A new iterative technique is employed to solve a system of nonlinear fractional partial differential equations. This new approach requires neither Lagrange multiplier like variational iteration method (VIM) nor polynomials like Adomian's decomposition method (ADM) so that can be more easily and effectively established for solving nonlinear fractional differential equations, and will overcome the limitations of these methods. The obtained numerical results show good agreement with those of analytical solutions. The fractional derivatives are described in Caputo sense.

In recent years, it has been turned out that fractional differential equations can be used successfully to model many phenomena in various fields such as fluid mechanics, viscoelasticity, physics, chemistry, and engineering. For instance, the fluid−dynamics traffic model with fractional derivatives [

In this work, we consider the solution of generalized Hirota−Satsuma coupled KdV of time−fractional order which is presented by a system of nonlinear partial differential equations, of the form:

In this section, there are some basic definitions and properties of the fractional calculus theory which are used in this paper.

A real function

The left−sided Riemann−Liouville fractional integral operator of order

The properties of the operator

The Riemann−Liouville derivative has certain disadvantages, in trying way to model real−world phenomena with fractional differential equations. Therefore, we will employ a modification of fractional differential operator

The (left−sided) Caputo fractional derivative of

Also, we need two of its basic properties.

If

The Caputo fractional derivative is considered here, because it allows traditional initial and boundary conditions to be included in the formulation of the problem.

In this work, we consider the one−dimensional linear nonhomogeneous fractional partial differential equations in fluid mechanics, where the unknown function

For

For more information on the mathematical properties of fractional derivatives and integrals, one can consult the mentioned references.

As pointed in [

Let

However (

In (

Now, we introduce a new convenient technique for controlling the convergence region and rate of solution series for this method. Assume that we gain a family of solution series in the auxiliary parameter

Having freedom for choosing the auxiliary function

In this section, we implement fractional iteration method to generalized Hirota−Satsuma coupled KdV of time−fractional order when

Choosing

In this section, four figures are presented corresponding to FIM results and exact solutions for the solitary wave solutions

Demonstrating the exactness of FIM, the numerical results are presented and only few iterations are required to achieve accurate solutions. The convergence of FIM for the generalized fractional−order Hirota−Satsuma−coupled KdV equation is controllable, using the so−called

The solitary wave solution of

The solitary wave solution of

The solitary wave solution of

Tables

Numerical values when

0.2 | 0 | 0.4935113355 | 0.4933513333 | 0.4933937249 | 0.4933513333 | 0.4933513226 | 0.4933513333 | 0.4933513225 |

0.25 | 0.4935979177 | 0.4934136490 | 0.4934546200 | 0.4934060408 | 0.4933937118 | 0.4933937581 | 0.4933937115 | |

0.5 | 0.4937081918 | 0.4935005536 | 0.4935399023 | 0.4934853752 | 0.4934607890 | 0.4934608711 | 0.4934607891 | |

0.75 | 0.4938416235 | 0.4936116158 | 0.4936491516 | 0.4935889426 | 0.4935522228 | 0.4935523392 | 0.4935522227 | |

1 | 0.4939975728 | 0.4937462882 | 0.4937818335 | 0.4937162326 | 0.4936675611 | 0.4936677111 | 0.4936675613 | |

0.4 | 0 | 0.4936853330 | 0.4934053333 | 0.4935033001 | 0.4934053333 | 0.4934051602 | 0.4934053333 | 0.4934051609 |

0.25 | 0.4938008948 | 0.4935090262 | 0.4935963610 | 0.4934954686 | 0.4934771397 | 0.4934775983 | 0.4934771401 | |

0.5 | 0.4939391903 | 0.4936368323 | 0.4937131147 | 0.4936097847 | 0.4935733947 | 0.4935741334 | 0.4935733945 | |

0.75 | 0.4940995646 | 0.4937881198 | 0.4938529939 | 0.4937477171 | 0.4936934501 | 0.4936944620 | 0.4936934499 | |

1 | 0.4942812668 | 0.4939621476 | 0.4940153259 | 0.4939085897 | 0.4938367205 | 0.4938379949 | 0.4938367203 | |

0.6 | 0 | 0.4938553137 | 0.4934953333 | 0.4936435681 | 0.4934953333 | 0.4934944586 | 0.4934953333 | 0.4934944625 |

0.25 | 0.4939921388 | 0.4936348749 | 0.4937639292 | 0.4936174202 | 0.4935955155 | 0.4935973485 | 0.4935955186 | |

0.5 | 0.4941508292 | 0.4937979026 | 0.4939071763 | 0.4937630802 | 0.4937202663 | 0.4937230371 | 0.4937202675 | |

0.75 | 0.4943306471 | 0.4939836133 | 0.4940726248 | 0.4939315968 | 0.4938681006 | 0.4938717813 | 0.4938681010 | |

1 | 0.4945307679 | 0.4941911034 | 0.4942594934 | 0.4941221502 | 0.4940383061 | 0.4940428589 | 0.4940383061 |

Numerical values when

0.2 | 0 | −3.004851885 | −3.009951963 | −3.010186962 | −3.011485031 | −3.013961813 | −3.013960000 | −3.013961811 |

0.25 | −2.999873282 | −3.004930481 | −3.005175951 | −3.006462592 | −3.008937820 | −3.008936014 | −3.008937819 | |

0.5 | −2.994919279 | −2.999927784 | −3.000183326 | −3.001457026 | −3.003927607 | −3.003925818 | −3.003927606 | |

0.75 | −2.989995777 | −2.994950010 | −2.995215173 | −2.996474487 | −2.998937350 | −2.998935586 | −2.998937348 | |

1 | −2.985108534 | −2.990003168 | −2.990277460 | −2.991521007 | −2.993973118 | −2.993971391 | −2.993973120 | |

0.4 | 0 | −2.998707345 | −3.001587258 | −3.003540427 | −3.004319121 | −3.007934500 | −3.007920000 | −3.007934475 |

0.25 | −2.993774914 | −2.996584581 | −2.998560913 | −2.999314737 | −3.002927784 | −3.002913367 | −3.002927763 | |

0.5 | −2.988873896 | −2.991611047 | −2.993607731 | −2.994336092 | −2.997942249 | −2.997927984 | −2.997942228 | |

0.75 | −2.984009951 | −2.986672646 | −2.988686777 | −2.989389199 | −2.992983946 | −2.992969899 | −2.992983924 | |

1 | −2.979188572 | −2.981775197 | −2.983803790 | −2.984479922 | −2.988058789 | −2.988045031 | −2.988058768 | |

0.6 | 0 | −2.994081735 | −2.994318388 | −2.997767879 | −2.997835535 | −3.001928965 | −3.001880000 | −3.001928766 |

0.25 | −2.989192682 | −2.989342884 | −2.992826759 | −2.992857834 | −2.996948389 | −2.996899772 | −2.996948197 | |

0.5 | −2.984339775 | −2.984405468 | −2.987918608 | −2.987913837 | −2.991996239 | −2.991948208 | −2.991996054 | |

0.75 | −2.979528475 | −2.979511953 | −2.983049116 | −2.983009390 | −2.987078404 | −2.987031190 | −2.987078224 | |

1 | −2.974764060 | −2.974667936 | −2.978223778 | −2.978150144 | −2.982200600 | −2.982154433 | −2.982200431 |

Numerical values when

0.2 | 0 | 1.507523900 | 1.504990747 | 1.504874024 | 1.504229289 | 1.502999100 | 1.503000000 | 1.502999100 |

0.25 | 1.509996714 | 1.507484860 | 1.507362938 | 1.506723878 | 1.505494461 | 1.505495357 | 1.505494461 | |

0.5 | 1.512457311 | 1.509969643 | 1.509842719 | 1.509210085 | 1.507982979 | 1.507983864 | 1.507982977 | |

0.75 | 1.514902760 | 1.512442049 | 1.512310343 | 1.511684858 | 1.510461580 | 1.510462458 | 1.510461582 | |

1 | 1.517330199 | 1.514899088 | 1.514762850 | 1.514145194 | 1.512927259 | 1.512928117 | 1.512927258 | |

0.4 | 0 | 1.510575823 | 1.509145401 | 1.508175285 | 1.507788516 | 1.505992798 | 1.506000000 | 1.505992810 |

0.25 | 1.513025704 | 1.511630175 | 1.510648553 | 1.510274137 | 1.508479570 | 1.508486739 | 1.508479588 | |

0.5 | 1.515459985 | 1.514100473 | 1.513108742 | 1.512746974 | 1.510955834 | 1.510962922 | 1.510955847 | |

0.75 | 1.517875852 | 1.516553321 | 1.515552926 | 1.515204040 | 1.513418571 | 1.513425547 | 1.513418581 | |

1 | 1.520270576 | 1.518985828 | 1.517978251 | 1.517642422 | 1.515864843 | 1.515871673 | 1.515864850 | |

0.6 | 0 | 1.512873311 | 1.512755768 | 1.511042443 | 1.511008840 | 1.508975680 | 1.509000000 | 1.508975778 |

0.25 | 1.515301647 | 1.515227046 | 1.513496642 | 1.513481209 | 1.511449479 | 1.511473624 | 1.511449571 | |

0.5 | 1.517712032 | 1.517679404 | 1.515934464 | 1.515936836 | 1.513909155 | 1.513933009 | 1.513909245 | |

0.75 | 1.520101750 | 1.520109958 | 1.518353088 | 1.518372820 | 1.516351785 | 1.516375237 | 1.516351875 | |

1 | 1.522468182 | 1.522515926 | 1.520749779 | 1.520786351 | 1.518774537 | 1.518797467 | 1.518774621 |

In this paper, the fractional iteration method (FIM) has been successfully applied to study Hirota−Satsuma−coupled KdV of time−fractional−order equation. FIM results are compared with the exact solutions and those obtained by Homotopy perturbation method [

The results show that fractional iteration method is a powerful and efficient technique in finding exact and approximate solutions for nonlinear partial differential equations of fractional order. The method provides the user with more realistic series solutions that converge very rapidly in real physical problems.

Compared with the ADM and VIM, the FIM has following advantages, [

The auxiliary parameter

The solution of a given nonlinear problem can be expressed by an infinite number of solution series and thus can be more efficiently approximated by a better selection of the auxiliary parameter values.

Unlike the ADM, the FIM method is free from the need to use Adomian polynomials.

This method has no need for the Lagrange multiplier, correction functional, stationary conditions, the variational theory, and so forth, which eliminates the complications that exist in the VIM.

The fractional iteration method can be easily comprehended with only a basic knowledge of fractional calculus.

Compared to the ADM and VIM, the presented method proves simpler in its principles and more convenient for computer algorithms.

In this work, we used Maple Package to calculate the series obtained by fractional iteration method.