Geometrical interpretation and approximate solution of non‐linear KdV equation

Abstract

PurposeThe purpose is to study an analytical solution of non‐linear Korteweg‐de Vries (KdV) equation by using the Adomian decomposition method (ADM).Design/methodology/approachThe solution is calculated in the form of a series with easily computable components. The non‐linear KdV equation has been considered and the analytic solution is compared with its numerical solution by using the ADM and Mathematica software program.FindingsThis approach to the non‐linear evolution equation was found to be valuable as a tool for scientists and applied mathematicians, because it provides immediate and visible symbolic terms of analytical solution as well as its numerical approximate solution to both linear and non‐linear problems without linearization or discretization.Research limitations/implicationsThis geometrical interpretation and the produced approximate solution of the non‐linear KdV equation illustrates the use of the ADM. Research using ADM is ongoing but already the numerical results obtained in this paper justify the advantages of this methodology, even in a few terms of approximation.Practical implicationsUsing the Mathematica software package the ADM was implemented for homogenous KdV equation as an illustrative example which has distinct applications for scientists and applied mathematicians.Originality/valueThis is an original study of the use of ADM for the solution of the non‐linear KdV equation. It also shows how the Mathematica software package can be used in such studies.

A reliable technique for solving third‐order dispersion equations

“…The application of the ADM for solving the nonlinear KdV equation (5.8) subject to the initial condition (1.2) has been undertaken by Wazwaz (2001) for H =0, ε =6, Bektas et al (2005) and Syam (2005) for H =0, ε =−6, Yan (2005) for H =0, ε =−6, −12, Cherruault et al (2002) for ε =1, and Kaya and Aassila (2002) for ε =±6.…”

On the solution of the nonlinear Korteweg–de Vries equation by the homotopy perturbation method

“…Other fluid dynamical applications have been studied by some researchers [7,8,21,25]. Using the Adomian decomposition method, an exact solution of the KdV equation was obtained by Cherruault et al [22] and Bektas et al [26]. Wazwaz [27] and Abassy et al [28] introduced the solution of the initial value problem for the KdV equation by the VIM.…”

Numerical solution of Korteweg–de Vries-Burgers equation by the modified variational iteration algorithm-II arising in shallow water waves

“…Recently, exact solution of coupled KdV equations based on Kudryashov technique demonstrated by [30]. Numerous approaches have been handled to these problems such as: finite difference scheme [31], ( ) ¢ G G -expansion method, finite volume scheme [32], homotopy analysis method [33], finite element scheme [34], decomposition method [35], spectral method [36], Wronskian form expansion method [37] Exp-function method, canonical formulation of Whitham's variational principle [38], residual power series method [39], tanh function method, variational iteration method [40], inverse scattering transform [41] and reduced differential transformation method [42].…”