Improvements in Newton-Rapshon Method for Nonlinear Equations Using Modified Adomian Decomposition Method

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1 International Journal of Mathematical Analysis Vol. 9, 2015, no. 39, HIKARI Ltd, Improvements in Newton-Rapshon Method for Nonlinear Equations Using Modified Adomian Decomposition Method Shin Min Kang Department of Mathematics and RINS Gyeongsang National University Jinju , Korea Waqas Nazeer Division of Science and Technology University of Education Lahore 54000, Pakistan Corresponding author Muhmmad Tanveer, Qaisar Mehmood and Kashif Rehman Department of Mathematics Lahore Leads University Lahore 54810, Pakistan Copyright c 2015 Shin Min Kang et al. This article is 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 In this paper, we present two new numerical algorithms for solving nonlinear equations based on Newton-Raphson method. New algorithms are constructed by using modified Adomian decomposition. The efficiency of new algorithms is shown by solving some numerical examples. Mathematics Subject Classification: 65H05, 65D32

2 1920 Shin Min Kang et al. Keywords: nonlinear equation, (modified) Adomian decomposition method, Newton-Rapshon method 1 Introduction Since in the begining of 1980 s the Adomian decomposition method has been applied to a wide class of functional equations ([3, 4, 5]). Adomian gives the solution as an infinite series converging to accurate solutions. To solve non linear equations we can use iterative methods such as Newton-Raphson method. Some progress to improve the order and accuracy of Newton-Raphson method by using Adomian decomposition technique. Adomian decomposition technique is much better than any other technique. Abbaoui and Cherruault [2] applied Adomian decomposition technique on simple iterative method to solve nonlinear equations. Babolian and Biazar [7, 8] modified the Adomian method. The convergence of decomposition have been investigated by many authors ([6, 7, 9, 10]). Theorem 1.1. ([7]) Suppose that g C p [a, b]. If g (k) (x) = 0 for k = 1, 2,..., p 1 and g (p) (x) 0, then the sequence x n is order p. We are going to develop new iterative schemes by improving Newton- Raphson method. These method are derived by expanding f(x) to third order and then applying modified Adomian decomposition technique. Some numerical examples are also solved to show that purposed algorithms behave better performance compared to other methods. 2 New iterative methods Consider the nonlinear equation The equation (2.1) is usually rewritten as f(x) = 0. (2.1) x = g(x), where (i) there exists [a, b] such that g(x) [a, b] for all x [a, b], (ii) there exists [a, b] such that g (x) L < 1 for all x [a, b]. Considering the following iteration scheme xn+1 = g, n 0,

3 Improvements in Newton-Rapshon method for nonlinear equations 1921 and starting with a suitable initial approximation x 0, we built up a sequence of approximations, say {x n }, for the solution of nonlinear equation, say α. the scheme will be converge to α, provided that (i) the initial approximation x 0 is chosen in the interval [a, b], (ii) g (x) < 1 for all x [a, b], (iii) a g(x) b for all x [a, b]. This is well known fixed point method (FPM) and has first order convergence. We assume that α is a simple root of (2.1) and γ is an initial guess sufficiently close to α. We can rewrite the nonlinear equation (2.1) as a coupled system: f + (x γ)f (x γ)2 f + where γ is the initial approximation for a zero of (2.1). We can rewrite (2.2) in the following form: (x γ)3 f = 0, (2.2) 6 x = γ f f 1 f (x γ)3 f (x γ)2 2 f 6 f. (2.3) We can also rewrite (2.3) as where and N(x) = 1 2 x = c + N(x), (2.4) c = γ f f f (x γ)3 f (x γ)2 f 6 f, where c is a constant and N(x) is a nonlinear function and for approximating x we can apply the Adomian decomposition method. The Adomian decomposition technique consist of representing the solution of (2.4) as a series. x = x n (2.5) n=0 and the nonlinear function is decomposed as N(x) = A n, (2.6) n=0

4 1922 Shin Min Kang et al. where A n are Adomian polynomials of x 0, x 1,..., x n given by d n A n = 1 n dλ n [ ( )] N λ i x i i=0 Substituting (2.5), (2.6) in (2.4), we have x i = c + n=0 The convergence of the series in (2.7) x 0 = c, λ=0, n 0. A n. (2.7) n=0 x n+1 = A n, n 0. The polynomials A n are generated for all kind of nonlinearity by Wazwaz [11]. The first few polynomials are given by A 0 = N(x 0 ), A 1 = x 1 N (x 0 ), A 2 = x 2 N (x 0 ) x2 1N (x 0 ). Other polynomials can be generated in a similar manner ([1, 2, 3, 4]). Let H m = x 0 + x x m = x 0 + A 0 + A A m 1 denote the (m + 1)-term approximation of x. Since the series converges very rapidly, the sum H m = m i=0 x i can serves as a practical solution in each iteration. We will show that the number of terms required to obtain accurate computable solution is very small. For m = 0, x = H 0 = x 0 = c = γ f f, x n+1 = x n f(x n) f, which is the Newton-Raphson method (NRM).

5 Improvements in Newton-Rapshon method for nonlinear equations 1923 For m = 1, For m = 2, x = H 1 = x 0 + x 1 = x 0 + A 0, x 0 = c = γ f f, A 0 = N(x 0 ) = 1 2 (x 0 γ) 2 f f (x 0 γ) 3 f 6 f = f 2 f + f 3 f, x = γ f f f 2 f x n+1 = x n f(x n) f f 2 f x = H 2 = x 0 + x 1 + x 2 = x 0 + A 0 + A 1, x 0 = c = γ f f, A 0 = N(x 0 ) = x 1 = f 2 f A 1 = x 1 N (x 0 ) = f 3 f 2 2f 5 + f 3 f, + f 3 f, + f 3 f. + 5f 4 f f 12f 6 x n+1 = x n f(x n) f f 2 f + f 3 f f 3 f 2 2f 5 + 5f 4 f f 12f 6 The above discussion suggest us following algorithms. Algorithm 2.1. Algorithm 2.2. x n+1 = x n f(x n) f f 2 f f 5 f 2 12f 7, f 5 f 2 12f 7. + f 3 f. (2.8) x n+1 = x n f(x n) f f 2 f + f 3 f f 3 f 2 2f 5 + 5f 4 f f 12f 6 f 5 f 2 12f 7. (2.9)

6 1924 Shin Min Kang et al. 3 Convergence analysis Theorem 3.1. Suppose that c is a root of the equation f(x) = 0. If f(x) is sufficiently smooth in the neighborhood of c, then the convergence order of Algorithm 2.1 given in (2.8) is at least three. Proof. To analysis the convergence of Algorithm 2.1 given in (2.8), let G = x f(x) f (x) f 2 (x)f (x) (x) + f 3 (x)f (x). (x) Let c be a simple zero of f and f(c) = 0. Then we can easily deduce by using the software Maple that G(c) = c, G (c) = 0, G (c) = 0, G (c) = 3(f (c)) 2 (f (c)) 2 0. (3.1) Now, from (3.1) it can be easily seen that G (c) 0, hence according to Theorem 1.1, the Algorithm 2.1 given in (2.8) has third order convergence. Theorem 3.2. Suppose that c is a root of the equation f(x) = 0. If f(x) is sufficiently smooth in the neighborhood of c, then the convergence order of Algorithm 2.2 given in (2.9) is at least four. Proof. To analysis the convergence of Algorithm 2.2 given in (2.9), let G = x f(x) f (x) f 2 (x)f (x) + f 3 (x)f (x) (x) f 3 (x)f 2 (x) (x) 2f 5 (x) + 5f 4 (x)f (x)f (x) 12f 6 (x) f 5 (x)f 2 (x) 12f 7. (x) Let c be a simple zero of f and f(c) = 0. Then we can easily deduce by using the software Maple that G(c) = c, G (c) = 0, G (c) = 0, G (c) = 0, G (4) (c) = 15f 3 (c) + f (4) (c)f 2 (c) f 3 (c) 0. (3.2) Now, from (3.2) it can be easily seen that G (4) (c) 0, hence according to Theorem 1.1, the Algorithm 2.2 given in (2.9) has fourth order convergence.

7 Improvements in Newton-Rapshon method for nonlinear equations Numerical applications In this section we consider some numerical examples to demonstrate the performance of the new developed iterative methods. Example 4.1. Consider sin 2 x x = 0 with initial guess x 0 = 2, the following table shows that fixed point method give root after 5 iterations, Newton-Raphson method give root after 4 iterations, while our Algorithm 2.1 and Algorithm 2.2 converge after 3 and 2 iterations, respectively. n FPM NRM Alogrithem 2.1 Alogrithem Example 4.2. Consider f(x) = ln x + x = 0 with initial guess x 0 = 0.5, the following table shows that fixed point method give root after 18 iterations, Newton-Raphson method give root after 6 iterations, while our Algorithm 2.1 and Algorithm 2.2 converge after 2 and 2 iterations, respectively. n FPM NRM Algorithm 2.1 Algorithm Example 4.3. Consider f(x) = e x 5x 2 = 0 with initial guess x 0 = 0.5, the following table shows that fixed point method give root after 9 iterations, Newton-Raphson method give root after 3 iterations, while our Algorithm 2.1

8 1926 Shin Min Kang et al. and Algorithm 2.2 converge after 2 and 2 iterations, respectively. n FPM NRM Algorithm 2.1 Algorithm Example 4.4. Consider f(x) = x 3 10 = 0 with initial guess x 0 = 2, the following table shows that fixed point method give root after 11 iterations, Newton-Raphson method give root after 3 iterations, while our Algorithm 2.1 and Algorithm 2.2 converge after 2 and 2 iterations, respectively. n FPM NRM Algorithm 2.1 Algorithm Example 4.5. Consider f(x) = x 3 + x 2 2 = 0 with initial guess x 0 = 1.5, the following table shows that fixed point method give root after 9 iterations, Newton-Raphson method give root after 4 iterations, while our Algorithm 2.1 and Algorithm 2.2 converge after 3 and 2 iterations, respectively. n FPM NRM Algorithm 2.1 Algorithm

9 Improvements in Newton-Rapshon method for nonlinear equations 1927 Example 4.6. Consider f(x) = cos x x = 0 with initial guess x 0 = 1, the following table shows that fixed point method give root after 29 iterations, Newton-Raphson method give root after 3 iterations, while our Algorithm 2.1 and Algorithm 2.2 converges after 2 and 2 iterations, respectively. n FPM NRM Algorithm 2.1 Algorithm Conclusions A new Algorithm 2.1 and Algorithm 2.2 for solving nonlinear equations is established. By using some examples the performance of the Algorithm 2.1 and Algorithm 2.2 is also discussed. The Algorithm 2.1 and Algorithm 2.2 are performing very well in comparison to the fixed point method and Newton- Raphson method. Algorithm 2.1 and Algorithm 2.2 can be studied for functional equations, and can be extended to a system of nonlinear equations.

10 1928 Shin Min Kang et al. References [1] S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations by modified Aodmian decomposition method, Appl. Math. Comput., 145 (2003), [2] K. Abbaoui and Y. Cherruault, Convergence of Adomian s method applied to nonlinear equations, Math. Comput. Modelling, 20 (1994), [3] G. Adomian, Nonlinear Stochastic Systems and Applications to Physics, Kluwer Academic Publishers, Dorerecht, [4] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, [5] G. Adomian and R. Rach, On the solution of algebraic equations by the decomposition method, J. Math. Anal. Appl., 105 (1985), [6] K. Abbaoui and Y. Cherruault, New ideas for proving convergence of decomposition methods, Comput. Math. Appl., 29 (1995), [7] E. Baboliana and J. Biazar, On the order of convergence of Adomian method, Appl. Math. Comput., 130 (2002), [8] E. Babolian and J. Biazar, Solution of nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput., 132 (2002), [9] Y. Cherruault, Convergence of Adomian s method, Kybernetes, 18 (1989), [10] Y. Cherruault and G. Adomian, Decomposition methods: A new proof of convergence, Math. Comput. Modelling, 18 (1993), [11] A. M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput., 111 (2000), Received: April 14, 2015; Published: July 23, 2015