International Journal of Differential Equations
Volume 2011 (2011), Article ID 989065, 10 pages
doi:10.1155/2011/989065
Research Article

Convergence of the New Iterative Method

Sachin Bhalekar1 and Varsha Daftardar-Gejji2

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.

Abstract

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.

1. Introduction

A variety of problems in physics, chemistry, biology, and engineering can be formulated in terms of the nonlinear functional equation 𝑢 = 𝑓 + 𝑁 ( 𝑢 ) , ( 1 . 1 ) where 𝑓 is a given function, and 𝑁 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) [1–3], 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 [8] 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.

2. Preliminaries

Let 𝑋 , 𝑌 be Banach spaces and 𝐹 ∶ 𝑋 → 𝑌 a map. 𝐿 ( 𝑋 , 𝑌 ) denotes the set of all linear maps from 𝑋 to 𝑌 . 𝐿 ( 𝑋 , 𝑌 ) is also a Banach space.

Definition 2.1 (see [9]). 𝐹 is said to be Fréchet differentiable at 𝑥 ∈ 𝑋 if there exists a continuous linear map 𝐴 ∶ 𝑋 → 𝑌 such that 𝐹 ( 𝑥 + ℎ ) − 𝐹 ( 𝑥 ) = 𝐴 ℎ + 𝑤 ( 𝑥 , ℎ ) , ( 2 . 1 ) where l i m ‖ ℎ ‖ → 0 ‖ 𝑤 ( 𝑥 , ℎ ) ‖ ‖ ℎ ‖ = 0 . ( 2 . 2 ) 𝐴 is called the Fréchet derivative of 𝐹 at 𝑥 and is also denoted by 𝐹 ( 𝑥 ) . Its value at ℎ is denoted by 𝐹 ( 𝑥 ) ( ℎ ) .
Note that 𝐹 is a linear map from 𝑋 to 𝐿 ( 𝑋 , 𝑌 ) .

Definition 2.2 (see [9]). 𝐹 is said to be twice differentiable if the map 𝐹 ∶ 𝑋 → 𝐿 ( 𝑋 , 𝑌 ) is Fréchet differentiable. The second derivative of 𝐹 is denoted by 𝐹 and is a linear map from 𝑋 to 𝐿 ( 𝑋 , 𝐿 ( 𝑋 , 𝑌 ) ) .
Note that 𝐿 ( 𝑋 , 𝐿 ( 𝑋 , 𝑌 ) ) is isomorphic to 𝐿 ( 𝑋 × 𝑋 , 𝑌 ) .

Theorem 2.3 (see [9]). The map 𝐹 ( 𝑥 ) ∈ 𝐿 ( 𝑋 2 , 𝑌 ) is symmetric, that is, 𝐹 ( 𝑥 ) ( 𝑥 1 , 𝑥 2 ) = 𝐹 ( 𝑥 ) ( 𝑥 2 , 𝑥 1 ) , 𝑥 1 , 𝑥 2 ∈ 𝑋 .

In this manner, 𝐹 ( 3 ) ( 𝑥 ) , 𝐹 ( 4 ) ( 𝑥 ) , … are inductively defined and 𝐹 ( 𝑛 ) ( 𝑥 ) ∈ 𝐿 ( 𝑋 𝑛 , 𝑌 ) is multilinear and symmetric map.

Theorem 2.4 ([9, Taylor's theorem]). Suppose that 𝐹 ∈ 𝐶 𝑛 ( 𝑈 ) , where 𝑈 is an open subset of 𝑋 containing the line segment from 𝑥 0 to ℎ , then 𝐹 𝑥 0 + ℎ = 𝐹 𝑥 0 + 𝐹 𝑥 0 ( ℎ ) + 1 2 ! 𝐹 𝑥 0 ( ℎ , ℎ ) + ⋯ + 1 ( 𝑛 − 1 ) ! 𝐹 ( 𝑛 − 1 ) 𝑥 0 ( ℎ , … , ℎ ) 𝑛 − 1 t i m e s + 1 ( 𝑛 − 1 ) ! 1 0 ( 1 − t ) n − 1 𝐹 ( 𝑛 ) 𝑥 0 + 𝑡 ℎ ( ℎ , … , ℎ ) 𝑛 t i m e s 𝑑 𝑡 = 𝑛 𝑘 = 0 1 𝑘 ! 𝐹 ( 𝑘 ) 𝑥 0 ( ℎ , … , ℎ ) 𝑘 t i m e s + 𝑞 ( 𝑥 ) , ( 2 . 3 ) where 𝑞 ( 𝑥 ) is such that ‖ 𝑞 ( 𝑥 ) ‖ = 𝑂 ( ‖ 𝑥 ‖ 𝑛 ) .
Since 𝐹 ( 𝑘 ) ( 𝑥 ) is symmetric, we denote ( ℎ , … , ℎ ) 𝑘 t i m e s by ℎ 𝑘 .

3. An Iterative Method

Daftardar-Gejji and Jafari [8] have considered the following functional equation: 𝑢 = 𝑓 + 𝑁 ( 𝑢 ) , ( 3 . 1 ) where 𝑁 is a nonlinear operator from a Banach space 𝐵 → 𝐵 , and 𝑓 is a given element of the Banach space 𝐵 . 𝑢 is assumed to be a solution of (3.1) having the series form 𝑢 = ∞ 𝑖 = 0 𝑢 𝑖 . ( 3 . 2 ) The nonlinear operator 𝑁 is decomposed as 𝑁 ( 𝑢 ) = 𝑁 𝑢 0 + 𝑁 𝑢 0 + 𝑢 1 − 𝑁 𝑢 0 + 𝑁 𝑢 0 + 𝑢 1 + 𝑢 2 − 𝑁 𝑢 0 + 𝑢 1 + ⋯ . ( 3 . 3 ) Let 𝐺 0 = 𝑁 ( 𝑢 0 ) and 𝐺 𝑛 = 𝑁 ⎛ ⎜ ⎝ 𝑛 𝑖 = 0 𝑢 𝑖 ⎞ ⎟ ⎠ − 𝑁 ⎛ ⎜ ⎝ 𝑛 − 1 𝑖 = 0 𝑢 𝑖 ⎞ ⎟ ⎠ , 𝑛 = 1 , 2 , … , ( 3 . 4 ) Then 𝑁 ( 𝑢 ) = ∑ ∞ 𝑖 = 0 𝐺 𝑖 .

Set 𝑢 0 = 𝑓 ( 3 . 5 ) 𝑢 𝑛 = 𝐺 𝑛 − 1 , 𝑛 = 1 , 2 , … , ( 3 . 6 ) then 𝑢 = ∞ 𝑖 = 0 𝑢 𝑖 ( 3 . 7 ) is a solution of (3.1).

3.1. Taylor Series and DJM

Using Theorem 2.4, 𝐺 1 = 𝑁 𝑢 0 + 𝑢 1 − 𝑁 𝑢 0 = 𝑁 𝑢 0 + 𝑁 𝑢 0 𝑢 1 + 𝑁 𝑢 0 𝑢 2 1 2 ! + ⋯ − 𝑁 𝑢 0 = ∞ 𝑘 = 1 𝑁 ( 𝑘 ) 𝑢 0 𝑢 𝑘 1 𝑘 ! , ( 3 . 8 ) 𝐺 2 = 𝑁 𝑢 0 + 𝑢 1 + 𝑢 2 − 𝑁 𝑢 0 + 𝑢 1 = 𝑁 𝑢 0 + 𝑢 1 𝑢 2 + 𝑁 𝑢 0 + 𝑢 1 𝑢 2 2 2 ! + ⋯ = ∞ 𝑗 = 1 ⎡ ⎢ ⎣ ∞ 𝑖 = 0 𝑁 ( 𝑖 + 𝑗 ) 𝑢 0 𝑢 𝑖 1 𝑖 ! ⎤ ⎥ ⎦ 𝑢 𝑗 2 𝑗 ! , ( 3 . 9 ) 𝐺 3 = ∞ 𝑖 3 = 1 ∞ 𝑖 2 = 0 ∞ 𝑖 1 = 0 𝑁 ( 𝑖 1 + 𝑖 2 + 𝑖 3 ) 𝑢 0 𝑢 𝑖 3 3 𝑖 3 ! 𝑢 𝑖 2 2 𝑖 2 ! 𝑢 𝑖 1 1 𝑖 1 ! . ( 3 . 1 0 ) In general, 𝐺 𝑛 = ∞ 𝑖 𝑛 = 1 ∞ 𝑖 𝑛 − 1 = 0 ⋯ ∞ 𝑖 1 = 0 ⎡ ⎢ ⎢ ⎣ 𝑁 ( ∑ 𝑛 𝑘 = 1 𝑖 𝑘 ) 𝑢 0 ⎛ ⎜ ⎜ ⎝ 𝑛 𝑗 = 1 𝑢 𝑖 𝑗 𝑗 𝑖 𝑗 ! ⎞ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎦ . ( 3 . 1 1 ) In view of (3.3)–(3.11), 𝑁 ( 𝑢 ) = 𝐺 0 + 𝐺 1 + 𝐺 2 + 𝐺 3 + ⋯ = 𝑁 𝑢 0 + ∞ 𝑘 = 1 𝑁 ( 𝑘 ) 𝑢 0 𝑢 𝑘 1 𝑘 ! + ∞ 𝑗 = 1 ⎡ ⎢ ⎣ ∞ 𝑖 = 0 𝑁 ( 𝑖 + 𝑗 ) 𝑢 0 𝑢 𝑖 1 𝑖 ! ⎤ ⎥ ⎦ 𝑢 𝑗 2 𝑗 ! + ∞ 𝑖 3 = 1 ∞ 𝑖 2 = 0 ∞ 𝑖 1 = 0 𝑁 ( 𝑖 1 + 𝑖 2 + 𝑖 3 ) 𝑢 0 𝑢 𝑖 3 3 𝑖 3 ! 𝑢 𝑖 2 2 𝑖 2 ! 𝑢 𝑖 1 1 𝑖 1 ! ⋯ = 𝑁 𝑢 0 + 𝑁 𝑢 0 𝑢 1 + 𝑢 2 + 𝑢 3 + ⋯ + 𝑁 𝑢 0 ⎡ ⎢ ⎣ 𝑢 2 1 2 ! + ⎛ ⎜ ⎝ 𝑢 1 𝑢 2 + 𝑢 2 2 2 ! ⎞ ⎟ ⎠ + ⎛ ⎜ ⎝ 𝑢 3 𝑢 2 + 𝑢 3 𝑢 1 + 𝑢 2 3 2 ! ⎞ ⎟ ⎠ + ⋯ ⎤ ⎥ ⎦ + 𝑁 ( 3 ) 𝑢 0 ⎡ ⎢ ⎣ 𝑢 3 1 3 ! + ⎛ ⎜ ⎝ 𝑢 2 𝑢 2 1 2 ! + 𝑢 2 2 2 ! 𝑢 1 + 𝑢 3 2 3 ! ⎞ ⎟ ⎠ + ⋯ ⎤ ⎥ ⎦ + ⋯ . ( 3 . 1 2 ) Using (3.22), 𝑁 ( 𝑢 ) = 𝑁 𝑢 0 + 𝑁 𝑢 0 𝑢 1 + 𝑢 2 + 𝑢 3 + ⋯ + 𝑁 𝑢 0 2 ! 𝑢 1 + 𝑢 2 + 𝑢 3 + ⋯ 2 + 𝑁 ( 3 ) 𝑢 0 3 ! 𝑢 1 + 𝑢 2 + 𝑢 3 + ⋯ 3 + ⋯ . ( 3 . 1 3 ) Equation (3.13) is Taylor series expansion of 𝑁 ( 𝑢 ) around 𝑢 0 . Thus, DJM is equivalent to Taylor series expansion around 𝑢 0 . In Adomian decomposition method (ADM) [1], right hand side of (3.13) is written as 𝑁 𝑢 0 𝐴 0 + 𝑁 𝑢 0 𝑢 1 𝐴 1 + ⎛ ⎜ ⎝ 𝑁 𝑢 0 𝑢 2 1 2 ! + 𝑁 𝑢 0 𝑢 2 ⎞ ⎟ ⎠ 𝐴 2 + ⎛ ⎜ ⎝ 𝑁 ( 3 ) 𝑢 0 𝑢 3 1 3 ! + 𝑁 𝑢 0 𝑢 1 𝑢 2 + 𝑁 𝑢 0 𝑢 3 ⎞ ⎟ ⎠ 𝐴 3 + ⋯ , ( 3 . 1 4 ) where 𝐴 0 , 𝐴 1 , … are Adomian polynomials, and 𝑢 𝑛 + 1 = 𝐴 𝑛 , 𝑛 ≥ 0 .

3.2. Convergence of DJM

Now, we present the condition for convergence of DJM.

Theorem 3.1. If 𝑁 is 𝐶 ( ∞ ) in a neighborhood of 𝑢 0 and ‖ ‖ 𝑁 ( 𝑛 ) 𝑢 0 ‖ ‖ = S u p 𝑁 ( 𝑛 ) 𝑢 0 ℎ 1 , … , ℎ 𝑛 ∶ ‖ ‖ ℎ 𝑖 ‖ ‖ ≤ 1 , 1 ≤ 𝑖 ≤ 𝑛 ≤ 𝐿 , ( 3 . 1 5 ) for any 𝑛 and for some real 𝐿 > 0 and ‖ 𝑢 𝑖 ‖ ≤ 𝑀 < 1 / 𝑒 , 𝑖 = 1 , 2 , … , then the series ∑ ∞ 𝑛 = 0 𝐺 𝑛 is absolutely convergent, and moreover, ‖ ‖ 𝐺 𝑛 ‖ ‖ ≤ 𝐿 𝑀 𝑛 𝑒 𝑛 − 1 ( 𝑒 − 1 ) , 𝑛 = 1 , 2 , … . ( 3 . 1 6 )

Proof. In view of (3.11), ‖ ‖ 𝐺 𝑛 ‖ ‖ ≤ 𝐿 𝑀 𝑛 ∞ 𝑖 𝑛 = 1 ∞ 𝑖 𝑛 − 1 = 0 ⋯ ∞ 𝑖 1 = 0 ⎛ ⎜ ⎝ 𝑛 𝑗 = 1 1 𝑖 𝑗 ! ⎞ ⎟ ⎠ = 𝐿 𝑀 𝑛 𝑒 𝑛 − 1 ( 𝑒 − 1 ) . ( 3 . 1 7 ) Thus, the series ∑ ∞ 𝑛 = 1 ‖ 𝐺 𝑛 ‖ is dominated by the convergent series 𝐿 𝑀 ( 𝑒 − 1 ) ∑ ∞ 𝑛 = 1 ( 𝑀 𝑒 ) 𝑛 − 1 , where 𝑀 < 1 / 𝑒 . Hence, ∑ ∞ 𝑛 = 0 𝐺 𝑛 is absolutely convergent, due to the comparison test.
As it is difficult to show boundedness of ‖ 𝑢 𝑖 ‖ , f o r a l l 𝑖 , a more useful result is proved in the following theorem, where conditions on 𝑁 ( 𝑘 ) ( 𝑢 0 ) are given which are sufficient to guarantee convergence of the series.

Theorem 3.2. If 𝑁 is 𝐶 ( ∞ ) and ‖ 𝑁 ( 𝑛 ) ( 𝑢 0 ) ‖ ≤ 𝑀 ≤ 𝑒 − 1 , for all 𝑛 , then the series ∑ ∞ 𝑛 = 0 𝐺 𝑛 is absolutely convergent.

Proof. Consider the recurrence relation 𝜉 𝑛 = 𝜉 0 e x p 𝜉 𝑛 − 1 , 𝑛 = 1 , 2 , 3 , … , ( 3 . 1 8 ) where 𝜉 0 = 𝑀 . Define 𝜂 𝑛 = 𝜉 𝑛 − 𝜉 𝑛 − 1 , 𝑛 = 1 , 2 , 3 , … . Using (3.6), (3.11), and the hypothesis of Theorem 3.2, we observe that ‖ ‖ 𝐺 𝑛 ‖ ‖ ≤ 𝜂 𝑛 , 𝑛 = 1 , 2 , 3 , ⋯ . ( 3 . 1 9 ) Let 𝜎 𝑛 = 𝑛 𝑖 = 1 𝜂 𝑖 = 𝜉 𝑛 − 𝜉 0 . ( 3 . 2 0 ) Note that 𝜉 0 = 𝑒 − 1 > 0 , 𝜉 1 = 𝜉 0 e x p ( 𝜉 0 ) > 𝜉 0 , a n d 𝜉 2 = 𝜉 0 e x p ( 𝜉 1 ) > 𝜉 0 e x p ( 𝜉 0 ) = 𝜉 1 . In general, 𝜉 𝑛 > 𝜉 𝑛 − 1 > 0 . Hence, ∑ 𝜂 𝑛 is a series of positive real numbers. Note that 0 < 𝜉 0 = 𝑀 = 𝑒 − 1 < 1 , 0 < 𝜉 1 = 𝜉 0 e x p 𝜉 0 < 𝜉 0 𝑒 1 = 𝑒 − 1 𝑒 1 = 1 , 0 < 𝜉 2 = 𝜉 0 e x p 𝜉 1 < 𝜉 0 𝑒 1 = 1 . ( 3 . 2 1 ) In general, 0 < 𝜉 𝑛 < 1 . Hence, 𝜎 𝑛 = 𝜉 𝑛 − 𝜉 0 < 1 . This implies that { 𝜎 𝑛 } ∞ 𝑛 = 1 is bounded above by 1, and hence convergent. Therefore, ∑ 𝐺 𝑛 is absolutely convergent by comparison test.

3.3. Illustrative Example

Consider the nonlinear IVP, 𝑦 ( 𝑡 ) = 1 2 + 1 8 𝑦 2 ( 𝑡 ) , 𝑦 ( 0 ) = 1 2 , 𝑡 ∈ [ 0 , 1 ] . ( 3 . 2 2 ) Integrating (3.22), we get 𝑦 ( 𝑡 ) = 1 2 + 𝑡 2 + 1 8 𝑡 0 𝑦 2 ( 𝑠 ) 𝑑 𝑠 . ( 3 . 2 3 ) Note that 𝑦 0 = ( 1 + 𝑡 ) / 2 and 𝑁 ( 𝑦 ) ( 𝑡 ) = ( 1 / 8 ) ∫ 𝑡 0 𝑦 2 ( 𝑠 ) 𝑑 𝑠 then [10] 𝑁 ( 𝑦 ) 𝑧 ( 𝑡 ) = 1 8 𝑡 0 𝜕 𝜕 𝑦 𝑦 2 𝑧 𝑑 𝑠 ( 3 . 2 4 ) = 1 8 𝑡 0 2 𝑦 ( 𝑠 ) 𝑧 ( 𝑠 ) 𝑑 𝑠 . ( 3 . 2 5 ) Similarly, ( 𝑁 ( 𝑦 ) ( 𝑧 1 , 𝑧 2 ) ( 𝑡 ) = ( 𝑡 / 4 ) 𝑧 1 ( 𝑡 ) 𝑧 2 ( 𝑡 ) and 𝑁 ( 𝑘 ) ( 𝑦 ) = 0 for 𝑘 ≥ 3 . Since 𝑡 ∈ [ 0 , 1 ] , ‖ ‖ 𝑁 𝑦 0 ( 𝑡 ) ‖ ‖ = ‖ ‖ ‖ 1 8 𝑡 0 ( 1 + 𝑠 ) 2 4 𝑑 𝑠 ‖ ‖ ‖ ≤ 7 6 4 < 1 𝑒 , ‖ ‖ 𝑁 𝑦 0 ( 𝑡 ) ‖ ‖ = ‖ ‖ ‖ 1 4 𝑡 0 ( 1 + 𝑠 ) 2 𝑑 𝑠 ‖ ‖ ‖ ≤ 3 1 6 < 1 𝑒 , ‖ ‖ 𝑁 𝑦 0 ( 𝑡 ) ‖ ‖ = ‖ ‖ ‖ 𝑡 4 ‖ ‖ ‖ ≤ 1 4 < 1 𝑒 , ‖ ‖ 𝑁 ( 𝑘 ) 𝑦 0 ( 𝑡 ) ‖ ‖ = 0 , 𝑘 ≥ 3 . ( 3 . 2 6 ) As the conditions of Theorem 3.2 are satisfied, the solution series 𝑦 = ∑ 𝑦 𝑖 obtained by DJM is convergent for 𝑡 ∈ [ 0 , 1 ] . The terms of the series are given by 𝑦 1 = 𝑁 𝑦 0 = 1 3 2 𝑡 + 𝑡 2 + 𝑡 3 3 , 𝑦 2 = 𝑁 𝑦 0 + 𝑦 1 − 𝑁 𝑦 0 = 𝑡 2 5 1 2 + 6 5 𝑡 3 2 4 5 7 6 + 6 7 𝑡 4 4 9 1 5 2 + 3 7 𝑡 5 1 2 2 8 8 0 + 𝑡 6 7 3 7 2 8 + 𝑡 7 5 1 6 0 9 6 , ( 3 . 2 7 ) and so on. In Figure 1, a seven-term approximate solution obtained by DJM (dashed line) is compared with exact solution (solid line) [ 8 s i n ( 𝑡 / 4 ) + 2 c o s ( 𝑡 / 4 ) ] / [ 4 c o s ( 𝑡 / 4 ) − s i n ( 𝑡 / 4 ) ] .

989065.fig.001
Figure 1: Solution of (3.22): dashed = DJM, solid = exact.
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 𝑓 ( 𝑥 ) = 0 . Further, they have shown that this method has fourth-order convergence [13]. Some new methods [14, 15] are proposed by these authors using DJM.

Mohyud-Din et al. [16] solved Hirota-Satsuma coupled KdV system 𝑢 𝑡 − 1 2 𝑢 𝑥 𝑥 𝑥 + 3 𝑢 𝑢 𝑥 − 3 ( 𝑣 𝑤 ) 𝑥 = 0 , 𝑣 𝑡 − 𝑣 𝑥 𝑥 𝑥 − 3 𝑢 𝑣 𝑥 = 0 , 𝑤 𝑡 + 𝑤 𝑥 𝑥 𝑥 − 3 𝑢 𝑤 𝑥 = 0 , ( 3 . 2 8 ) using DJM. These authors [17] also have applied DJM in solutions of some fifth-order boundary value problems 𝑦 ( 𝑣 ) ( 𝑥 ) = 𝑔 ( 𝑥 ) 𝑦 + 𝑞 ( 𝑥 ) , ( 3 . 2 9 ) with boundary conditions 𝑦 ( 𝑎 ) = 𝐴 1 , 𝑦 ( 𝑎 ) = 𝐴 2 , 𝑦 ( 𝑎 ) = 𝐴 3 , 𝑦 ( 𝑏 ) = 𝐵 1 , 𝑦 ( 𝑏 ) = 𝐵 2 . Noor and Din [18] have used DJM to solve Helmholtz equations.

A variety of fractional-order differential equations such as diffusion-wave equations [19], boundary value problems [20], partial differential equations [21, 22], and evolution equations [23] are solved successfully by Daftardar-Gejji and Bhalekar using DJM. Further Jafari et al. [24] have solved nonlinear diffusion-wave equations using DJM. Fard and Sanchooli [25] have used DJM for solving linear fuzzy Fredholm integral equations.

Recently, Srivastava and Rai [26] 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.

Acknowledgment

V. Daftardar-Gejji acknowledges the Department of Science and Technology, New Delhi, India for the Research Grants (Project no. SR/S2/HEP-024/2009).

References

  1. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. View at Zentralblatt MATH
  2. G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 508–518, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Y. Cherruault, “Convergence of Adomian's method,” Kybernetes, vol. 18, no. 2, pp. 31–38, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. K. Abbaoui and Y. Cherruault, “New ideas for proving convergence of decomposition methods,” Computers & Mathematics with Applications, vol. 29, no. 7, pp. 103–108, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H.-W. Choi and J.-G. Shin, “Symbolic implementation of the algorithm for calculating Adomian polynomials,” Applied Mathematics and Computation, vol. 146, no. 1, pp. 257–271, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. H. Jafari and V. Daftardar-Gejji, “Solving a system of nonlinear fractional differential equations using Adomian decomposition,” Journal of Computational and Applied Mathematics, vol. 196, no. 2, pp. 644–651, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. V. Daftardar-Gejji and H. Jafari, “An iterative method for solving nonlinear functional equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 753–763, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. C. Joshi and R. K. Bose, Some Topics in Nonlinear Functional Analysis, A Halsted Press Book, John Wiley & Sons, New York, NY, USA, 1985.
  10. A. A. M. Cuyt, “Padé-approximants in operator theory for the solution of nonlinear differential and integral equations,” Computers & Mathematics with Applications, vol. 8, no. 6, pp. 445–466, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. A. Noor and K. I. Noor, “Three-step iterative methods for nonlinear equations,” Applied Mathematics and Computation, vol. 183, no. 1, pp. 322–327, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. A. Noor, K. I. Noor, S. T. Mohyud-Din, and A. Shabbir, “An iterative method with cubic convergence for nonlinear equations,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 1249–1255, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. K. I. Noor and M. A. Noor, “Iterative methods with fourth-order convergence for nonlinear equations,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 221–227, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  14. M. A. Noor, “New iterative schemes for nonlinear equations,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 937–943, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. A. Noor, K. I. Noor, E. Al-Said, and M. Waseem, “Some new iterative methods for nonlinear equations,” Mathematical Problems in Engineering, vol. 2010, Article ID 198943, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. S. T. Mohyud-Din, A. Yildirim, and S. M. M. Hosseini, “Numerical comparison of methods for Hirota-Satsuma model,” Applications and Applied Mathematics, vol. 5, no. 10, pp. 1554–1563, 2010. View at Zentralblatt MATH
  17. S. T. M. Din, A. Yildirim, and M. M. Hosseini, “An iterative algorithm for fifth-order boundary value problems,” World Applied Sciences Journal, vol. 8, no. 5, pp. 531–535, 2010.
  18. M. A. Noor and S. T. M. Din, “An iterative method for solving Helmholtz equations,” Arab Journal of Mathematics and Mathematical Sciences, vol. 1, no. 1, pp. 13–18, 2007.
  19. V. Daftardar-Gejji and S. Bhalekar, “Solving fractional diffusion-wave equations using a new iterative method,” Fractional Calculus & Applied Analysis, vol. 11, no. 2, pp. 193–202, 2008. View at Zentralblatt MATH
  20. V. Daftardar-Gejji and S. Bhalekar, “Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1801–1809, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. S. Bhalekar and V. Daftardar-Gejji, “New iterative method: application to partial differential equations,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 778–783, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. V. Daftardar-Gejji and S. Bhalekar, “An Iterative method for solving fractional differential equations,” Proceedings in Applied Mathematics and Mechanics, vol. 7, no. 1, pp. 2050017–2050018, 2008. View at Publisher · View at Google Scholar
  23. S. Bhalekar and V. Daftardar-Gejji, “Solving evolution equations using a new iterative method,” Numerical Methods for Partial Differential Equations, vol. 26, no. 4, pp. 906–916, 2010. View at Zentralblatt MATH
  24. H. Jafari, S. Seifi, A. Alipoor, and M. Zabihi, “An Iterative Method for solving linear and nonlinear fractional diffusion-wave equation,” International e-Journal of Numerical Analysis and Related Topics, vol. 3, pp. 20–32, 2009.
  25. O. S. Fard and M. Sanchooli, “Two successive schemes for numerical solution of linear fuzzy Fredholm integral equations of the second kind,” Australian Journal of Basic and Applied Sciences, vol. 4, no. 5, pp. 817–825, 2010. View at Scopus
  26. V. Srivastava and K. N. Rai, “A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 616–624, 2010. View at Zentralblatt MATH