Adomian Decomposition Method

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

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 Fy(x)=g(x) , where F represents a general nonlinear ordinary or partial differential operator including both linear and nonlinear terms .The linear terms are decomposed intoL+R , where L is easily invertible (usually the highest order derivative) and R is the remained of the linear operator.

Considering these components solution can be approximated as y(x)=∑_(n=0)^∞▒y_n =y_0+y_1+y_2+y_3+⋯ y(x)=x^3/3+x^7/63+(2x^11)/2079+⋯ 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