Abstract: the adomian polynomials, which are tailored to

 

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.

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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).