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International Journal of Differential Equations
Volume 2011 (2011), Article ID 989065, 10 pages
Convergence of the New Iterative Method
1Department of Mathematics, Shivaji University, Kolhapur 416004, India
2Department of Mathematics, University of Pune, Pune 411007, India
Received 7 May 2011; Accepted 21 August 2011
Academic Editor: Dexing Kong
Copyright © 2011 Sachin Bhalekar and Varsha Daftardar-Gejji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A new iterative method introduced by Daftardar-Gejji and Jafari (2006) (DJ Method) is an efficient technique to solve nonlinear functional equations. In the present paper, sufficiency conditions for convergence of DJM have been presented. Further equivalence of DJM and Adomian decomposition method is established.
A variety of problems in physics, chemistry, biology, and engineering can be formulated in terms of the nonlinear functional equation where is a given function, and is the nonlinear operator. Equation (1.1) represents integral equations, ordinary differential equations (ODEs), partial differential equations (PDEs), differential equations involving fractional order, systems of ODE/PDE, and so on. Various methods such as Laplace and Fourier transform and Green's function method have been used to solve linear equations. For solving nonlinear equations, however, one has to resort to numerical/iterative methods. Adomian decomposition method (ADM) has proved to be a useful tool for solving functional equation (1.1) [1####^~^~^~^~^~^####x2013;3], since it offers certain advantages over numerical methods. This method yields solutions in the form of rapidly converging infinite series which can be effectively approximated by calculating only first few terms. In the last two decades, extensive work has been done using ADM as it provides analytical approximate solutions for nonlinear equations without linearization, perturbation, or discretization. Though Adomian's technique is simple in principle, it involves tedious calculations of Adomian polynomials [4, 5]. Researchers have explored symbolic computational packages such as Mathematica for finding Adomian polynomials [6, 7].
As a pursuit of this, Daftardar-Gejji and Jafari  have introduced a new decomposition method (DJM) to solve (1.1) which is simple and easy to implement. It is economical in terms of computer power/memory and does not involve tedious calculations such as Adomian polynomials. DJM has been employed successfully to solve a variety of problems.
Present paper analyzes convergence of DJM in detail and establishes its equivalence to ADM.
Let , be Banach spaces and a map. denotes the set of all linear maps from to . is also a Banach space.
Definition 2.1 (see ). is said to be Fr####^~^~^~^~^~^####xe9;chet differentiable at if there exists a continuous linear map such that
is called the Fr####^~^~^~^~^~^####xe9;chet derivative of at and is also denoted by . Its value at is denoted by .
Note that is a linear map from to .
Definition 2.2 (see ). is said to be twice differentiable if the map is Fr####^~^~^~^~^~^####xe9;chet differentiable. The second derivative of is denoted by and is a linear map from to .
Note that is isomorphic to .
Theorem 2.3 (see ). The map is symmetric, that is, , .
In this manner, are inductively defined and is multilinear and symmetric map.
Theorem 2.4 ([9, Taylor's theorem]). Suppose that , where is an open subset of containing the line segment from to , then
where is such that .
Since is symmetric, we denote by .
3. An Iterative Method
Daftardar-Gejji and Jafari  have considered the following functional equation: where is a nonlinear operator from a Banach space , and is a given element of the Banach space . is assumed to be a solution of (3.1) having the series form The nonlinear operator is decomposed as Let and Then .
Set then is a solution of (3.1).
3.1. Taylor Series and DJM
Using Theorem 2.4, In general, In view of (3.3)####^~^~^~^~^~^####x2013;(3.11), Using (3.22), Equation (3.13) is Taylor series expansion of around . Thus, DJM is equivalent to Taylor series expansion around . In Adomian decomposition method (ADM) , right hand side of (3.13) is written as where are Adomian polynomials, and .
3.2. Convergence of DJM
Now, we present the condition for convergence of DJM.
Theorem 3.1. If is in a neighborhood of and for any and for some real and , , then the series is absolutely convergent, and moreover,
Proof. In view of (3.11),
Thus, the series is dominated by the convergent series , where . Hence, is absolutely convergent, due to the comparison test.
As it is difficult to show boundedness of , a more useful result is proved in the following theorem, where conditions on are given which are sufficient to guarantee convergence of the series.
Theorem 3.2. If is and , for all , then the series is absolutely convergent.
Proof. Consider the recurrence relation where . Define . Using (3.6), (3.11), and the hypothesis of Theorem 3.2, we observe that Let Note that . In general, . Hence, is a series of positive real numbers. Note that In general, . Hence, . This implies that is bounded above by 1, and hence convergent. Therefore, is absolutely convergent by comparison test.
3.3. Illustrative Example
Consider the nonlinear IVP, Integrating (3.22), we get Note that and then  Similarly, and for . Since , As the conditions of Theorem 3.2 are satisfied, the solution series obtained by DJM is convergent for . The terms of the series are given by and so on. In Figure 1, a seven-term approximate solution obtained by DJM (dashed line) is compared with exact solution (solid line) .
3.4. Applications of DJM
DJM has been further explored by many researchers. Several numerical methods with higher-order convergence can be generated using DJM. M. A. Noor and K. I. Noor [11, 12] have developed a three-step predictor-corrector method for solving nonlinear equation . Further, they have shown that this method has fourth-order convergence . Some new methods [14, 15] are proposed by these authors using DJM.
Mohyud-Din et al.  solved Hirota-Satsuma coupled KdV system using DJM. These authors  also have applied DJM in solutions of some fifth-order boundary value problems with boundary conditions . Noor and Din  have used DJM to solve Helmholtz equations.
A variety of fractional-order differential equations such as diffusion-wave equations , boundary value problems , partial differential equations [21, 22], and evolution equations  are solved successfully by Daftardar-Gejji and Bhalekar using DJM. Further Jafari et al.  have solved nonlinear diffusion-wave equations using DJM. Fard and Sanchooli  have used DJM for solving linear fuzzy Fredholm integral equations.
Recently, Srivastava and Rai  have proposed a new mathematical model for oxygen delivery through a capillary to tissues in terms of multiterm fractional diffusion equation. They have solved the multiterm fractional diffusion equation using DJM and ADM and have shown that the results are in perfect agreement.
V. Daftardar-Gejji acknowledges the Department of Science and Technology, New Delhi, India for the Research Grants (Project no. SR/S2/HEP-024/2009).
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