# On Adomian Decomposition Method (ADM) for numerical solution of ordinary differential equations.

Introduction

Adomian Decomposition Method (ADM) is one of the new methods for solving Initial Value Problem in Ordinary Differential Equations of various kinds arising not only in the field of medicine, physical and biological science but also in the area of engineering. It is important to note that a large amount of research works has been devoted to the application of (ADM) to wide class of linear and nonlinear ordinary or partial differential equations.

In this paper, we present the theoretical analysis and practical applications of (ADM). Infact, we present a further insight into the use of (ADM) for solving first order ordinary differential equations of the form y' = f(x,y), y(a) - [y.sub.0].(1)

We compare the result obtained with the exact or theoretical solution.

The Basic Concepts of Adomian Decomposition Method (ADM)

For our construction, we shall refer to our equation (1). We apply (ADM) to find a numerical solution of the special first order Ordinary Differential Equations of the form (1).

We further assume that y(x) is sufficiently differentiable and that the solution of (1) exists and satisfy the Lipschitz condition. It must be stated here that the method can handle effectively higher order initial value problems. (ADM) usually defines a equation in an operator form by considering the highest-ordered derivative in the problem (A.M W azwat)

In an operator form, equation (1) can be written as

Ly=f(x,y) (2)

Where the differential operator L is given as,

L=d./dx (3)

The inverse operator [L.sup.-1] is considered a one fold integral operator defined by,

[L.sup.-1]= [[integral].sup.x.sub.0] (*)dx (4)

If we operate [L.sup.-1] on the right hand side of (2) and use the initial condition y(a) = [y.sub.0], we have,

y(x)= [y.sub.0] + [L.sup.-1] f(x,Y) (5)

The ADM introduce the solution y(x) in an infinite series form as,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where the components [y.sub.n](x) will be determined recursively. Moreover, the method defined the nonlinear function f(x,y) by the inifinite series of the form;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

If we now use equation (6) and (7) in (5) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The next step is to seek a way to determine the component [y.sub.n](x) for which n 0. W e first identify the zeroth component [y.sub.0](x) by the term that arise from the initial condition. The remaining component is determined by using the preceding component (G.Adoman 1994)

Each term of the series in (6) is given by the recurrent relation,

[y.sub.0](x) = [y.sub.0] (9)

[y.sub.n+1](x) = [L.sup.-1]([A.sub.n]), n[greater than or equal to]0) (10)

We must state here that in practice all term of the series in (6) cannot be determined and the solution will be approximated by series of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

With (11), we obtain series solution for our system (1) (A.M W azwaz 2002). The method reduces significantly the massive computation which may arise if discretization methods are used for the solution of non linear problems.

Implementation Procedure

For an illustration of our discussion, we use two test problems. We shall also consider the performance of the method discussed with the theoretical solution.

Problem 1

We consider the system

[y.sup.1]=[y.sup.2], y(0) =1 (12)

with the theoretical solution given as

y(x)=1/1-x, 0[less than or equal to]x<1 (13)

we apply ADM operator to equation (12) to produce

Ly = [y.sup.2] (14)

By finding the inverse operator and imposing initial condition we obtain

y(x) = y(0) + [L.sup.-1]([y.sup.2]) (15)

Substituting the series in (6) for y(x) and the series polynomial we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Where the Adomian Polynomial An can be derived as follows,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

If we collect and rearrange like terms we get the followings Adomian Polynimials.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can then proceed to compute [y.sub.1], [y.sub.2], [y.sub.3] ... ..

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently, [y.sub.4](x)= [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For application purpose, only few terms of the series will be computed. Tables (1) compare the (ADM) result with the theoretical solutions.

Remark

It must be stated here that (ADM) will only work for this particular text problem if Ox< 1 and that the efficiency of the approach can be enhanced by computing further terms of the series.

Problem 2

Let us consider the differential equation dN/dt=rN N(0)= 1000, which represent the growth of bac a colony. We assume that the model grows continuously and without restriction. One may ask that how many Bacteria are in the colony after some hours if an individual produces an average of 0.2 offspring every hour where N(t) is the population size at time t. the above equation can be solved analytically to give

N(t) = N(0)[e.sup.rt]

Where r = 0.2 and N(0) = 1000

N(t) = 1000[e.sup.0.2t]

By using the (ADM), we have

LN = r N (18)

By finding the inverse operator and imposing the initial condition we have

N(t) = 1000 + r[L.sup.-1](N) [N.sub.0](t) = 1000 [N.sub.n+1] (t) = r[L.sup.-1] (Nn) (19)

Hence;

[N.sub.1](t) (1000t)

[N.sub.2](t) = [r.sup.2] (1000[t.sup.2])/2!

[N.sub.3](t) = [r.sup.3] (1000[t.sup.3])/3!

[N.sub.n] (t) = [r.sup.n] (1000[t.sup.n])/n!

Where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Table (2) compares the result obtained using (ADM) with that of theoretical solution. Only the ten terms of the decomposition series were used in evaluating the numerical result.

Conclusion

The results obtained from the test problems have shown that (ADM) is a powerful and efficient technique in finding an approximate solution to both linear and non-linear first-order Ordinary Initial Value Problems which occur most often in physical and Biological sciences. It is interesting to note that the (ADM) results tend to the theoretical solution by increasing the number of terms computed.

References

G. Adoman, 1994. Solving frontier problems of physics. The Decomposition method, kluwer, Bostrun 1994.

G. Adonman, 1988. A review of the decomposition method in applied mathematics, J. math. Anal, Apple, 135: 501-504.

G. Adomian and R. Rach, 1992. Noise terms in decomposition series solution, Comput. Math. Appl., 24: 61.

Chawla, M.M. and C.P. Katti, 1984. A finite-difference method for a class of singular boundary value problem, IMAJ. Numerical Anal., 4: 457.

Van kampen, N.G., 1993. J. stat. phys., 70: 103-106.

Wazwat, A.M., 2000. A new algorithm for calculating Adomian polynomials for non linear operations, Apply Math. Compute, III: 53-69.

E.A. Ibijola, B.J. Adegboyegun and O.Y. Halid: On Adomian Decomposition Method (ADM) for Numerical Solution of Ordinary Differential Equations: Adv. in Nat. Appl. Sci., 2(3):165-169, 2008 Department of Mathematical Sciences, University of Ado-ekiti, Nigeria

Corresponding Author: E.A. Ibijola, Department of Mathematical Sciences, University of Ado-ekiti, Nigeria E-mail: ibjemm@yahoo.com

Adomian Decomposition Method (ADM) is one of the new methods for solving Initial Value Problem in Ordinary Differential Equations of various kinds arising not only in the field of medicine, physical and biological science but also in the area of engineering. It is important to note that a large amount of research works has been devoted to the application of (ADM) to wide class of linear and nonlinear ordinary or partial differential equations.

In this paper, we present the theoretical analysis and practical applications of (ADM). Infact, we present a further insight into the use of (ADM) for solving first order ordinary differential equations of the form y' = f(x,y), y(a) - [y.sub.0].(1)

We compare the result obtained with the exact or theoretical solution.

The Basic Concepts of Adomian Decomposition Method (ADM)

For our construction, we shall refer to our equation (1). We apply (ADM) to find a numerical solution of the special first order Ordinary Differential Equations of the form (1).

We further assume that y(x) is sufficiently differentiable and that the solution of (1) exists and satisfy the Lipschitz condition. It must be stated here that the method can handle effectively higher order initial value problems. (ADM) usually defines a equation in an operator form by considering the highest-ordered derivative in the problem (A.M W azwat)

In an operator form, equation (1) can be written as

Ly=f(x,y) (2)

Where the differential operator L is given as,

L=d./dx (3)

The inverse operator [L.sup.-1] is considered a one fold integral operator defined by,

[L.sup.-1]= [[integral].sup.x.sub.0] (*)dx (4)

If we operate [L.sup.-1] on the right hand side of (2) and use the initial condition y(a) = [y.sub.0], we have,

y(x)= [y.sub.0] + [L.sup.-1] f(x,Y) (5)

The ADM introduce the solution y(x) in an infinite series form as,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where the components [y.sub.n](x) will be determined recursively. Moreover, the method defined the nonlinear function f(x,y) by the inifinite series of the form;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

If we now use equation (6) and (7) in (5) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The next step is to seek a way to determine the component [y.sub.n](x) for which n 0. W e first identify the zeroth component [y.sub.0](x) by the term that arise from the initial condition. The remaining component is determined by using the preceding component (G.Adoman 1994)

Each term of the series in (6) is given by the recurrent relation,

[y.sub.0](x) = [y.sub.0] (9)

[y.sub.n+1](x) = [L.sup.-1]([A.sub.n]), n[greater than or equal to]0) (10)

We must state here that in practice all term of the series in (6) cannot be determined and the solution will be approximated by series of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

With (11), we obtain series solution for our system (1) (A.M W azwaz 2002). The method reduces significantly the massive computation which may arise if discretization methods are used for the solution of non linear problems.

Implementation Procedure

For an illustration of our discussion, we use two test problems. We shall also consider the performance of the method discussed with the theoretical solution.

Problem 1

We consider the system

[y.sup.1]=[y.sup.2], y(0) =1 (12)

with the theoretical solution given as

y(x)=1/1-x, 0[less than or equal to]x<1 (13)

we apply ADM operator to equation (12) to produce

Ly = [y.sup.2] (14)

By finding the inverse operator and imposing initial condition we obtain

y(x) = y(0) + [L.sup.-1]([y.sup.2]) (15)

Substituting the series in (6) for y(x) and the series polynomial we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Where the Adomian Polynomial An can be derived as follows,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

If we collect and rearrange like terms we get the followings Adomian Polynimials.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can then proceed to compute [y.sub.1], [y.sub.2], [y.sub.3] ... ..

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently, [y.sub.4](x)= [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For application purpose, only few terms of the series will be computed. Tables (1) compare the (ADM) result with the theoretical solutions.

Remark

It must be stated here that (ADM) will only work for this particular text problem if Ox< 1 and that the efficiency of the approach can be enhanced by computing further terms of the series.

Problem 2

Let us consider the differential equation dN/dt=rN N(0)= 1000, which represent the growth of bac a colony. We assume that the model grows continuously and without restriction. One may ask that how many Bacteria are in the colony after some hours if an individual produces an average of 0.2 offspring every hour where N(t) is the population size at time t. the above equation can be solved analytically to give

N(t) = N(0)[e.sup.rt]

Where r = 0.2 and N(0) = 1000

N(t) = 1000[e.sup.0.2t]

By using the (ADM), we have

LN = r N (18)

By finding the inverse operator and imposing the initial condition we have

N(t) = 1000 + r[L.sup.-1](N) [N.sub.0](t) = 1000 [N.sub.n+1] (t) = r[L.sup.-1] (Nn) (19)

Hence;

[N.sub.1](t) (1000t)

[N.sub.2](t) = [r.sup.2] (1000[t.sup.2])/2!

[N.sub.3](t) = [r.sup.3] (1000[t.sup.3])/3!

[N.sub.n] (t) = [r.sup.n] (1000[t.sup.n])/n!

Where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Table (2) compares the result obtained using (ADM) with that of theoretical solution. Only the ten terms of the decomposition series were used in evaluating the numerical result.

Conclusion

The results obtained from the test problems have shown that (ADM) is a powerful and efficient technique in finding an approximate solution to both linear and non-linear first-order Ordinary Initial Value Problems which occur most often in physical and Biological sciences. It is interesting to note that the (ADM) results tend to the theoretical solution by increasing the number of terms computed.

References

G. Adoman, 1994. Solving frontier problems of physics. The Decomposition method, kluwer, Bostrun 1994.

G. Adonman, 1988. A review of the decomposition method in applied mathematics, J. math. Anal, Apple, 135: 501-504.

G. Adomian and R. Rach, 1992. Noise terms in decomposition series solution, Comput. Math. Appl., 24: 61.

Chawla, M.M. and C.P. Katti, 1984. A finite-difference method for a class of singular boundary value problem, IMAJ. Numerical Anal., 4: 457.

Van kampen, N.G., 1993. J. stat. phys., 70: 103-106.

Wazwat, A.M., 2000. A new algorithm for calculating Adomian polynomials for non linear operations, Apply Math. Compute, III: 53-69.

E.A. Ibijola, B.J. Adegboyegun and O.Y. Halid: On Adomian Decomposition Method (ADM) for Numerical Solution of Ordinary Differential Equations: Adv. in Nat. Appl. Sci., 2(3):165-169, 2008 Department of Mathematical Sciences, University of Ado-ekiti, Nigeria

Corresponding Author: E.A. Ibijola, Department of Mathematical Sciences, University of Ado-ekiti, Nigeria E-mail: ibjemm@yahoo.com

Table 1: X ADOMAIN EXACT ERROR 0.00 1.000000000 1.000000000 0.000000000 0.10 1.111111164 1.111111164 0.000000000 0.20 1.249999881 1.250000000 0.000000119 0.30 1.428571224 1.428571463 0.000000238 0.40 1.666666746 1.666666627 0.000000119 0.50 2.000000000 2.000000000 0.000000000 0.60 2.500000000 2.500000238 0.000000238 0.70 3.333333969 3.333333969 0.000000000 0.80 5.000000000 5.000001907 0.000001907 0.90 10.000004768 10.000009537 0.000004768 Table 2: t ADOMAIN EXACT ERROR 0.00 1000.000000 1000.000000 0.000000 0.50 1105.170898 1105.171021 0.000122 1.00 1221.402832 1221.402710 0.000122 1.50 1349.858765 1349.858765 0.000000 2.00 1491.824707 1491.824707 0.000000 2.50 1648.721313 1648.721313 0.000000 3.00 1822.118774 1822.118774 0.000000 3.50 2013.752808 2013.752686 0.000122 4.00 2225.540771 2225.540771 0.000000 4.50 2459.602783 2459.603027 0.000244 5.00 2718.281982 2718.281982 0.000000

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Title Annotation: | Original Article |
---|---|

Author: | Ibijola, E.A.; Adegboyegun, B.J.; Halid, O.Y. |

Publication: | Advances in Natural and Applied Sciences |

Geographic Code: | 6NIGR |

Date: | Sep 1, 2008 |

Words: | 1423 |

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