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

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.

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) [13], 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 𝐹𝑋𝑌 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 𝑥𝑋 if there exists a continuous linear map 𝐴𝑋𝑌 such that 𝐹(𝑥+)𝐹(𝑥)=𝐴+𝑤(𝑥,),(2.1) where lim0𝑤(𝑥,)=0.(2.2)𝐴 is called the Fréchet derivative of 𝐹 at 𝑥 and is also denoted by 𝐹(𝑥). Its value at is denoted by 𝐹(𝑥)().
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 𝐹𝑋𝐿(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(𝑋2,𝑌) is symmetric, that is, 𝐹(𝑥)(𝑥1,𝑥2)=𝐹(𝑥)(𝑥2,𝑥1), 𝑥1,𝑥2𝑋.

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

Theorem 2.4 ([9, Taylor's theorem]). Suppose that 𝐹𝐶𝑛(𝑈), where 𝑈 is an open subset of 𝑋 containing the line segment from 𝑥0 to , then 𝐹𝑥0+=𝐹𝑥0+𝐹𝑥0()+12!𝐹𝑥0(,)++1(𝑛1)!𝐹(𝑛1)𝑥0(,,)𝑛1times+1(𝑛1)!10(1t)n1𝐹(𝑛)𝑥0+𝑡(,,)𝑛times𝑑𝑡=𝑛𝑘=01𝑘!𝐹(𝑘)𝑥0(,,)𝑘times+𝑞(𝑥),(2.3) where 𝑞(𝑥) is such that 𝑞(𝑥)=𝑂(𝑥𝑛).
Since 𝐹(𝑘)(𝑥) is symmetric, we denote (,,)𝑘times by 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 𝐵𝐵, and 𝑓 is a given element of the Banach space 𝐵. 𝑢 is assumed to be a solution of (3.1) having the series form𝑢=𝑖=0𝑢𝑖.(3.2) The nonlinear operator 𝑁 is decomposed as 𝑁(𝑢)=𝑁𝑢0+𝑁𝑢0+𝑢1𝑁𝑢0+𝑁𝑢0+𝑢1+𝑢2𝑁𝑢0+𝑢1+.(3.3) Let 𝐺0=𝑁(𝑢0) and 𝐺𝑛=𝑁𝑛𝑖=0𝑢𝑖𝑁𝑛1𝑖=0𝑢𝑖,𝑛=1,2,,(3.4) Then 𝑁(𝑢)=𝑖=0𝐺𝑖.

Set 𝑢0=𝑓(3.5)𝑢𝑛=𝐺𝑛1,𝑛=1,2,,(3.6) then 𝑢=𝑖=0𝑢𝑖(3.7) is a solution of (3.1).

3.1. Taylor Series and DJM

Using Theorem 2.4,𝐺1=𝑁𝑢0+𝑢1𝑁𝑢0=𝑁𝑢0+𝑁𝑢0𝑢1+𝑁𝑢0𝑢212!+𝑁𝑢0=𝑘=1𝑁(𝑘)𝑢0𝑢𝑘1𝑘!,(3.8)𝐺2=𝑁𝑢0+𝑢1+𝑢2𝑁𝑢0+𝑢1=𝑁𝑢0+𝑢1𝑢2+𝑁𝑢0+𝑢1𝑢222!+=𝑗=1𝑖=0𝑁(𝑖+𝑗)𝑢0𝑢𝑖1𝑖!𝑢𝑗2𝑗!,(3.9)𝐺3=𝑖3=1𝑖2=0𝑖1=0𝑁(𝑖1+𝑖2+𝑖3)𝑢0𝑢𝑖33𝑖3!𝑢𝑖22𝑖2!𝑢𝑖11𝑖1!.(3.10) In general, 𝐺𝑛=𝑖𝑛=1𝑖𝑛1=0𝑖1=0𝑁(𝑛𝑘=1𝑖𝑘)𝑢0𝑛𝑗=1𝑢𝑖𝑗𝑗𝑖𝑗!.(3.11) In view of (3.3)–(3.11),𝑁(𝑢)=𝐺0+𝐺1+𝐺2+𝐺3+=𝑁𝑢0+𝑘=1𝑁(𝑘)𝑢0𝑢𝑘1𝑘!+𝑗=1𝑖=0𝑁(𝑖+𝑗)𝑢0𝑢𝑖1𝑖!𝑢𝑗2𝑗!+𝑖3=1𝑖2=0𝑖1=0𝑁(𝑖1+𝑖2+𝑖3)𝑢0𝑢𝑖33𝑖3!𝑢𝑖22𝑖2!𝑢𝑖11𝑖1!=𝑁𝑢0+𝑁𝑢0𝑢1+𝑢2+𝑢3++𝑁𝑢0𝑢212!+𝑢1𝑢2+𝑢222!+𝑢3𝑢2+𝑢3𝑢1+𝑢232!++𝑁(3)𝑢0𝑢313!+𝑢2𝑢212!+𝑢222!𝑢1+𝑢323!++.(3.12) Using (3.22), 𝑁(𝑢)=𝑁𝑢0+𝑁𝑢0𝑢1+𝑢2+𝑢3++𝑁𝑢02!𝑢1+𝑢2+𝑢3+2+𝑁(3)𝑢03!𝑢1+𝑢2+𝑢3+3+.(3.13) Equation (3.13) is Taylor series expansion of 𝑁(𝑢) around 𝑢0. Thus, DJM is equivalent to Taylor series expansion around 𝑢0. In Adomian decomposition method (ADM) [1], right hand side of (3.13) is written as𝑁𝑢0𝐴0+𝑁𝑢0𝑢1𝐴1+𝑁𝑢0𝑢212!+𝑁𝑢0𝑢2𝐴2+𝑁(3)𝑢0𝑢313!+𝑁𝑢0𝑢1𝑢2+𝑁𝑢0𝑢3𝐴3+,(3.14) where 𝐴0,𝐴1, are Adomian polynomials, and 𝑢𝑛+1=𝐴𝑛,𝑛0.

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 𝑢0 and 𝑁(𝑛)𝑢0=Sup𝑁(𝑛)𝑢01,,𝑛𝑖1,1𝑖𝑛𝐿,(3.15) for any 𝑛 and for some real 𝐿>0 and 𝑢𝑖𝑀<1/𝑒, 𝑖=1,2,, then the series 𝑛=0𝐺𝑛 is absolutely convergent, and moreover, 𝐺𝑛𝐿𝑀𝑛𝑒𝑛1(𝑒1),𝑛=1,2,.(3.16)

Proof. In view of (3.11), 𝐺𝑛𝐿𝑀𝑛𝑖𝑛=1𝑖𝑛1=0𝑖1=0𝑛𝑗=11𝑖𝑗!=𝐿𝑀𝑛𝑒𝑛1(𝑒1).(3.17) Thus, the series 𝑛=1𝐺𝑛 is dominated by the convergent series 𝐿𝑀(𝑒1)𝑛=1(𝑀𝑒)𝑛1, where 𝑀<1/𝑒. Hence, 𝑛=0𝐺𝑛 is absolutely convergent, due to the comparison test.
As it is difficult to show boundedness of 𝑢𝑖,forall𝑖, a more useful result is proved in the following theorem, where conditions on 𝑁(𝑘)(𝑢0) are given which are sufficient to guarantee convergence of the series.

Theorem 3.2. If 𝑁 is 𝐶() and 𝑁(𝑛)(𝑢0)𝑀𝑒1, for all 𝑛, then the series 𝑛=0𝐺𝑛 is absolutely convergent.

Proof. Consider the recurrence relation 𝜉𝑛=𝜉0exp𝜉𝑛1,𝑛=1,2,3,,(3.18) where 𝜉0=𝑀. Define 𝜂𝑛=𝜉𝑛𝜉𝑛1,𝑛=1,2,3,. Using (3.6), (3.11), and the hypothesis of Theorem 3.2, we observe that 𝐺𝑛𝜂𝑛,𝑛=1,2,3,.(3.19) Let 𝜎𝑛=𝑛𝑖=1𝜂𝑖=𝜉𝑛𝜉0.(3.20) Note that 𝜉0=𝑒1>0,𝜉1=𝜉0exp(𝜉0)>𝜉0,and𝜉2=𝜉0exp(𝜉1)>𝜉0exp(𝜉0)=𝜉1. In general, 𝜉𝑛>𝜉𝑛1>0. Hence, 𝜂𝑛 is a series of positive real numbers. Note that 0<𝜉0=𝑀=𝑒1<1,0<𝜉1=𝜉0exp𝜉0<𝜉0𝑒1=𝑒1𝑒1=1,0<𝜉2=𝜉0exp𝜉1<𝜉0𝑒1=1.(3.21) In general, 0<𝜉𝑛<1. Hence, 𝜎𝑛=𝜉𝑛𝜉0<1. This implies that {𝜎𝑛}𝑛=1 is bounded above by 1, and hence convergent. Therefore, 𝐺𝑛 is absolutely convergent by comparison test.

3.3. Illustrative Example

Consider the nonlinear IVP,𝑦(𝑡)=12+18𝑦2(𝑡),𝑦(0)=12,𝑡[0,1].(3.22) Integrating (3.22), we get𝑦(𝑡)=12+𝑡2+18𝑡0𝑦2(𝑠)𝑑𝑠.(3.23) Note that 𝑦0=(1+𝑡)/2 and 𝑁(𝑦)(𝑡)=(1/8)𝑡0𝑦2(𝑠)𝑑𝑠 then [10] 𝑁(𝑦)𝑧(𝑡)=18𝑡0𝜕𝜕𝑦𝑦2𝑧𝑑𝑠(3.24)=18𝑡02𝑦(𝑠)𝑧(𝑠)𝑑𝑠.(3.25) Similarly, (𝑁(𝑦)(𝑧1,𝑧2)(𝑡)=(𝑡/4)𝑧1(𝑡)𝑧2(𝑡) and 𝑁(𝑘)(𝑦)=0 for 𝑘3. Since 𝑡[0,1],𝑁𝑦0(𝑡)=18𝑡0(1+𝑠)24𝑑𝑠764<1𝑒,𝑁𝑦0(𝑡)=14𝑡0(1+𝑠)2𝑑𝑠316<1𝑒,𝑁𝑦0(𝑡)=𝑡414<1𝑒,𝑁(𝑘)𝑦0(𝑡)=0,𝑘3.(3.26) As the conditions of Theorem 3.2 are satisfied, the solution series 𝑦=𝑦𝑖 obtained by DJM is convergent for 𝑡[0,1]. The terms of the series are given by 𝑦1=𝑁𝑦0=132𝑡+𝑡2+𝑡33,𝑦2=𝑁𝑦0+𝑦1𝑁𝑦0=𝑡2512+65𝑡324576+67𝑡449152+37𝑡5122880+𝑡673728+𝑡7516096,(3.27) and so on. In Figure 1, a seven-term approximate solution obtained by DJM (dashed line) is compared with exact solution (solid line) [8sin(𝑡/4)+2cos(𝑡/4)]/[4cos(𝑡/4)sin(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 𝑢𝑡12𝑢𝑥𝑥𝑥+3𝑢𝑢𝑥3(𝑣𝑤)𝑥=0,𝑣𝑡𝑣𝑥𝑥𝑥3𝑢𝑣𝑥=0,𝑤𝑡+𝑤𝑥𝑥𝑥3𝑢𝑤𝑥=0,(3.28) 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),(3.29) with boundary conditions 𝑦(𝑎)=𝐴1,𝑦(𝑎)=𝐴2,𝑦(𝑎)=𝐴3,𝑦(𝑏)=𝐵1,𝑦(𝑏)=𝐵2. 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).

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