_{1}

^{*}

In this paper, the Adomian decomposition method with Green’s function (Standard Adomian and Modified Technique) is applied to solve linear and nonlinear tenth-order boundary value problems with boundary conditions defined at any order derivatives. The numerical results obtained with a small amount of computation are compared with the exact solutions to show the efficiency of the method. The results show that the decomposition method is of high accuracy, more convenient and efficient for solving high-order boundary value problems.

In the beginning of the 1980’s, Adomian [

The lower-order boundary value problems have been vastly examined, analytically and numerically, in the literature. In contrast, higher-order boundary value problems have not been studied to the same extent that lower- order equations have been investigated. Nowadays, higher-order boundary value problems receive an increased interest due to the fact that they are noted in many mathematical physics applications. A class of characteristic- value problems of high order (as high as twenty four) are known to arise in hydrodynamic and hydromagnetic stability [

Theorems which list the conditions for the existence and uniqueness of solutions of BVPs of higher order are thoroughly investigated in a book by Agarwal [

Different numerical and semi analytical methods have been proposed by various authors to solve tenth-order boundary-value problems. A few of them are: Tenth degree spline method [

The main objective of this paper is to apply the Standard Adomian with Green’s function (SAwGF) and Modified Technique with Green’s function (MTwGF) to linear and nonlinear tenth-order boundary value problems with boundary conditions defined at any order derivatives.

Let us consider the general BVP of tenth-order

with boundary conditions

where

fined in the interval

Applying the decomposition method as in [

where

where

The Adomian’s technique consists of approximating the solution of (1) as an infinite series

and decomposing the nonlinear operator N as

where

The proofs of the convergence of the series

and (5) into (3) yields

From (6), the iterates defined using the Standard Adomian Method are determined in the following recursive way:

and the iterates defined using the Modified Technique [

Thus all components of

to the solution

In this section, the ADM with the Green’s function (Standard Adomian and Modified Technique) for solving linear and nonlinear tenth-order boundary value problems is illustrated in the following examples. To show the high accuracy of the solution results compared with the exact solution, we give the maximum absolute error and the maximum residual error. The computations associated with the examples were performed using a Maple 13 package with a precision of 40 dígits.

Consider the following linear BVP of tenth-order [

with boundary conditions

The exact solution of (7), (8) is

Applying the decomposition method, Equation (7) can be written as

where

where

and

where

Substituting (4) in (9), the iterates defined using the Standard Adomian Method are determined in the following recursive way:

and the iterates defined using the Modified Technique [

In

Consider the following linear BVP of tenth-order [

. Comparison of maximum errors for example 1

SAwGF | MTwGF | DTM [9] | EDSM [10] | NPSM [11] | VIT [12] | |
---|---|---|---|---|---|---|

1.99E-16 | 6.69E-14 | 2.70E-08 | 3.28E-06 | 4.72E-06 | 1.97E-06 | |

1.41E-16 | 4.56E-14 | - | - | - | - |

. Maximum residual error for example 1

SAwGF | MTwGF | ||||
---|---|---|---|---|---|

2 | 4.51E-09 | 2.94E-09 | 5.02E-06 | 3.98E-06 | |

3 | 4.64E-17 | 3.11E-17 | 2.23E-14 | 1.48E-14 | |

4 | 4.17E-25 | 2.71E-25 | 4.63E-22 | 3.68E-22 |

with boundary conditions

The exact solution of (10), (11) is

Applying the decomposition method, Equation (10) can be written as

where

where

and

and the iterates defined using the Modified Technique [

In

Consider the following linear BVP of tenth-order [

with boundary conditions

The exact solution of (13), (14) is

Applying the decomposition method, Equation (13) can be written as

where

where

. Comparison of maximum errors for example 2

SAwGF | MTwGF | DTM [9] | EDSM [10] | NPSM [11] | VIT [12] | |
---|---|---|---|---|---|---|

4.98E-14 | 4.80E-12 | 1.12E-06 | 8.85E-08 | 4.67E-07 | 4.24E-07 | |

3.66E-14 | 3.52E-12 | - | - | - | - |

. Maximum residual error for example 2

SAwGF | MTwGF | |||||
---|---|---|---|---|---|---|

2 | 2.46E-07 | 1.81E-07 | 2.37E-05 | 1.74E-05 | ||

3 | 4.98E-14 | 3.66E-14 | 4.80E-12 | 3.52E-12 | ||

4 | 1.01E-20 | 7.38E-21 | 9.69E-19 | 7.11E-19 |

and

In

Finally, we consider the following nonlinear BVP of tenth-order [

with boundary conditions

The exact solution of (16), (17) is

. Comparison of maximum errors for example 3

SAwGF | MTwGF | EDSM [10] | |||
---|---|---|---|---|---|

7.05E-16 | 2.29E-23 | 2.75E-14 | 1.16E-21 | 3.73E-08 | |

2.56E-16 | 1.05E-23 | 1.86E-14 | 8.60E-22 | - |

. Maximum residual error for example 3

SAwGF | MTwGF | |||||
---|---|---|---|---|---|---|

2 | 2.13E-08 | 1.47E-08 | 1.13E-06 | 8.17E-07 | ||

3 | 4.16E-16 | 2.56E-16 | 1.79E-14 | 1.06E-14 | ||

4 | 1.17E-23 | 7.77E-24 | 5.74E-22 | 3.88E-22 |

Applying the decomposition method, Equation (16) can be written as

where

where

and

where

Substituting (4) and (5) in (18), the iterates defined using the Standard Adomian Method are determined in the following recursive way:

For the nonlinear term

. Comparison of maximum errors for example 4

SAwGF | MDM [8] | HPM [13] | ||
---|---|---|---|---|

4.90E-10 | 1.34E-14 | 4.58E-06 | 1.45E-05 | |

3.47E-10 | 9.48E-15 | - | - |

. Maximum residual error for example 4

2 | 4.59E-05 | 3.25E-05 |

3 | 1.30E-09 | 8.89E-10 |

4 | 4.04E-14 | 2.69E-14 |

In

The ADM with Green’s function (Standard Adomian and Modified Technique) has been applied for solving linear and nonlinear tenth-order boundary value problems with boundary conditions defined at any order derivatives. Comparison of the results obtained by the present method with those obtained by the Tenth degree spline method, Modified decomposition method with the inverse operator, Differential transform method, Eleventh degree spline method, Non-polynomial spline method, Variational iteration technique and Homotopy perturbation method has revealed that the present method is superior because of the lower error and fewer required iterations. It has been shown that error is monotonically reduced with the increment of the integer n.