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

Authors: Shams A. Ahmed; Mohamed Elbadri; Mohamed Z. Mohamed
Journal: Abstract and Applied Analysis
 (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.


---

## Body

## 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 [1–5]. 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 [10–12], 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 [17–30] 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 [31–35]. 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]:(1)θt+θθρ+αθσ=1Rθρρ+θσσ,αt+θαρ+αασ=1Rαρρ+ασσ,with the initial conditions:(2)θρ,σ,0=wρ,σ,ρ,σ∈E,αρ,σ,0=hρ,σ,ρ,σ∈E,and the boundary conditions:(3)θρ,σ,t=w1ρ,σ,t,ρ,σ∈∂E,αρ,σ,t=h1ρ,σ,t,ρ,σ∈∂E,where E=ρ,σa≤ρ≤b,a≤σ≤b and ∂E is its boundary, θρ,σ,t and αρ,σ,t are the velocity components to be determined, w,h,w1, and h1 are the known functions, and R 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) gives(4)Sθρ,σ,u=wρ,σ−uSθθρ+αθσ+uS1R∇2θ,Sαρ,σ,u=hρ,σ−uSθαρ+αασ+uS1R∇2α,where ∇2=∂2/∂ρ2+∂2/∂σ2. Apply the inverse operator S−1 to both sides of the equation (4), and it gives(5)θρ,σ,t=wρ,σ−S−1uSθθρ+αθσ+S−1uS1R∇2θ,αρ,σ,t=hρ,σ−S−1uSθαρ+αασ+S−1uS1R∇2α.The Adomian decomposition method suggests that the linear termsθρ,σ,t and αρ,σ,t and the nonlinear terms θθρ,αθσ,θαρ,andαασ are decomposed by an infinite series of components:(6)θρ,σ,t=∑n=0∞θnρ,σ,t,θθρ=∑n=0∞An,αθσ=∑n=0∞Bn,αρ,σ,t=∑n=0∞αnρ,σ,t,θαρ=∑n=0∞Cn,αασ=∑n=0∞Cn.For some Adomian polynomials,Anθ are given by(7)Anθ0,θ1,θ2,…,θn=1n!dndλnN∑n=0∞λnθnλ=0,n=0,1,2,….Substituting equation (6) into both sides of equation (5) leads to(8)∑n=0∞θnρ,σ,t=wρ,σ−S−1uS∑n=0∞An+∑n=0∞Bn+S−1uS1R∇2∑n=0∞θnρ,σ,t,∑n=0∞αnρ,σ,t=hρ,σ−S−1uS∑n=0∞Cn+∑n=0∞Dn+S−1uS1R∇2∑n=0∞αnρ,σ,t.To construct the recursive relation needed for the determination of the componentsθ0,θ1,θ2,…,θn and α0,α1,α2,…,αn, it is important to note that the Adomian method suggests that the zeroth components θ0 and α0 are usually defined by the functions wρ,σ and hρ,σ.Accordingly, the formal recursive relation is defined in (Figures1 and 2).(9)θ0ρ,σ,t=wρ,σ,θk+1ρ,σ,t=−S−1uSAk+Bk+S−1uS1R∇2θk,k≥0,α0ρ,σ,t=hρ,σ,αk+1x,y,t=−S−1uSCk+Dk+S−1uS1R∇2αk,k≥0.Figure 1
Distributions of approximation solutions forθρ,σ,t at (a)t=0.01 and (b) t=0.5 with R=100, for Example 2.
(a)
(b)Figure 2
Distributions of approximation solutions forαρ,σ,t at (a)t=0.01 and (b) t=0.5 with R=100, for Example 2.
(a)
(b)Having determined these components, substitute it intoθρ,σ,t=∑n=0∞θnρ,σ,t and αρ,σ,t=∑n=0∞αnρ,σ,t 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]:(10)θρ,σ,0=ρ+σ,ρ,σ∈E,αρ,σ,0=ρ−σ,ρ,σ∈E.Solution.
Subsequent to the discussion presented above, the system of equation (8) becomes(11)∑n=0∞θnρ,σ,t=ρ+σ−S−1uS∑n=0∞An+∑n=0∞Bn+S−1uS1R∇2∑n=0∞θnρ,σ,t,∑n=0∞αnρ,σ,t=ρ−σ−S−1uS∑n=0∞Cn+∑n=0∞Dn+S−1uS1R∇2∑n=0∞αnρ,σ,t.
The recursive relation can be constructed from equation (11) given by(12)θ0ρ,σ,t=ρ+σ,θk+1ρ,σ,t=−S−1uSAk+Bk+S−1uS1R∇2θk,k≥0,α0ρ,σ,t=ρ−σ,αk+1x,y,t=−S−1uSCk+Dk+S−1uS1R∇2αk,k≥0.
We get the next couple of components, and upon settingR=1, we have(13)θ0,α0=ρ+σ,ρ−σ,θ1,α1=−2ρt,−2σt,θ2,α2=2ρt2+2σt2,2ρt2−2σt2,θ3,α3=−4ρt3,−4σt3,θ4,α4=4ρt4+4σt4,4ρt4−4σt4,and so on. Consequently, the solution in a series form is given by(14)θ,α=ρ1+2t2+4t4+⋯−2ρt1+2t2+⋯+σ1+2t2+4t4+⋯,ρ1+2t2+4t4+⋯−2σt1+2t2+⋯−σ1+2t2+4t4+⋯,and in a closed form it is(15)θρ,σ,t,αρ,σ,t=ρ+σ−2ρt1−2t2,ρ−σ−2σt1−2t2,which 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]:(16)θρ,σ,0=34−141+eRσ−ρ/8,αρ,σ,0=34+141+eRσ−ρ/8,with the exact solutions:(17)θρ,σ,t=34−141+eR4σ−4ρ−t/32,αρ,σ,t=34+141+eR4σ−4ρ−t/32.Solution.
Using the previous aforesaid discussion, we get(18)θ0ρ,σ,t=34−141+eR−ρ+σ/8,α0ρ,σ,t=34+141+eR−ρ+σ/8,θ1ρ,σ,t=−e1/8R−ρ+σRt641+e1/8R−ρ+σ3−e1/4R−ρ+σRt641+e1/8R−ρ+σ3+e1/8R−ρ+σRt1281+e1/8R−ρ+σ2,α1ρ,σ,t=e1/8R−ρ+σRt641+e1/8R−ρ+σ3+e1/4R−ρ+σRt641+e1/8R−ρ+σ3−e1/8R−ρ+σRt1281+e1/8R−ρ+σ2,θ2ρ,σ,t=−eRρ/8+Rσ/8−eRρ/8+eRσ/8R2t28192eRρ/8+eRσ/83,α2ρ,σ,t=eRρ/8+Rσ/8−eRρ/8+eRσ/8R2t28192eRρ/8+eRσ/83,θ3ρ,σ,t=−eRρ/8+Rσ/8eRρ/4+eRσ/4−4eRρ/8+Rσ/8R3t3786432eRρ/8+Rσ/84,α3ρ,σ,t=eRρ/8+Rσ/8eRρ/4+eRρ/4−4eRρ/8+Rσ/8R3t3786432eRρ/8+Rσ/84.
Therefore, the solutionθρ,σ,t and αρ,σ,t in the series form is given by(19)θρ,σ,t=θ0ρ,σ,t+θ1ρ,σ,t+θ2ρ,σ,t+θ3ρ,σ,t,αρ,σ,t=α0ρ,σ,t+α1ρ,σ,t+α2ρ,σ,t+α3ρ,σ,t.
Numerical outcomes shown in Tables1–4 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, R, and t.Table 1
The (SDM) results forθρ,σ,t for first four approximations for R = 1, with mesh points ρ  = 0.1 and σ  = 0.1, for Example 2.
t
Exactθρ,σ,t
SDMθρ,σ,t
θ
Exact
−
θ
SDM
0.05
0.6249023437698682
0.6249023437698682
0
0.1
0.6248046876589456
0.6248046876589457
1.1102230
E
−
16
0.15
0.6247070317864406
0.6247070317864418
1.2212453
E
−
15
0.2
0.6246093762715608
0.6246093762715658
4.9960036
E
−
15
0.25
0.6245117212335117
0.6245117212335268
1.5099033
E
−
14
0.3
0.6244140667914967
0.6244140667915343
3.7636561
E
−
14
0.35
0.6243164130647162
0.6243164130647978
8.1601392
E
−
14
0.4
0.6242187601723671
0.6242187601725259
1.5876189
E
−
13
0.45
0.6241211082336422
0.6241211082339287
2.8654856
E
−
13
0.5
0.6240234573677299
0.624023457368215
4.8505644
E
−
13Table 2
The (SDM) results forαρ,σ,t for first four approximations for R = 1, with mesh points ρ  = 0.1 and σ  = 0.1, for Example 2.
t
Exactαρ,σ,t
SDMαρ,σ,t
α
Exact
−
α
SDM
0.05
0.875097656230132
0.875097656230132
0
0.1
0.875195312341054
0.875195312341054
1.1102230
E
−
16
0.15
0.875292968213559
0.875292968213558
1.2212453
E
−
15
0.2
0.875390623728439
0.875390623728434
4.9960036
E
−
15
0.25
0.875488278766488
0.875488278766473
1.5099033
E
−
14
0.3
0.875585933208503
0.875585933208466
3.7636561
E
−
14
0.35
0.875683586935284
0.875683586935202
8.1601392
E
−
14
0.4
0.875781239827633
0.875781239827474
1.5887291
E
−
13
0.45
0.875878891766358
0.875878891766071
2.8654856
E
−
13
0.5
0.87597654263227
0.875976542631785
4.8505644
E
−
13Table 3
The (SDM) results forθρ,σ,t, for first four approximations, and for R = 1, with mesh points ρ  = 0.3 and σ  = 0.1, for Example 2.
t
Exactθρ,σ,t
SDMθρ,σ,t
θ
Exact
−
θ
SDM
0.05
0.6233399413556532
0.6233399413556536
4.4408921
E
−
16
0.1
0.6232423033622646
0.6232423033622709
6.3282712
E
−
15
0.15
0.6231446675140766
0.6231446675141092
3.2529535
E
−
14
0.2
0.6230470339301989
0.6230470339303031
1.0424994
E
−
13
0.25
0.6229494027297301
0.6229494027299877
2.5757174
E
−
13
0.3
0.6228517740317571
0.6228517740322975
5.4045657
E
−
13
0.35
0.6227541479553547
0.6227541479563675
1.0127454
E
−
12
0.4
0.6226565246195848
0.6226565144551626
1.7474910
E
−
12
0.45
0.622558904143496
0.6225588896714487
2.8308467
E
−
12
0.5
0.6224612866461229
0.622461286650486
4.3630655
E
−
12Table 4
The (SDM) results forαρ,σ,t, for first four approximations, and for R = 1, with mesh points ρ  = 0.3 and σ  = 0.1, for Example 2.
t
Exactαρ,σ,t
SDMαρ,σ,t
α
Exact
−
α
SDM
0.05
0.876660058644347
0.876660058644346
4.4408921
E
−
16
0.1
0.876757696637735
0.876757696637729
6.3282712
E
−
15
0.15
0.876855332485923
0.876855332485891
3.2529535
E
−
14
0.2
0.876952966069801
0.876952966069697
1.0424994
E
−
13
0.25
0.87705059727027
0.877050597270012
2.5757174
E
−
13
0.3
0.877148225968243
0.877148225967703
5.4045657
E
−
13
0.35
0.877245852044645
0.877245852043633
1.0127454
E
−
12
0.4
0.877343475380415
0.877343485544838
1.7474910
E
−
12
0.45
0.877441095856504
0.877441095853673
2.8309577
E
−
12
0.5
0.877538713353877
0.877538713349514
4.3630655
E
−
12


## 4. Conclusion

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



---

*Source: 7413859-2020-02-11.xml*