In this work, a modified residual power series method is implemented for providing efficient analytical and approximate solutions for a class of coupled system of nonlinear fractional integrodifferential equations. The proposed algorithm is based on the concept of residual error functions and generalized power series formula. The fractional derivative is described under the Caputo concept. To illustrate the potential, accuracy, and efficiency of the proposed method, two numerical applications of the coupled system of nonlinear fractional integrodifferential equations are tested. The numerical results confirm the theoretical predictions and depict that the suggested scheme is highly convenient, is quite effective, and practically simplifies computational time. Consequently, the proposed method is simple, accurate, and convenient in handling different types of fractional models arising in the engineering and physical systems.

The theory of fractional calculus is indeed a generalization of the standard calculus that deals with differentiation and integration of a noninteger order, which is utilized to describe various real-world phenomena arising in natural sciences, applied mathematics, and engineering fields with great applications for these tools, including nonlinear oscillation of earthquakes, fractional fluid-dynamic traffic, economics, solid mechanics, viscoelasticity, and control theory [

This paper aims to introduce a recent analytical as well as numerical method based on the use of fractional residual power series (RPS) technique for obtaining the approximate solution for a class of coupled system of fractional integrodifferential equations in the following form:

This is subject to the following initial conditions:

The residual power series method (RPSM) has a wide range of applications, especially in simulating nonlinear issues in a fractional meaning, which has been developed and modified over recent years as a powerful mathematical treatment indispensable in dealing with the emerging realistic system in physics, engineering, and natural sciences [

Freihet et al. [

The outline of this paper is organized as follows: In the next section, we review some basic definitions and theories related to fractional differentiation and fractional power series representations. In Section

In this section, basic definitions and results related to fractional calculus are given and fractional power series concept is also represented.

The Riemann–Liouville fractional integral operator of order

For

For

For

The following are some interesting properties of the operator

For any constant

A fractional power series (FPS) representation at

Suppose that

The purpose of this section is to construct FPS solution for the coupled system of nonlinear fractional integrodifferential equations (

The RPS algorithm proposing the solution of equations (

The truncated series form of equation (

Clearly, if

Define the residual function for equations (

Also, define the

According to the RPS algorithm [

For obtaining the coefficients

To find the coefficients

Step 1: substitute expansion (

Step 2: find the relation of fractional formula

Step 3: for

Step 4: solve the obtained algebraic fractional system

Step 5: substitute the values of

This section aims to test two applications of the system of nonlinear fractional IDEs to show the efficiency, accuracy, and applicability of the proposed method. In this section, all calculations are preformed using Wolfram-Mathematica 10.

Consider the following nonlinear fractional integrodifferential equation:

This is subject to the following initial conditions:

The exact solution of this coupled system is

Using the RPS algorithm, the

Consequently,

Numerical simulation specializes in advanced numeric or approximate methods for finding digital or approximate solutions and estimating errors of these approximations. For such purpose, a few decimal numbers are often recorded to calculate the absolute error, which produces some vague estimates related to significance and units, as well as does not provide clear and explicit evidence regarding the subject matter of the study. Therefore, the comparison of absolute error with the exact value leads to the determination of the relative error as a ratio between the value of absolute error and the exact, which gives some importance and reduces ambiguity for a deeper understanding of the behavior of the approximate solutions. By using the RPS method of Example

Tables

Numerical results of exact

0 | 1.0 | 1.0 | 0.0 |

0.1 | 1.1051709 | 1.1051708 | |

0.2 | 1.2214028 | 1.2214000 | |

0.3 | 1.3498588 | 1.3498375 | |

0.4 | 1.4918247 | 1.4917333 | |

0.5 | 1.6487213 | 1.6484375 | |

0.6 | 1.8221188 | 1.8214000 | |

0.7 | 2.0137527 | 2.0121708 | |

0.8 | 2.2255409 | 2.2224000 | |

0.9 | 2.4596031 | 2.4538375 | |

1.0 | 2.7182818 | 2.7083333 |

Numerical results of exact

0 | 1.0 | 1.0 | 0.0 |

0.1 | 0.9048374 | 0.9048375 | |

0.2 | 0.8187308 | 0.8187333 | |

0.3 | 0.7408182 | 0.7408375 | |

0.4 | 0.67033201 | 0.6704000 | |

0.5 | 0.6065307 | 0.6067708 | |

0.6 | 0.5488116 | 0.5494000 | |

0.7 | 0.4965853 | 0.4978375 | |

0.8 | 0.4493290 | 0.4517333 | |

0.9 | 0.4065697 | 0.4108375 | |

1.0 | 0.3678794 | 0.3750000 |

Numerical results of sixth approximate solution

0.0 | 1.0 | 1.0 | 1.0 | 1.0 |

0.1 | 1.0510833 | 1.0733961 | 1.1073884 | 1.1601510 |

0.2 | 1.1246667 | 1.1716823 | 1.2392129 | 1.3365098 |

0.3 | 1.2212500 | 1.2956381 | 1.3974229 | 1.5359285 |

0.4 | 1.3413333 | 1.4446154 | 1.5801578 | 1.7560644 |

0.5 | 1.4854167 | 1.6181748 | 1.7860776 | 1.9951767 |

0.6 | 1.6540000 | 1.8160345 | 2.0142021 | 2.2519736 |

0.7 | 1.8475833 | 2.0380217 | 2.2637874 | 2.5254605 |

0.8 | 2.0666667 | 2.2840401 | 2.5342526 | 2.8148458 |

0.9 | 2.3117500 | 2.5540495 | 2.8251333 | 3.1194830 |

1.0 | 2.5833333 | 2.8480518 | 3.1360514 | 3.4388329 |

Numerical results of sixth approximate solution

0.0 | 1.0 | 1.0 | 1.0 | 1.0 |

0.1 | 0.9048333 | 0.8780779 | 0.8460694 | 0.8088467 |

0.2 | 0.8186666 | 0.7855478 | 0.7499395 | 0.7123611 |

0.3 | 0.7404999 | 0.7071736 | 0.6734520 | 0.6391004 |

0.4 | 0.6693333 | 0.6386210 | 0.6084174 | 0.5772768 |

0.5 | 0.6041666 | 0.5772055 | 0.5505285 | 0.5214466 |

0.6 | 0.5439999 | 0.5209163 | 0.4969643 | 0.4684158 |

0.7 | 0.4878333 | 0.4681027 | 0.4456563 | 0.4160595 |

0.8 | 0.4346666 | 0.4173375 | 0.3949800 | 0.3628482 |

0.9 | 0.3835000 | 0.3673449 | 0.3435991 | 0.3076204 |

1.0 | 0.3333333 | 0.3169592 | 0.2903784 | 0.2494581 |

Consider the following fractional integrodifferential equation:

This is subject to the following initial conditions:

The exact solution of this coupled system is

The absolute errors are listed in Tables

Numerical results of exact

0.1 | 0.0998334 | 0.0998257 | |

0.2 | 0.1986693 | 0.1986624 | |

0.3 | 0.2955202 | 0.2955216 | |

0.4 | 0.3894183 | 0.3894152 | |

0.5 | 0.4794255 | 0.4794232 | |

0.6 | 0.5646425 | 0.5646492 | |

0.7 | 0.6442177 | 0.6442199 | |

0.8 | 0.7173561 | 0.7173533 | |

0.9 | 0.7833269 | 0.7833286 | |

1.0 | 0.8414710 | 0.8414475 |

Numerical results of exact

0.1 | 0.9950042 | 0.9950103 | |

0.2 | 0.9800666 | 0.9800597 | |

0.3 | 0.9553365 | 0.9553270 | |

0.4 | 0.9210610 | 0.9210410 | |

0.5 | 0.8775826 | 0.8775771 | |

0.6 | 0.8253356 | 0.8253269 | |

0.7 | 0.7648422 | 0.7645622 | |

0.8 | 0.6967067 | 0.6966984 | |

0.9 | 0.6216100 | 0.6215986 | |

1.0 | 0.5403023 | 0.5402669 |

Numerical results of approximate solution

0.0 | 0.0 | 0.0 | 0.0 |

0.1 | 0.0749501 | 0.1027865 | 0.1422048 |

0.2 | 0.1553618 | 0.2021055 | 0.2644892 |

0.3 | 0.2451462 | 0.3086629 | 0.3897369 |

0.4 | 0.3445577 | 0.4231060 | 0.5195960 |

0.5 | 0.4535802 | 0.5454302 | 0.6544017 |

0.6 | 0.5721434 | 0.6754941 | 0.7941689 |

0.7 | 0.7001695 | 0.8131291 | 0.9388129 |

0.8 | 0.8375854 | 0.9581693 | 1.0882176 |

0.9 | 0.9843269 | 1.1104609 | 1.2422602 |

1.0 | 1.1403382 | 1.2698635 | 1.4008207 |

Numerical results of approximate solution

0.0 | 1.0 | 1.0 | 1.0 |

0.1 | 1.0036839 | 0.9995798 | 0.9902709 |

0.2 | 0.9918177 | 0.9770693 | 0.9526430 |

0.3 | 0.9678131 | 0.9409453 | 0.9018037 |

0.4 | 0.9329689 | 0.8938412 | 0.8413882 |

0.5 | 0.8881139 | 0.8372777 | 0.7733064 |

0.6 | 0.8338592 | 0.7722922 | 0.6987749 |

0.7 | 0.7706901 | 0.6996578 | 0.6186510 |

0.8 | 0.6990108 | 0.6199832 | 0.5335787 |

0.9 | 0.6191685 | 0.5337664 | 0.4440635 |

1.0 | 0.5314684 | 0.4414262 | 0.3505153 |

In this work, a class of a coupled system of nonlinear fractional integrodifferential equations of fractional order has been discussed by using the RPS method under the Caputo fractional derivative. The RPS algorithm has been given to optimize the approximate solution by minimizing a residual error function with the help of generalized Taylor formula. To demonstrate the consistency with the theoretical framework, two illustrative examples have been provided. The obtained results indicated that the approximate solutions are coinciding with the exact solution and with each other for different values of the fractional order over the selected nods and parameters. From our results, we can conclude that the RPS method is a systematic and suitable scheme to address many fractional initial value problems with great potential in scientific applications. The calculations have been performed by using Wolfram-Mathematica 10.

The data used to support the findings of this study are available from the corresponding author upon request.

The author declares that there are no conflicts of interest.