Abstract and Applied Analysis

Abstract and Applied Analysis / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 7413859 | 7 pages | https://doi.org/10.1155/2020/7413859

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

Academic Editor: Kunquan Lan
Received29 Oct 2019
Revised29 Dec 2019
Accepted07 Jan 2020
Published11 Feb 2020

Abstract

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. This 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 two-dimensional nonlinear Burger’s differential equations. To clarify the facility and accuracy of the strategy, two examples are provided.

1. Introduction

Burger’s equation is one of the foremost necessary partial differential equations in fluid mechanics. This equation demonstrates the coupling between diffusion and convection processes. Burger’s equation describes the structure of shock waves, traffic flow, and acoustic transmission. Additionally, like this, it also appears in varied areas of applied mathematics and physics, such as modelling of gas dynamics [15]. Recently, many numerical and analytical methods have been used to study the two-dimensional Burger’s equation such as the differential transformation method [6], homotopy perturbation method [7], homotopy analysis method [8], variational iteration method [9], Adomian decomposition method [1012], cubic B-spline differential quadrature method [13], finite difference method [14], finite element [15], and local discontinuous Galerkin finite element method [16] and also mathematicians have used transform methods coupled with analytical methods [1730] to solve PDEs. The Sumudu decomposition method (SDM) is one of these methods, and it has been successfully used to solve intricate problems in engineering mathematics and applied science [3135]. The SDM was first introduced by Kumar [36], to solve nonlinear partial differential equations that show in all aspects of applied science and engineering. This method is an elegant combination of the Sumudu transform method and the Adomian decomposition method. The 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 two-dimensional nonlinear Burger’s equations [9]:with the initial conditions:and the boundary conditions:where and is its boundary, and are the velocity components to be determined, , and are the known functions, and is the Reynolds number.

The 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. This work is organized as follows: the analysis of the method is given in Section 2. The application of SDM to two examples is given in Section 3. Concluding remarks are given in the last section.

2. Analysis of the Method

Now, to obtain the approximate solution of equation (1), apply the Sumudu transformation to equation (1) and using the given condition (2) giveswhere . Apply the inverse operator to both sides of the equation (4), and it gives

The Adomian decomposition method suggests that the linear terms and and the nonlinear terms are decomposed by an infinite series of components:

For some Adomian polynomials, are given by

Substituting equation (6) into both sides of equation (5) leads to

To construct the recursive relation needed for the determination of the components and , it is important to note that the Adomian method suggests that the zeroth components and are usually defined by the functions and .

Accordingly, the formal recursive relation is defined in (Figures 1 and 2).

Having determined these components, substitute it into and to obtain the solution in a series form.

3. 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) becomesThe recursive relation can be constructed from equation (11) given byWe get the next couple of components, and upon setting , we haveand so on. Consequently, the solution in a series form is given byand in a closed form it iswhich is the exact solution of two-dimensional Burger’s equations [9].

Example 2. Consider another system of Burger’s equations (1), with the following initial conditions [9]:with the exact solutions:

Solution. Using the previous aforesaid discussion, we getTherefore, the solution and in the series form is given byNumerical outcomes shown in Tables 14 illustrate that the accuracy of SDM agrees good with the exact solutions of the system of two-dimensional Burger’s equation, and absolute errors are very small for the present choice of ρ, σ, , and .


tExact SDM

0.050
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5


tExact SDM

0.050
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5


tExact SDM

0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5


tExact SDM

0.05
0.1
0.15
0.2
0.25</