ADMOdinary Diff

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Solution of the system of ordinarydifferential equations by Adomian

decomposition method

J. Biazar a, E. Babolian b, R. Islam c,*

a Guilan University, P.O. Box 1914, Rasht, Iranb Teacher Training University, P.C. 15618, Tehran, Iran

c Department of Civil Engineering, Dalhousie University, D510, 1360 Barrington Street,

P.O. Box 1000, Halifax Nova Scotia B3J 2X4, Canada

Abstract

In this article we use Adomian decomposition method, which is a well-knownmethod for solving functional equations now-a-days, to solve systems of differential

equations of the first order and an ordinary differential equation of any order by con-

verting it into a system of differential of the order one. Theoretical considerations are

being discussed, and convergence of the method for theses systems is addressed. Some

examples are presented to show the ability of the method for linear and non-linear

systems of differential equations.

2002 Elsevier Inc. All rights reserved.

Keywords:Adomian decomposition method; Systems of ordinary differential equations

1. Introduction

A system of ordinary differential equations of the first order can be con-

sidered as:

*Corresponding author.

E-mail address: [email protected](R. Islam).

0096-3003/$ - see front matter 2002 Elsevier Inc. All rights reserved.

doi:10.1016/S0096-3003(02)00806-8

Applied Mathematics and Computation 147 (2004) 713719

www.elsevier.com/locate/amc
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


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y01 f1x;y1;. . .;yny0

2 f

2x;y

1;. . .;y

n

.

.

.

y0n fnx;y1;. . .;yn

8>>>>>: 1

where each equation represents the first derivative of one of the unknown

functions as a mapping depending on the independent variable x, and n un-

known functions f1;. . .;fn.Since every ordinary differential equation of order n can be written as a

system consisting ofn ordinary differential equation of order one, we restrict

our study to a system of differential equations of the first order.

2. Using Adomian decomposition method to solve (1)

We can present the system (1), by using the ith equation as:

Lyi fix;y1;. . .;yn i 1; 2;. . .; n 2

where L is the linear operator d=dx with the inverse L1 Rx

0: dx. Applying

the inverse operator on (2) we get the following canonical form, which is

suitable for applying Adomian decomposition method.

yi yi0

Z x0

fix;y1;. . .;yn dx i 1; 2;. . .; n 3

As usual in Adomian decomposition method the solution of Eq. (3) is

considered to be as the sum of a series:

yi X1j0

fi;j 4

And the integrand in the Eq. (3), as the sum of the following series:

fix;y1;. . .;yn X1j0

Ai;jfi;0;fi;1;. . .;fi;j 5

where Ai;jfi;0;fi;1;. . .;fi;n are called Adomian polynomials. Substituting (4)and (5) into (3) we get:

X1j0

fi;j yi0

Z x0

Xnj0

Ai;jfi;0;. . .;fi;j

yi0 X

1

j0

Z x

0

Ai;jfi;0;. . .;fi;j 6

714 J. Biazar et al. / Appl. Math. Comput. 147 (2004) 713719


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From which we define:

fi;0 yi0

fi;n1

Z x0

Ai;nfi;0;. . .;fi;n dx n 0; 1; 2;. . . 7

3. Convergence of the method

Since after the first step of the mentioned procedure we derive the Eq. (3),

which is a system of Volterra integral equations of the second kind, for theconvergence of the method, we refer the reader to [1] in which the problem of

convergence has been discussed briefly.

4. Numerical examples

In this part we present three examples. The first and the second examples are

considered to illustrate the method for linear and non-linear systems of ordi-

nary differential equations of order one. While in third example we solve a

differential equation of order three by transforming it into a system of differ-

ential equations of the first order.

Example 1.Consider the following system of differential equations, with initial

values y10 1, y20 0, and y30 2.

y01 y3 cosx

y02 y3 ex

y03 y1 y2

8

Using the inverse operator L1 Rx

0: dx we get:

y1 1

Z x0

cosx dx

Z x0

y3 dx

y2

Z x0

ex dx

Z x0

y3 dx

y3 2

Z x0

y1;n y2;n dx

Using the alternate algorithm for computing the Adomian polynomials

[2,3], the Adomian procedure would be as the following:

J. Biazar et al. / Appl. Math. Comput. 147 (2004) 713719 715


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y1;0 1 sinx y1;n1 Z x

0

y3;n dx

y2;0 1 ex y2;n1

Z x0

y3;n dx

y3;0 2 y3;n1

Z x0

y1;n y2;n dx

n 0; 1; 2;. . .

After two steps we get the exact solutions y1 ex, y2 sinx, y3 e

x cosx.

Example 2. In this example we solve the following non-linear system of

differential equations, with exact solutionsy1 e2x, y2x e

x, andy3x xex.

y01 2y22

y02 exy1

y03 y2 y3

8>: 9

Applying the inverse operator L

1

Rx

0 : dx, we get:

y1 1 2

Z x0

y22dx

y2 1

Z x0

exy1 dx

y3

Z x0

y2 y3 dx

10

Computing the Adomian polynomials by the algorithm presented in [2],

Adomian method leads to the following scheme:

y1;0 1 y1;n1 2

Z x0

Xnk0

y2;ky2;nk

!dx

y2;0 1 y2;n1

Z x0

exy1;n dx

y3;0 0 y2;n1Z

x

0

y2;n y3;n dx



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Approximations to the solutions with five terms are as follows:

y1x 426x 31ex 4x2 16x 15e2x

3:11111e3x 58:6666x 134:888

y2x 48x 7ex 82x 5e2x

0:11111112x 13e3x 11:5555

y3x 12x 3ex 2x 7:5e2x

0:111111e3x 0:00833333x5

0:0833333x4 0:5x3 x2 4:66666x 10:3888

Some numerical values of these solutions are presented in Table 1.

Example 3.Consider the following non-linear ordinary differential equation of

order 3, with the initial conditions y0 0, y00 1 and y000 2, and theexact solution yx xex.

y000 1

x

yy0 11

Considering three functions, y1x yx, y2x y0x, and y3x y

00x,we can convert (11) into the following non-linear system of three differential

equation of order one.

y01 y2

y02 y3

y031

xy1 y3

12

As in the previous examples if we apply the inverse operator, and using the

alternate algorithm for computing Adomian polynomials [2], we would have

the following scheme:

Table 1

Numerical values of the solutions of Example 2

xi y1xi ey1xi y2xi ey2xi y3xi ey3xi

0 1.00008 0 1 0 0 0

0.1 1.22132 1.65353E-5 1.10516 2.93237E-6 0.110517 0

0.2 1.49186 5.33750E-5 1.22139 1.12113E-5 0.244275 0

0.3 1.82161 5.97402E-4 1.34974 1.14097E-4 0.404906 5.11652E-5

0.4 2.22249 3.13158E-3 1.49125 5.72981E-4 0.59456 2.73288E-4

0.5 2.70702 1.13417E-2 1.64676 1.95913E-3 0.823372 9.88533E-4

0.6 3.28813 3.20685E-2 1.81686 5.24972E-3 1.09046 2.80769E-3

0.7 3.97860 7.66600E-2 2.00184 1.19097E-2 1.40286 6.76093E-3

0.8 4.79050 1.62539E-1 2.20161 2.39291E-2 1.76603 1.14401E-2

0.9 5.73528 3.14450E-1 2.41576 4.38414E-2 2.18568 2.79635E-2



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y1;0 0 y1;n1 Z x

0

y2;n dx

y2;0 1 y2;n1

Z x0

y3;n dx n 0; 1;. . .

y3;0 2 y3;n1

Z x0

1

xy1;n

y3;n

dx

13

Let yp y1;0 y1;1 y1;p is a notation for an approximation to thesolution with p 1 term. From (12) some computed approximations are asfollows:

y3 x 1

x x2

3

y4 x 1

x

x2

2

x3

12

y5 x 1

x

x2

2

x3

6

x4

60

y6 x 1

x

x2

2

x3

6

7x4

180

x5

360

y7 x 1

x x2

2 x3

6 x4

24 31x5

4320 x7

2520

y8 x 1

x

x2

2

x3

6

x4

24

x5

120

167x7

151; 200

x8

20; 160

.

.

.

It seems to be reasonable to conclude the exact solution, yx xex.

5. Conclusion

Adomian decomposition method has been known to be a powerful device

for solving many functional equations as algebraic equations, ordinary and

partial differential equations, integral equations and so on. Here we used this

method for solving systems of differential equations. It is demonstrated that

this method has the ability of solving systems of both linear and non-linear

differential equations. In Example 1, the system was a linear system and we

derived the exact solutions. There is another linear example in which we ob-

tained an exact solution. For non-linear systems, we usually derive a very good

approximations to the solutions, as in Example 2, and some times the exact

solutions can be found, as in Example 3. Extension of the method for solving



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systems of partial differential equations offers an excellent oppurtunity for

future research.

Acknowledgements

This study was possible due to funding provided by NSERC Operating

Grant and a joint NSERC/ACPI CRD research grant. This funding is grate-

fully acknowledged.

References

[1] E. Babolian, J. Biazar, Solution of a system of nonlinear Volterra integral equations of the

second kind, Far East J. Math. Sci. 2 (6) (2000) 935945.

[2] J. Biazar, E. Babolian, A. Nouri, R. Islam, An alternate algorithm for computing Adomian

decomposition method in special cases, Appl. Math. Comput. 138 (2003) 17.

[3] J. Biazar, M. Tango, E. Babolian, R. Islam, Solution of the kinetic modeling of lactic acid

fermentation using Adomian decomposition method, Appl. Math. Comput. 139 (2003) 249258.


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