# On the theory and application of Adomian Decomposition method for numerical solution of second-order ordinary differenctial equation.

Introduction

It is a well known and documented fact that many phenomena in Engineering, Science, Management and Economics can be modelled using the theory of derivatives and integrals. It is also interesting to say that solutions to most of differential equations that arise from the above model cannot be easily obtained by analytical means. Therefore, an approximate solutions are needed which are generated by numerical techniques.

Some of the existing methods are based on discretization and they only allow the solutions to a given ordinary differential equations at a given interval. The above deficiency leads to a situation where some fundamental phenomena are easily avoided. The ADM is a relatively new approach, which provides an analytic approximation to linear and none linear problems. The method is quantitative rather than qualitative. It is analytic and it requires neither linearization nor perturbation. It is also continuous with no resort to discretization. The method provides the solution as an infinite series in which each term can be determined. Throughout, we shall consider equation of the form;

[y.sup.ii] = f(x,y), y(0)[y.sub.0],[y.sup.i](0) = y,x [member of] [0,b] (1)

We shall proceed to discuss the basic theory and concepts of Adomian Decomposition Method (ADM)

The theory and concepts of adomian decomposition method (adm)

The method consists of splitting the given equation into linear and non-linear parts, inverting the highest order derivative operator contained in the linear operator on both sides identifying the initial conditions and the terms involving the independent variables alone as initial approximation, decomposing the unknown function into a series whose components can be easily computed, decomposing the non-linear function in terms if polynomial called Adomian's polynomials, and finding the successive terms of the series solution by recurrent relation using the polynomials obtained (cf. Adomian 1988)

To solve problems of the form (1), we write it in an operator form as

Ly = f(x, y) (2)

Where the differential operator L is given as

L = [d.sup.2]/d[x.sup.2]

The inverse operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

If we operate on both sides of (2) and impose the initial conditions we obtain

y(w) = [y.sub.0] + [y.sub.i]x + [L.sup.-1] (f (x, y)) (4)

The Adomian decomposition method introduces the solution y(x) by an infinite series of components

y(x) = [[infinity].summation over (n-0)] An (5)

and the non-linear function f(x, y) by an infinite series of polynomials

f (x, y) = [[infinity].summation over (n-0)] An (6)

Where the components [y.sub.n](x) f he solution y(x) rill be determined recurrently, and the Adomian's Polynomial s [A.sub.n] can be calculated for various classes of non-linearity according to algorithms recently set by G.Adomian and R. Rach (1992).

If we substitute (5) and (6) into (4), we obtain

[[infinity].summation over (n-0)][y.sub.n](x) = [y.sub.0] + [y.sub.1]x + [L.sup.1] [[[infinity].summation over (n-0)] An] (7)

We next determine the components [y.sub.n](x) for which n [greater than or equal to] 0. We first identify the zeroth component [y.sub.0] (x) by all terms that arise from the initial conditions. The remaining components are determined by using the preceding component. Each term of the series (5) is given by the recurrect relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

It must be stated here that all terms of series (5) cannot be computed and the solution of (1) will be approximated by series of the form [[PHI].sub.N](x) = [N-1.summation over (n-0)] [y.sub.n](x) (9)

The method reduces significantly the massive computation which may arise if discretization methods are used for the solution of non non-linear problems.

Applications and results

Example

Consider the linear equation

[y.sup.[??]] = x + y, [y.sub.(0)] = 1, [y.sup.[??]] (0), x [member of] [0, 5] (10)

With the theoretical solution

y(x) = [e.sup.x] - x

We apply ADM operator to equation (10) to produce

Ly = x + y (11)

Operating [L.sup.-1] on both sides of (11) and use the initial conditions, we obtain

y(x) = 1 + [[??].sup.-1](x) + [[??].sup.1]y (12)

By using (7), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The ADM introduces the recursive relation

[y.sub.0](x) = 1 + [L.sup.-1](x) = 1 + [x.sup.3]/6

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

We can then proceed to compute the first few terms of the series.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

For application purpose, only the first ten terms of the series is computed. Table (1) compare the result obtained using ADM with exact solution. It is obvious that the result is in agreement with the exact solution. Higher accuracy can be obtained by evaluating more components of the series (15)

Example 2

Let us consider the equation

[x.sup.II] (t) + [e.aup.-rt] = 0, , x(f) = 0, x(0) = 1, [x.sup.[??]] (0) = 0 t [member of] [0,1] (16)

where x(t) is the displacement at time and is a positive constant which represent the model for a spring-mass system. For [gamma] = 0, the equation reduces to

[x.sup.II](t) + x(t) = 0, x(0) = 1, [x.sup.[??]](0) = 0 (17)

The exact solution of (17) is cost t.

In an operator form, (16) becomes

Lx = -x (18)

Operating [L.sup.-1] on both sides of (18) and using the initial conditions, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Here, only the first ten terms of the decomposition series were used in evaluating the approximate solution for the table (2). The efficiency of this approach can be drastically enhanced by computing further terms of the series.

Concluding remark

In this section, a simple proof of convergence of Adomian's technique is presented. The ADM introduces the solution y(x) of (1) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Where [A.sub.n] 's are polynomials in [y.sub.0], [y.sub.1] ... [y.sub.n] called determining the sequence

[S.sub.[??]] = [y.sub.0] + [y.sub.1] + [y.sub.2] + [y.sub.3] ... [y.sub.n] (22)

For the study of the numerical resolution of (1)

Cherrault and Rach (1995) used fixed-point theorem. Theorem (Cherrault and Rach 1995)

Let N be an operator from a Hilbert space H into H and Y be the exact solution of (1) then

[[infinity].summation over (n=0)] [y.sub.n](x)

which is obtained by ADM, converges to y when there exist [alpha] [member of] [0,1] uch that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

And we show that [{[S.sub.n]}.sup.[infinity].sub.n=0] a Cauchy sequence in Hilbert H for this purpose, consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

But for every n.m [member of] N.n [greater than or equal to] m we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

hence,

lim [??] [S.sub.n] - [S.sub.m] [??] = 0

n, m [right arrow] s + [infinity]

i.e [{[S.sub.n]}.sup.+[infinity].sub.+0] Cauchy sequence in the Hilbert space H and it implies that there exist S, SEH such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

.e. S = [[infinity].summation over (n=0)] [y.sub.n]

We have been able to present and apply the ADM to second order differential equations. We have presented and compared the numerical solution using ADM with the theoretical solution. From our findings, we observed that more accuracy can be obtained by accommodating more terms in our decomposition series. One of the advantages of ADM is that it generates solutions over infinite intervals.

References

Adomian, G., 1988. A review of decomposition method in applied mathematics. J. Mathr. Anal. Appl.

Wa2wat, A.M., 2001. A new algorithms for solving differential equations of Lane-Emden type, Appl. Math. Camp.

Rach, R., 1992. Noise terms in decomposition series solution, Comput. Math. Appl.

Wa2waz, A.M., 2000. A new algorithm for calculating Adomian polynomial for non-linear equations, Appl. Math. Comput.

Cherrualt, Y., G. Adomian, 1993. Decomposition Methods; a new proof of convergence, Math. Comput. Model. Nagle, R.K., E.B. Saff, 1994. Fundamentals of Differential Equations, third ed., Addison-Wesly Publishers.

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

E.A. Ibijola and B.J. Adegboyegun

Department of mathematical sciences, university of ado-ekiti, nigeria.

E.A. Ibijola and B.J. Adegboyegun: On the Theory and Application of Adomian Decomposition Method for Numerical Solution of Second-order Ordinary Differenctial Equations: Adv. in Nat. Appl. Sci., 2(3): 208-213, 2008

It is a well known and documented fact that many phenomena in Engineering, Science, Management and Economics can be modelled using the theory of derivatives and integrals. It is also interesting to say that solutions to most of differential equations that arise from the above model cannot be easily obtained by analytical means. Therefore, an approximate solutions are needed which are generated by numerical techniques.

Some of the existing methods are based on discretization and they only allow the solutions to a given ordinary differential equations at a given interval. The above deficiency leads to a situation where some fundamental phenomena are easily avoided. The ADM is a relatively new approach, which provides an analytic approximation to linear and none linear problems. The method is quantitative rather than qualitative. It is analytic and it requires neither linearization nor perturbation. It is also continuous with no resort to discretization. The method provides the solution as an infinite series in which each term can be determined. Throughout, we shall consider equation of the form;

[y.sup.ii] = f(x,y), y(0)[y.sub.0],[y.sup.i](0) = y,x [member of] [0,b] (1)

We shall proceed to discuss the basic theory and concepts of Adomian Decomposition Method (ADM)

The theory and concepts of adomian decomposition method (adm)

The method consists of splitting the given equation into linear and non-linear parts, inverting the highest order derivative operator contained in the linear operator on both sides identifying the initial conditions and the terms involving the independent variables alone as initial approximation, decomposing the unknown function into a series whose components can be easily computed, decomposing the non-linear function in terms if polynomial called Adomian's polynomials, and finding the successive terms of the series solution by recurrent relation using the polynomials obtained (cf. Adomian 1988)

To solve problems of the form (1), we write it in an operator form as

Ly = f(x, y) (2)

Where the differential operator L is given as

L = [d.sup.2]/d[x.sup.2]

The inverse operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

If we operate on both sides of (2) and impose the initial conditions we obtain

y(w) = [y.sub.0] + [y.sub.i]x + [L.sup.-1] (f (x, y)) (4)

The Adomian decomposition method introduces the solution y(x) by an infinite series of components

y(x) = [[infinity].summation over (n-0)] An (5)

and the non-linear function f(x, y) by an infinite series of polynomials

f (x, y) = [[infinity].summation over (n-0)] An (6)

Where the components [y.sub.n](x) f he solution y(x) rill be determined recurrently, and the Adomian's Polynomial s [A.sub.n] can be calculated for various classes of non-linearity according to algorithms recently set by G.Adomian and R. Rach (1992).

If we substitute (5) and (6) into (4), we obtain

[[infinity].summation over (n-0)][y.sub.n](x) = [y.sub.0] + [y.sub.1]x + [L.sup.1] [[[infinity].summation over (n-0)] An] (7)

We next determine the components [y.sub.n](x) for which n [greater than or equal to] 0. We first identify the zeroth component [y.sub.0] (x) by all terms that arise from the initial conditions. The remaining components are determined by using the preceding component. Each term of the series (5) is given by the recurrect relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

It must be stated here that all terms of series (5) cannot be computed and the solution of (1) will be approximated by series of the form [[PHI].sub.N](x) = [N-1.summation over (n-0)] [y.sub.n](x) (9)

The method reduces significantly the massive computation which may arise if discretization methods are used for the solution of non non-linear problems.

Applications and results

Example

Consider the linear equation

[y.sup.[??]] = x + y, [y.sub.(0)] = 1, [y.sup.[??]] (0), x [member of] [0, 5] (10)

With the theoretical solution

y(x) = [e.sup.x] - x

We apply ADM operator to equation (10) to produce

Ly = x + y (11)

Operating [L.sup.-1] on both sides of (11) and use the initial conditions, we obtain

y(x) = 1 + [[??].sup.-1](x) + [[??].sup.1]y (12)

By using (7), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The ADM introduces the recursive relation

[y.sub.0](x) = 1 + [L.sup.-1](x) = 1 + [x.sup.3]/6

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

We can then proceed to compute the first few terms of the series.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

For application purpose, only the first ten terms of the series is computed. Table (1) compare the result obtained using ADM with exact solution. It is obvious that the result is in agreement with the exact solution. Higher accuracy can be obtained by evaluating more components of the series (15)

Example 2

Let us consider the equation

[x.sup.II] (t) + [e.aup.-rt] = 0, , x(f) = 0, x(0) = 1, [x.sup.[??]] (0) = 0 t [member of] [0,1] (16)

where x(t) is the displacement at time and is a positive constant which represent the model for a spring-mass system. For [gamma] = 0, the equation reduces to

[x.sup.II](t) + x(t) = 0, x(0) = 1, [x.sup.[??]](0) = 0 (17)

The exact solution of (17) is cost t.

In an operator form, (16) becomes

Lx = -x (18)

Operating [L.sup.-1] on both sides of (18) and using the initial conditions, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Here, only the first ten terms of the decomposition series were used in evaluating the approximate solution for the table (2). The efficiency of this approach can be drastically enhanced by computing further terms of the series.

Concluding remark

In this section, a simple proof of convergence of Adomian's technique is presented. The ADM introduces the solution y(x) of (1) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Where [A.sub.n] 's are polynomials in [y.sub.0], [y.sub.1] ... [y.sub.n] called determining the sequence

[S.sub.[??]] = [y.sub.0] + [y.sub.1] + [y.sub.2] + [y.sub.3] ... [y.sub.n] (22)

For the study of the numerical resolution of (1)

Cherrault and Rach (1995) used fixed-point theorem. Theorem (Cherrault and Rach 1995)

Let N be an operator from a Hilbert space H into H and Y be the exact solution of (1) then

[[infinity].summation over (n=0)] [y.sub.n](x)

which is obtained by ADM, converges to y when there exist [alpha] [member of] [0,1] uch that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

And we show that [{[S.sub.n]}.sup.[infinity].sub.n=0] a Cauchy sequence in Hilbert H for this purpose, consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

But for every n.m [member of] N.n [greater than or equal to] m we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

hence,

lim [??] [S.sub.n] - [S.sub.m] [??] = 0

n, m [right arrow] s + [infinity]

i.e [{[S.sub.n]}.sup.+[infinity].sub.+0] Cauchy sequence in the Hilbert space H and it implies that there exist S, SEH such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

.e. S = [[infinity].summation over (n=0)] [y.sub.n]

We have been able to present and apply the ADM to second order differential equations. We have presented and compared the numerical solution using ADM with the theoretical solution. From our findings, we observed that more accuracy can be obtained by accommodating more terms in our decomposition series. One of the advantages of ADM is that it generates solutions over infinite intervals.

References

Adomian, G., 1988. A review of decomposition method in applied mathematics. J. Mathr. Anal. Appl.

Wa2wat, A.M., 2001. A new algorithms for solving differential equations of Lane-Emden type, Appl. Math. Camp.

Rach, R., 1992. Noise terms in decomposition series solution, Comput. Math. Appl.

Wa2waz, A.M., 2000. A new algorithm for calculating Adomian polynomial for non-linear equations, Appl. Math. Comput.

Cherrualt, Y., G. Adomian, 1993. Decomposition Methods; a new proof of convergence, Math. Comput. Model. Nagle, R.K., E.B. Saff, 1994. Fundamentals of Differential Equations, third ed., Addison-Wesly Publishers.

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

E.A. Ibijola and B.J. Adegboyegun

Department of mathematical sciences, university of ado-ekiti, nigeria.

E.A. Ibijola and B.J. Adegboyegun: On the Theory and Application of Adomian Decomposition Method for Numerical Solution of Second-order Ordinary Differenctial Equations: Adv. in Nat. Appl. Sci., 2(3): 208-213, 2008

Table 1 At H = 0.5 X ADOMIAN EXACT ERROR 0.00 1.00000000 1.00000000 0.00000000 0.50 1.14872134 1.14872122 0.00000012 1.00 1.71828175 1.71828175 0.00000000 1.50 2.98168921 2.98168898 0.00000024 2.00 5.38905621 5.38905621 0.00000000 2.50 9.68249416 9.68249416 0.00000000 3.00 17.08553696 17.08553696 0.00000000 3.50 29.61545563 29.61545181 0.00000381 4.00 50.59815216 50.59814835 0.00000381 4.50 85.51713562 85.51712799 0.00000763 5.00 143.41316223 143.41316223 0.00000000 At h = 0.2 0.00 1.00000000 1.00000000 0.00000000 0.10 1.00517094 1.00517094 0.00000000 0.20 1.02140284 1.02140272 0.00000012 0.30 1.04985881 1.04985881 0.00000000 0.40 1.09182465 1.09182477 0.00000012 0.50 1.14872134 1.14872122 0.00000012 0.60 1.22211874 1.22211885 0.00000012 0.70 1.31375277 1.31375265 0.00000012 0.80 1.42554104 1.42554104 0.00000000 0.90 1.55960333 1.55960321 0.00000012 1.00 1.71828198 1.71828210 0.00000012 Table 2 H = 0.2 X ADOMIAN EXACT ERROR 0.00 1.00000000 1.00000000 0.00000000 0.20 0.98006660 0.98006660 0.00000000 0.40 0.92106104 0.92106098 0.00000000 0.60 0.82533562 0.82533562 0.00000000 0.80 0.69670671 0.69670671 0.00000000 1.00 0.54030228 54030228 0.00000000 1.20 0.36235774 0.36235771 0.00000003 1.40 0.16996713 0.16996706 0.00000007 1.60 -0.02919978 -0.02919967 0.00000012 1.80 -0.22721598 -0.22720228 0.00001369 2.00 -0.4163340 -0.41614705 0.00018698 X ADOMIAN EXACT ERROR 0.00 1.00000000 1.00000000 0.00000000 0.10 0.99500418 0.99500418 0.00000000 0.20 0.95533645 0.98006660 0.00000000 0.40 0.92106104 0.95533651 0.00000006 0.50 0.87758261 0.92106098 0.00000006 0.60 0.82533562 0.87758255 0.00000006 0.70 0.76484221 0.82533562 0.00000000 0.80 0.69670671 0.69670665 0.00000006 0.90 0.62160987 0.62160987 0.00000000 1.00 0.54030216 0.54030222 0.00000006

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

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

Publication: | Advances in Natural and Applied Sciences |

Geographic Code: | 6NIGR |

Date: | Sep 1, 2008 |

Words: | 1845 |

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