APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNS...
Am26242246
1. P. R. Mistry, V. H. Pradhan / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.242-246
Analytic Solution Of Burger’s Equations By Variational Iteration
Method
P. R. Mistry* and V. H. Pradhan**
*
(Department of Applied Mathematics & Humanities, S.V.N.I.T., Surat-395007, India)
**
(Departments of Applied Mathematics & Humanities, S.V.N.I.T., Surat-395007, India)
Abstract
By means of variational iteration method the 2 VARIATIONAL ITERATION METHOD:
solutions of (1+1), (1+2) and (1+3) dimensional Consider the following differential equation:
Burger equations are exactly obtained. In this Lu Nu h ( x, t ) (2.1)
paper, He's variational iteration method is where L is a linear operator, N a nonlinear
introduced to overcome the difficulty arising in
calculating Adomian polynomials. operator, and an h ( x, t ) is the source
inhomogeneous term. The VIM was proposed by
Key words: Burger’s equation, Nonlinear time “He”, where a correctional functional for equation
dependent partial differential equations, Variational (1.4.1) can be written as
iteration method and Lagrange multiplier. un 1 (t ) un (t )
t
, n 0 (2.2)
1 Introduction
We often come with non linear partial
( Lu ( ) Nu ( ) h ( )) d
0
n
n
differential equations obtained through
Where is a general Lagrange multiplier
mathematical models of scientific phenomena.
which can be identified optimally via the variational
There are some methods to obtain approximate theory. The subscript n indicates the n th
solution of this kind of equation. Some of them are approximation and
u n is considered as a restricted
numerical methods, homotopy analysis, Exp-
variation, i.e. un 0 . It is clear that the
function method, and linearization of the equation
successive approximations un , n 0 can be
[1, 6, 7]. In 1999, the variational iteration method
established by determining , so we first
was developed by mathematician “He”. This method
determine the Lagrange multiplier that will be
is used for solving linear and non linear differential identified optimally via integration by parts. The
equations. The method introduces a reliable and successive approximation un ( x, t ), n 0 of the
efficient process for a wide variety of scientific and solution u( x, t ) will be readily obtained using the
engineering applications [1, 2, 4, 5,]. It is based on Lagrange multiplier obtained and by using any
Lagrange multiplier and it has the merits of selective function u 0 . The initial values u( x,0)
simplicity and easy execution. This method avoids and ut ( x,0) are usually used for selecting the
linearization of the problem. zeroth approximation u 0 . With determined, then
In this paper exact solution of (1+1), (1+2) several approximation un ( x, t ), n 0 , follow
and (1+3) dimensional Burger equation [3] has been immediately. Consequently the exact solution may
obtained by variational iteration method. The be obtained by using
mentioned problem has been solved by u lim un (2.3)
n
N.Taghizadeh etc. [3] by Homotopy Perturbation
Method and Reduced Differential Transformation 3. APPLICATION OF VIM FOR
BURGER’S EQUATION:
Method here in the present paper we have solved the 3.1 (1+1)-Dimensional Burgers equation
same problem by variational iteration mehod.. u u 2u
u 2 0 (3.1.1)
t x x
242 | P a g e
2. P. R. Mistry, V. H. Pradhan / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.242-246
u4 {{{{x(1 t t 2 2 t 3 3 t 4 4
I.C.: u x,0 x (3.1.2)
13t 5 5 2t 6 6 29t 7 7 71t 8 8
Now following the variational iteration 15 3 63 252
method given in the above section we get the 86t
9 9
22t
10 10
5t
11 11
t12 12
following functional
567 315 189 126
un 1 ( x, t ) un ( x, t ) t
13 13
t
14 14
t
15 15
)}}}} (3.1.9)
un 2u 567 3969 59535
u
t
(3.1.3)
0 t
un n
x
2n d
x
u5 {{{{{x(1 t t 2 2 t 3 3 t 4 4
43t 6 6 13t 7 7 943t 8 8
t 5 5
Stationary conditions can be obtained as follows: 45 15 1260
3497t 9 9 27523t10 10 1477t11 11
' 0
5670 56700 4050
(3.1.4)
1 17779t 12 12
13141t 13 13
1019t14 14
t
68040 73710 8820
The Lagrange multiplier can therefore be 63283t 15 15
43363t 16 16
1080013t17 17
simply identified as 1 , and substituting this
893025 1058400 48580560
value of Lagrange multiplier into the functional
(3.1.3) gives the following iteration equation. 2588t 18 18
162179t 19 19
16511t 20 20
229635 30541455 7144200
un 1 ( x, t ) un ( x, t ) 207509t 21 21
557t 22 22
2447t 23 23
t
un u 2 un (3.1.5) 225042300 1666980 22504230
t un xn
0
2
x
d
16927t 24 24 5309t 25 25 t 26 26
540101520 675126900 595350
As stated before, we can select Initial 2t 27 27 13t 28 28 t 29 29
condition given in the equation (3.1.2) and using
this selection in (3.1.5) we obtain the following 6751269 315059220 236294415
successive approximations: t 30 30 t 31 31
)}}}}}(3.1.10)
3544416225 109876902975
u1 {x tx } (3.1.6)
1 u x, t u1 u2 u3 .... (3.1.11)
u2 {{x (tx t 2 x t 3 x 2 )}} (3.1.7)
3 x
u x, t (3.1.12)
1
u3 {{{x tx t 2 x 2 t 3 x 3 1 t
3
3.2 (2+1)-Dimensional Burger’s equation
1 3 2
(tx t 2 x t x )
3 u u u
(3.1.8)
2 4 3 u u
(tx t x t x t x
2 3 2
t x y
3 (3.2.1)
2u 2u
1 1 1
t 5 x 4 t 6 x 5 t 7 x 6 )}}} 2 2 0
3 9 63 x y
I.C.: u x, y,0 x y (3.2.2)
Now following the variational iteration method, we
get the following functional
243 | P a g e
3. P. R. Mistry, V. H. Pradhan / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.242-246
un 1 ( x, t ) un ( x, t ) u4 {{{{x y 2tx
un u u 2ty 4t 2 x 2
un n un n
t
t x y (3.2.3) 4t 2 y 2 8t 3 x 3
2u 2u d
8t 3 y 3 16t 4 x 4
0
2n 2n
x y
16t 4 y 4
416 5 5
t x
15
Stationary conditions can be obtained as follows: 416 5 5 128 6 6
t y t x
' 0
15 3
128 6 6 3712 7 7
1
(3.2.4)
t y t x
t 3 63
3712 7 7 4544 8 8
The Lagrange multiplier can therefore be t y t x
simply identified as 1 , and substituting this 63 63
value of Lagrange multiplier into the functional 4544 8 8 44032 9 9
(3.2.3) gives the following iteration equation. t y t x
63 567
un 1 ( x, t ) un ( x, t ) 44032 9 9 22528 10 10
t y t x
567 315
un u u
un n un n
22528 10 10 10240 11 11
t y t x
t x y
t
(3.2.5)
2u 2u d
315 189
10240 11 11 2048 12 12
0
2n 2n t y t x
x y
189 63
2048 12 12 8192 13 13
As stated before, we can select Initial t y t x
condition given in the equation (3.2.2) and using 63 567
this selection in (3.2.5) we obtain the following
8192 13 13 16384t14 x 14
successive approximations: t y
567 3969
u1 {x y 2t ( x y ) } (3.2.6) 16384t y
14 14
32768t15 x 15
u2 {{x y 2tx 2ty 3969 59535
8 32768t y
15 15
4t 2 x 2 4t 2 y 2 t 3 x 3 (3.2.7) }}}} (3.2.9)
3 59535
8
t 3 y 3}}
3
u3 {{{x y 2tx 2ty
4t 2 x 2 4t 2 y 2 8t 3 x 3 8t 3 y 3
32 32 32
t 4 x 4 t 4 y 4 t 5 x 5 (3.2.8)
3 3 3
32 64 64
t 5 y 5 t 6 x 6 t 6 y 6
3 9 9
128 7 7 128 7 7
t x t y }}}
63 63
244 | P a g e
4. P. R. Mistry, V. H. Pradhan / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.242-246
u5 {{{{{x y 2tx 2ty un 1 ( x, t ) un ( x, t )
4t 2 x 2 4t 2 y 2 8t 3 x 3 un u u u
un n un n un n
8t y 16t x 16t y
3 3 4 4 4 4 t
t x y z
(3.3.3)
2752 6 6 2u 2u 2u d
32t 5 x 5 32t 5 y 5 t x 0
2 2 2
n n n
45 x y z
2752 6 6 1664 7 7
t y t x
45 15 Stationary conditions can be obtained as follows:
1664 7 7 60352 8 8
t y t x ' 0
15 315 (3.3.4)
60352 8 8 895232t 9 x 9 1 t
t y
315 2835
The Lagrange multiplier can therefore be
895232t y 9 9
7045888t10 x 10 simply identified as 1 , and substituting this
2835 14175 value of Lagrange multiplier into the functional
(3.3.3) gives the following iteration equation.
7045888t y 10 10
1512448t11 x 11
14175 2025 un 1 ( x, t ) un ( x, t )
1512448t y 11 11
9102848t12 x 12 un
u u u
2025 8505 un n un n un n
t x y z
t
(3.3.5)
9102848t y 53825536t13 x 13 2u 2u 2u d
12 12
8505 36855 0
2 2 2
n n n
x y z
53825536t y 13 13
4173824t14 x 14
36855 2205 As stated before, we can select Initial condition
4173824t y 14 14
2073657344t15 x 15 given in the equation (3.3.2) and using this selection
in (3.3.5) we obtain the following successive
2205 893025 approximations:
2073657344t y 15 15
893025 u1 {x y z 3t ( x y z ) } (3.3.6)
88807424t16 x 16 u2 {{x y z 3tx 3ty
.......}}}}} (3.2.10)
33075 3tz 9t 2 x 2 9t 2 y 2
(3.3.7)
9t 2 z 2 9t 3 x 3
u x, t u1 u2 u3 .... (3.2.11)
x y 9t 3 y 3 9t 3 z 3}}
u x, y , t (3.2.12) u3 {{{x y z 3tx 3ty
1 2 t
3tz 9t 2 x 2 9t 2 y 2
3.3 (3+1)-Dimensional Burger’s equation
u u u u 9t 2 z 2 27t 3 x 3 27t 3 y 3
u u u
t x y z 27t 3 z 3 54t 4 x 4 54t 4 y 4
(3.3.1)
2u 2u 2u 54t 4 z 4 81t 5 x 5 81t 5 y 5 (3.3.8)
2 2 2 0
x y z 81t z 81t x 81t y
5 5 6 6 6 6
I.C.: u x, y, z,0 x yz (3.3.2) 81t 6 z 6
243 7 7 243 7 7
t x t y
Now following the variational iteration method, we 7 7
get the following functional 243 7 7
t z }}}
7
245 | P a g e
5. P. R. Mistry, V. H. Pradhan / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.242-246
u4 {{{{x y z 3tx 3ty iteration method is a powerful mathematical tool to
solving Burger’s equations. In our work, we use the
3tz 9t 2 x 2 9t 2 y 2 9t 2 z 2 Mathematica Package to calculate the series
obtained from the variational iteration method.
27t 3 x 3 27t 3 y 3 27t 3 z 3
81t 4 x 4 81t 4 y 4 81t 4 z 4 References
1. Sadighi A and Ganji D, Exact solution of
1053 5 5 1053 5 5
t x t y nonlinear diffusion equations by variational
5 5 iteration method, Commuters and
Mathematics with Applications, 54, (2007),
1053 5 5
t z 486t 6 x 6 pp. 1112-1121.
5 2. He.J.H. Variational iteration method -- a
1053 5 5 kind of non-linear analytical technique:
t z 486t 6 x 6 some examples, Internat. J. Nonlinear
5 Mech. 34, (1999).,pp. 699–708
486t 6 y 6 486t 6 z 6 3. N.Taghizadeh, M.Akbari and
A.Ghelichzadeh, Exact Solution of Burgers
7047 7 7 7047 7 7
t x t y Equations by Homotopy Perturbation
7 7 Method and Reduced Differential
Transformation Method, Australian Journal
7047 7 7 51759 8 8
t z t x of Basic and Applied Sciences, 5(5):
7 28 (2011),pp. 580-589,
...........}}}} (3.3.9) 4. Salehpoor E. and Jafari H., Variational
iteration method: A tools for solving partial
u x, y, z, t u1 u2 u3 ... (3.3.10) differential equations, The journal of
Mathematics and Computer Science, 2,
x yz
u x, y , z , t (3.3.11) (2011)pp. 388-393.
1 3 t 5. Wazwaz A. M.., The variational iteration
method: A Powerful scheme for handling
3.4 (n+1)-Dimensional Burger’s equation linear and nonlinear diffusion equations.
Computer and Mathematics with
u u u u u Applications 54, (2007) pp.933-939.
u u u ...... u 6. M.A. Abdou and A.A. Soliman,
t x1 x2 x3 xn Variational iteration method for solving
Burger’s and coupled Burger’s equations
2u 2u 2u 2u Journal of Computational and Applied
2 2 2 .... 0 Mathematics 181 (2005), pp. 245 – 251.
x1 x2 x3 xn 2
7. Junfeng Lu, Variational iteration method
(3.4.1) for solving a nonlinear system of second-
u x1 , x2 , x3 ,.........., xn , 0 order boundary value problems, Computers
I.C.: (3.4.2) and Mathematics with Applications 54
x1 x2 ...... xn (2007), pp.1133–1138.
Similarly, we apply VIM on the equation (3.4.1) we
get the following exact solution.
u x1 , x2 , x3 ,.............., xn , t
x1 x2 x3 x4 ............ xn (3.4.3)
1 n t
4. Conclusion
In this paper, the variational iteration
method has been successfully applied to finding the
solution of (1+1), (1+2) and (1+3) dimensional
Burger’s equations. The solution obtained by the
variational iteration method is an infinite power
series for appropriate initial condition, which can, in
turn, be expressed in a closed form, the exact
solution. The results show that the variational
246 | P a g e