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Abstract
In this paper, Adomian decomposition method (ADM) is implemented to approximate the solution of the Korteweg-de Vries (KdV) equations of seventh order, which are Kaup-Kuperschmidt equation and seventh order Kawahara equation. The results obtained by the ADM are compared with the exact solutions. It is found that the ADM is very efficient and convenient and can be applied to a large class of problems. The conservation properties of solution are examined by calculating the first three invariants.
Highlights
The general seventh-order Korteweg-de Vries (KdV) equation reads ut + au3ux bu 3 x+ cuuxuxx du 2 u xxx eu xx u xxx (1)+ fuxuxxxx + guuxxxxx + uxxxxxxx = 0, where a, b, c, d, e, f and g are nonzero parameters
+147u ux xxxx + 42uuxxxxx + uxxxxxxx = 0, Another form of the seventh-order KdV equation is called seventh order Kawahara equation [2] which can be shown in the form
The paper is arranged in the following manner: in Section 2, we present the Adomian decomposition method (ADM); Section 3 presents the Conservation laws (CLaws) for (KK) and Kawahara seventh-order KdV equations [13]
Summary
These equations were introduced initially by Pomeau et al [3] for discussing the structural stability of KdV equation under a singular perturbation. Adomian in the 1980’s [4] [5] [6] [7] This technique has been shown to solve effectively, and accurately a large class of linear and nonlinear, ordinary or partial, deterministic or stochastic differential equations with approximates which converge rapidly to accurate solutions. This method is well-suited to physical problems since it makes the unnecessary linearization, perturbation problem being solved, sometimes seriously.
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