The fundamental objective of this work is to construct a comparative study of some modified methods with Sumudu transform on fractional delay integro-differential equation. The existed solution of the equation is very accurately computed. The aforesaid methods are presented with an illustrative example.

Recently, a variety of transformations were applied by some mathematicians which include widespread tremendous progress in the study of linear and nonlinear equations. A similar great transform to Laplace, Fourier and other transformations are so-called Sumudu transform [

In order to solve various linear and nonlinear equations approximately, many researchers incorporate traditional methods into transformations, such as Laplace, Sumudu, differential transform method and others [

Henceforth, we focus our attention on the combination of the homotopy perturbation (HPM), series solution, homotopy analysis (HAM), Adomian decomposition method (ADM) and variational iteration methods (VIM) with Sumudu transform, then compares the previous composition in order to ensure the high accuracy and reliability and convergence for the offered methods. The above mentioned combination is namely modified methods. One can see those modified methods are considered to be a new powerful development technique to solve several problems. For more details, we point out those modified methods can be used to solve some non-linear fractional delay integro-differential equations. Nowadays, comparison between the preceding mentioned methods plays a vital role in the study of fractional delay and nonlinear integro-differential equation.

The plan of this discussion as follows. In Section one, the main concepts that are applied in our paper are considered. In Section two, the modified methods are mentioned. In Section three, computational scheme for the proposed problem discussed. In Section four, special cases are stated. Finally, the comparative study is concluded in Section five.

Here, we now briefly recall some necessary definitions, preliminaries and properties are used further in this work as follows:

_{1}, _{2}).

The relation between Laplace and Sumudu transforms are as

For further properties of the Sumudu transform

where

[3] Linearity of the Sumudu transform, if

[4] For

[5] The Sumudu transform of the Riemann-Liouville fractional integral of order

[6] The Caputo fractional derivative of order

We describe the following fractional derivative in the Caputo sense

under the initial condition:^{+}), _{1}, _{2} are real finite constants.

Taking the Sumudu operator

With the help of the linearity of Sumudu operator,

Now, applying the property of the differentiation for Sumudu transform

where

further, the solution

substitute

On comparing of the two both sides of

in the same manner, we can calculate _{n},

Evidently, on continuing the same fourth steps from

By the comparison of the coefficients on two both sides

In this section, Sumudu transform directly coupled with a homotopy analysis method. For an embedding parameter

By HAM, the deformation equation of order zero is constructed as follows:

where _{0}(_{1}(

by Taylor series, we can expand

where

where

And

By means of the inverse Sumudu transform operator ^{−1}

The solution can be written as

This section is based on the combination of variational iteration method with Sumudu transform.

Let us apply the inverse Sumudu transform of both sides

Now, differential the preceding equations with respect to

Due to the variational iteration method, the correct function can be rewritten as:

So, the limit of {_{m}(_{m≥0} is equivalent to the exact solution.

Upon using the Sumudu decomposition method, definition of the solution _{j}, _{j} and Z_{j} are the Adomian polynomials of _{0}, _{1}, … , _{j}.

Now, the Adomian polynomials for the nonlinear functions are written as

The new recursive relation becomes

Finally,

Substitute

By Comparison of two both sides

In general form,

Suggest that

Generally,

Example:

Solution:

Taking Sumudu transform to both sides of

From

That is, for a special value

By homotopy perturbation Sumudu transform method, it follows immediately that,

i.e.,

Repeat the above procedure until

Throughout method of series solution in _{i} = 1

_{0} =

Put

As a result

The n-th order deformation equation is obtained as

Especially, if

For more generality un approaches to _{0} =

We construct the iteration formula as

In particular,

Hence, the general term um is obtained as _{m} =

Obviously,

For more generality,

From Adomian decomposition, we have

Then it follows:

The estimate we will obtain in this section seems to be independent of interest. Without restriction of generality, we can assume

_{exact}(

Exact value _{exact} ( |
|||||
---|---|---|---|---|---|

0.1 | 0.1 | 0.35682 | 0.1 | 0.02379 | 0.09500 |

0.2 | 0.2 | 0.50463 | 0.2 | 0.06728 | 0.18000 |

0.3 | 0.3 | 0.61804 | 0.3 | 0.12361 | 0.25500 |

0.4 | 0.4 | 0.71365 | 0.4 | 0.19031 | 0.32000 |

0.5 | 0.5 | 0.79788 | 0.5 | 0.26596 | 0.37500 |

0.6 | 0.6 | 0.87404 | 0.6 | 0.34962 | 0.42000 |

0.7 | 0.7 | 0.94407 | 0.7 | 0.44057 | 0.45500 |

0.8 | 0.8 | 1.00925 | 0.8 | 0.53827 | 0.48000 |

0.9 | 0.9 | 1.07047 | 0.9 | 0.64229 | 0.49500 |

1 | 1 | 1.12838 | 1 | 0.75225 | 0.50000 |

1.1 | 1.1 | 1.18345 | 1.1 | 0.8787 | 0.60500 |

1.2 | 1.2 | 1.23608 | 1.2 | 0.98886 | 0.72000 |

1.3 | 1.3 | 1.28655 | 1.3 | 1.11501 | 0.84500 |

1.4 | 1.4 | 1.33512 | 1.4 | 1.24611 | 0.98000 |

1.5 | 1.5 | 1.38198 | 1.5 | 1.38198 | 1.12500 |

1.6 | 1.6 | 1.42730 | 1.6 | 1.52245 | 1.28000 |

1.7 | 1.7 | 1.47123 | 1.7 | 1.67739 | 1.44500 |

1.8 | 1.8 | 1.51388 | 1.8 | 1.81666 | 1.62000 |

1.9 | 1.9 | 1.55536 | 1.9 | 1.97013 | 1.80500 |

2.0 | 2.0 | 1.59577 | 2.0 | 2.12769 | 2 |

It is evident that this manuscript is more generalized from analogues papers. More remarkable special cases which are covered by a lot of last papers.

The benefit of modified several methods is that it successfully contributed to the rapidly progressing of the exact, approximate and numerical solutions. Also, modified methods are considered as a new powerful technique to solve a large range of equations such as differential, integral and integro-differential. At a certain value of

The authors are very grateful to the editors and reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper.