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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.

The general seventh-order KdV equation (gsKdV) reads

u t + a u 3 u x + b u x 3 + c u u x u x x + d u 2 u x x x + e u x x u x x x + f u x u x x x x + g u u x x x x x + u x x x x x x x = 0 , (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) [

u t + 2016 u 3 u x + 630 u x 3 + 2268 u u x u x x + 504 u 2 u x x x + 252 u x x u x x x + 147 u x u x x x x + 42 u u x x x x x + u x x x x x x x = 0 , (2)

Another form of the seventh-order KdV equation is called seventh order Kawahara equation [

u t + 6 u u x + u x x x − u x x x x x + α u x x x x x x x = 0 , (3)

where α is a nonzero constant. These equations were introduced initially by Pomeau et al. [

The Adomian decomposition method (ADM) was first proposed by George Adomian in the 1980’s [

Consider the (gsKdV) equation in an operator form

L t ( u ) + a ( K u ) + b ( M u ) + c ( N u ) + d ( P u ) + e ( Q u ) + f ( R u ) + g ( V u ) + L 7 x ( u ) = 0 , (4)

where the notations K u = u 3 u x , M u = u x 3 , N u = u u x u x x , P u = u 2 u x x x ,

Q u = u x x u x x x , R u = u x u x x x x and V u = u u x x x x x symbolize the nonlinear terms,

respectively. Also, the notation L t = ∂ ∂ t and L 7 x = ∂ 7 ∂ 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 = ∫ 0 t ( . ) d t (5)

Thus, applying the inverse operator L t − 1 to (4) yields

u ( x , t ) = h ( x ) − a L t − 1 ( K u ) − b L t − 1 ( M u ) − c L t − 1 ( N u ) − d L t − 1 ( P u ) − e L t − 1 ( Q u ) − f L t − 1 ( R u ) − g L t − 1 ( V u ) − L t − 1 ( L 7 x u ) . (6)

The standard ADM [

u ( x , t ) = ∑ n = 0 ∞ u n ( x , t ) , (7)

with u 0 identified as u ( x ,0 ) . The nonlinear terms Ku, Mu, Nu, Pu, Qu, Ru and Vu can be decomposed into infinite series of polynomial given by

K u = u 2 u x = ∑ n = 0 ∞ A n , (8)

M u = u x u x x = ∑ n = 0 ∞ B n , (9)

N u = u u x x x = ∑ n = 0 ∞ C n , (10)

P u = u u x x x = ∑ n = 0 ∞ D n , (11)

Q u = u u x x x = ∑ n = 0 ∞ E n , (12)

R u = u u x x x = ∑ n = 0 ∞ F n , (13)

V u = u u x x x = ∑ n = 0 ∞ 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 , ⋯ , u n defined by equation

P n = 1 n ! d n d λ n [ N ( ∑ i = 0 ∞ λ i u i ( x , t ) ) ] λ = 0 , n ≥ 0. (15)

The components u n ( x , t ) can be determined sequentially by the standard recursion scheme as:

( u 0 ( x , t ) = h ( x ) , u n + 1 = − a L t − 1 ( A n ) − b L t − 1 ( B n ) − c L t − 1 ( C n ) − d L t − 1 ( D n ) − e L t − 1 ( E n ) − f L t − 1 ( F n ) − g L t − 1 ( G n ) − L t − 1 ( L 7 x u n ) , n ≥ 0. (16)

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

1) For KK equation Equation (2), the conservative quantities I i ( i = 1 , 2 , 3 ) can be written as

I 1 = ∫ − ∞ ∞ u d x , I 2 = ∫ − ∞ ∞ ( u 3 − 1 8 u x 2 ) d x , I 3 = ∫ − ∞ ∞ ( u 4 − 3 4 u u x 2 + 1 48 u x x 2 ) d x , (17)

2) For seventh-order Kawahara equation Equation (3), the conservative quantities I i ( i = 1 , 2 , 3 ) can be written as

I 1 = ∫ − ∞ ∞ u d x , I 2 = ∫ − ∞ ∞ u 2 d x , I 3 = ∫ − ∞ ∞ ( − u 3 + 1 2 ( u x ) 2 − 1 2 ( u x x ) 2 + 1 2 α ( u x x x ) 2 ) d x , (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.

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

u ( x ,0 ) = 1 3 k 2 − 1 2 k 2 tanh 2 ( k x ) ,

By ADM the recursive relations are

{ u 0 = 1 3 k 2 − 1 2 k 2 tanh 2 ( k x ) , u n + 1 = − 2016 L t − 1 ( A n ) − 630 L t − 1 ( B n ) − 2268 L t − 1 ( C n ) − 504 L t − 1 ( D n ) − 252 L t − 1 ( E n ) − 147 L t − 1 ( F n ) − 42 L t − 1 ( G n ) − L t − 1 ( L 7 x u n ) , n ≥ 0.

The first few components are thus determined as follows:

{ u 0 = 1 3 k 2 − 1 2 k 2 tanh 2 ( k x ) , u 1 = − 4 k 9 t sinh ( k x ) 3 cosh 3 ( k x ) , u 2 = 8 k 16 t 2 ( 2 cosh 2 ( k x ) − 3 ) 9 cosh 4 ( k x ) ,

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

u ( x , t ) = u 0 + u 1 + u 2 + ⋯

and in a closed form u ( x , t ) = 1 3 k 2 − 1 2 k 2 tanh 2 ( k ( x + 4 3 k 6 t ) ) .

The results produced by the proposed method with only few components (n = 5) are compared with the exact solution and listed in

x | Exact | ADM | Absolute Error |
---|---|---|---|

0.10 | 0.00333283 | 0.00333283 | 9.98865803e^{−19} |

0.20 | 0.00333133 | 0.00333133 | 1.99693372e^{−18} |

0.30 | 0.00332884 | 0.00332884 | 2.99340524e^{−18} |

0.40 | 0.00332534 | 0.00332534 | 3.98748557e^{−18} |

0.50 | 0.00332085 | 0.00332085 | 4.97838399e^{−18} |

I_{1} | I_{2} | I_{3} | |||||||
---|---|---|---|---|---|---|---|---|---|

t/x | 0.1 | 0.3 | 0.5 | 0.1 | 0.3 | 0.5 | 0.1 | 0.3 | 0.5 |

0.1 | 3.333e^{−4} | 9.996e^{−4} | 1.665e^{−3 } | 3.703e^{−9 } | 1.108e^{−8 } | 1.840e^{−8} | 3.316e^{−11} | 9.910e^{−11} | 1.639e^{−10} |

0.2 | 3.333e^{−4} | 9.996e^{−4 } | 1.665e^{−3 } | 3.703e^{−9 } | 1.108e^{−8 } | 1.840e^{−8 } | 3.316e^{−11 } | 9.910e^{−11} | 1.639e^{−10 } |

0.3 | 3.333e^{−4} | 9.996e^{−4 } | 1.665e^{−3 } | 3.703e^{−9 } | 1.108e^{−8 } | 1.840e^{−8 } | 3.316e^{−11 } | 9.910e^{−11} | 1.639e^{−10 } |

0.4 | 3.333e^{−4} | 9.996e^{−4 } | 1.665e^{−3 } | 3.703e^{−9 } | 1.108e^{−8 } | 1.840e^{−8 } | 3.316e^{−11 } | 9.910e^{−11} | 1.639e^{−10 } |

0.5 | 3.333e^{−4} | 9.996e^{−4 } | 1.665e^{−3 } | 3.703e^{−9 } | 1.108e^{−8 } | 1.840e^{−8 } | 3.316e^{−11 } | 9.910e^{−11} | 1.639e^{−10 } |

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

u ( x , 0 ) = ω sech 6 ( k x ) ,

By ADM the recursive relations are

{ u 0 = ω sech 6 ( k x ) , u n + 1 = − 6 L t − 1 ( A n ) − L t − 1 ( L 3 x u n ) + L t − 1 ( L 5 x u n ) − α L t − 1 ( L 7 x u n ) , n ≥ 0.

The first few components are thus determined as follows:

{ u 0 = ω sech 6 ( k x ) , u 1 = 1 cosh 13 ( k x ) ( 12 t k ω sinh ( k x ) ( 23328 α k 6 cosh 6 ( k x ) − 215488 α k 6 cosh 4 ( k x ) − 648 k 4 cosh 6 ( k x ) + ⋯ ) ) , u 2 = 1 cosh 20 ( k x ) ( 12 t 2 k 2 ω ( 108783285811200 α 2 k 12 cosh 6 ( k x ) − 175649727052800 α 2 k 12 cosh 4 ( k x ) + 138322888704000 α 2 k 12 cosh 2 ( k x ) + ⋯ ) ) ,

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

u ( x , t ) = u 0 + u 1 + u 2 + ⋯

and in a closed form u ( x , t ) = ω sech 6 ( k ( x − x 0 t ) ) .

The results produced by the proposed method with only few components (n = 5) are compared with the exact solution and listed in

x | Exact | ADM | Absolute Error |
---|---|---|---|

0.10 | 0.14648359 | 0.14648359 | 1.24475423e^{−11} |

0.20 | 0.14639977 | 0.14639977 | 2.70176889e^{−11} |

0.30 | 0.14617341 | 0.14617341 | 4.13404041e^{−11} |

0.40 | 0.14580523 | 0.14580523 | 5.52856164e^{−11} |

0.50 | 0.14529642 | 0.14529642 | 6.87280812e^{−11} |

I_{1} | I_{2} | I_{3} | |||||||
---|---|---|---|---|---|---|---|---|---|

t/x | 0.1 | 0.3 | 0.5 | 0.1 | 0.3 | 0.5 | 0.1 | 0.3 | 0.5 |

0.1 | 1.465e^{−2} | 4.390e^{−2} | 7.300e^{−2} | 2.146e^{−3} | 6.424e^{−3} | 1.066e^{−2} | −3.245e^{−4} | −9.697e^{−4} | −1.602e^{−3} |

0.2 | 1.465e^{−2} | 4.391e^{−2} | 7.304e^{−2} | 2.146e^{−3} | 6.428e^{−3} | 1.067e^{−2} | −3.245e^{−4} | −9.708e^{−4} | −1.606e^{−3} |

0.3 | 1.465e^{−2} | 4.392e^{−2} | 7.308e^{−2} | 2.145e^{−3} | 6.430e^{−3} | 1.068e^{−2} | −3.244e^{−4} | −9.716e^{−4} | −1.609e^{−3} |

0.4 | 1.464e^{−2} | 4.393e^{−2} | 7.311e^{−2} | 2.145e^{−3} | 6.432e^{−3} | 1.069e^{−2} | −3.242e^{−4} | −9.721e^{−4} | −1.612e^{−3} |

0.5 | 1.464e^{−2} | 4.393e^{−2} | 7.313e^{−2} | 2.143e^{−3} | 6.433e^{−3} | 1.070e^{−2} | −3.239e^{−4} | −9.722e^{−4} | −1.614e^{−3} |

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Alzaid, N.A. and Alrayiqi, B.A. (2019) Approximate Solution Method of the Seventh Order KdV Equations by Decomposition Method. Journal of Applied Mathematics and Physics, 7, 2148-2155. https://doi.org/10.4236/jamp.2019.79147