Abstract:

This

paper presents the Adomian Decomposition Method for the solution of second

order linear and first order non-linear differential equations with the initial

conditions and hence comparison of Adomian solution with exact solution for the

second order linear differential equation. It is important to note that a large

amount of research work has been devoted to the application of the Adomian

decomposition method to a wide class of linear, nonlinear ordinary and partial

differential equations .The adomian decomposition method provides the solution

as an infinite series in which each term can be easily determined. A key

notation is the adomian polynomials, which are tailored to the particular

nonlinearity to solve nonlinear operator euation. This Adomian polynomials allow, for solution

convergence of the non-linear portion of the equation without simply

linearizing the system.

Key

words: Adomian decomposition method, Linear, Non-linear, Ordinary

differential equation, Initial value problem.

1. Introduction:

Most of the engineering problems are nonlinear and therefore some of them are

solved using numerical methods and some are solved using the different analytic

methods. One of semi-exact methods which does not need linearization or

discretization is Adomian Decomposition Method (ADM) see Bellman and Adomian

1;Adomian(1994). .The objective of the

decomposition method is to make physically realistic solutions of complex

systems without the usual modeling and solution compromises to achieve

tractability. This method is a powerful technique, which provides an efficient

algorithm for analytic approximate solutions and numeric simulations for

real-world applications in the applied science and engineering, particularly in

the practical solution of the linear or nonlinear and deterministic or

stochastic operator equations, including ordinary and partial differential

equation, integral equations, integro-differential equations, etc. Adomian

decomposition method has been employed by Gejji and Jafari 2 to obtain

solutions of a system of fractional differential equations and also discussed

the convergence of the method.

2 .Adomian Decomposition Method (ADM)

:

Consider

the equation , where represents a general nonlinear ordinary or

partial differential operator including both linear and nonlinear terms .The

linear terms are decomposed into , where is easily invertible (usually the highest

order derivative) and is the remained of the linear operator.

Thus, the equation can

be written as

(1)

Where indicates the nonlinear terms.

By solving this

equation for , since is invertible, we can write (2)

If is a second-order operator, is a twofold indefinite integral. By solving

Eq. (2), we have

(3)

Where and are constants of integration and can be found

from the boundary or initial conditions. Adomian decomposition method assumes

the solution that can be expanded into infinite series as

(4)

Also, the non linear

term will be written as

(5)

Where are the special Adomian polynomials. By substituting

Eqs. (4) and (5) in Eq. (3), we get

(6)

By specified , next components of can be determined

Finally after some

iterations and getting sufficient accuracy, the solution can be expressed by

Eq.(4).

In Eq. (6), the Adomian

polynomials can be generated by several means. Here are two ways

,

(

OR )

Continuing this, we can

get the other adomian polynomials.

3.

Solving Differential Equations by Adomian Decomposition Method :

Problem

1:

Consider the linear

ordinary differential equation .

Exact solution of this

O.D.E is

Given D.E equation can

be written as

(3.1)

Where is the differential operator and . Assume the inverse of

the operator exists and it can be integrated from to i.e. .

After operating on Eq. (3.1) ,we have

The Adomian

decomposition introduces the following expressions

Thus we obtain

Where

Therefore, we have

and so on. Considering

these components solution can be approximated as

Following table

compares the ADM solution with the exact solution.

Exact

ADM

0

0.1

0.2

0.3

0.4

0.5

4

4.1342

4.3345

4.5983

4.9248

5.3140

4

4.1342

4.3345

4.5984

4.9251

5.3150

The accuracy of ADM

solution increases by increasing the number of terms.

Problem

2: Consider

the non linear ordinary differential equation

.

This differential

equation can be written as (3.2)

Where is the differential operator, and . . Assume the inverse

of the operator exists and it can be integrated from to i.e. .

After operating on Eq. (3.2) ,we have

The Adomian

decomposition introduces the following expressions

Thus we obtain

Where

Therefore, we have

and so on. Considering

these components solution can be approximated as

This is the solution of

taken non linear differential equation. The accuracy of ADM solution increases

by increasing the number of terms.

4.

Conclusion: It was observed that solutions of the

first order linear and second order nonlinear differential equations with

initial conditions are obtained by the powerful and efficient Adomian

decomposition method. Also, we compared the Adomian solution of the linear

differential equation with exact solution, it shows that adomian solution is

very close to exact solution. Better accuracy can be obtained for the adomian

solution by accommodating more terms in our decomposition series.

References:

1. Bellman,

R.E.,Adomian, G.:Partial Differential Equations: New Methods for their

Treatment and Solution. D. Reidal, Dordrecht(1985) .

2.

Gejji, V.D.,Jafari,H.: Adomian Decomposition: A Tool for Solving a System of

Fractional Differential Equations.J.Math.Anal.Appl.301(2),508-518(2005).

3. G. Nhawu, p. Mafuta,

J. Mushanyu.: The Adomian Decomposition Method for Numerical Solution of First

Order Differential Equations.J.Math.Comput.Sci.3.307-314(2016).

4. j. Biazar and S.M. Shafiof:

A Simple Algorithm for Calculating Adomian

Polynomials.Int.J.comtemp.Math.Sciences.20,

975-982(2007).

5. Ch. RamReddy, T. Pradeepa,

Ch. Venkata Rao , O. Surendar,M.Chitra.: Analytical Solution of Mixed

Convection Flow of a Newtonian Fluid between Vertical Parallel Plates with

Soret, Hall and Ion-slip Effects: Adomian Decomposition Method.Int.J.Appl.Comput.Math.(2015).