# The Variational Homotopy Perturbation Method for Solving((n×n)+1) Dimensional Burgers’ Equations

Authors: F. A. Hendi; B. S. Kashkari; A. A. Alderremy
Journal: Journal of Applied Mathematics
 (2016)

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## Abstract

The variational homotopy perturbation method VHPM is used for solvingn-dimensional Burgers’ system. Some examples are examined to validate that the method reduced the calculation size, treating the difficulty of nonlinear term and the accuracy.


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## Body

## 1. Introduction

The variational iteration method VIM and the homotopy perturbation method HPM were proposed by He in [1–6]. Many researchers used these methods in a variety of scientific fields of partial differential equations PDEs including Burgers’ equation which arises in many of physically important phenomena [7–9]. It was shown that the methods are stronger than other techniques such as the Adomian decomposition method [10–18]. In our work n-dimensional Burgers’ equation is solved by the variational homotopy perturbation method VHPM which is combination of VIM and HPM. The VHPM was proposed in [19–21]. Vector Burgers’ system is given by [22](1)Ut+U·∇U=μΔU,where u1,u2,…,un are the velocity components and μ is the kinematic viscosity. t is time and Δ and ∇ are (2)Δ=∂2x12+∂2x22+⋯+∂2xn2,∇=∂x1+∂x2+⋯+∂xn.Equation (1) can be written as(3)∂ui∂t+∑j=1nuj∂ui∂xj=μΔui,i=1,2,…,n.


## 2. Variational Iteration Method

According to the variational iteration method [2, 3, 10–14] we can write the correction functional for (3) as (4)un+1=un+∫0tλiξ∂ui∂ξ+∑j=1nuj~∂ui~∂xj-μΔui~dξ,where i=1,2,…,n, u=u(xi,ξ), λ is a general Lagrangian multiplier which can be found via variational theory, and ui~ are restricted variation which means δui~=0. The solution is given by (5)uixj,t=limn→∞⁡ui,nxj,t,j=1,2,…,n.


## 3. Homotopy Perturbation Method

Applying HPM according to [4–6, 15–17] for (3), we construct the following homotopy: (6)1-p∂ui,k∂t-∂ui,0∂t+p∑j=1nuj,k∂ui,k∂xj-μΔui,k=0or (7)∂ui,k∂t-∂ui,0∂t+p∂ui,0∂t+∑j=1nuj,k∂ui,k∂xj-μΔui,k=0,where i=1,2,…,n, k=1,2,…,p∈[0,1] is an embedding parameter, while u(1,0)=f1xj,0,u2,0=f2xj,0,…,u(n,0)=fn(xj,0) are initial approximations of (3). Assume the solution of (3) has the form (8)ui=∑l=0∞plui,lxj,t,i,j=1,2,…,n.Now, substituting ui from (8) in (7) and comparing coefficients of terms with identical powers of p we get (9)p0:∂ui,0∂t-∂ui,0∂t=0,p1:∂ui,0∂t+uj,0∂ui,0∂xj-μΔui,0,p2:uj,1∂ui,1∂xj-μΔui,1,p3:uj,2∂ui,2∂xj-μΔui,2,⋮The solution of (7) is (10)uixj,t=ui,0+ui,1+ui,2+⋯.


## 4. Variational Homotopy Perturbation Method

Consider (3) according to [19–21]. In HPM, assume that the solution of (3) has the form (11)ui=∑l=0∞plui,lxj,t=vi,i,j=1,2,…,n,uj=∑l=0∞pluj,lxj,t=vj,i,j=1,2,…,n.From (11), (3) can be written as (12)∂vi∂t+∑j=1nvj∂vi∂xj=μΔvi,i=1,2,…,n.In VIM, from the correction functional for (12) we can write (13)vi+1=v0+p∫0tλiξ-∑j=1nvj~∂vi~∂xj+μΔvi~dξ,where i=1,2,…,n, v=v(xi,ξ); from (11) in (13) and by comparing the coefficients  of like powers of p, we get (14)p0:ui,0xj,0=fixj,0,p1:ui,1xj,t=∫0tλiξ-uj,0xj,ξ∂ui,0xj,ξ∂xj+μΔui,0xj,ξdξ,p2:ui,2xj,t=∫0tλiξ-uj,1xj,ξ∂ui,1xj,ξ∂xj+μΔui,1xj,ξdξ,p3:ui,3xj,t=∫0tλiξ-uj,2xj,ξ∂ui,2xj,ξ∂xj+μΔui,2xj,ξdξ,⋮The approximations solution is given by (15)uixj,t=ui,0+ui,1+ui,2+⋯.To demonstrate the efficiency of the methods we have solved some examples by VHPM as (1+1)-dimensional, (1+2)-dimensional, (1+3)-dimensional, and 2-dimensional. Then we can generalize it for (n+1)-dimensional or n-dimensional.


## 5. Application

Example 1.
Consider (1+1)-dimensional Burgers’ equation [17] (16)∂u∂t+u∂u∂x=∂2u∂x2with the initial condition (17)ux,0=u0x=2x.The correction functional for (16) is(18)Un+1=Un+p∫0tλξ∂U∂t+U~nx,ξ∂U~nx,ξ∂x-∂2U~nx,ξ∂x2dξ.The general Lagrangian multiplier λ can be found as follows: (19)λ′=0,1+λξξ=t=0.Then, λ=-1.
Equation (11) can be written as (20)U=∑l=0∞plulx,t.Applying VHPM, we have (21)u0+pu1+p2u2+⋯=2x-p∫0tu0+pu1+p2u2+⋯∂∂xu0+pu1+p2u2+⋯dξ-p∫0t-∂2∂x2u0+pu1+p2u2+⋯dξ.Comparing the coefficient of like powers of p, we get (22)p0:u0=2x,p1:u1=-∫0tu0∂u0∂x-∂2∂x2u0dξ=-4xt,p2:u2=-∫0tu1∂u1∂x-∂2∂x2u1dξ=8xt2,p3:u3=-∫0tu2∂u2∂x-∂2∂x2u2dξ=-16xt2,⋮The approximations solution is given by (23)ux,t=u0+u1+u2+u3+⋯.Exact solution is (u∗=2x/(1+2t)).
The results are in Table1.Table 1
Comparison of VHPM solutions with exact solution att=0.01 (Example 1).
x
u
∗
x
,
y
,
t
u
x
,
y
,
t
u
∗
-
u
0.1
0.1960784314
0.1960784314
0
0.2
0.3921568628
0.3921568627
1
×
1
0
-
10
0.3
0.5882352942
0.5882352941
1
×
1
0
-
10
0.4
0.7843137254
0.7843137254
0
0.5
0.9803921568
0.9803921568
0
0.6
1.176470588
1.176470588
0
0.7
1.372549020
1.372549020
0
0.8
1.568627451
1.568627451
0
0.9
1.764705882
1.764705882
0Example 2.
Consider (1+2)-dimensional Burgers’ equation [17] (24)∂u∂t=u∂u∂x+∂2u∂x2+∂2u∂y2with the initial condition (25)ux,y,0=u0x,y=x+y.As above, we have (26)Un+1=u0-p∫0t-Unx,y,ξ∂Unx,y,ξ∂x-∂2Unx,y,ξ∂x2-∂2Unx,y,ξ∂y2dξ,u0+pu1+p2u2+⋯=x+y-p∫0t-u0+pu1+p2u2+⋯∂∂xu0+pu1+p2u2+⋯dξ-p∫0t-∂2∂x2u0+pu1+p2u2+⋯-∂2∂y2u0+pu1+p2u2+⋯dξ.Comparing the coefficient of like powers of p, we get (27)p0:u0=x+y,p1:u1=x+yt,p2:u2=x+yt2,p3:u3=x+yt3,⋮The approximations solution is given by (28)ux,y,t=u0+u1+u2+u3+⋯.Exact solution is (u∗=(x+y)/(1-t)).
The results are in Table2.Table 2
Comparison of VHPM solutions with exact solution atx=0.1 and y=0.1 (Example 2).
t
u
∗
(
x
,
y
,
t
)
u
(
x
,
y
,
t
)
|
u
∗
-
u
|
0.01
0.2020202020
0.2020202020
0
0.02
0.2040816327
0.2040816326
1
×
1
0
-
10
0.03
0.2061855670
0.2061855669
1
×
1
0
-
10
0.04
0.2083333333
0.2083333325
8
×
1
0
-
10
0.05
0.2105263158
0.2105263125
3.3
×
1
0
-
9
0.06
0.2127659574
0.2127659475
9.9
×
1
0
-
9
0.07
0.2150537634
0.2150537381
2.53
×
1
0
-
8
0.08
0.2173913043
0.2173912474
5.69
×
1
0
-
8
0.09
0.2197802198
0.2197801030
1.168
×
1
0
-
7
0.10
0.2222222222
0.2222220000
2.222
×
1
0
-
7Example 3.
Consider (1+3)-dimensional Burgers’ equation [17] (29)∂u∂t=u∂u∂x+∂2u∂x2+∂2u∂y2+∂2u∂z2with the initial condition (30)ux,y,z,0=u0x,y,z=x+y+z.We have (31)Un+1=u0-p∫0t-Unx,y,ξ∂Unx,y,ξ∂x-∂2Unx,y,ξ∂x2-∂2Unx,y,ξ∂y2-∂2Unx,y,ξ∂z2dξ,u0+pu1+p2u2+⋯=x+y+z-p∫0t-u0+pu1+p2u2+⋯∂∂xu0+pu1+p2u2+⋯dξ-p∫0t-∂2∂x2u0+pu1+p2u2+⋯-∂2∂y2u0+pu1+p2u2+⋯-∂2∂z2u0+pu1+p2u2+⋯dξ.Comparing the coefficient of like powers of p, we get (32)p0:u0=x+y+z,p1:u1=x+y+zt,p2:u2=x+y+zt2,p3:u3=x+y+zt3,⋮The approximations solution is given by (33)ux,y,z,t=u0+u1+u2+u3+⋯.Exact solution is (u∗=(x+y+z)/(1-t)).
The results are in Table3.Table 3
Comparison of VHPM solutions with exact solution atx=0.1, y=0.1, and z=0.1 (Example 3).
t
u
∗
(
x
,
y
,
t
)
u
(
x
,
y
,
t
)
|
u
∗
-
u
|
0.01
0.3030303030
0.3030303030
0
0.02
0.3061224490
0.3061224490
0
0.03
0.3092783505
0.3092783503
2
×
1
0
-
10
0.04
0.3125000000
0.3124999987
1.3
×
1
0
-
9
0.05
0.3157894737
0.3157894688
4.9
×
1
0
-
9
0.06
0.3191489362
0.3191489213
1.49
×
1
0
-
8
0.07
0.3225806452
0.3225806072
3.80
×
1
0
-
8
0.08
0.3260869565
0.3260868710
8.55
×
1
0
-
8
0.09
0.3296703297
0.3296701545
1.752
×
1
0
-
7
0.10
0.3333333333
0.3333330000
3.333
×
1
0
-
7Example 4.
Consider two-dimensional Burgers’ equations [23] (34)∂u∂t+u∂u∂x+v∂u∂y=1R∂2u∂x2+∂2u∂y2,∂v∂t+u∂v∂x+v∂v∂y=1R∂2v∂x2+∂2v∂y2with the initial conditions: (35)u0=ux,y,0=34-141+ey-xR/8,v0=vx,y,0=34+141+ey-xR/8.The correction functional for (34) is(36)Un+1=Un+p∫0tλ1ξ∂U∂t+U~nx,y,ξ∂U~nx,y,ξ∂x+V~nx,y,ξ∂U~nx,y,ξ∂xdξ+p∫0tλ1ξ-1R∂2U~nx,y,ξ∂x2+∂2U~nx,y,ξ∂y2dξ,Vn+1=Vn+p∫0tλ2ξ∂V∂t+V~nx,y,ξ∂V~nx,y,ξ∂x+U~nx,y,ξ∂V~nx,y,ξ∂xdξ+p∫0tλ2ξ-1R∂2V~nx,y,ξ∂x2+∂2V~nx,y,ξ∂y2dξ.The general Lagrangian multipliers are λ1=-1=λ2.Equation (11) can be written as (37)U=∑l=0∞plulx,y,t,V=∑l=0∞plvlx,y,t.By VHPM, we have (38)u0+pu1+p2u2+⋯=u0-p∫0t-u0+pu1+p2u2+⋯∂∂xu0+pu1+p2u2+⋯dξ-p∫0tv0+pv1+p2v2+⋯∂∂xu0+pu1+p2u2+⋯dξ-p∫0t1R∂2∂x2u0+pu1+p2u2+⋯+∂2∂y2u0+pu1+p2u2+⋯dξ,v0+pv1+p2v2+⋯=v0-p∫0t-u0+pu1+p2u2+⋯∂∂xv0+pv1+p2v2+⋯dξ-p∫0tv0+pv1+p2v2+⋯∂∂xv0+pv1+p2v2+⋯dξ-p∫0t1R∂2∂x2v0+pv1+p2v2+⋯+∂2∂y2v0+pv1+p2v2+⋯dξ.Comparing the coefficient of like powers of p, we get (39)p0:u0=34-141+ey-xR/8v0=34+141+ey-xR/8,p1:u1=1128Re-1/8-y+xRt1+e-1/8-y+xR2v1=-1128Re-1/8-y+xRt1+e-1/8-y+xR2,p2:u2=-12048R2e-1/4-y+xRt21+e-1/8-y+xR4v2=12048R2e-1/4-y+xRt21+e-1/8-y+xR4,p3:u3=132768R3e-3/8-y+xRt31+e-1/8-y+xR6v3=-132768R3e-3/8-y+xRt31+e-1/8-y+xR6,⋮The approximations solution is given by (40)ux,y,t=u0+u1+u2+u3+⋯,vx,y,t=v0+v1+v2+v3+⋯.Exact solution u∗=3/4-1/41+e(4y-4x-t)R/32; v∗=3/4+1/41+e(4y-4x-t)R/32.The results are in Table4.Table 4
Comparison of VHPM solutions with exact solution atx=1, y=0, and R=100 (Example 4).
t
u
∗
(
x
,
y
,
t
)
u
(
x
,
y
,
t
)
|
u
∗
-
u
|
v
∗
(
x
,
y
,
t
)
v
(
x
,
y
,
t
)
|
v
∗
-
v
|
0.01
0.5000009030
0.5000009571
5.41
×
1
0
-
8
0.9999990970
0.9999990321
6.49
×
1
0
-
8
0.02
0.5000008752
0.5000009862
1.110
×
1
0
-
7
0.9999991248
0.9999990031
1.217
×
1
0
-
7
0.03
0.5000008483
0.5000010153
1.670
×
1
0
-
7
0.9999991517
0.9999989741
1.776
×
1
0
-
7
0.04
0.5000008222
0.5000010445
2.223
×
1
0
-
7
0.9999991778
0.9999989451
2.327
×
1
0
-
7
0.05
0.5000007969
0.5000010736
2.767
×
1
0
-
7
0.9999992031
0.9999989161
2.870
×
1
0
-
7
0.06
0.5000007724
0.5000011027
3.303
×
1
0
-
7
0.9999992276
0.9999988871
3.405
×
1
0
-
7
0.07
0.5000007486
0.5000011318
3.832
×
1
0
-
7
0.9999992514
0.9999988581
3.933
×
1
0
-
7
0.08
0.5000007256
0.5000011609
4.353
×
1
0
-
7
0.9999992744
0.9999988281
4.463
×
1
0
-
7
0.09
0.5000007032
0.5000011900
4.868
×
1
0
-
7
0.9999992968
0.9999987991
4.977
×
1
0
-
7
0.10
0.5000006816
0.5000012191
5.375
×
1
0
-
7
0.9999993184
0.9999987701
5.483
×
1
0
-
7


## 6. Conclusion

In this work, the approximate solutions ofn-dimensional Burgers’ equations are obtained by combination of two powerful methods VIM and HPM in VHPM. The examples have shown the efficiency and accuracy of the VHPM; it reduces the size of computation without the restrictive assumption to handle nonlinear terms and it gives the solutions rapidly.



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