Approximate Solution Method of the Seventh Order KdV Equations by Decomposition Method

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1. Introduction

The general seventh-order KdV equation (gsKdV) reads

$\begin{array}{l}{u}_{t}+a{u}^{3}{u}_{x}+b{u}_{x}^{3}+cu{u}_{x}{u}_{xx}+d{u}^{2}{u}_{xxx}+e{u}_{xx}{u}_{xxx}\\ +f{u}_{x}{u}_{xxxx}+gu{u}_{xxxxx}+{u}_{xxxxxxx}=0,\end{array}$ (1)

where a, b, c, d, e, f and g are nonzero parameters. One of the well-known particular cases of Equation (1) is called seventh order Kaup Kuperschmidt equation (KK) [1] which can be shown in the form

$\begin{array}{l}{u}_{t}+2016{u}^{3}{u}_{x}+630{u}_{x}^{3}+2268u{u}_{x}{u}_{xx}+504{u}^{2}{u}_{xxx}+252{u}_{xx}{u}_{xxx}\\ +147{u}_{x}{u}_{xxxx}+42u{u}_{xxxxx}+{u}_{xxxxxxx}=0,\end{array}$ (2)

Another form of the seventh-order KdV equation is called seventh order Kawahara equation [2] which can be shown in the form

${u}_{t}+6u{u}_{x}+{u}_{xxx}-{u}_{xxxxx}+\alpha {u}_{xxxxxxx}=0,$ (3)

where $\alpha $ is a nonzero constant. These equations were introduced initially by Pomeau et al. [3] for discussing the structural stability of KdV equation under a singular perturbation. These equations play an important role in mathematical physics, engineering and applied sciences for investigating travelling solitary wave solutions.

The Adomian decomposition method (ADM) was first proposed by George Adomian in the 1980’s [4] [5] [6] [7] . This technique has been shown to solve effectively, easily, 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. Conservation laws (CLaws) are of basic importance in the study of evolution equations because they provide physical, conserved quantities for all solutions $u\left(x\mathrm{,}t\right)$ , and they can be used to check the accuracy of numerical solution methods [8] [9] [10] [11] [12] . The paper is arranged in the following manner: in Section 2, we present the ADM; Section 3 presents the CLaws for (KK) and Kawahara seventh-order KdV equations [13] [14] ; in Section 4, the ADM is implemented to some problems in addition to studying the properties of CLaws; finally, a brief conclusion is given in Section 5.

2. The Method of Solution

Consider the (gsKdV) equation in an operator form

$\begin{array}{l}{L}_{t}\left(u\right)+a\left(Ku\right)+b\left(Mu\right)+c\left(Nu\right)+d\left(Pu\right)+e\left(Qu\right)\\ +f\left(Ru\right)+g\left(Vu\right)+{L}_{7x}\left(u\right)=0,\end{array}$ (4)

where the notations $Ku={u}^{3}{u}_{x}$ , $Mu={u}_{x}^{3}$ , $Nu=u{u}_{x}{u}_{xx}$ , $Pu={u}^{2}{u}_{xxx}$ ,

$Qu={u}_{xx}{u}_{xxx}$ , $Ru={u}_{x}{u}_{xxxx}$ and $Vu=u{u}_{xxxxx}$ symbolize the nonlinear terms,

respectively. Also, the notation ${L}_{t}=\frac{\partial}{\partial t}$ and ${L}_{7x}=\frac{{\partial}^{7}}{\partial {x}^{7}}$ symbolize the linear differential operators. Assuming ${L}_{t}^{-1}$ the inverse of operator of ${L}_{t}$ exists and conveniently by

${L}_{t}^{-1}={\displaystyle \underset{0}{\overset{t}{\int}}\left(\mathrm{.}\right)\text{d}t}$ (5)

Thus, applying the inverse operator ${L}_{t}^{-1}$ to (4) yields

$\begin{array}{c}u\left(x,t\right)=h\left(x\right)-a{L}_{t}^{-1}\left(Ku\right)-b{L}_{t}^{-1}\left(Mu\right)-c{L}_{t}^{-1}\left(Nu\right)-d{L}_{t}^{-1}\left(Pu\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-e{L}_{t}^{-1}\left(Qu\right)-f{L}_{t}^{-1}\left(Ru\right)-g{L}_{t}^{-1}\left(Vu\right)-{L}_{t}^{-1}\left({L}_{7x}u\right).\end{array}$ (6)

The standard ADM [15] defines the solution $u\left(x\mathrm{,}t\right)$ by the decomposition series

$u\left(x,t\right)={\displaystyle \underset{n=0}{\overset{\infty}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{u}_{n}\left(x,t\right),$ (7)

with ${u}_{0}$ identified as $u\left(x\mathrm{,0}\right)$ . The nonlinear terms Ku, Mu, Nu, Pu, Qu, Ru and Vu can be decomposed into infinite series of polynomial given by

$Ku={u}^{2}{u}_{x}={\displaystyle \underset{n=0}{\overset{\infty}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{A}_{n},$ (8)

$Mu={u}_{x}{u}_{xx}={\displaystyle \underset{n=0}{\overset{\infty}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{B}_{n},$ (9)

$Nu=u{u}_{xxx}={\displaystyle \underset{n=0}{\overset{\infty}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{C}_{n},$ (10)

$Pu=u{u}_{xxx}={\displaystyle \underset{n=0}{\overset{\infty}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{D}_{n},$ (11)

$Qu=u{u}_{xxx}={\displaystyle \underset{n=0}{\overset{\infty}{\sum}}}\text{\hspace{0.05em}}{E}_{n},$ (12)

$Ru=u{u}_{xxx}={\displaystyle \underset{n=0}{\overset{\infty}{\sum}}}\text{\hspace{0.05em}}{F}_{n},$ (13)

$Vu=u{u}_{xxx}={\displaystyle \underset{n=0}{\overset{\infty}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{G}_{n},$ (14)

where ${A}_{n}$ , ${B}_{n}$ , ${B}_{n}$ , ${D}_{n}$ , ${E}_{n}$ , ${F}_{n}$ and ${G}_{n}$ are the so-called Adomian polynomials of ${u}_{0},{u}_{1},\cdots ,{u}_{n}$ defined by equation

${P}_{n}=\frac{1}{n!}\frac{{\text{d}}^{n}}{\text{d}{\lambda}^{n}}{\left[N\left({\displaystyle \underset{i=0}{\overset{\infty}{\sum}}}\text{\hspace{0.05em}}{\lambda}^{i}{u}_{i}\left(x,t\right)\right)\right]}_{\lambda =0},n\ge 0.$ (15)

The components ${u}_{n}\left(x\mathrm{,}t\right)$ can be determined sequentially by the standard recursion scheme as:

$\begin{array}{l}(\begin{array}{l}{u}_{0}\left(x,t\right)=h\left(x\right),\\ {u}_{n+1}=-a{L}_{t}^{-1}\left({A}_{n}\right)-b{L}_{t}^{-1}\left({B}_{n}\right)-c{L}_{t}^{-1}\left({C}_{n}\right)-d{L}_{t}^{-1}\left({D}_{n}\right)-e{L}_{t}^{-1}\left({E}_{n}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}-f{L}_{t}^{-1}\left({F}_{n}\right)-g{L}_{t}^{-1}\left({G}_{n}\right)-{L}_{t}^{-1}\left({L}_{7x}{u}_{n}\right),n\ge 0.\end{array}\hfill \end{array}$ (16)

3. Conservation Laws

The conservation properties of the solution are examined by calculating the Claws.

1) For KK equation Equation (2), the conservative quantities ${I}_{i}\left(i=1,2,3\right)$ can be written as

$\begin{array}{l}{I}_{1}={\displaystyle {\int}_{-\infty}^{\infty}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}u\text{d}x,\\ {I}_{2}={\displaystyle {\int}_{-\infty}^{\infty}}\left({u}^{3}-\frac{1}{8}{u}_{x}^{2}\right)\text{d}x,\\ {I}_{3}={\displaystyle {\int}_{-\infty}^{\infty}}\left({u}^{4}-\frac{3}{4}u{u}_{x}^{2}+\frac{1}{48}{u}_{xx}^{2}\right)\text{d}x,\end{array}$ (17)

2) For seventh-order Kawahara equation Equation (3), the conservative quantities ${I}_{i}\left(i=1,2,3\right)$ can be written as

$\begin{array}{l}{I}_{1}={\displaystyle {\int}_{-\infty}^{\infty}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}u\text{d}x,\\ {I}_{2}={\displaystyle {\int}_{-\infty}^{\infty}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{u}^{2}\text{d}x,\\ {I}_{3}={\displaystyle {\int}_{-\infty}^{\infty}}\left(-{u}^{3}+\frac{1}{2}{\left({u}_{x}\right)}^{2}-\frac{1}{2}{\left({u}_{xx}\right)}^{2}+\frac{1}{2}\alpha {\left({u}_{xxx}\right)}^{2}\right)\text{d}x\text{,}\end{array}$ (18)

Since the conservation constants are expected to remain constant during the run of the algorithm to have accurate numerical scheme, conservation constants will be monitored. As various problems of science were modeled by non linear partial differential equations and since therefore the seventh order KdV equation is of high importance, the following examples have been considered.

4. Numerical Examples

Example 1. Consider the seventh-order (KK) equation Equation (2) with initial condition

$u\left(x\mathrm{,0}\right)=\frac{1}{3}{k}^{2}-\frac{1}{2}{k}^{2}{\mathrm{tanh}}^{2}\left(kx\right),$

By ADM the recursive relations are

$\{\begin{array}{l}{u}_{0}=\frac{1}{3}{k}^{2}-\frac{1}{2}{k}^{2}{\mathrm{tanh}}^{2}\left(kx\right),\\ {u}_{n+1}=-2016{L}_{t}^{-1}\left({A}_{n}\right)-630{L}_{t}^{-1}\left({B}_{n}\right)-2268{L}_{t}^{-1}\left({C}_{n}\right)-504{L}_{t}^{-1}\left({D}_{n}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}-252{L}_{t}^{-1}\left({E}_{n}\right)-147{L}_{t}^{-1}\left({F}_{n}\right)-42{L}_{t}^{-1}\left({G}_{n}\right)-{L}_{t}^{-1}\left({L}_{7x}{u}_{n}\right),n\ge 0.\end{array}$

The first few components are thus determined as follows:

$\{\begin{array}{l}{u}_{0}=\frac{1}{3}{k}^{2}-\frac{1}{2}{k}^{2}{\mathrm{tanh}}^{2}\left(kx\right),\\ {u}_{1}=\frac{-4{k}^{9}t\mathrm{sinh}\left(kx\right)}{3{\mathrm{cosh}}^{3}\left(kx\right)},\\ {u}_{2}=\frac{8{k}^{16}{t}^{2}\left(2{\mathrm{cosh}}^{2}\left(kx\right)-3\right)}{9{\mathrm{cosh}}^{4}\left(kx\right)},\end{array}$

and so on. Consequently, the solution in a series form is given by

$u\left(x,t\right)={u}_{0}+{u}_{1}+{u}_{2}+\cdots $

and in a closed form $u\left(x,t\right)=\frac{1}{3}{k}^{2}-\frac{1}{2}{k}^{2}{\mathrm{tanh}}^{2}\left(k\left(x+\frac{4}{3}{k}^{6}t\right)\right)$ .

The results produced by the proposed method with only few components (n = 5) are compared with the exact solution and listed in Table 1, also the Claws for the seventh-order (KK) equation are given in Table 2. The profile of the solitary wave at $t=0.3$ is displayed in Figure 1.

Table 1. Comparison between exact solution $u\left(x,t\right)$ and approximate solution using ADM where $k=0.1$ .

Table 2. Computed quantities ${I}_{1}\mathrm{,}{I}_{2}\mathrm{,}{I}_{3}$ for the seventh-order KK equation by ADM.

Figure 1. Comparison between exact solution $u\left(x,t\right)$ and approximate solution using ADM.

Example 2. Consider the seventh-order Kawahara equation Equation (3) with initial condition

$u\left(x,0\right)=\omega {\text{sech}}^{6}\left(kx\right),$

By ADM the recursive relations are

$\{\begin{array}{l}{u}_{0}=\omega {\text{sech}}^{6}\left(kx\right),\\ {u}_{n+1}=-6{L}_{t}^{-1}\left({A}_{n}\right)-{L}_{t}^{-1}\left({L}_{3x}{u}_{n}\right)+{L}_{t}^{-1}\left({L}_{5x}{u}_{n}\right)-\alpha {L}_{t}^{-1}\left({L}_{7x}{u}_{n}\right),n\ge 0.\end{array}$

The first few components are thus determined as follows:

$\{\begin{array}{l}{u}_{0}=\omega {\text{sech}}^{6}\left(kx\right),\\ {u}_{1}=\frac{1}{{\mathrm{cosh}}^{13}\left(kx\right)}(12tk\omega \mathrm{sinh}\left(kx\right)(23328\alpha {k}^{6}{\mathrm{cosh}}^{6}\left(kx\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-215488\alpha {k}^{6}{\mathrm{cosh}}^{4}\left(kx\right)-648{k}^{4}{\mathrm{cosh}}^{6}\left(kx\right)+\cdots )),\\ {u}_{2}=\frac{1}{{\mathrm{cosh}}^{20}\left(kx\right)}(12{t}^{2}{k}^{2}\omega (108783285811200{\alpha}^{2}{k}^{12}{\mathrm{cosh}}^{6}\left(kx\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-175649727052800{\alpha}^{2}{k}^{12}{\mathrm{cosh}}^{4}\left(kx\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+138322888704000{\alpha}^{2}{k}^{12}{\mathrm{cosh}}^{2}\left(kx\right)+\cdots )),\end{array}$

and so on. Consequently, the solution in a series form is given by

$u\left(x,t\right)={u}_{0}+{u}_{1}+{u}_{2}+\cdots $

and in a closed form $u\left(x,t\right)=\omega {\text{sech}}^{6}\left(k\left(x-{x}_{0}t\right)\right)$ .

The results produced by the proposed method with only few components (n = 5) are compared with the exact solution and listed in Table 3, also the Claws for the seventh-order Kawahara equation are given in Table 4. The profile of the solitary wave at $t=0.3$ is displayed in Figure 2.

Table 3. Comparison between exact solution $u\left(x\mathrm{,}t\right)$ and approximate solution using ADM where $\alpha =\frac{769}{2500}$ , $\omega =\frac{86625}{591361}$ , $k=\frac{5}{\sqrt{1538}}$ and ${x}_{0}=\frac{180000}{591361}$ .

Table 4. Computed quantities ${I}_{1}\mathrm{,}{I}_{2}\mathrm{,}{I}_{3}$ for the seventh-order Kawahara equation by ADM.

Figure 2. Comparison between exact solution $u\left(x\mathrm{,}t\right)$ and approximate solution using ADM.

5. Conclusion

In this paper, the ADM was used to solving seventh order KdV equations with initial conditions. We have found out that this method is applicable and efficient technique. All the numerical results obtained by using ADM show very good agreement with the exact solutions for a few terms. The conservation laws are used to assess the accuracy and the efficiency of the method. We have noticed that the method accomplished the aim of preserving conserved quantities, as we saw all invariants were almost constant.

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