An analytical approach for systems of fractional differential equations by means of the innovative homotopy perturbation method

We have applied the new approach of homotopy perturbation method (NAHPM) for partial differential system equations featuring time-fractional derivative. The Caputo-type of fractional derivative is considered in this paper. A combination of NAHPM and multiple fractional power series form has been used the first time to present analytical solution. In order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples have been given. All numerical calculations in this manuscript have been carried out with Mathematica.


Introduction
In this research work, it has been proposed that the new HPM based on the multiple fractional power series can be engaged to solution of partial differential system equations featuring time-fractional derivative (FPDSEs).
This system equation has frequently appeared in different fields of science and engineering such as; physics, optics, plasma physics, superconductivity and quantum mechanics [1].
There are some more books related to fractional calculus for interested readers [2,3].It should be noted that there are no accurate analytical solutions for most fractional differential equations.Consequently, for such equations we have to employ some direct and iterative methods.Researchers have used various methods to solve systems equations in recent years.Some familiar methods are: Adomian's decomposition method [4,5,6], homotopy perturbation method [7,8], homotopy analysis method [9,10,11] and so on [12,13,14,15,16,17,18].This paper is organized: in Section 2, fundamental idea of the new method is presented.We explained convergence of this method in section 3.In Section 4, the application of innovative HPM to FPDSEs is illustrated, and some numerical examples are presented.And conclusions are drawn in Section 5.

Fundamental idea of the new method
In this part of the paper, we present and define Riemann-Liouville fractional integral and Caputo's fractional derivative that are presented [2].Then the new approach of homotopy perturbation method (NAHPM) is introduced and explained in detail.
Definition 2.1.A real function f (x), x > 0, is considered to be in the space C ν , (ν ∈ R), if there exists a real number n(> ν), so that f (x) = x n f 1 (x), where f 1 (x) ∈ C[0, ∞), and it is said to be in the space C k ν if and only if To describe the fundamental ideas of the NAHPM method for partial differential system equations featuring time-fractional derivative: ), in which n − 1 < µ ≤ n and i = 1, . . ., k, with the following initial conditions for i = 1, 2, . . ., k and j = 1, 2, . . ., n − 1: ). where N 1 , . . ., N k are nonlinear operators, which usually depend on the functions u i and derivatives, D µ denotes that Caputo fractional and h 1 , . . ., h k , are inhomogeneous term.

Convergence analysis
A large number of problems can be treated by NAHPM through applying the methodology that has been elaborated in the previous sections.Theorem 3.1.Presume that S and T are Banach spaces and A : S → T is a contractive nonlinear mapping which is Then due to Banach's fixed point theorem A has a unique fixed point u, Assume that the sequence provided by new HP M is stated that assume that w 0 = υ 0 ∈ B r (υ), where B r (υ) = {υ ∈ S; υ − υ < r}, then we have Proof.(i) By inductive way featuring k = 1 Allow that w k−1 − υ ≤ λ k−1 υ 0 − υ , while induction hypothesis, hence

Test examples
Now, we apply NAHPM based on the multiple fractional power series to solve FPDSEs.All of the plots and computations for this equations have been done with M athematica .
Example 4.1.We purpose the following FPDSEs: The estimate answers featuring µ = 2 acquired for several amounts of x and t applying NAHPM, is shown in Table 1.featuring the primary conditions: , v t (x, 0) = 0.
In Figure 4.1, we may view the estimate answers featuring µ = 2 which is concluded for several amounts of t and x utilizing NAHPM.
Example 4.2.Next, we consider the system of coupled fractional equations [19]: along with: With considering U 1 (x, t) and V 1 (x, t), the result shows The estimate solution for µ = 1.
In Figure 2, we can view the approximate answers featuring µ = 1.
In Table 2, we may view the approximate answers featuring µ = 1, which is concluded for several amounts of t and x applying NAHPM.
We can see the analytical answers toward µ = 1, in figure 3. The analytical answers featuring µ = 1 acquired for several amounts of x, and t applying NAHPM, can be seen in Table 3.

Conclusion
In this paper, we have successfully applied NAHPM to obtain series solution of partial differential system equations featuring time-fractional derivative.The result indicated that a few iteration of NAHPM will result in some solution.
Finally, it should be added that the suggested approach has the potentials to be applied in solving other similar nonlinear problems in partial differential equations of fractional order.

Table 1 .
Approximate result of test example 4.1.

Table 2 .
Approximate result of test example 4.2.We choose the non-linear system of inhomogeneous F P DEs:

Table 3 .
Approximate result of test example 4.3.tx u N AHP M =u Exact v N AHP M =v Exact w N AHP M =w Exact