We study higher-order boundary value problems (HOBVP) for higher-order nonlinear differential equation. We make comparison among differential transformation method (DTM), Adomian decomposition method (ADM), and exact solutions. We provide several examples in order to compare our results. We extend and prove a theorem for nonlinear differential equations by using the DTM. The numerical examples show that the DTM is a good method compared to the ADM since it is effective, uses less time in computation, easy to implement and achieve high accuracy. In addition, DTM has many advantages compared to ADM since the calculation of Adomian polynomial is tedious. From the numerical results, DTM is suitable to apply for nonlinear problems.

Recently, many researchers use ADM to approximate numerical solutions. In [

In addition, Wazwaz [

The Adomian decomposition method is widely used in applied science to compute the series solution accurately because it provides rapid convergent series to solve the problem. The big application of ADM in research area is stochastic and deterministic problems involving differential, integrodifferential, integral, differential delay, and systems of such equations; for example, see [

Further, Ray and Bera in [

In [

Ayaz in [

Recently, Arikoglu and Ozkol solved fractional differential equations by using differential transform method. They applied fractional differential equations to various types of problems such as the Bagley-Torvik, Ricatti, and composite fractional oscillation equations; see [

In this study, we make comparison among differential transformation method, Adomian decomposition method and exact solutions. We prove that DTM is more powerful technique than ADM and can be applied to nonlinear problems easily.

Suppose that the function

The differential transform of the function

The following theorems are operations of differential transforms.

If

If

If

If

If

If

If

If

If

If

If

See the details in [

Now we prove the following generalized theorem.

The general differential transformation for nonlinear

We prove this theorem by induction method. First of all, we prove the generalization of the differential equation

For

Now assume that for

Note that the theorem holds for

By using DTM, we solve the following nonlinear equation of order three for fifth-order BVP:

By using the transformed equation (

Finally, the following series solution can be formed by applying the inverse transformation in (

Next, we solve the following fourth-order nonlinear for fifth-order BVP by using DTM:

By using the transformed equation (

Finally, the following series solution can be formed by applying the inverse transformation in (

We perform the following third order of nonlinear equation for sixth-order BVP by using DTM:

By using the transformed equation (

Consequently, the following series solution can be formed by applying the inverse transformation equation in (

Consider the differential equation

The series solution of

From the above equations, we observe that the component

For comparison purpose, we solve the boundary value problems in Examples

By using ADM, we solve the following nonlinear equation of order three for fifth-order BVP:

Now, we solve the following fourth-order nonlinear for fifth-order boundary value problems by using ADM:

Finally, we perform third order of nonlinear function for sixth-order BVP for ADM as follows:

We provide the results of the given examples in Tables

Numerical result for Example

Exact solution | DTM ( | ADM ( | |
---|---|---|---|

0.0 | 1 | 1 | 1 |

0.1 | 1.051271096 | 1.051278920 | 1.051304392 |

0.2 | 1.105170918 | 1.105220776 | 1.105376925 |

0.3 | 1.161834243 | 1.161964144 | 1.162354804 |

0.4 | 1.221402759 | 1.221630872 | 1.222288426 |

0.5 | 1.284025416 | 1.284337420 | 1.285197759 |

0.6 | 1.284025416 | 1.284337420 | 1.351122659 |

0.7 | 1.284025416 | 1.284337420 | 1.420166655 |

0.8 | 1.284025416 | 1.284337420 | 1.492533672 |

0.9 | 1.284025416 | 1.284337420 | 1.568557204 |

1.0 | 1.284025416 | 1.284337420 | 1.648721285 |

Numerical result for Example

Exact solution | DTM ( | ADM ( | |
---|---|---|---|

0.0 | 1 | 1 | 1 |

0.1 | 0.9672161006 | 0.9672221453 | 0.9667810534 |

0.2 | 0.9355069849 | 0.9672221453 | 0.9321799874 |

0.3 | 0.9048374181 | 0.9049350590 | 0.8944787522 |

0.4 | 0.8751733191 | 0.8753424236 | 0.8533754418 |

0.5 | 0.8464817250 | 0.8467098288 | 0.8103487343 |

0.6 | 0.8187307532 | 0.8189816371 | 0.7687007110 |

0.7 | 0.7918895662 | 0.7921124409 | 0.7332405365 |

0.8 | 0.7659283385 | 0.7660753950 | 0.7095695543 |

0.9 | 0.7408182206 | 0.7408702826 | 0.7029247972 |

1.0 | 0.7165313107 | 0.7165313107 | 0.7165313263 |

Numerical result for Example

Exact solution | DTM ( | ADM ( | |
---|---|---|---|

0.0 | 1 | 1 | 1 |

0.1 | 0.9512294245 | 0.9492075127 | 0.9492075127 |

0.2 | 0.9048374181 | 0.8916268943 | 0.8916268943 |

0.3 | 0.8607079765 | 0.8251427077 | 0.8251427077 |

0.4 | 0.8187307532 | 0.7534730280 | 0.7534730280 |

0.5 | 0.7788007831 | 0.6839803183 | 0.6839803183 |

0.6 | 0.7408182206 | 0.6254839642 | 0.6254839642 |

0.7 | 0.7046880897 | 0.7046783358 | 0.5860748693 |

0.8 | 0.6703200461 | 0.6703157625 | 0.5709325210 |

0.9 | 0.6376281517 | 0.6376273947 | 0.5801448253 |

1.0 | 0.6065306598 | 0.6065306599 | 0.6065306590 |

From the results, the proposed method, as well as the Differential Transformation Method, is more accurate than the Adomian Decomposition Method. The errors between the solutions of Differential Transformation Method and the exact solutions are smaller compared to the errors between the solutions of Adomian Decomposition Method and the exact solutions. In addition, Differential Transformation Method also shows less computational effort because it needs less time in calculation. Besides that, it is hard to calculate Adomian polynomials. From the results we obtained, it can reinforce conclusion made by many researchers that Differential Transformation Method is more efficient and accurate than Adomian Decomposition Method. Therefore, we can conclude that Differential Transformation Method is applicable for such problems in the bounded domains. The computations in all examples were performed by using Maple 13.

The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the Research University Grant scheme 05-01-09-0720RU and Fundamental Research Grant scheme 01-11-09-723FR. The authors also thank the referee(s) for very constructive comments and suggestions that improved the paper.