^{1}

^{1}

^{2}

In this article, variational iteration method (VIM) and homotopy perturbation method (HPM) solve the nonlinear initial value problems of first-order fractional quadratic integro-differential equations (FQIDEs). We use the Caputo sense in this article to describe the fractional derivatives. The solutions of the problems are derived by infinite convergent series, and the results show that both methods are most convenient and effective.

The fractional calculus has appeared in many areas during the recent decades. Some scientists use approximation and numerical methods because there are almost no exact solutions of the fractional differential equations. He has pro- posed the VIM and HPM to solve the problems of linear and nonlinear [

Our goal for this article is extending the analysis of VIM and HPM to construct the approximate solutions of the following nonlinear initial value problems for first-order fractional quadratic integro-differential equations.

subject to the following initial condition:

In this section, we intend to present some basic definitions and properties of fractional calculus theory which are further used in this article.

Definition 1.

Areal function

Definition 2.

The Riemann-Liouville fractional integral operator of order

Some properties of the operator

for

1.

2.

3.

Definition 3. The fractional derivative of

for

Lemma 1.

If

1.

2.

The basic concept of the VIM is constructing the correction functional for the fractional quadratic integro-differential equation sees Equation (1) with initial conditions,

taking the variation of Equation (7) to the independent variable

to make the previous equation stationary, we gain the following stationary conditions:

finally, the Lagrange multiplier is:

We achieve the following iteration formula by substitution of (10) into the functional (6)

the initial approximation

The main concept of the HPM is constructing the homotopy for fractional quadratic integro-differential equation sees Equation (1),

and when

The solution of Equation (1) can be considered as a power series in p which is the basic assumption of HPM :

when

First, substitute the relation (16) in the Equation (15). Second, equate the terms which have the same power’s of p which yield to the following series of equations:

and so on, the functions

n this section, we apply VIM and HPM to first-order nonlinear (FQIDEs).

Example 1.

Consider the following nonlinear first-order (FQIDEs):

subject to the following initial condition

According to VIM, the expression of the iteration formula (12) for Equation (21) can be observed in the following form:

To avoid the difficulty of fractional integration, for the exponential term we

take the truncated taylor expansion in (23), e.g.,

tisfy the initial condition (22), we assume that the initial approximation has the following form

According to HPM, we build the following homotopy:

First, substitute the relation (16) in the Equation (26).

Second, equate the terms which have the same power’s of p which yield to the following series of Equations:

x | ||||
---|---|---|---|---|

0.10 | 3.46688592 | 2.27052430 | 1.65826130 | 1.33251002 |

0.20 | 4.93376139 | 3.18886968 | 2.27376989 | 1.74152708 |

0.30 | 6.70750098 | 4.28829701 | 3.00556169 | 2.24764961 |

0.40 | 8.84373442 | 5.65002513 | 3.90802714 | 2.87616667 |

0.50 | 11.37418953 | 7.33008584 | 5.02934761 | 3.65730794 |

0.60 | 14.32676071 | 9.37815747 | 6.41827073 | 4.62649375 |

0.70 | 17.72917203 | 11.84228561 | 8.12594765 | 5.82458502 |

0.80 | 21.60990786 | 14.77061532 | 10.20665676 | 7.29813333 |

0.90 | 25.99846722 | 18.21217939 | 12.71816657 | 9.09963086 |

1.00 | 30.92541601 | 22.21731293 | 15.72195568 | 11.28776041 |

and so on, apply the operator

and so on, by taking the truncated taylor expansions for the exponential term in

(29, 30): e.g.,

tion, thus by solving Equations (28, 29, 30), we obtain

The two terms approximation are formed as the following Equation

Example 2.

Consider the following (FQIDEs):

According to VIM, the expression of the iteration Formula (12) for Equation (34) can be observed in the following form:

To avoid the difficulty of fractional integration, for the exponential term we

take the truncated taylor expansion in (35), e.g.,

the initial condition, we assume that the initial approximation has the following form

x | ||||
---|---|---|---|---|

0.10 | 3.34976072 | 2.25905581 | 1.65729056 | 1.33243542 |

0.20 | 4.53980075 | 3.12399413 | 2.26453456 | 1.74033333 |

0.30 | 5.90653782 | 4.10952104 | 2.97106726 | 2.24160625 |

0.40 | 7.51861405 | 5.28303363 | 3.82016538 | 2.85706667 |

0.50 | 9.41603098 | 6.68897807 | 4.84789742 | 3.61067708 |

0.60 | 11.63265252 | 8.36684782 | 6.09010254 | 4.52980000 |

0.70 | 14.20081997 | 10.35549050 | 7.58435402 | 5.64544792 |

0.80 | 17.15275198 | 12.69459789 | 9.37077013 | 6.99253333 |

0.90 | 20.52106372 | 15.42533768 | 11.49244150 | 8.61011875 |

1.00 | 24.33898202 | 18.59065973 | 13.99570082 | 10.54166666 |

According to HPM, we build the following homotopy:

First, substitute the relation (16) in the Equation (37).

Second, equate the terms which have the same power’s of p which yield to the following series of Equations:

and so on, applying the operator

x | ||||
---|---|---|---|---|

0.10 | 1.74557818 | 1.39713379 | 1.20769182 | 1.10549607 |

0.20 | 2.06685872 | 1.63644354 | 1.37989631 | 1.22394067 |

0.30 | 2.39681774 | 1.88439871 | 1.56340949 | 1.35821602 |

0.40 | 2.74634447 | 2.15443603 | 1.76686736 | 1.51115026 |

0.50 | 3.11743203 | 2.45118250 | 1.99492282 | 1.68553370 |

0.60 | 3.51027500 | 2.77680334 | 2.25082546 | 1.88413224 |

0.70 | 3.92459385 | 3.13252560 | 2.53716249 | 2.10969920 |

0.80 | 4.35998072 | 3.51915865 | 2.85615873 | 2.36498657 |

0.90 | 4.81599699 | 3.93730820 | 3.20982456 | 2.65275652 |

1.00 | 5.29219083 | 4.38747370 | 3.60003936 | 2.97579365 |

and so on, by taking the truncated taylor expansions for the exponential term in

(40, 41): e.g.,

To avoid the difficulty of fractional integration, thus by solving Equations (39, 40, 41), we obtain

the two terms approximation are formed as the following Equation

x | ||||
---|---|---|---|---|

0.10 | 1.74531342 | 1.39709144 | 1.20768552 | 1.10549521 |

0.20 | 2.06407267 | 1.63581216 | 1.37976305 | 1.22391477 |

0.30 | 2.38608229 | 1.88141089 | 1.56263514 | 1.35803058 |

0.40 | 2.71887729 | 2.14558172 | 1.76420974 | 1.51041158 |

0.50 | 3.06116974 | 2.43083965 | 1.98807470 | 1.68339534 |

0.60 | 3.40994267 | 2.73693555 | 2.23607620 | 1.87906377 |

0.70 | 3.76162440 | 3.06236426 | 2.50903647 | 2.09921280 |

0.80 | 4.11218308 | 3.40478991 | 2.80699405 | 2.34530601 |

0.90 | 4.45689697 | 3.76108758 | 3.12924403 | 2.61840230 |

1.00 | 4.78998433 | 4.12719346 | 3.47425744 | 2.91904762 |

Example 3.

Consider the following nonlinear (FQIDEs)

For

We can take an initial approximation

The first two iterations are easily obtained from (47) and are given by:

Therefore, we obtain the exact solution,

According to HPM, we construct the following homotopy:

and continuously trace an implicity defined curve from starting point

and so on, we obtain

obtained readily by

In this paper, we have applied the VIM and HPM to find the solution of nonlinear initial value problem of fractional quadratic integro-differential equations for the first order. The methods do not require any linearization, perturbation or restrictive assumptions, we have observed that the VIM and HPM is a very powerful and effective tool for finding the solutions of the fractional quadratic integro-differential Equation. We use the Maple package (2015) in calculations.

Alhendi, F., Shammakh, W. and Al-Badrani, H. (2017) Numerical Solutions for Quadratic Integro- Differential Equations of Fractional Or- ders. Open Journal of Applied Sciences, 7, 157-170. https://doi.org/10.4236/ojapps.2017.74014