Application of the Fourier transform via Adomian ’ s decomposition method

In this work, the Adomian Decomposition Method (ADM) is applied to compute the Fourier transform (FT) of functions on R. The basic properties of the Fourier transform are again obtained by ADM and examples assuring the applicability of ADM are presented.


Introduction
One of the interesting transforms in theory of differential equations (DEs) is Adomian decomposition Method .Adomian decomposition Method applied for inital value , boundary value problems, linear or nonlinear, ordinary and partial differential equations.At the beginning of the 1980, the Adomain decomposition method has been applied to a wide class of functional equations [6,7,8,9,15].The topic of the Adomian's decomposition method is discussed by Babolian [11,12,13] and Abbasbandy [1,2,3].H. Rouhparvar, S. Abbasbandy, T. Allahviranloo,in 2010 Showed that existence and uniqueness of solution an uncertain characteristic Cauchy reaction-diffusion equation by Adomian decomposition method [14].As well as in 2009 M. A. Fariborzi Araghi and Sh.Sadigh Behzadi, solving nonlinear volterra fredholm integro-differential equations using the modified Adomian decomposition method [10].Also, there exist some recently published papers with some modifications about application of the Adomain decomposition method to solve a differential equation [5,16,17,18].Adomain gives the solution as an infinite series usually converging to an accurate solution.The method devoted by scientists, because this method continuously deforms a simple problem easy to be solved into the difficult problem under study.A considerable amount of research work has been invested recently in applying this method to a wide class of linear and nonlinear ordinary, partial differential equations, and integral equations as well.In this work, we intend to use ADM for computiing the Fourier Transform (FT) Method of functions defined on R. Generally the first-order differential equation can be transformed into integrals, which are always very difficult to solve numerically or analytically.So, we consider the general forms of the first-order differential equations for illustrating the basic idea of the FT, Where i 2 = −1, s ∈ R, and f (x), is a real valued function on R. The general solution of (1.1) is given by: Where Than right hand side of the above equation is called Fourier transform of function f .So, we can conclude that for computing dx the Fourier transform of function f , is we can ues where u(x) is the solution (1.1).We solve (1.1) via ADM and then apply (1.3) to obtain the Fourier transform of function f .This paper representation that Fourier transform can be easily calculated by Adomain decomposition method.The structure of the paper is organized as follows: Adomian decomposition method described in Section 2. In Section 3, we present some application of ADM to Fourier transform indicated.Also, we present this method for the existence of solution for (1.1).In Section 4, we will restrict our attention to a few aspects of the theory.Finally, in Section 5, we illustrate the accuracy of methods by solving numerical examples ,and a brief in conclusion is given in Section 6 .

Adomian decomposition method (ADM)
The Adomian decomposition method is a numerical method for solving a wide variety of effectively, easily and accurately a large class of linear or nonlinear, ordinary , partial, deterministic , stochastic, differential , integral equations and usually gets the solution in a series form (see for instance [5,7,9,17]).The method is well-suited to physical problems since it does not require unnecessary linearization, perturbation and other restrictive methods and assumptions which may change the problems being solved sometimes seriously.For the aim of illustration of the methodology to the proposed method, using ADM, we begin by considering the differential equation.
(2.4) whereL is the invertible operator of highest-order derivative with respect to x j (1 ≤j≤n) let the order of this operator is k, N is the nonlinear operator, R is a the remainder linear operator and g(x 1 , x 2 , . . ., x n ) is a given function.Let the inverse operator of L is defined as L −1 (.) = ∫∫ . . .

∫ (.).
Where k is defined as the integration level, therefore we can get the solution of Eq. (2.4) as the following (2.5) Where f (x 1 , x 2 , . . ., x n ) is appearing using integrating the source function g(x 1 , x 2 , . . ., x n ) and the given initial conditions, namely And ψ is the solution of the following homogeneous differential equation And the nonlinear operator Nu(x 1 , x 2 , . . ., x n ) can be decomposed of the from (2.9)

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Each term A M is called Adomian's polynomial and it is given by Where λ is a parameter inserted for convenience.At last, the approximated solution of Eq. (2.5) is deduced by the following recurrence relation ) 12) The efficiency of method can be improved by taking further components of the solution series.Finally, we approximation the solution by the truncated series ϕ defined by ) The convergence of the decomposition series have investigated by several authors( For more details see [8,9]).

Application of ADM to Fourier transform
Now the solution series given by ADM is applied to solving the Fourier transform.Consider the Fourier transform (1.2) with initial condition (1.3).The Eq. (1.1) can be written in an operator from Where the differential operator L is L ≡ d dt .It is assumed that the inverse operator L −1 is an integral operator given by L −1= ∫ t 0 (.)ds.According to the previous section.The ADM [7,15,8,9] assumes that the unknown function u(x) can be represented by series of the from.

Main features of FT reviewed
In Fourier analysis there is no need to calculate the transform for every function, instead some fundamental rules are used to relate the new function with the transform at hand.Here we present some well-known properties of the Fourier transforms and prove them in the farm work of ADM.Throughout this section f , f 1 , f 2 , and g, will be functions on R with the corresponding Fourier transforms F, F 1 , F 2 , and G.If we obtain g from some modification of f , there will be a corresponding modification of F, that produces G.Likewise, if we obtain g from some combination of f 1 , and f 2 , then there will be a corresponding combination of F 1 , and F 2 , that produces G.
which exact solution FT is should be ( sinπs πs ) 2 .We have, otherwise.

Conclusion
Adomian decomposition Method,is an elegant method which is easy to use.Its applicability in computing integral transforms such as Laplace transform have been proved by scientists.In the paper, we have verified its efficiency in Fourier transform which is the most important integral transform.The classic calculation of Fourier transform involves a computation of an infinite range of definite integral.In stead, the proposed method based on ADM uses differentiations, so it can be used as an alternative.Using these properties and the presented examples, it would by easy to calculate the Fourier transform of a large number of functions.It would be a good idea to use ADM to evaluate more integral transforms and specially two-dimensional cases of the mentioned ones.