Abstract

In this research, systems of linear and nonlinear differential equations of fractional order are solved analytically using the novel interesting method: ARA- Residual Power Series (ARA-RPS) technique. This approach technique is based on the combination of the residual power series scheme with the ARA transform to establish analytical approximate solutions in a fast convergent series representation using the concept of the limit. The proposed method needs less time and effort compared with the residual power series technique. To prove the simplicity, applicability, and reliability of the presented method, three numerical examples are proposed and simulated. The obtained results show that the ARA-RPS technique is applicable, simple, and effective to get solutions to linear and nonlinear engineering and physical problems.

1. Introduction

Numerous phenomena in various fields of science can be fruitfully formulated by the use of fractional derivatives. This is because the sensible modeling for a physical phenomenon depends on instantaneous time as well as on prior time history; hence, we may use fractional calculus to deal with these problems [1–10].

Many physical and engineering problems can be formulated by fractional differential equations (FDEs) and obtaining the solutions of these equations have been the theme of many interesting investigations and have attracted the attention of researchers. Recently, there have been a large number of techniques dedicated to get solutions of FDEs [11–16]. These techniques can be classified into two categories, approximate and analytical, such as transform methods, homotopy analysis methods, and residual power series methods. [17–25].

On the other hand, there are a lot of contributions related to solving systems of FDEs and since the finding of the analytical solution for a system of FDEs sometimes required a complex computation, analytical, and numerical techniques were established and developed to search for solutions of linear and nonlinear FDEs, see [26–36].

In the literature, various integral transforms such as Laplace, Fourier, Sumudu, and ARA transforms have been introduced to deal with differential equations. ARA transform was introduced first in 2020, which is a powerful technique in solving differential equations and systems. One advantage of this transform is that it can solve differential equations with singular points around zero. Also, ARA transform is applicable for some functions that the Laplace transform cannot be applied to; see [37].

In this paper, we employ the ARA-RPS method to investigate analytical and approximate solutions of linear and nonlinear systems of FDEs. The ARA-RPS technique that has been introduced in [39] is a combination of the ARA transform [37–39] and the residual power series method [40–47].

To achieve our goal of finding the solutions of systems of FDEs, we operate the ARA transform on the original system and implement appropriate series expansions to solve the new system. The expansion coefficients are determined based on the concepts of limits at infinity, unlike the fractional power series (FPS) method which may take more time in the solution procedure.

This work is organized as follows: in the second section, we introduce some basic definitions and theorems of ARA transform and fractional derivatives. The methodology of the ARA-RPS method is presented in section three. Numerical examples and simulations are introduced in section four.

2. Basic Definitions and Theorems

In this section, we introduce the definition of the fractional operators Riemann–Liouville and Caputo. Definition of the ARA transform and some properties and theorems of the fractional ARA-Tayler’s series representations are also revisited.

Definition 1. The Riemann–Liouville fractional integral of the function of order is defined bywhere is the gamma function.

Definition 2. The Caputo fractional derivative of the function of order is defined bywhere is a nonnegative integer.

Definition 3. [37] The ARA transform of order of the continuous function on the interval , is defined byThe inverse of the ARA transform is given bywhere andIn the following arguments, we present some basic properties of the ARA transform [39] that are essential in our research.
Let and be two continuous functions on the interval for which the ARA transform exists. Then we have for (1) where and are nonzero constants.(2)(3)(4)(5)(6)(7)(8)(9)

Theorem 1. [26] Suppose that the FPS representation of the function at has the formIf and are continuous on , for each , then the coefficients have the form
for ,
where ( times).

Theorem 2. [39] Let be a continuous function on every finite interval in which the ARA transform of order two exists and has the FPS representation:Then

Remarks(1)The inverse ARA transform of order two for the FPS representation in Theorem 2 has the following form(2)The ARA transform of order one of the function has the following presentation

For the convergence analysis and conditions of the previous theorem, we propose the following results.

Theorem 3. [39] Let be a continuous function on every finite interval in which the ARA transform exists. Assume that has the following series representation:where

If on , then the remainder , of the above series, when we consider only the first n terms, satisfies the following inequality:

3. Methodology of the ARA-RPSM

Many phenomena in the fields of engineering and physics could be regulated by fractional initial value problems to simulate and describe some features of these phenomena. Analytical methods are used to get approximate solutions for some fractional initial value problems that we could not find out their exact solutions for. One of these techniques is the ARA-RPS method that depends on coupling the residual power series method with the ARA transform.

To perform the ARA-RPS method, consider the following system of FDEs in the sense of the Caputo derivative

with the initial conditions (ICs)where are continuous real valued functions defined on and are unknown analytical functions to be determined.

Firstly, we apply the ARA transform of order two on both sides of each equation in system (14)

Using property 8 and the ICs in (15), we getwhere

Hence (17) can be written as

Assuming that the ARA-RPS solutions of equations in (18) have the series expansions

Using the fact thatwe have for and so the k^th truncated series expansions of (19) and (20) have the form

To find the coefficients of the series expansions in (22) and (23), we define the ARA-residual functions of the (18) as follows:and the k^th ARA-residual functions are

The following facts are necessary to obtain the ARA-RPS solutions:(i)(ii), for .(iii) and .(iv), for , .

To find the coefficients ’s in the series expansion (24) and (25), we solve the system

for and .

Then the obtained coefficients are substituted in the series solution (23) and then apply the inverse ARA transform of order two to get the solution of system (14) in the original space.

4. Numerical Examples

In this section, we introduce three interesting examples on systems of linear and nonlinear FDEs. These examples demonstrate the applicability and efficiency of the presented method.

Example 1. Consider the linear system of FDEswith the ICsSolution: Applying the ARA transform of order two on both sides of the equations in system (27), we haveRunning the ARA transform on the equations in system (29), we haveSubstituting the ICs (28) and simplifying the equations in system (30), we getAssume that the ARA-RPS solutions of (31) have the following series representations:Using the fact in property 9we get the coefficients , and so the k^th ARA-RPS solutions of system (31) have the formDefine the ARA-residual functions of the equations in system (31) as follows:and the k^th ARA-residual functions of system (35)To find the first unknown coefficients and of the series expansions (34), we substitute the first truncated expansions , , , and into the first ARA-residual functions and in system (36) to getwhich can be written asSimplifying the equations in system (38) and multiplying the resulting equations by , we obtainTaking the limit as to both sides of equations in system (39), we haveTo find the second unknown coefficients and of the series expansions (34), we substitute the second truncated expansion , , , and into the second ARA-residual functions and in system (39) to getSimplifying the equations in system (41) and multiplying both sides by , we getSubstituting the values of and then taking the limit as to both sides of equations in system (42), we haveBy following the same steps as before, and based on the fact
for .
We can conclude thatConsequently, the ARA-RSP solutions of system (31) areFurthermore, to get the series solutions in the original space of system (27), we apply the inverse ARA transform of order two to both sides of equations in system (45) to getFor , the series expansions (46) have the form, and which are coincide with the exact solutions for the ordinary system (27) and (28).
Table 1, shows a comparison of the ARA-RPS solution and the exact solution of Example 1 with at various values of and . The absolute error is also presented in Table 1. Numerical results of Example 1 with , and different values of are listed in Table 2. In addition, the comparison of the results obtained for the exact solution corresponding to, and the numerical solutions given by the ARA-RPS method for different values of , , and are plotted in Figure 1. Figure 2 portrays a very precise agreement of the exact solutions and the ARA-RPS solutions of Example 1 at different time levels with fixed . The obtained results show that the ARA-RPS method is efficient in obtaining approximate solutions of systems of FDEs. This is obvious in the following tables, which illustrate the small errors between the ARA-RPS solutions and the exact ones.