Microsoft Word - DECOMPOSITION METHOD COMPARISON BURGERS_normeratio172.doc


Ratio mathematica 18 
(2008), 51 - 61 

51 

DECOMPOSITION METHOD IN 
COMPARISON WITH NUMERICAL 

SOLUTIONS OF BURGERS EQUATION 

Christos Mamaloukas* and Stefanos Spartalis** 

ABSTRACT – This paper presents a solution of the one-dimension Burgers 
equation using Decomposition Method and compares this solution to the 
analytic solution [Cole] and solutions obtained with other numerical methods. 
Even though decomposition method is a non-numerical method, it can be 
adapted for solving nonlinear differential equations. The advantage of this 
methodology is that it leads to an analytical continuous approximated solution 
that is very rapidly convergent [2,7,8]. This method does not take any help of 
linearization or any other simplifications for handling the non-linear terms. 
Since the decomposition parameter, in general, is not a perturbation parameter, 
it follows that the non-linearities in the operator equation can be handled easily, 
and accurate solution may be obtained for any physical problem. 

1. Introduction 
Many problems in Fluid Mechanics and in Physics are governed generally by 
the Navier-Stokes equations. These equations can show the behaviour of a 
certain attribute (e.g. momentum, heat) in space and time. The one-dimension 
non-linear differential equation which is used as a model for these problems is 
Burgers equation. This equation is applied to laminar and turbulence flows as 
                                                        
* Athens University of Economics and Business, Dept. of Informatics, 76 
Patision Str, 10434 Athens, Greece, email: mamkris@aueb.gr 
 
** Democritus University of Thrace, Dept. of Production Engineering and 
Management, School of Engineering, University Library Building, Kimeria 
67100 Xanthi, Greece, email: sspart@pme.duth.gr 
 



 

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   Num.18 -2008 Ratio Mathematica  C. Mamaloukas, S. Spartalis 

well. The Burgers equation which is the one-dimension nonlinear Diffusion 
Equation is similar to the one dimension Navier-Stokes equation without the 
stress term. Many researchers tried to find analytic and numerical solutions of 
this equation using the appropriate initial and boundary conditions. 
Characteristically in Benton and Platzman [10] are mentioned almost 35 
distinct solutions of Burger equation but only the half of them are having 
physical interest. Agas [9] tried to get approximate solution of Burger equation 
using a new numerical solution which is called “Group of Explicit” Method. He 
also tried the method of Finite Differences and the method of Lines in Finite 
Elements. The problem he faced was that these methods could not give 
solutions for big values of the Reynolds number. He also found some problems 
in convergence. 
In this paper, a solution obtained by the Adomian's Decomposition Method 
(ADM), which is described briefly in this paper and was used by Mamaloukas 
[12, 13] for the numerical solution of the one-dimensional Kortweg-de Vries 
equation and the pulsatile flow of an incompressible viscous fluid through a 
circular rigid tube provided with constriction, is compared numerically and 
graphically to the analytic and to some others numerical methods. As it is 
shown in the diagrams at the end of this paper this method gives a computable 
and accurate solution of the problem using only a small number of terms. 
 

2. Formulation of the Problem 
Consider the Burgers equation with the following form 

2

2

u u u
u

t x x


  
 

  
                (1) 

with boundary conditions:    (0, ) (1, ) 0 0u t u t for t                 
(2) 
and initial condition:             ( , 0) 4 (1 ) sinu x x x or x                
(3) 
 

3. Brief Description of ADM 

Let 
2

2andt xxL Lt x
 

 
 

. Then the equation (1) takes the form: 

t xxL u Nu L u        (4) 

 



 

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Num.18 – 2008 Ratio Mathematica C. Mamaloukas,  et al 

where the first term is the linear, the third is the highest order term and the 
second is the non-linear term given by 

u
Nu u

x





       (5) 

Now, solving (4) for and  t xxL u L u  correspondingly we have  

t xxL u L u Nu        (6) 

      1xx tL u L u Nu                     
(7) 

By defining the one and twofold right-inverse operators 1 1andt xxL L
  , given by 

the form    1 1  and  t xxL dt L dx dx      , we can formally obtain from (6) 
and (7) 

 1 1t t t xxL L u L L u Nu         (8) 

 1 1 1xx xx xx tL L u L L u Nu         (9) 

 
From relations (8) and (9) we obtain 

   1 1 10
1
2 xx t t xx

u u L L u Nu L L u Nu             (10) 

where the term 0u  is to be determined from the initial conditions, so, is 
           0 4 (1 ) sinu x x or x                 
(11) 
 

4. Solution of Burger's Equation with ADM 
Now, we introduce a formal counting parameter λ to write equation (10) in the 
following form 

   1 1 10
1
2 xx t t xx

u u L L u Nu L L u Nu              (12) 

The equation (12) is called parameterized equation and the parameter λ inserted 
here is not a perturbation parameter; it is used only for grouping the terms.  
The u and the nonlinear term Nu are decomposed into the following 
parameterised forms 



 

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0

n
n

n
u u





                  (13) 

0

n
n

n
N u A





                  (14) 

where nA  are the Adomian’s special polynomials [1,2] for the specific non-
linearity to be determined by expanding Nu in the ascending power of λ and 
equating the terms of like powers of λ from both sides of (12). These special 
polynomials depend only on the 0u  to nu  components. 
Substituting the expressions (13) and (14) into (12) and then equating the like 
power terms from both sides of the resulting expression we have 
 

   

   

   

1 1 1
1 0 0 0 0

1 1 1
2 1 1 1 1

1 1 1
1

1
2
1
2

...............................................................................
1

, 0,
2

xx t t xx

xx t t xx

n xx t n n t xx n n

u L L u A L L u A

u L L u A L L u A

u L L u A L L u A n

 

 

 

  

  

  


     

     

       1, 2,..., n

 (15) 

 
All components are determinable since 0A  depends only on 0u , 1A  depends 
only on 0, 1u u ,…, nA  depends only on 0 1, , ..., nu u u . So, in order to determine 
Adomian’s special polynomials, from (5) and (14) we write 

0

n
n

n

u
A u

x









       (16) 

Substituting (13) into (16) and then comparing like-power terms of λ on both 
sides of resulting expression we obtain the following polynomials. 

0
0 0

01
1 0 1

02 1
2 0 1 2

..................................................

..................................................

u
A u

x
uu

A u u
x x

uu u
A u u u

x x x







 
 

 
  

  
     (17) 



 

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Using the initial condition (11) 0A  can be calculated from expression (17). 
Substituting the result in the expression of 1u  (15) and then performing all 
necessary calculations and integrations with respect to t and x respectively, we 
have 1u  which is: 
 

3 5
2 3 4

1
4 4

4 8 24 16 2
3 5
v x v x

u v t t x t x t x v x                      

(18) 
 
and 2u  which is 
 

3
2 2 2 2 2 2 2 2 2 2

2

5
3 2 3 2 3 4 4 2 4

2 5 2 6 2 7 2 8
5 6

2
16 4 16 32 16 96

3
32 148 2

16 160 80
3 3 5

8 248 10 248 4
15 5 9 15 5 5

v x
u v t v t x t x v t x v x v t x t x

v x
v t x v t x t x v x v t x t x

v x v x v x v x
v t x v t x

        

       

     

           

(19) 
 

If we suggest as a solution of u an approximation of only two or three terms 
then from (11), (18) and (19) we have the solution of (1): 
 

0 1u u u   or 0 1 2u u u u    
 
As an example, if we give the values 0.25, 0.05, 0.001x t v    we get 

0.712314u   with two terms and 0.712791u   with three terms. 
 

5. Tables of Results and Diagrams 

For the solution of this equation the initial conditions ( , 0) 4 (1 )u x x x   and 
( , 0) sinu x x  were used without restricting generality. The boundary 

conditions were (0, ) (1, ) 0 0u t u t for t   . The compared methods are 
Analytic Solution, the Implicit method, the Explicit method, the methods of 
lines with Gauss-Legendre and Hermite, the Group of Explicit method and the 
Decomposition method. For comparison reasons with the results of other 
published papers, 0.25x   and time amplitude 0.01 0.25t   were used. 
 



 

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In the above diagrams numerical results of Burger equation are registered for 
different values of v. For comparison reasons the viscosity values 

1, 0.1, 0.01, 0.001v v v v     were used. 
 
We give below the Tables of Results and some diagrams only for the first initial 
condition ( ) 4 (1 )u x x x   for values of x = 0.25, 0.5, 0.75 for t = 0.01, 0.05 
(0.05) 0.25 and for v = 1, 0.1, 0.01, 0.001. As for the second initial condition 
we get similar results. For comparison reasons we also use as a solution of u, 
two and three terms of Decomposition method. 
 



 

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Table 1: Comparison results for Burger equation for initial condition 4 (1 )x x  with 

1, 0.1, 0.01, 0.001, 0.25, 0.05v x t      

x=.25 v=0.1   4x(1-x)          x=.25 v=1             
t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp  t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp 

0,01 0,7422 0,7276 0,7236 0,7274 0,7273 0,7272 0,7399  0,01 0,6724 0,6592 0,6489 0,6584 0,6583 0,6574 0,7163 

0,05 0,6621 0,6745 0,67 0,6453 0,6452 0,6471 0,6939  0,05 0,4356 0,4224 0,4213 0,3503 0,3254 0,4206 0,5263 

0,1 0,584 0,5675 0,5548 0,5608 0,5592 0,5671 0,6364  0,1 0,2751 0,2619 0,2603 0,4527 -0,053 0,2601 0,2888 

0,15 0,5189 0,5043 0,5042 0,4931 0,4853 0,5039 0,5789  0,15 0,1794 0,1662 0,1642 -0,934 -0,408 0,1644 0,0513 

0,2 0,4681 0,4535 0,4536 0,4362 0,4213 0,4531 0,5214  0,2 0,1191 0,1059 0,1052   -0,7593 0,1041 -0,1862 
0,25 0,4265 0,4119 0,4118 0,4263 0,3652 0,4115 0,4639  0,25 0,0807 0,0675 0,0639   -1,1234 0,0657 -0,4237 

                                 
x=.50/t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp  x=.50/t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp 

0,01 0,9917 0,9911 0,9901 0,9916 0,9923 0,9914 1,0027  0,01 0,9184 0,9194 0,9188 0,9197 0,9204 0,918 1,0267 

0,05 0,9533 0,9527 0,9423 0,9516 0,9524 0,953 0,9867  0,05 0,639 0,64 0,6438 0,5978 0,5983 0,6386 0,8667 

0,1 0,8993 0,8987 0,8815 0,893 0,8933 0,899 0,9667  0,1 0,4019 0,4029 0,4023 0,1275 0,2 0,4015 0,6667 

0,15 0,8434 0,8428 0,8326 0,8317 0,8313 0,8431 0,9467  0,15 0,2524 0,2534 0,2514 -0,1408 -0,2023 0,252 0,4667 

0,2 0,7889 0,7883 0,7935 0,7352 0,7714 0,7886 0,9267  0,2 0,1585 0,1595 0,1583   -0,6192 0,1581 0,2667 

0,25 0,7375 0,7369 0,7328 0,5198 0,7143 0,7372 0,9067  0,25 0,0914 0,1065 0,0916   -1,0642 0,0991 0,0667 
                                 

x=.75/t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp  x=.75/t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp 

0,01 0,7417 0,7571 0,7517 0,7567 0,7572 0,757 0,7654  0,01 0,7677 0,6928 0,6913 0,6818 0,6824 0,6927 0,837 

0,05 0,7663 0,7818 0,7823 0,778 0,7793 0,7816 0,7795  0,05 0,5065 0,4917 0,4924 0,3355 0,3772 0,4915 0,707 

0,1 0,7882 0,8038 0,7906 0,7892 0,7934 0,8035 0,7969  0,1 0,3239 0,309 0,3106 -0,8926 -0,046 0,3089 0,5445 

0,15 0,7999 0,8154 0,8013 0,778 0,7923 0,8152 0,8145  0,15 0,2069 0,1921 0,1926 -0,602 -0,5021 0,1919 0,382 

0,2 0,802 0,8093 0,8179 0,1649 0,7782 0,8173 0,832  0,2 0,1342 0,1188 0,1171   -0,992 0,1192 0,2195 

0,25 0,7955 0,8067 0,8116 0,206 0,7553 0,8108 0,8495  0,25 0,0892 0,0735 0,0793   -1,5243 0,0742 0,057 
                 

x=.25 v=0.01              x=.25 v=0.001             
t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp  t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp 

0,01 0,7492 0,7346 0,7342 0,7344 0,734 0,7342 0,7422  0,01 0,7349 0,701 0,6945 0,7351 0,7352 0,735 0,7425 

0,05 0,746 0,6766 0,6748 0,675 0,6745 0,6755 0,7106  0,05 0,6746 0,6432 0,5932 0,6778 0,6765 0,6782 0,7123 

0,1 0,742 0,6122 0,6087 0,608 0,6075 0,6104 0,6711  0,1 0,6075 0,6015 0,6014 0,6126 0,6126 0,6144 0,6746 

0,15 0,738 0,5562 0,5512 0,5494 0,5514 0,5537 0,6316  0,15 0,5538 0,5843 0,5885 0,5555 0,5571 0,5582 0,6369 

0,2 0,734   0,5016 0,4998 0,5043 0,5046 0,5921  0,2 0,4979     0,5061 0,5112 0,5094 0,5992 

0,25 0,73     0,4534 0,4642 0,4652 0,5526  0,25 0,4543     0,463 0,4743 0,4673 0,5615 
                 

x=.50/t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp  x=.50/t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp 

0,01 0,9992 0,9986 0,9992 0,9984 0,999 0,9989 1,0003  0,01 0,9982 0,9325 0,9632 0,9991 0,9991 0,9996 1 

0,05 0,996 0,9873 0,9901 0,9854 0,9875 0,9887 0,9987  0,05 0,9864 0,9432 0,9736 0,9888 0,9901 0,9921 0,9998 

0,1 0,992 0,9611 0,9662 0,9556 0,9573 0,9636 0,9967  0,1 0,9558 0,9228 0,9323 0,9617 0,9643 0,9699 0,9997 

0,15 0,988 0,9233 0,9299 0,9144 0,9183 0,9263 0,9947  0,15 0,9243 0,8647 0,8842 0,9223 0,9283 0,9343 0,9995 

0,2 0,984   0,8835 0,8667 0,8734 0,8796 0,9927  0,2 0,8663     0,8754 0,8872 0,8871 0,9993 

0,25 0,98     0,7908 0,8283 0,8256 0,9907  0,25 0,7906     0,8209 0,8453 0,8306 0,9991 
                                 

x=.75/t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp  x=.75/t Analytic Implicit Explicit L-Gauss L-Hermite Group Decomp 

0,01 0,7556 0,7644 0,7517 0,7643 0,7651 0,7645 0,7583  0,01 0,7652 0,6946 0,7432 0,765 0,7652 0,7651 0,7576 

0,05 0,7722 0,8241 0,7823 0,8206 0,8243 0,824 0,7867  0,05 0,8204 0,8417 0,8436 0,8249 0,8284 0,8285 0,7874 

0,1 0,8469 0,9027 0,7906 0,886 0,8954 0,902 0,8222  0,1 0,8859 0,9632 0,9348 0,895 0,9062 0,9138 0,8247 

0,15 0,925 0,9843 0,8013 0,9381 0,9544 0,9832 0,8577  0,15 0,9379 1,023 1,018 0,9507 0,971 1,0049 0,862 

0,2 0,9945   0,8179 0,9631 0,9931 1,0671 0,8932  0,2 0,9628     0,9817 1,013 1,1005 0,8993 

0,25 0,9996   0,8116 0,8109 1,005 1,1513 0,9287  0,25 0,8115     0,9514 1,024 1,1988 0,9366 



 

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Diagram 1: Comparison results with 1, 0.25, 0.01 0.25v x t     using 2 

and 3 terms 
 

 
 
Diagram 2: Comparison results with 1, 0.5, 0.01 0.25v x t     using 2 and 
3 terms 



 

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Num.18 – 2008 Ratio Mathematica C. Mamaloukas,  et al 

 
 
Diagram 3: Comparison results with 0.1, 0.75, 0.01 0.25v x t     using 2 
and 3 terms 
 

 
 
Diagram 4: Comparison results with 0.01, 0.5, 0.01 0.25v x t     and 2 
terms 



 

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Diagram 5: Comparison results with 0.001, 0.75, 0.01 0.25v x t     and 2 
terms 

6. Discussions 
The analytic solution as it is described by Cole [13] is liable to restrictions 

concerning the values of the coefficient 
1

e

v
R

 . For example, if the value of the 

Reynolds number is greater than 1000 then we can not find any solution 
because Fourier series do not converge. For this reason we try numerical 
approaches, like finite differences and finite elements.  
Concerning finite differences the explicit method give us adequate results if and 
only if 1 / 2  . Otherwise results did not converge. With the implicit method 
we do not need the covenant 1 / 2  , but we need a large number of 
calculations. Finally, the group of explicit methods gives us adequate results 
with few calculations and the method is more stable.  
Concerning finite elements, the method of lines with Gauss and Hermite was 
used with initial and boundary conditions from Madsen and Sincovec [14]. 
These methods, using great values of Reynolds number, gave us adequate 
results without the limitations for x  and t , with small number of repetitions 
and without stability limitations. However, the errors depend first on the choice 
of the polynomial and second on the choice of x  and t  



 

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Concerning the decomposition method from the above diagrams it is obvious 
how powerful this method is. Using only two terms we can obtain similar results 
with the other numerical methods and the analytic solution. Of course, in some 
cases the present solutions deviate from the solutions given in the table. The 
decomposition solution can be further improved if more-term approximations of 
the solution are obtained. 
As far as accurate results are concerned, computational experience has shown 
that they can be obtained easily by taking half a dozen terms. In case we do not 
have a sufficiently high precision by using a few of the nA , then accordingly to 
Rach R. [15] there are two alternatives. One is to compute additional terms by 
any of the available procedures. The second approach is to use the Adomian-
Malakian ''convergence acceleration'' procedure [16]. This unique approach 
conveniently yields the error-damping effect of calculating many more terms of 
the An to determine whether further calculation is required. 

7. Conclusions 
The great advantage of the decomposition method is that of avoiding 
simplifications and restrictions which change the non-linear problem into a 
mathematically tractable one, whose solution is not consistent to physical 
solution. 
Further study on the stability and the convergence of the solutions will prove the 
accuracy of the above method. 

BIBLIOGRAPHY 

1. G. ADOMIAN, Nonlinear Stochastic Operator Equations, Academic 
Press, 1986. 

2. G. ADOMIAN, Nonlinear Stochastic Systems Theory and Applications to 
Physics, Kluwer Academic Publishers, 1989. 

3. G. ADOMIAN, J. Math. Anal. Appl., 119, (1986), 340-360. 
4. G. ADOMIAN, Appld. Math. Lett., 6, No5, (1993), 35-36. 
5. G. ADOMIAN, Rach R., On the Solution of Nonlinear Differential 

Equations with Convolution Product Non-linearities, J. Math. Anal. 
Appl., 114, (1986), 171-175. 

6. G. ADOMIAN, Solving Frontier Problems of Physics: The Decomposition 
Method, Kluwer Academic Publishers, 1994. 

7. Y. CHERRUAULT, Kybernetes, 18, No2, (1989), 31-39. 
8. Y. CHERRUAULT, Math. Comp. Modeling, 16, No2, (1992), 85-93. 
9. C. AGAS, The Effect of Kinematic Viscosity in the Numerical Solution of 

Burger Equation, Thessaloniki, 1998. 



 

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10. E.R. BENTON & G.W. PLATZMAN, A Table of Solutions of the one-
dimensional Burgers Equation, Quart. Appl. Math., 1972. 

11. J. M. BURGERS, The Nonlinear Diffusion Equation, D. Reidel Publishing 
Company, Univ. of Maryland, USA, 1974. 

12. C. MAMALOUKAS, Numerical Solution of one dimensional Kortweg-de 
Vries Equation, BSG Proceedings 6, Global Analysis, Differential 
Geometry and Lie Algebras, 6, (2001), 130-140. 

13. C. MAMALOUKAS, Haldar K., Mazumdar H. P., Application of double 
decomposition to pulsatile flow, Journal of Computational & Applied 
Mathematics, 10 , Issue 1-2, (2002), 193–207. 

14. J.D. COLE, On a Quasilinear Parabolic Equation Occurring in 
Aerodynamics, A.Appl.  Maths, 9, (1951), 225-236. 

15. N. K. MADSEN and R. F. SINCOVEC, General Software for Partial 
Differential Equations in Numerical Methods for Differential System, Ed. 
Lapidus L., and Schiesser W. E., Academic Press, Inc., 1976. 

16. R. RACH, A Convenient Computational Form of the Adomian 
Polynomials, J. Math. Anal. Appl., 102, (1984), 415-419. 

17. G. ADOMIAN and MALAKIAN, Self-correcting approximate solutions 
by the iterative method for nonlinear Stochastic Differential Equations, J. 
Math. Anal. Appl., 76, (1980), 309-327. 

18. G. ADOMIAN and MALAKIAN, Inversion of Stochastic Partial 
Differential Operators-The Linear Case, J. Math. Anal. Appl., 77, (1980), 
505-512. 

19. G. ADOMIAN and MALAKIAN, Existence of the Inverse of a Linear 
Stochastic Operator, J. Math. Anal. Appl., 114, (1986), 55-56. 

20. G. ADOMIAN and R. RACH, Inversion of Nonlinear Stochastic 
Operators, J. Math. Anal. Appl., 91, (1983), 39-46.