A New Efficient Method for Solving Two-Dimensional Nonlinear System of Burger’s Differential Equations

In this work, the Sumudu decomposition method (SDM) is utilized to obtain the approximate solution of two-dimensional nonlinear system of Burger’s differential equations. )is method is considered to be an effective tool in solving many problems. Our results have shown that the SDM offers a much better approximation for solving several numbers of systems of twodimensional nonlinear Burger’s differential equations. To clarify the facility and accuracy of the strategy, two examples are provided.


Introduction
Burger's equation is one of the foremost necessary partial differential equations in fluid mechanics.
is method is an elegant combination of the Sumudu transform method and the Adomian decomposition method. e SDM method generates the solution in a series form whose components are determined by a recursive relationship.
In the current study, we consider the system of twodimensional nonlinear Burger's equations [9]: with the initial conditions: and the boundary conditions: where E � (ρ, σ) | a ≤ ρ ≤ b , a ≤ σ ≤ b and zE is its boundary, θ(ρ, σ, t) and α(ρ, σ, t) are the velocity components to be determined, w, h, w 1 , and h 1 are the known functions, and R is the Reynolds number. e major objective of this work is to get analytical and numerical solutions of the system of two-dimensional nonlinear Burger's equations (1) by using SDM. is work is organized as follows: the analysis of the method is given in Section 2. e application of SDM to two examples is given in Section 3. Concluding remarks are given in the last section.
Accordingly, the formal recursive relation is defined in (Figures 1 and 2).
Having determined these components, substitute it into to obtain the solution in a series form.

Application
In this part, two examples are provided to illustrate the method.
Example 1. Consider the system of two-dimensional Burger's equation (1), with the following initial conditions [9]: Solution. Subsequent to the discussion presented above, the system of equation (8) becomes e recursive relation can be constructed from equation (11) given by We get the next couple of components, and upon setting R � 1, we have Abstract and Applied Analysis and so on. Consequently, the solution in a series form is given by (θ, α) � ρ 1 + 2t 2 + 4t 4 + · · · − 2ρt 1 + 2t 2 + · · · + σ 1 + 2t 2 + 4t 4 + · · · , and in a closed form it is which is the exact solution of two-dimensional Burger's equations [9].

Conclusion
In this paper, SDM had been successfully applied to find the solutions of the system of two-dimensional nonlinear Burger's equations. e numerical studies showed that SDM offers accurate results for two-dimensional nonlinear Burger's equations in comparison with another analytical methods. is fact is shown in the second example. erefore, this method may be a favourable method to solve other nonlinear partial differential equations.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.