A Comparative Study of the Fractional-Order System of Burgers Equations

: This paper is related to the fractional view analysis of coupled Burgers equations, using innovative analytical techniques. The fractional analysis of the proposed problems has been done in terms of the Caputo-operator sense. In the current methodologies, ﬁrst, we applied the Elzaki transform to the targeted problem. The Adomian decomposition method and homotopy perturbation method are then implemented to obtain the series form solution. After applying the inverse transform, the desire analytical solution is achieved. The suggested procedures are veriﬁed through speciﬁc examples of the fractional Burgers couple systems. The current methods are found to be effective methods having a close resemblance with the actual solutions. The proposed techniques have less computational cost and a higher rate of convergence. The proposed techniques are, therefore, beneﬁcial to solve other systems of fractional-order problems.


Introduction
Fractional calculus (FC) has become an important mathematical approach for explaining non-local behavioural models. Fractional derivatives have mathematically interpreted many physical problems in recent decades; these representations have produced excellent results in real-world modelling issues. Coimbra, Riemann-Liouville, Riesz, Weyl, Hadamard, Liouville-Caputo, Grunwald-Letnikov, Caputo-Fabrizio, Atangana-Baleanu, among others, gave many basic definitions of fractional operators [1,2]. A wide variety of non-linear equations have been developed and commonly implemented in numerous non-linear physical sciences such as biology, chemistry, applied mathematics and various branches of physics such as plasma physics, condensed matter physics, fluid mechanics, field theory, and non-linear optics over the past few years. The exact result of non-linear equations plays a vital role in deciding the characteristics and behaviour of physical processes. A differential equation symmetry is a transformation that makes the differential equation invariant. The existence of such symmetries may aid in the solution of the differential equation. A scheme of differential equations line symmetry is a continous symmetry of a scheme of differential equations. Solving a linked set of ordinary differential equations can reveal symmetries. It is sometimes easier to solve these equations than it is to solve the original differential equations. The symmetry structure of the system consists of integer partial differential equations and fractional-order partial differential equations with the fractional Caputo derivative. Many effective techniques have been used to solved nonlinear FPDEs, for example, the homotopy perturbation transformation technique [3,4], the homotopy analysis transformation technique [5,6], reduced differential transformation technique [7,8], the finite element method [9], the finite difference method [10], and so on. Definition 1. The fractional-order Caputo derivative of h( ), h ∈ C ω −1 , ω ∈ N, ω > 0, is given as Definition 2. The fractional-order Caputo derivative of Elzaki transform is define as Definition 3. The fractional-order Riemann-Liouville of integral α > 0, of a function h ∈ C ω , is given as Basic properties:

Basic Concept of Elzaki Transformation
The Elzaki transformation is a new transformation described for functions of exponential order. We recognize functions in the set A, described as: For a given function in the set, the constant M must be a finite number, and the constants k 1 and k 2 must be finite or infinite. The Elzaki transformation, as described by the integral equation We can obtain the basic solutions

Definition 4. The inverse Elzaki transform is given as
The inverse Elzaki transform of some of the functions are given by cos a Theorem 1. If T(s) is the Elzaki transformation of h( ), one can take into consideration the Elzaki transformation of the Riemann-Liouville derivative as follows:

Proof. The Laplace transformation
Therefore, when we put 1 s for s 2 , the Elzaki transformation of fractional-order of h( ) is as below:

The General Methodology of HPTM
In this section, the HPTM for the solution of fractional partial differential equations [35,36] D α u(ξ, ζ, ) + Mu(ξ, ζ, ) + Nu(ξ, ζ, ) = h(ξ, ζ, ), the initial condition is M is linear and N non-linear functions. Using the Elzaki transform of Equation (1) Now, by applying inverse transformation, we get where The perturbation methodology in based on power series with parameter p is now described as where perturbation term p and p ∈ [0, 1]. The non-linear functions can be defined as where H ω are He's polynomials of u 0 , u 1 , u 2 , · · · , u ω , and can be determined as putting Equations (7) and (8) in Equation (5), we have Both sides comparison coefficient of p, we have

The Methodology of EDM
Consider the general procedure of EDM to solve the fractional partial differential equation.
with the initial condition Where is D α = ∂ α ∂ α the Caputo fractional derivative of order α, L and N are linear and non-linear functions, respectively and q is source function.
Applying the Elzaki transform to Equation (12), Using the differentiation property, we get The infinite series solution of u(ξ, ζ, ) The nonlinear terms of N to solve with the help of Adomian polynomials is defined as Putting Equation (16) and Equation (17) into (15), Now using EDM, we have Generally, we can write Implemented the inverse Elzaki transform of Equations (20) and (21), we get Example 1. Consider the fractional-order system of Burgers equations with initial conditions Re denotes the Reynolds number. Now, by applying ET to Equation (23), we obtain the following outcome Define the non-linear operator as By the above equation, we get Apply inverse ET on Equation (27) and then reduces to Now we implement HPM With the help of He's polynomials H ω (u) and H ω (v), the nonlinear terms can be found He's polynomials are defined as Comparing p-like coefficients, we get , , , .

Provides the series form solution is
Now we apply the EDM Assume that the infinite series solution of the unknown functions u(ξ, ζ, ) and v(ξ, ζ, ) respectively as follows D ω are the Adomian polynomials and they signifying the non-linear terms. Using the these terms, we can rewrite Equation (28) as Both sides comparing of Equation (33), we can be written as . .
Continuing in the same manner, the remaining components of the Elzaki decomposition method solution u ω and v ω (ω ≥ 3) can be achieved smoothly. As a result, we arrive at the series solution as The exact results for Equation (23) In Figure 1, show that the Elzaki decomposition method and Homotopy perturbation transform method show that the close contact with each other of Example 1. In Figure 2, represent the different fractional-order behaviour of u(ξ, ζ, ). Similarly, in Figure 3, show that the Elzaki decomposition method and Homotopy perturbation transform method show that the close contact with each other of Example 1. In Figure 4, represent the different fractional-order behaviour of v(ξ, ζ, ). In Tables 1 and 2 show that the absolute error with different fractional-order with respect to α and β.

Conclusions
In the present article, the homotopy perturbation method and Elzaki decomposition method are applied for the solution of coupled systems of fractional Burger equations. The graphical and tabular representations of the derived results have been done. These representations of the obtained results have clearly confirmed the higher accuracy of the suggested methods. The solutions are obtained for fractional systems which are closely related to their actual solutions. The convergence of fractional solutions to integer order solution has been shown. The fewer calculations and higher accuracy are the valuable themes of the present methods. The researchers then modified it to solve other systems with fractional partial differential equations.