Convergence of the New Iterative Method

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)=𝐴ℎ+𝑤(𝑥,ℎ),(2.1) where lim_{‖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(𝑋²,𝑌) is symmetric, that is, 𝐹^(𝑥)(𝑥₁,𝑥₂)=𝐹^(𝑥)(𝑥₂,𝑥₁), 𝑥₁,𝑥₂∈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 𝑥₀ to ℎ, then 𝐹𝑥₀+ℎ=𝐹𝑥₀+𝐹^𝑥₀(ℎ)+12!𝐹^𝑥₀(ℎ,ℎ)+⋯+1(𝑛−1)!𝐹^(𝑛−1)𝑥₀(ℎ,…,ℎ)_{𝑛−1times}+1(𝑛−1)!¹₀(1−t)ⁿ⁻¹𝐹^(𝑛)𝑥₀+𝑡ℎ(ℎ,…,ℎ)_{𝑛times}𝑑𝑡=^𝑛_𝑘=01𝑘!𝐹^(𝑘)𝑥₀(ℎ,…,ℎ)_{𝑘times}+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)_{𝑘times} 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 form𝑢=^∞_𝑖=0𝑢_𝑖.(3.2) The nonlinear operator 𝑁 is decomposed as 𝑁(𝑢)=𝑁𝑢₀+𝑁𝑢₀+𝑢₁−𝑁𝑢₀+𝑁𝑢₀+𝑢₁+𝑢₂−𝑁𝑢₀+𝑢₁+⋯.(3.3) Let 𝐺₀=𝑁(𝑢₀) and 𝐺_𝑛=𝑁⎛⎜⎝^𝑛_𝑖=0𝑢_𝑖⎞⎟⎠−𝑁⎛⎜⎝^𝑛−1_𝑖=0𝑢_𝑖⎞⎟⎠,𝑛=1,2,…,(3.4) Then 𝑁(𝑢)=∑^∞_𝑖=0𝐺_𝑖.

Set 𝑢₀=𝑓(3.5)𝑢_𝑛=𝐺_𝑛−1,𝑛=1,2,…,(3.6) then 𝑢=^∞_𝑖=0𝑢_𝑖(3.7) is a solution of (3.1).

3.1. Taylor Series and DJM

Using Theorem 2.4,𝐺₁=𝑁𝑢₀+𝑢₁−𝑁𝑢₀=𝑁𝑢₀+𝑁^𝑢₀𝑢₁+𝑁^𝑢₀𝑢²₁2!+⋯−𝑁𝑢₀=^∞_𝑘=1𝑁^(𝑘)𝑢₀𝑢^𝑘₁𝑘!,(3.8)𝐺₂=𝑁𝑢₀+𝑢₁+𝑢₂−𝑁𝑢₀+𝑢₁=𝑁^𝑢₀+𝑢₁𝑢₂+𝑁^𝑢₀+𝑢₁𝑢²₂2!+⋯=^∞_𝑗=1⎡⎢⎣^∞_𝑖=0𝑁^(𝑖+𝑗)𝑢₀𝑢^𝑖₁𝑖!⎤⎥⎦𝑢^𝑗₂𝑗!,(3.9)𝐺₃=^∞_𝑖3=1^∞_𝑖2=0^∞_𝑖1=0𝑁^{(𝑖1+𝑖2+𝑖3)}𝑢₀𝑢^𝑖3₃𝑖₃!𝑢^𝑖2₂𝑖₂!𝑢^𝑖1₁𝑖₁!.(3.10) In general, 𝐺_𝑛=^∞_𝑖𝑛=1^∞_{𝑖𝑛−1=0}⋯^∞_𝑖1=0⎡⎢⎢⎣𝑁^{(∑𝑛𝑘=1𝑖𝑘)}𝑢₀⎛⎜⎜⎝^𝑛_𝑗=1𝑢^𝑖𝑗_𝑗𝑖_𝑗!⎞⎟⎟⎠⎤⎥⎥⎦.(3.11) In view of (3.3)–(3.11),𝑁(𝑢)=𝐺₀+𝐺₁+𝐺₂+𝐺₃+⋯=𝑁𝑢₀+^∞_𝑘=1𝑁^(𝑘)𝑢₀𝑢^𝑘₁𝑘!+^∞_𝑗=1⎡⎢⎣^∞_𝑖=0𝑁^(𝑖+𝑗)𝑢₀𝑢^𝑖₁𝑖!⎤⎥⎦𝑢^𝑗₂𝑗!+^∞_𝑖3=1^∞_𝑖2=0^∞_𝑖1=0𝑁^{(𝑖1+𝑖2+𝑖3)}𝑢₀𝑢^𝑖3₃𝑖₃!𝑢^𝑖2₂𝑖₂!𝑢^𝑖1₁𝑖₁!⋯=𝑁𝑢₀+𝑁^𝑢₀𝑢₁+𝑢₂+𝑢₃+⋯+𝑁^𝑢₀⎡⎢⎣𝑢²₁2!+⎛⎜⎝𝑢₁𝑢₂+𝑢²₂2!⎞⎟⎠+⎛⎜⎝𝑢₃𝑢₂+𝑢₃𝑢₁+𝑢²₃2!⎞⎟⎠+⋯⎤⎥⎦+𝑁⁽³⁾𝑢₀⎡⎢⎣𝑢³₁3!+⎛⎜⎝𝑢₂𝑢²₁2!+𝑢²₂2!𝑢₁+𝑢³₂3!⎞⎟⎠+⋯⎤⎥⎦+⋯.(3.12) Using (3.22), 𝑁(𝑢)=𝑁𝑢₀+𝑁^𝑢₀𝑢₁+𝑢₂+𝑢₃+⋯+𝑁^𝑢₀2!𝑢₁+𝑢₂+𝑢₃+⋯²+𝑁⁽³⁾𝑢₀3!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 as𝑁𝑢₀_{𝐴0}+𝑁^𝑢₀𝑢_{1𝐴1}+⎛⎜⎝𝑁^𝑢₀𝑢²₁2!+𝑁^𝑢₀𝑢₂⎞⎟_{𝐴2}⎠+⎛⎜⎝𝑁⁽³⁾𝑢₀𝑢³₁3!+𝑁^𝑢₀𝑢₁𝑢₂+𝑁^𝑢₀𝑢₃⎞⎟_{𝐴3}⎠+⋯,(3.14) where 𝐴₀,𝐴₁,… 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 𝑢₀ and ‖‖𝑁^(𝑛)𝑢₀‖‖=Sup𝑁^(𝑛)𝑢₀ℎ₁,…,ℎ_𝑛∶‖‖ℎ_𝑖‖‖≤1,1≤𝑖≤𝑛≤𝐿,(3.15) for any 𝑛 and for some real 𝐿>0 and ‖𝑢_𝑖‖≤𝑀<1/𝑒, 𝑖=1,2,…, then the series ∑^∞_𝑛=0𝐺_𝑛 is absolutely convergent, and moreover, ‖‖𝐺_𝑛‖‖≤𝐿𝑀^𝑛𝑒^𝑛−1(𝑒−1),𝑛=1,2,….(3.16)

Proof. In view of (3.11), ‖‖𝐺_𝑛‖‖≤𝐿𝑀^𝑛∞_𝑖𝑛=1^∞_{𝑖𝑛−1=0}⋯^∞_𝑖1=0⎛⎜⎝^𝑛_𝑗=11𝑖_𝑗!⎞⎟⎠=𝐿𝑀^𝑛𝑒^𝑛−1(𝑒−1).(3.17) Thus, the series ∑^∞_𝑛=1‖𝐺_𝑛‖ is dominated by the convergent series 𝐿𝑀(𝑒−1)∑^∞_𝑛=1(𝑀𝑒)^𝑛−1, where 𝑀<1/𝑒. Hence, ∑^∞_𝑛=0𝐺_𝑛 is absolutely convergent, due to the comparison test.
As it is difficult to show boundedness of ‖𝑢_𝑖‖,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)‖≤𝑀≤𝑒⁻¹, for all 𝑛, then the series ∑^∞_𝑛=0𝐺_𝑛 is absolutely convergent.

Proof. Consider the recurrence relation 𝜉_𝑛=𝜉₀exp𝜉_𝑛−1,𝑛=1,2,3,…,(3.18) where 𝜉₀=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,⋯.(3.19) Let 𝜎_𝑛=^𝑛_𝑖=1𝜂_𝑖=𝜉_𝑛−𝜉₀.(3.20) Note that 𝜉₀=𝑒⁻¹>0,𝜉₁=𝜉₀exp(𝜉₀)>𝜉₀,and𝜉₂=𝜉₀exp(𝜉₁)>𝜉₀exp(𝜉₀)=𝜉₁. In general, 𝜉_𝑛>𝜉_𝑛−1>0. Hence, ∑𝜂_𝑛 is a series of positive real numbers. Note that 0<𝜉₀=𝑀=𝑒⁻¹<1,0<𝜉₁=𝜉₀exp𝜉₀<𝜉₀𝑒¹=𝑒⁻¹𝑒¹=1,0<𝜉₂=𝜉₀exp𝜉₁<𝜉₀𝑒¹=1.(3.21) In general, 0<𝜉_𝑛<1. Hence, 𝜎_𝑛=𝜉_𝑛−𝜉₀<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)=12+18𝑦²(𝑡),𝑦(0)=12,𝑡∈[0,1].(3.22) Integrating (3.22), we get𝑦(𝑡)=12+𝑡2+18^𝑡₀𝑦²(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)=18^𝑡₀𝜕𝜕𝑦𝑦²𝑧𝑑𝑠(3.24)=18^𝑡₀2𝑦(𝑠)𝑧(𝑠)𝑑𝑠.(3.25) Similarly, (𝑁^(𝑦)(𝑧₁,𝑧₂)(𝑡)=(𝑡/4)𝑧₁(𝑡)𝑧₂(𝑡) and 𝑁^(𝑘)(𝑦)=0 for 𝑘≥3. Since 𝑡∈[0,1],‖‖𝑁𝑦₀(𝑡)‖‖=‖‖‖18^𝑡₀(1+𝑠)²4𝑑𝑠‖‖‖≤764<1𝑒,‖‖𝑁^𝑦₀(𝑡)‖‖=‖‖‖14^𝑡₀(1+𝑠)2𝑑𝑠‖‖‖≤316<1𝑒,‖‖𝑁^𝑦₀(𝑡)‖‖=‖‖‖𝑡4‖‖‖≤14<1𝑒,‖‖𝑁^(𝑘)𝑦₀(𝑡)‖‖=0,𝑘≥3.(3.26) 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 𝑦₁=𝑁𝑦₀=132𝑡+𝑡²+𝑡³3,𝑦₂=𝑁𝑦₀+𝑦₁−𝑁𝑦₀=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)].

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−12𝑢_𝑥𝑥𝑥+3𝑢𝑢_𝑥−3(𝑣𝑤)𝑥=0,𝑣_𝑡−𝑣_𝑥𝑥𝑥−3𝑢𝑣_𝑥=0,𝑤_𝑡+𝑤_𝑥𝑥𝑥−3𝑢𝑤_𝑥=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).