Approximate Analytical Solution for the Forced Korteweg-de Vries Equation

Vincent Daniel David,Faisal Salah,Zainal Abdul Aziz,Mojtaba Nazari,Vahid Barati

doi:10.1155/2013/795818

Abstract

The forced Korteweg-de Vries (fKdV) equations are solved using Homotopy Analysis Method (HAM). HAM is an approximate analytical technique which provides a novel way to obtain series solutions of such nonlinear problems. It has the auxiliary parameterℏ, where it is easy to adjust and control the convergence region of the series solution. Some examples of forcing terms are employed to analyse the behaviours of the HAM solutions for the different fKdV equations. Finally, this form of HAM solution is compared with the analytical soliton-type solution of fKdV equation as derived by Zhao and Guo. The results is found to be in good agreement with Zhao and Guo.

Highlights

IntroductionAn analytical model of Tsunami propagation was proposed by Pelinovsky et al [1] as follows:
An analytical model of Tsunami propagation was proposed by Pelinovsky et al [1] as follows: ∂η ∂t + c ∂η ∂x αη β ∂3u ∂x3 = ∂f ∂x (1)with α = 3c, β = ch02, f = −cz, (2)2h0 where η = η(x, t) refers to the elevation of free water surface, z = z(x, t) represents the solid bottom, h is assumed to be the constant mean water depth, and c ≈ √gh is the long wave speed with g being gravity acceleration
The forcing term in the forced Korteweg-de Vries (KdV) (fKdV) can be assumed to be derivable from atmospheric disturbances

Summary

Introduction

An analytical model of Tsunami propagation was proposed by Pelinovsky et al [1] as follows:. The forcing term in the fKdV can be assumed to be derivable from atmospheric disturbances Various forms of this equation have been extensively studied (see Grimshaw et al [6], Pelinovsky et al [7]) and numerical results show that the solution contains the set of solitary waves. The main reason of this work is to solve fKdV equation by using the homotopy analysis method (HAM) for various forcing terms, including in [3]. This approximate analytical HAM solution will be used to be compared with the analytical soliton-type solution of Zhao and Guo [3].

Basic Ideas of HAM

HAM Solution on KdV

Approximate Analytical Solution for fKdV Using HAM

Conclusion