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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 dis·cret·i·za·tion  
n.
The act of making mathematically discrete.
 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 See add/drop multiplexer.

(language) ADM - A picture query language, extension of Sequel2.

["An Image-Oriented Database System", Y. Takao et al, in Database Techniques for Pictorial Applications, A. Blaser ed, pp. 527-538].
 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 In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential  nor perturbation perturbation (pŭr'tərbā`shən), in astronomy and physics, small force or other influence that modifies the otherwise simple motion of some object. The term is also used for the effect produced by the perturbation, e.g. . It is also continuous with no resort to discretization. The method provides the solution as an infinite series infinite series

In mathematics, the sum of infinitely many numbers, whose relationship can typically be expressed as a formula or a function. An infinite series that results in a finite sum is said to converge (see convergence). One that does not, diverges.
 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 The Adomian decomposition method (ADM) is a non-numerical method for solving nonlinear differential equations, both ordinary and partial. The general direction of this work is towards a unified theory for Partial Differential Equations (PDE).  (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 polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a0xn+a  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 differential operator

In mathematics, any combination of derivatives applied to a function. It takes the form of a polynomial of derivatives, such as D2xxD2xyD
 L is given as

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

The inverse operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (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 summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument)  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 according to
prep.
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

3.
 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 (jargon) zeroth - First.

Since zero is the lowest value of an unsigned binary integer, which is one of the most fundamental types in programming and hardware design, it is often natural to count from zero rather than one, especially when the integer is actually an index, as
 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 See recursion.

recursive - recursion
 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 In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. . Theorem (Cherrault and Rach 1995)

Let N be an operator from a Hilbert space Noun 1. Hilbert space - a metric space that is linear and complete and (usually) infinite-dimensional
metric space - a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the
 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 (mathematics) Cauchy sequence - A sequence of elements from some vector space that converge and stay arbitrarily close to each other (using the norm definied for the space).  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 SEH Structured Exception Handling
SEH Societas Europaea Herpetologica
SEH Société d'Ecologie Humaine
SEH St Elizabeths Hospital (Anacostia, Washington, DC)
SEH Safety, Environment and Health
SEH St.
 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 constraint satisfaction, a decomposition method translates a constraint satisfaction problem into another constraint satisfaction problem that is binary and acyclic. Decomposition methods work by grouping variables into sets, and solving a subproblem for each set.  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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