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In this paper, the Adomian Decomposition Method (ADM) and the Differential Transform Method (DTM) are applied to solve the multi-pantograph delay equations. The sufficient conditions are given to assure the convergence of these methods. Several examples are presented to demonstrate the efficiency and reliability of the ADM and the DTM; numerical results are discussed, compared with exact solution. The results of the ADM and the DTM show its better performance than others. These methods give the desired accurate results only in a few terms and in a series form of the solution. The approach is simple and effective. These methods are used to solve many linear and nonlinear problems and reduce the size of computational work.

Pantograph is a device located on the electriclocomotive. The first time, electric locomotive was made in America in 1851. It was commissioned in 1895. Mathematical model of pantograph was first developed by Taylor and Ockendon (1971) [

In recent years, the multi-pantograph delay differential equations were studied by many authors. For examples, Li and Liu [

A numerical method based on the Adomian Decomposition Method (ADM) which has been used from the 1970s to the 1990s by George Adomian [

ADM and DTM have been shown to solve effectively, easily and accurately a large class of linear and nonlinear, ordinary, partial, deterministic or stochastic differential equations with approximate solutions which converge rapidly to accurate solutions [

This study is presented as follows: In second section, we start by presenting ADM and DTM to solve multi-pantograph delay differential equations. In third section, we continue to the presentation of the convergence of ADM with Theorem 3.1 and Definition 3.2. In fourth section, these methods are shown and compared by four examples by taking various values for t and error evaluation is made. Also, we have plotted the graphs for numerical solutions of ADM and DTM and exact solution.

We examined that multi-pantograph delay differential equations are solved by several methods. Thus, we wanted to show up that may be more efficient, simpler and reliable the solution treatment of the ADM for multi- pantograph delay differential equations. The results show that the ADM is more powerful method than other methods for multi-pantograph delay differential equations.

In this paper, we consider the following multi-pantograph equations [

Using the ADM, the differential operator L is given by

The inverse operator

operating with

where the method defines

where

Operating with

to determine the components

First, we identify the zero component

and Equation (8) gives for

The Adomian decomposition method assumes that the unknown function

so that the components

Differential transform of function

In Equation (11),

From Equations (12) and (11), we obtain

Equation (13) implies that concept of differential transform is derived from Taylor series expansion, but the method does not evaluate the derivatives symbolically.

In actual applications, the function

Equation (13) implies that is

is negligibly small. In fact, m is decided by the convergence of natural frequency in this study.

The following theorems that can be deduced from Equations (11) and (12) are given below, see [

Theorem 1 If

Theorem 2 If

Theorem 3 If

Theorem 4 If

Theorem 5 If

The Adomian Decomposition Method is equivalent to the sequence defined as follows [

by using the iterative scheme

and related to the functional equation

The numerical solution of Equation (15) was used fixed-point theorem by Cherruault [

Theorem 3.1 Let N be an operator from Hilbert space H in to H and y be the exact solution of functional equation.

Proof See [

Definition 3.2 For every

Corollary 3.3 In Theorem 4.1,

In this section, four experiments of multi-pantograph delay differential equations are given to illustrate the efficiency of the ADM and the DTM. The examples are computed using Maple 15. Results obtained by the methods are compared with the exact solution of each example and found to be good agreement with each other. The absolute errors in tables are given at selected points.

Example 4.1 Consider the following linear multi-pantograph delay equation of the first-order [

which has the exact solution,

Using the ADM, we get according to Equations (3)-(10), we obtain recursive formula for

Thus, we obtain

The solution by DTM method:

By using Theorems of DTM, we have following recurrence relation:

Utilizing the recurrence relation, we find

Finally, the differential inverse transform of

we obtain the following series solution

the closed form of above solution is

which is exactly the same as the exact solution.

The obtained results (ADM and DTM) are exactly the same with the one found by exact solution. It is clear from

Using our methods, we choose 6 points on [0, 1] respectively. The numerical results are given in the following

Example 4.2 Solve the following nonlinear pantograph delay equation of first-order [

which has the exact solution,

The solution by ADM method:

By applying the ADM, according to Equations (3)-(10), we obtain

t | Solution | |||
---|---|---|---|---|

Error | ||||

0 | 1 | 1 | 0 | |

0.2 | 4.238932099 | 4.238932099 | 0 | |

0.4 | 9.789234568 | 9.789234568 | 0 | |

0.6 | 18.10116667 | 18.10116667 | 0 | |

0.8 | 29.62498766 | 29.62498766 | 0 | |

1 | 44.8109568 | 44.8109568 | 0 | |

We have solved this problem using the proposed method. Recursive formula and the sequence of approximate solution are obtained as follows:

thus, we obtain:

Using to convergence of ADM’s method,

Here, the values of

The solution by DTM method:

By using Theorems of DTM, we have following recurrence relation:

Utilizing the recurrence relation, we find

Finally, the differential inverse transform of

we obtain the following series solution

The obtained results (ADM and DTM) are exactly the same with each other. Increasing the approximation order up to the absolute differences between the numerical solutions are calculated for and comparisons have been made with known results as reported in

Example 4.3 Consider the following linear multi-pantograph delay equation of the first-order [

The solution by ADM method:

By applying the ADM, according to Equations (3)-(10), we obtain

We have solved this problem using the proposed method. Recursive formula and the sequence of approximate solution are obtained as follows:

t | Solution | ||
---|---|---|---|

0 | 0 | 0 | 0 |

0.2 | 0.198669331 | 0.198669 | 0.198669333 |

0.4 | 0.389418342 | 0.389418 | 0.389418342 |

0.6 | 0.564642446 | 0.564642 | 0.564642446 |

0.8 | 0.717355723 | 0.717356 | 0.717355723 |

1 | 0.841468254 | 0.841471 | 0.841468254 |

Thus, we obtain:

The solution by DTM method:

By using Theorems of DTM, we have following recurrence relation:

Using the recurrence relation, we find

Finally, the differential inverse transform of

we obtain the following series solution

The obtained results (ADM and DTM) are exactly the same with the one found by exact solution. It is clear from

Example 4.4 Consider the following linear multi-pantograph delay equation of the first-order [

The solution by ADM method:

By applying the ADM, according to Equations (3)-(10), we obtain

We have solved this problem using the proposed method. Recursive formula and the sequence of approximate solution are obtained as follows:

t | Solution | |
---|---|---|

0 | 2.127 | 2.127 |

0.2 | 1.601884802 | 1.601884802 |

0.4 | 1.307204736 | 1.307204736 |

0.6 | 1.219390558 | 1.219390558 |

0.8 | 1.444676306 | 1.444676306 |

1 | 2.219099301 | 2.219099301 |

Thus, we obtain:

Using to convergence of ADM’s method,

according to the obtained results ADM is convergent to the exact solution.

The solution by DTM method:

By using Theorems of DTM, we have following recurrence relation,

Utilizing the recurrence relation, we find

Finally, the differential inverse transform of

we obtain the following series solution

The obtained results (ADM and DTM) are exactly the same with the one found by exact solution. It is clear from

It has been the aim of this paper to show that it appears natural to approximate the solution of multi-pantograph delay differential equation by ADM and DTM. We obtain the high approximate solutions or the exact solutions within a few iterations. It is concluded from figures and tables that the successive approximations methods are an accurate and efficient method to solve multi-pantograph delay differential equations. Some numerical examples have been provided to illustrate that the present method is effective in accuracy and convergence speed. In a word, the ADM and DTM show that the techniques are reliable, powerful and promising methods for linear

t | Solution | |
---|---|---|

0.0 | 1 | 1 |

0.2 | 0.703233817 | 0.703233817 |

0.4 | 0.480505241 | 0.480505241 |

0.6 | 0.316847313 | 0.316847313 |

0.8 | 0.202197189 | 0.202197189 |

1.0 | 0.131396133 | 0.131396133 |

and nonlinear problems.

MusaCakir,DeryaArslan, (2015) The Adomian Decomposition Method and the Differential Transform Method for Numerical Solution of Multi-Pantograph Delay Differential Equations. Applied Mathematics,06,1332-1343. doi: 10.4236/am.2015.68126