}_{n x} - μ u_{n x x t}} d τ,$ where ${(δ ũ^{p})}_{n x}$ is considered as a

Case I

For the purpose of illustration of the variational iteration method for solving the GRLW equation, we consider the case for $p = 2$ . In this case, the solitary wave solution can be evaluated for the equation: $\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} + ν \frac{\partial (u^{2})}{\partial x} - μ \frac{\partial^{3} u}{\partial x^{2} \partial t} = 0,$ and the initial condition given in the form $u (x, 0) = A {sech}^{2} (K (x + x_{0})) .$

The solution of Eq. (8) can be evaluated by using formula (7), and we can calculate $u_{1}, u_{2}$ , the so on as:

$u_{1} (x, t) = A {sech}^{2} (K (x + x_{0})) + A K t (1 + 4 A ν + \cosh (2 K (x + x_{0}))) {sech}^{4} (K (x + x_{0})) \times \tanh (K (x + x_{0})),$ $u_{2} (x, t) = A {sech}^{2} ($

Conclusions

In this paper, the variational iteration method has been successfully applied to find the solution of the generalized regularized long wave equation. The solution obtained by the variational iteration method is an infinite power series for appropriate initial condition, which can be, in turn, expressed in a closed form, the exact solution. The results show that the variational iteration method is powerful mathematical tool for solving generalized regularized long wave equation, it is also a

Acknowledgements

The author would like to thank the referees for their comments and discussions.

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Numerical simulation of the generalized regularized long wave equation by He’s variational iteration method

Abstract

Introduction

Section snippets

Variational iteration method

Analysis of the method on GRLW equation

Case I

Conclusions

Acknowledgements

J. Comput. Phys.

J. Comput. Phys.

Appl. Math. Comput.

Comput. Meth. Appl. Mech. Eng.

Comput. Meth. Appl. Mech. Eng.

Int. J. Nonlinear Mech.

Commun. Nonlinear Sci. Numer. Simul.

Appl. Math. Comput.

J. Comput. Appl. Math.

Calculations of the development of an undular bore

J. Fluid Mech.