Advanced
NEW FAMILY OF ITERATIVE METHODS FOR SOLVING NON-LINEAR EQUATIONS USING NEW ADOMIAN POLYNOMIALS
NEW FAMILY OF ITERATIVE METHODS FOR SOLVING NON-LINEAR EQUATIONS USING NEW ADOMIAN POLYNOMIALS
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. Aug, 22(3): 231-243
Copyright © 2015, Korean Society of Mathematical Education
  • Received : January 28, 2015
  • Accepted : June 29, 2015
  • Published : August 31, 2015
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
Arif Rafiq
DEPARTMENT OF MATHEMATICS, LAHORE LEADS UNIVERSITY, LAHORE, PAKISTANEmail address:aarafiq@gmail.com
Ayesha Inam Pasha
CIIT, LAHORE, PAKISTANEmail address:ayeshams2@gmail.com
Byung-Soo Lee
DEPARTMENT OF MATHEMATICS, KYUNGSUNG UNIVERSITY, BUSAN 608-736, KOREAEmail address:bslee@ks.ac.kr

Abstract
We suggest and analyze a family of multi-step iterative methods for solving nonlinear equations using the decomposition technique mainly due to Rafiq et al. [13] .
Keywords
1. INTRODUCTION
In recent years, much attention has been given to develop several iterative methods for solving nonlinear equations (see for example [1 , 6 - 12 , 14] ). These methods can be classified as one-step and two-step methods.
Abbasbandy [1] and Chun [6] have proposed and studied several one-step and two-step iterative methods with higher order convergence by using the decomposition technique of Adomian [2] .
In [11] , Noor developed two-step and three-step iterative methods by using the Adomian decomposition technique and by combining the well-known Newton method with other one-step and two-step methods.
In [1 , 11 - 12] , the authors have used the higher order derivatives which is a drawback. To overcome this drawback, following the lines of [11] , we suggest and analyze a family of multi-step iterative methods which do not involve the high-order derivatives of the function for solving nonlinear equations using the decomposition technique mainly due to Rafiq et al. [13] . We also discuss the convergence of the new proposed methods. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative method. Our results can be considered as an improvement and refinement of the previous results.
2. ITERATIVE METHODS
Consider the nonlinear equation
PPT Slide
Lager Image
We assume that α is a simple root of (1) and β is an initial guess sufficiently close to α . We can rewrite (1) as a coupled system using the Taylor series.
PPT Slide
Lager Image
PPT Slide
Lager Image
We can rewrite (3) in the following form as
PPT Slide
Lager Image
We can rewrite (4) in the following equivalent form as
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and
PPT Slide
Lager Image
In order to prove the multi-step iterative methods, He [8] and Lao [10] have considered the case with the definition that
PPT Slide
Lager Image
and Noor and Noor [12] have considered the case
PPT Slide
Lager Image
which is actually
PPT Slide
Lager Image
and is the stronger one. We rectify this error and also remove such kind of conditions. For this purpose, we substituting (3) into (7) to obtain
PPT Slide
Lager Image
with
PPT Slide
Lager Image
We now construct a new family of iterative methods by using the following decomposition method mainly due to Rafiq et al. [13] . This decomposition of the nonlinear operator N ( x ) is quite different than that of Adomian decomposition. The main idea of this technique is to look for a solution of (5) having the series form
PPT Slide
Lager Image
The nonlinear operator N can be decomposed as
PPT Slide
Lager Image
where Ai are the functions which are known as the new Adomian polynomials depending on x 0 , x 1 , ⋯ ; given by a formula
PPT Slide
Lager Image
First few new Adomian polynomials are as follows
PPT Slide
Lager Image
Substituting (12) and (13) into (5), we obtain
PPT Slide
Lager Image
It follows from (6), (12) and (17), that
PPT Slide
Lager Image
This allows us to suggest the following one-step iterative method for solving (1).
Algorithm 1.
For a given x 0 , compute the approximate solution x n+1 by the iterative scheme
PPT Slide
Lager Image
which is known as “ Newton’s Method ” and it has the second order convergence .
From (10) and (17), we have
PPT Slide
Lager Image
Again using (12), (15), (16), (17) and (18) , we conclude that
PPT Slide
Lager Image
Using this relation, we can suggest the following two-step iterative methods for solving (1).
Algorithm 2.
For a given x 0 , compute the approximate solution x n+1 by the iterative scheme
Predictor-Step
PPT Slide
Lager Image
Corrector-Step
PPT Slide
Lager Image
This Algorithm is commonly known as “ Double-Newton Method ” with the third order convergence .
Again
PPT Slide
Lager Image
From (11), (17)-(21), we conclude that
PPT Slide
Lager Image
Using this, we can suggest and analyze the following two-step iterative method for solving (1).
Algorithm 3. (AP)
For a given x 0 , compute the approximate solution x n+1 by the iterative scheme
Predictor-Step
PPT Slide
Lager Image
Corrector-Step
PPT Slide
Lager Image
Algorithm 3 is called the two-step iterative method for solving (1).
Again using (11) and (16), we have
PPT Slide
Lager Image
From (11), (18) – (20), we have
PPT Slide
Lager Image
Using this, we can suggest and analyze the following iterative method for solving (1).
Algorithm 4.
For a given x 0 , compute the approximate solution x n+1 by the iterative scheme
Predictor-Step
PPT Slide
Lager Image
Corrector-Step
PPT Slide
Lager Image
If f' ( yn ) = 0 then Algorithm 3 reduces to the following method
PPT Slide
Lager Image
Now using finite difference approximation, we obtain
PPT Slide
Lager Image
Combining (22) and (24), we suggest the following new iterative method for solving (1) as follows
PPT Slide
Lager Image
Also we can suggest and analyze the following iterative method for solving (1).
Algorithm 5. (A)
For a given x 0 , compute the approximate solution x n+1 by the iterative scheme
Predictor-Step
PPT Slide
Lager Image
Corrector-Step
PPT Slide
Lager Image
3. CONVERGENCE ANALYSIS
Theorem 1. Let β I be a simple zero of sufficiently differentiable function f : I ⊆ ℝ → ℝ for an open interval I . If x 0 is sufficiently close to β , then the two-step iterative method defined by Algorithm 3 has the fourth-order convergence .
Proof . Let β I be a simple zero of f . Since f is sufficiently differentiable function, by expanding f ( xn ) and f' ( xn ) about β , we get
PPT Slide
Lager Image
PPT Slide
Lager Image
where
PPT Slide
Lager Image
, k = 1, 2, 3, · · · and en = xn β .
Now from (27) and (28), we have
PPT Slide
Lager Image
From (18) and (29), we get
PPT Slide
Lager Image
Now expanding f ( yn ) about β , we have
PPT Slide
Lager Image
From (28) and (31), we have
PPT Slide
Lager Image
Now expanding f' ( yn ) about β and using (32), we have
PPT Slide
Lager Image
From (31) and (34), we get
PPT Slide
Lager Image
From (29) and (35), we have
PPT Slide
Lager Image
From (22), (32) and (35), one obtains
PPT Slide
Lager Image
Hence it is proved.                                                                                          ☐
Theorem 2. Let β I be a simple zero of a sufficiently differentiable function f : I ⊆ ℝ → ℝ for an open interval I . If x 0 is sufficiently close to β , then the iterative method defined by Algorithm 4 has the fourth- order convergence .
Proof. From (28) and (33), we have
PPT Slide
Lager Image
From (35) and (37), we get
PPT Slide
Lager Image
From (23), (32), (35) and (38), one obtains
PPT Slide
Lager Image
Hence it is proved.                                                                                           ☐
Theorem 3. Let β I be a simple zero of a sufficiently differentiable function f : I ⊆ ℝ → ℝ for an open interval I . If x 0 is sufficiently close to β , then iterative method defined by Algorithm 5 has the third- order convergence .
Proof . From (27) and (31), we have
PPT Slide
Lager Image
From (32) and (40), we get
PPT Slide
Lager Image
From (26), (30), (32) and (41), one obtains
PPT Slide
Lager Image
Hence it is proved.                                                                                           ☐
4. NUMERICAL EXAMPLES
We provide some examples to illustrate the efficiency of the new developed iterative methods. Put ϵ = 10 −15 .
The following stopping criteria is used for the computer programs
  • (1) |xn+1−xn| <ϵ,
  • (2) |f(xn+1)| <ϵ.
The examples are the same as in Chun [6] :
  • F1(x) =sin2x−x2+ 1,
  • F2(x) =x2−ex− 3x+ 2,
  • F3(x) =cosx−x,
  • F4(x) = (x− 1)3− 1,
  • F5(x) =x3− 10,
  • F6(x) =x·ex2−sin2x+ 3cosx+ 5,
  • F7(x) =ex2+7x−30− 1.
Also for the convergence criteria, it was required that the distance of two consecutive approximations δ for the zero was less than 10 −15 . Also displayed are the number of iterations (IT) to approximate the zero, the approximate zero x 0 , the value f ( x 0 ) and δ . We compare the Newton method (NM), the Double Newton method (DNM), the method of Noor (NR) [12] and the method (AP) , introduced in the Algorithm 3 (see Table 1 ).
PPT Slide
Lager Image
Now we compare the Newton method (NM), the Double Newton method (DNM), the method of Weerakoon and Fernando [14] , the method of Frontini and Sormani [7] , the method of Homeier [9] , the method of Noor (NR) [12] and the method (A), introduced in the Algorithm 5 (see Table 2 ).
PPT Slide
Lager Image
5. CONCLUSION
We have suggested a family of two-step iterative methods for solving nonlinear equations by using a new decomposition technique mainly due to Rafiq et al. [13] . It is important to note that the implementation of these methods does not require the computation of higher order derivatives compared to most other methods of the same order.
2010 Mathematics Subject Classification. 26A18, 39B12.
References
Abbasbandy S. 2003 Improving Newton Raphson method for nonlinear equations by modified Adomian decomposition method Appl. Math. Comput. 145 887 - 893    DOI : 10.1016/S0096-3003(03)00282-0
Adomian G. 1989 Nonlinear Stochastic Systems and Applications to Physic Kluwer Academic Publishers Dordrecht
Adomian G. 1992 A review of the decomposition method and some recent results for nonlinear equations Math. Comput. Model 13 (7) 17 - 43
Adomian G. 1994 Solving Frontier problems of Physics: The Decomposition Method Kluwer Boston, MA
Adomian G. , Rach R. 1992 Noise terms in decomposition series solution Comput. Math. Appl. 24 (11) 61 - 64
Chun C. 2005 Iterative methods improving Newton method by the decomposition method Comput. Math. Appl. 50 1559 - 1568    DOI : 10.1016/j.camwa.2005.08.022
Frontini M. , Sormani E. 2003 Some variants of Newton method with third order convergence J. Comput. Appl. Math. 140 419 - 426    DOI : 10.1016/S0096-3003(02)00238-2
He J.H. 2003 A new iteration method for solving algebraic equations Appl. Math. Comput. 135 81 - 84    DOI : 10.1016/S0096-3003(01)00313-7
Homeier H.H. 2005 On Newton-type methods with cubic convergence J. Comput. Appl. Math. 176 425 - 432    DOI : 10.1016/j.cam.2004.07.027
Luo X. 2005 A note on the new iteration for solving algebraic equations Appl. Math. Comput. 171 1177 - 1183    DOI : 10.1016/j.amc.2005.01.124
Noor M.A. 2007 New family of iterative methods for nonlinear equations Appl. Math. Comput. 190 553 - 558    DOI : 10.1016/j.amc.2007.01.045
Noor M.A. , Noor K.I. 2006 Some iterative schemes for nonlinear equations Appl. Math. Comput. 183 774 - 779    DOI : 10.1016/j.amc.2006.05.084
Rafiq A. , Lee B.S. , Pasha A.I. , Malik M.Y. 2011 On Adomian type decomposition method and its convergence Nonlinear Analysis Forum 16 169 - 181
Weerakoon S. , Fernando G.I. 2000 A variant of Newton method with accelerated third order convergence Appl. Math. Lett. 17 87 - 93