Combination of Laplace transform and residual power series techniques to solve autonomous n-dimensional fractional nonlinear systems


 In this work, a new iterative algorithm is presented to solve autonomous n-dimensional fractional nonlinear systems analytically. The suggested scheme is combination of two methods; the Laplace transform and the residual power series. The methodology of this algorithm is presented in details. For the accuracy and effectiveness purposes, two numerical examples are discussed. Finally, the impact of the fractional order acting on these autonomous systems is investigated using graphs and tables.


Introduction
Extending the ordinary-partial di erential equations into fractional di erential equations has been attracted by many researchers since the fractional derivatives are more general, applicable and more e cient for real world phenomena, especially when the dynamics of a given mathematical model is a ected by constraints inherent to the system [1]. Since it is di cult to nd explicit solutions to these fractional problems, it is necessary to use, alternative methods, numerical and approximate techniques.
In this work, we are interested in introducing a new analytical scheme to solve nonlinear fractional autonomous dynamical systems. Dynamical systems describe the prediction of future states follow from the current state. Di erent numerical and analytical algorithms were used to solve fractional dynamical systems, such as, Homotopy analysis method, the variational iteration method, Ritz method and the explicit one-step method [7][8][9][10]. In this context, we present a new algorithm constructed by combining the Laplace transform method and the residual power series method (LRPS). This new technique has been recently proposed for the rst time in [11] and used in [12].
The organization of the paper is the following: In Section 2 we present the steps of applying the LRPS method to solve n × n autonomous fractional dynamical systems. In Section 3 we study two examples of order × and × , and also provide graphical analysis. Finally, the conclusion is given in Section 4.

Description of LRPS
In this section, we present in details the steps of applying the LRPS scheme in solving the following autonomous fractional system: subject to the initial conditions where < α ≤ , ≤ t ≤ , hm are suitable functions, and D α t is the Caputo-derivative.
Applying the Laplace transform to (1) we get where , We assume that Vm(s), m = , , , . . . , n have fractional power series representation, i.e., Next, we let V k m (s) denote the k-th truncated series of Vm(s), i.e., Now, we de ne the Laplace residual function and the k-th Laplace residual function to (3), respectively, as: To determine the coe cients cmr , m = , , , . . . , n and r = , , , . . . , k, we substitute (6) into (8), then multiply the resulting equation by s kα+ , and next we solve the following iterative equation for the unknowns cmr : k = , , , .... Finally, we apply the Laplace inverse to V k m (s) to obtain the k-th approximate solution ν k m (t).

Numerical Problems and concluding remarks
In this section we present two examples of fractional nonlinear autonomous systems of order × and × . The steps of implementing the LRPS will be clari ed, and the e ectiveness of the proposed method will be tested by providing a graphical analysis.

. Example 1
Consider the following × nonlinear autonomous system [7,8] subject to Applying the Laplace transform to (10)−(11), we get We assume that both V (s) and V (s) have fractional power series representation as and we assume that the k-th truncated series of V (s) and It is clear that the Laplace residual function for both V k (s) and V k (s) are Accordingly, the k-th Laplace residual functions, LRes k , are To determine a and b , we consider As V (s) = s + a s α+ and V (s) = b s α+ , we get Multiply (18) by s α+ , we obtain that Finally, we solve the following system which gives that In a similar manner, to nd a and b , we consider As V (s) = s + s α+ + a s α+ and V (s) = s α+ + b s α+ , we obtain Multiply (23) we deduce that Hence, the 2nd-approximate LRPS solution of V (s) and V (s) are Proceeding as the above illustrated steps in determining the unknown functions a k and b k , one can easily reach the following results: and Therefore, V (s) = s + s α+ + s α+ + s α+ + s α+ + s α+ + ..., Consequently, the solution of (10)- (11) is We point out that the exact solutions to the system in Example 1 for the case of α = are ν (t) = e t and ν (t) = te t [7,8]. We consider ϕ (t) = i= a i t iα to be the LRPS approximation of ν (t), and ϕ (t) = i= b i t iα to be the LRPS approximation of ν (t). In Figure 1, the rst sub-gure represents the values of ϕ (t) for di erent values of the fractional order α and values of ν (t), while the second sub-gure represents the absolute error |ν (t) − ϕ (t)| when α = . In Figure 2, the rst sub-gure represents the values of ϕ (t) for di erent values of the fractional order α and values of ν (t), while the second sub-gure represents the absolute error |ν (t) − ϕ (t)| when α = . From these plots, we observe that the curves of ϕ i (t) : i = , , are mapping continuously and gradually as α varies from to and converges to ν i (t) : i = , , when α = . Also, we observe that the approximations ϕ i (t) : i = , , are in excellent agreement with ν i (t) : i = , , when α = .
On the other side, as shown in Table 1, we provide numerical investigations on the accuracy of LRPS applied to Example 1. While as, in Table 2, we present the impact of the fractional order α acting on the values of the unknowns eld functions ν i (t) : i = , .   Table 1: Numerical values of ϕ (t) and ϕ (t) for α = . , . , . to Example 1.

. Example 2
Let us consider the following × nonlinear autonomous system [9,10] subject to (32) Apply the Laplace transform to (31)-(32), we get Accordingly, the k-th Laplace residual functions, LRes k , are To determine a , b and c , we substitute (35) in (37) with k = to get Multiply (38) by s α+ , Proceeding as the above illustrated steps in determining the unknown functions a k , b k and c k , one can easily verify the following results: Therefore, V (s) = s + s α+ + s α+ + s α+ + s α+ + s α+ + ..., Consequently, the solution of (31)-(32) is It is worth mentioning that the exact solutions to the system in Example 2 for the case α = are ν (t) = e t , ν (t) = e t and ν (t) = e t − [9,10]. We consider ψ (t) = i= a i t iα to be the LRPS approximation of ν (t), ψ (t) = i= b i t iα to be the LRPS approximation of ν (t) and ψ (t) = i= c i t iα to be the LRPS approximation of ν (t). In Figure 3, the rst subgure represents the values of ψ (t) for di erent values of the fractional order α and values of ν (t), while the second sub-gure represents the absolute error |ν (t)−ψ (t)| when α = . In Figure 4, the rst sub-gure represents the values of ψ (t) for di erent values of the fractional order α and values of ν (t), while the second sub-gure represents the absolute error |ν (t) − ψ (t)| when α = . In Figure 5, the rst sub-gure represents the values of ψ (t) for di erent values of the fractional order α and values of ν (t), while the second sub-gure represents the absolute error |ν (t) − ψ (t)| when α = . For this × system, one can observe the same ndings depicted for Example 1. Finally, as shown in Table 3, we provide numerical investigations on the accuracy of LRPS applied to Example 2.    Table 3: Absolute errors: |ψ (t) − ν (t)|, |ψ (t) − ν (t)| and |ψ (t) − ν (t)| to Example 2.

Conclusion
A combination of two schemes; the Laplace transform and the residual power series, is adapted to solve nonlinear Caputo-fractional autonomous dynamical systems. The methodology, reliability and the accuracy of the new technique are introduced by solving × and × systems. The role of the fractional derivative is investigated by using graphical analysis. Finally, the advantage of the current method was depicted as converting the whole fractional problem into pure algebraic computational scheme which can be executed using any available computational softwares.
As a future work, the authors plan to extend the use of LRPS to solve multi-dimensional various fractional problems arising in Engineering and Science.

Funding information:
The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Con ict of interest:
The authors state no con ict of interest.