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New iterative methods for nonlinear equations by modified HPM
Applied Mathematics and Computation 191 (2007) 122–127 www.elsevier.com/locate/amc
New iterative methods for nonlinear equations by modified HPM A. Golbabai, M. Javidi
*
Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844, Iran
Abstract A numerical method based on modified homotopy perturbation method (HPM) is proposed for solving nonlinear algebric equations. It is shown that the proposed method has third-order convergence. To assess its validity and accuracy, the method is applied to solve several test problems. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Modified homotopy perturbation method; Nonlinear algebric equations; Iterative method
1. Introduction The development of numerical techniques for solving nonlinear algebric equations is a subject of considerable interest. There are many papers that deal with nonlinear algebric equations, e.g. Golbabai and Javidi [1], Chun [2], Noor and Noor [3,4], Basto et al. [5], Abbasbandy [6], Babolian and Biazar [7], Jafari and Gejji [8], He [9]. A more extensive list of references as well as a survey on progress made on this class of problems may be found in Noor [10]. In the recent paper, a numerical method based on modified homotopy perturbation method is proposed for solving real and complex zeroes of nonlinear equation f ðxÞ ¼ 0. The proposed method is applied to solve test problems in order to assess its validity and accuracy. 2. Modified homotopy perturbation method The application of homotopy perturbation method in linear and nonlinear problems has been devoted by scientists and engineers, because this method is to continuously deform a simple problem which is easy to solve into the under study problem which is difficult to solve. This method was proposed first by He in 1997 and systematical description in 2000 which is, in fact, a coupling of the traditional perturbation method and homotopy in topology [11]. This method was further developed and improved by He and applied to non-linear *
Corresponding author. E-mail address: [email protected] (M. Javidi).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.02.070
A. Golbabai, M. Javidi / Applied Mathematics and Computation 191 (2007) 122–127
123
oscillators with discontinuities [12], non-linear wave equations [13], asymptotology [14], boundary value problem [15], limit cycle and bifurcation of nonlinear problems [16] and many other subjects. Thus, He’s method is a universal one which can solve various kinds of nonlinear equations. After that many researchers applied the method to various linear and nonlinear problems: Abbasbandy [17], Ariel et al. [18], Ganji and Sadighi [19], Rafei and Ganji [20], Siddiqui et al. [21], Ghasemi et al. [22], to El-Shahed [23], Javidi and Golbabai [24]. To illustrate basic ideas of modified homotopy perturbation method, we consider the following nonlinear algebric equation: f ðxÞ ¼ 0;
x 2 R:
ð1Þ
We construct a homotopy ðR ½0; 1Þ R ! R which satisfies H ðx; p; aÞ ¼ pf ðxÞ þ ð1 pÞ½f ðxÞ f ðx0 Þ þ pð1 pÞa ¼ 0;
a; x 2 R; p 2 ½0; 1;
ð2Þ
or H ðx; p; aÞ ¼ f ðxÞ f ðx0 Þ þ pf ðx0 Þ þ pð1 pÞa ¼ 0;
a; x 2 R; p 2 ½0; 1;
ð3Þ
where a is an unknown real number and p is embedding parameter, x0 is an initial approximation of Eq. (1). It is obvious that H ðx; 0Þ ¼ f ðxÞ f ðx0 Þ ¼ 0;
ð4Þ
H ðx; 1Þ ¼ f ðxÞ ¼ 0:
ð5Þ
The embedding parameter p monotonically increases from zero to unit as trivial problem H ðx; 0Þ ¼ f ðxÞ f ðx0 Þ ¼ 0 is continuously deformed to original problem H ðx; 1Þ ¼ f ðxÞ ¼ 0. The modified HPM uses the homotopy parameter p as an expanding parameter to obtain [10–20]: x ¼ x0 þ px1 þ p2 x2 þ . . . :
ð6Þ
The approximate solution of Eq. (1), therefore, can be readily obtained: x ¼ lim x ¼ x0 þ x1 þ x2 þ . . . : p!1
ð7Þ
The convergence of the series (7) has been proved by He in this paper [25]. For the application of modified HPM to (1) we can write (3) as follows by expanding f ðxÞ into a Taylor series around x0 : 00 f 0 ðx0 Þ 2 f ðx0 Þ þ ðx x0 Þ þ . . . f ðx0 Þ þ pf ðx0 Þ þ pð1 pÞa ¼ 0: f ðx0 Þ þ ðx x0 Þ ð8Þ 1! 2! Substitution of (6) into (8) yields 00 f 0 ðx0 Þ 2 f ðx0 Þ þ ðx0 þ px1 þ p2 x2 þ . . . x0 Þ þ ... f ðx0 Þ þ ðx0 þ px1 þ p2 x2 þ . . . x0 Þ 1! 2! f ðx0 Þ þ pf ðx0 Þ þ pð1 pÞa ¼ 0:
ð9Þ
By equating the terms with identical powers of p, we have p0 : f ðx0 Þ f ðx0 Þ ¼ 0; 1
0
p : x1 f ðx0 Þ þ f ðx0 Þ þ a ¼ 0; 1 p2 : x2 f 0 ðx0 Þ þ x21 f 00 ðx0 Þ a ¼ 0; 2 1 3 0 p : x3 f ðx0 Þ þ x1 x2 f 00 ðx0 Þ þ x31 f 000 ðx0 Þ ¼ 0: 6
ð10Þ ð11Þ ð12Þ ð13Þ
We try to find parameter a, such that x2 ¼ 0:
ð14Þ
124
A. Golbabai, M. Javidi / Applied Mathematics and Computation 191 (2007) 122–127
0 Þþa Hence, by substituting x1 ¼ f fðx0 ðx from (11) into (12), we have 0Þ 2 1 f ðx0 Þ þ a f 00 ðx0 Þ a ¼ 0: x2 f 0 ðx0 Þ þ 2 f 0 ðx0 Þ
ð15Þ
We can rewrite (15) as follows: x2 f 0 ðx0 Þ þ Aa2 þ ð2Af ðx0 Þ 1Þa þ Af 2 ðx0 Þ ¼ 0;
ð16Þ
00 1 f ðx0 Þ . 2 f 02 ðx0 Þ
Setting x2 ¼ 0 into (16) and solve it, we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 02 ðx0 Þ f ðx0 Þf 00 ðx0 Þ f 0 ðx0 Þ f 02 ðx0 Þ 2f ðx0 Þf 00 ðx0 Þ a¼ f 00 ðx0 Þ
where A ¼
By substituting (17) into (11), we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 00 0 x1 ¼
f ðx0 Þf ðx0 Þf ðx0 Þf ðx0 Þ f 02 ðx0 Þ2f ðx0 Þf 00 ðx0 Þ f 00 ðx0 Þ f 0 ðx0 Þ
ð17Þ
þ f ðx0 Þ
ð18Þ
By substituting (18) into (7), we can obtain the zero of Eq. (1) as follows pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 02 ðx0 Þf ðx0 Þf 00 ðx0 Þf 0 ðx0 Þ f 02 ðx0 Þ2f ðx0 Þf 00 ðx0 Þ þ f ðx0 Þ f 00 ðx0 Þ þ : x ¼ x0 þ x1 þ x2 þ ¼ x0 0 f ðx0 Þ
ð19Þ
This formulations allows us to suggest the following iterative method for solving nonlinear Eq. (1). Algorithm 2.1. For a given z0, calculate the approximation solution znþ1 by the iterative scheme pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 0 ðzn Þ þ f 02 ðzn Þ 2f ðzn Þf 00 ðzn Þ : znþ1 ¼ zn f 00 ðzn Þ
ð20Þ
Algorithm 2.2. For a given z0, calculate the approximation solution znþ1 by the iterative scheme znþ1 ¼ zn
f 0 ðzn Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 02 ðzn Þ 2f ðzn Þf 00 ðzn Þ : f 00 ðzn Þ
ð21Þ
We consider the convergence of Algorithm 2.2. Definition 2.1. Let en ¼ zn r be the truncation error in the nth iterate. If there exists a number k P 1 and a constant c 6¼ 0 such that lim
n!1
jenþ1 j jen jk
¼ c;
ð22Þ
then k is called the order of convergence of the method. Theorem 2.1. Consider the nonlinear equation f ðxÞ ¼ 0. Suppose f is sufficiently differentiable. Then for the iterative method defined by Eq. (21), the convergence is at least of order 3. Proof. Let r be a simple zero of f. Since, f is sufficiently differentiable, by expanding f ðzn Þ; f 0 ðzn Þ and f 00 ðzn Þ around r, we get f ðzn Þ ¼ f 0 ðrÞ½en þ c2 e2n þ c3 e3n þ c4 e4n þ c5 e5n . . .; f 0 ðzn Þ ¼ f 0 ðrÞ½1 þ 2c2 en þ 3c3 e2n þ 4c4 e3n þ 5c5 e4n þ 6c6 e5n . . .; 00
0
f ðzn Þ ¼ f ðrÞ½2c2 þ 6c3 en þ where cn ¼
ðnÞ 1 f ðrÞ ; n! f 0 ðrÞ
12c4 e2n
þ
20c5 e3n
þ
n ¼ 1; 2; 3; . . . and en ¼ zn r.
30c6 e4n
þ
42c7 e5n
ð23Þ . . .;
A. Golbabai, M. Javidi / Applied Mathematics and Computation 191 (2007) 122–127
Now from (23) and
125
pffiffiffiffiffiffiffiffiffiffiffi 1 x ’ 1 12 x 18 x2 , we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 02 ðzn Þ 2f ðzn Þf 00 ðzn Þ ¼ f 0 ðrÞ 1 ½6c3 e2n þ ð20c4 þ 4c2 c3 Þe3n þ ;
ð24Þ
1 1 2 ’ 1 f6c3 e2n þ ð20c4 þ 4c2 c3 Þe3n þ g f6c3 e2n þ ð20c4 þ 4c2 c3 Þe3n þ g : 2 8 Substituting (23), (24) into (21) yields: enþ1 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 02 ðzn Þ 2f ðzn Þf 00 ðzn ÞÞ 2f ðzn Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2c3 e3n þ Oðe4n Þ f 0 ðzn Þ þ f 02 ðzn Þ 2f ðzn Þf 00 ðzn Þ
en ðf 0 ðzn Þ þ
From the above term, we have lim
n!1
jenþ1 j 3
jen j
¼ 2c3 ¼
1 f 000 ðrÞ 3 f 0 ðrÞ
ð25Þ
which shows that Algorithm 2.2 is at least a third order convergent method, the required result. 3. Applications We present some examples to illustrate the efficiency of the developed method in this paper. We apply the Algorithm 2.1 (A1) and Algorithm 2.2 (A2) for the following test problems.
Table 1 Numerical results for Example 1 Method
x0
n
xn
jf ðxn Þj
f1 A1 A1 A2 A2
10 1,000,000 10 1,000,000
1 1 1 1
0 1.00000000000000i 0 1.00000000000000i 0 + 1.00000000000000i 0 + 1.00000000000000i
0 0 0 0
f2 A1 A2
10 10
1 1
0.62500000000000 0.78062474979980i 0.62500000000000 + 0.78062474979980i
0 0
f3 A1 A1 A2 A2 A1
1 1 1 1 10
5 5 2 4 6
0.23673291566846 2.04159920739809i 0.23673290386456 2.04159922690925i 0.47347871922668 0.47346580772913 0.23673290386456 + 2.04159922690925i
2.0127e7 1.8310e15 6.0329e5 1.9984e15 1.8310e15
f4 A1 A1 A2 A2 A2 A2
10 10 10 10 5 5
3 4 3 4 4 5
0.90999663733283 0.91000757248871 0.45896226625028 0.45896226753695 3.73307902854733 0.00000000002014i 3.73307902863281
3.2539e5 4.4408e16 4.3562e9 2.2204e16 1.7046e9 7.1054e15
f5 A1 A1 A2 A2
10 10 10 10
5 6 5 7
2.05892687903733 2.06061115234403i 2.05892548616325 2.06061121597737i 0.11791163218484 + 0.00006005627555i 0.11785097232650
1.7224e5 3.5527e15 7.6693e4 1.1102e15
126
A. Golbabai, M. Javidi / Applied Mathematics and Computation 191 (2007) 122–127
Table 2 Numerical results for Example 2 Method
x0
n
xn
jf ðxn Þj
f1 A1 A1 A1 A2 A2 A2
0 0 0 0 0 0
1 2 3 8 9 10
0.25834261322606 0.25753028540925 0.25753028543986 2.08395653730930 2.73333363329271i 2.08438140711048 2.73300672534965i 2.08438140713089 2.73300672536096i
0.0030 1.1565e10 4.4408e16 0.0047 2.0636e10 5.0242e15
f2 A1 A1 A2 A2
0 0 0 0
6 7 5 6
1.33997399226311 0.75857399599429i 1.33996949441870 0.75857208933040i 1.20764782713135 + 0.00000000000093i 1.20764782713092
1.1282e4 4.8849e15 2.0881e11 3.5527e15
f3 A1 A1 A1 A1 A1 A1 A1 A2 A2 A2
3 3 3 8 8 8 8 8 8 8
2 3 4 5 6 7 8 8 9 10
3.14158789502121 3.14159265358979 3.14159265358979 9.02169881646105 + 0.04130522314755i 9.00077636954597 + 0.00002556098149i 8.99999999610981 0.00000000038504i 9.00000000000000 + 0.00000000000000i 9.42473099126943 0.00015790982573i 9.42477796077744 + 0.00000000000685i 9.42477796076938
0.1518 1.0265e11 3.9088e12 3.2753e14 5.9949e16 3.0280e21 1.2013e37 4.7879e17 3.0743e24 1.0677e28
Example 1. Consider the following nonlinear equations: f1 ¼ x2 þ 1 ¼ 0; f2 ¼ 4x2 5x þ 4 ¼ 0; f3 ¼ x3 þ 4x þ 2 ¼ 0;
ð26Þ
2
f4 ¼ expðxÞ 3x ¼ 0; f5 ¼ x3 þ 4x2 þ 8x 1 ¼ 0: In Table 1, we list the results obtained by modified homotopy perturbation method. Example 2. Consider the following nonlinear equations: f 1 ¼ x2 exp ðxÞ 3x þ 2 ¼ 0; 2
f2 ¼ x exp ðx2 Þ sin ðxÞ þ 3 cos ðxÞ þ 5 ¼ 0;
ð27Þ
f 3 ¼ log ðx þ 10Þexp3 ðxÞ sinðxÞ: In Table 2, we list the results obtained by modified homotopy perturbation method. 4. Conclusions A numerical method based modified homotopy perturbation method is proposed for solving nonlinear algebric equations. To check the numerical method, it is applied to solve different test problems. Convergence is also observed in the numerical solutions when the calculation is refined by increasing the iteration number used. The numerical results confirm the validity of the numerical method and suggest that it is an interesting and viable alternative to existing numerical methods for solving the problem under consideration. This method also calculate the complex and real zero’s of nonlinear algebric equations.
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127
References [1] A. Golbabai, M. Javidi, A new family of iterative methods for solving system of nonlinear algebric equations, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2007.02.055. [2] C. Chun, Iterative methods improving Newton’s method by the decomposition method, Comput. Math. Appl. 50 (2005) 1559–1568. [3] M. Aslam Noor, New family of iterative methods for nonlinear equations, Appl. Math. Comput. (2007), doi:10.1016/ j.amc.2007.01.045. [4] M. Aslam Noor, K. Inayat Noor, Modified iterative methods with cubic convergence for solving nonlinear equations, Appl. Math. Comput. (2006), doi:10.1016/j.amc:2006.05.155. [5] M. Basto, V. Semiao, F.L. Calheiros, A new iterative method to compute nonlinear equations, Appl. Math. Comput. 173 (2006) 468– 483. [6] S. Abbasbandy, Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 145 (2003) 887–893. [7] E. Babolian, J. Biazar, Solution of nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 132 (2002) 167–172. [8] H. Jafari, V.D. Gejji, Revised Adomian decomposition method for solving a system of nonlinear equations, Appl. Math. Comput. 175 (2006) 1–7. [9] J.H. He, A new iterative method for solving algebric equations, Appl. Math. Comput. 135 (2003) 81–84. [10] M.A. Noor, New iterative schemes for nonlinear equations, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.09.028. [11] J.H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech. 35 (1) (2000) 37–43. [12] J.H. He, The homotopy perturbation method for non-linear oscillators with discontinuities, Appl. Math. Comput. 151 (1) (2004) 287– 292. [13] J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fract. 26 (3) (2005) 695–700. [14] J.H. He, Asymptotology by homotopy perturbation method, Appl. Math. Comput. 156 (3) (2004) 591–596. [15] J.H. He, Homotopy perturbation method for solving boundary problems, Phys. Lett. A 350 (1-2) (2006) 87–88. [16] J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fract. 26 (3) (2005) 827–833. [17] S. Abbasbandy, Iterated He’s homotopy perturbation method for quadratic Riccati differential equation, Appl. Math. Comput. 175 (1) (2006) 581–589. [18] P.D. Ariel, T. Hayat, S. Asghar, Homotopy perturbation method and axisymmetric flow over a stretching sheet, Int. J. Nonlinear Sci. Numer. Simul. 7 (4) (2006) 399–406. [19] D.D. Ganji, A. Sadighi, Application of He’s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations, Int. J. Nonlinear Sci. Numer. Simul. 7 (4) (2006) 411–418. [20] M. Rafei, D.D. Ganji, Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method, Int. J. Nonlinear Sci. Numer. Simul. 7 (3) (2006) 321–328. [21] A.M. Siddiqui, R. Mahmood, Q.K. Ghori, Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder, Phys. Lett. A 352 (4–5) (2006) 404–410. [22] M. Ghasemi et al., Numerical solution of the nonlinear Voltra–Fredholm integral equations by using homotopy perturbation method, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.10.015. [23] M. El-Shahed, Application of He’s homotopy perturbation method to Voltra’s integro-differential equation, Int. J. Nonlinear Sci. Numer. Simul. 6 (2) (2005) 163–168. [24] M. Javidi, A. Golbabai, A numerical solution for solving system of Fredholm integral equations by using homotopy perturbation method, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2006.12.070. [25] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178 (3–4) (1999) 257–262.
New iterative methods for nonlinear equations by modified HPM A. Golbabai, M. Javidi
*
Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844, Iran
Abstract A numerical method based on modified homotopy perturbation method (HPM) is proposed for solving nonlinear algebric equations. It is shown that the proposed method has third-order convergence. To assess its validity and accuracy, the method is applied to solve several test problems. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Modified homotopy perturbation method; Nonlinear algebric equations; Iterative method
1. Introduction The development of numerical techniques for solving nonlinear algebric equations is a subject of considerable interest. There are many papers that deal with nonlinear algebric equations, e.g. Golbabai and Javidi [1], Chun [2], Noor and Noor [3,4], Basto et al. [5], Abbasbandy [6], Babolian and Biazar [7], Jafari and Gejji [8], He [9]. A more extensive list of references as well as a survey on progress made on this class of problems may be found in Noor [10]. In the recent paper, a numerical method based on modified homotopy perturbation method is proposed for solving real and complex zeroes of nonlinear equation f ðxÞ ¼ 0. The proposed method is applied to solve test problems in order to assess its validity and accuracy. 2. Modified homotopy perturbation method The application of homotopy perturbation method in linear and nonlinear problems has been devoted by scientists and engineers, because this method is to continuously deform a simple problem which is easy to solve into the under study problem which is difficult to solve. This method was proposed first by He in 1997 and systematical description in 2000 which is, in fact, a coupling of the traditional perturbation method and homotopy in topology [11]. This method was further developed and improved by He and applied to non-linear *
Corresponding author. E-mail address: [email protected] (M. Javidi).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.02.070
A. Golbabai, M. Javidi / Applied Mathematics and Computation 191 (2007) 122–127
123
oscillators with discontinuities [12], non-linear wave equations [13], asymptotology [14], boundary value problem [15], limit cycle and bifurcation of nonlinear problems [16] and many other subjects. Thus, He’s method is a universal one which can solve various kinds of nonlinear equations. After that many researchers applied the method to various linear and nonlinear problems: Abbasbandy [17], Ariel et al. [18], Ganji and Sadighi [19], Rafei and Ganji [20], Siddiqui et al. [21], Ghasemi et al. [22], to El-Shahed [23], Javidi and Golbabai [24]. To illustrate basic ideas of modified homotopy perturbation method, we consider the following nonlinear algebric equation: f ðxÞ ¼ 0;
x 2 R:
ð1Þ
We construct a homotopy ðR ½0; 1Þ R ! R which satisfies H ðx; p; aÞ ¼ pf ðxÞ þ ð1 pÞ½f ðxÞ f ðx0 Þ þ pð1 pÞa ¼ 0;
a; x 2 R; p 2 ½0; 1;
ð2Þ
or H ðx; p; aÞ ¼ f ðxÞ f ðx0 Þ þ pf ðx0 Þ þ pð1 pÞa ¼ 0;
a; x 2 R; p 2 ½0; 1;
ð3Þ
where a is an unknown real number and p is embedding parameter, x0 is an initial approximation of Eq. (1). It is obvious that H ðx; 0Þ ¼ f ðxÞ f ðx0 Þ ¼ 0;
ð4Þ
H ðx; 1Þ ¼ f ðxÞ ¼ 0:
ð5Þ
The embedding parameter p monotonically increases from zero to unit as trivial problem H ðx; 0Þ ¼ f ðxÞ f ðx0 Þ ¼ 0 is continuously deformed to original problem H ðx; 1Þ ¼ f ðxÞ ¼ 0. The modified HPM uses the homotopy parameter p as an expanding parameter to obtain [10–20]: x ¼ x0 þ px1 þ p2 x2 þ . . . :
ð6Þ
The approximate solution of Eq. (1), therefore, can be readily obtained: x ¼ lim x ¼ x0 þ x1 þ x2 þ . . . : p!1
ð7Þ
The convergence of the series (7) has been proved by He in this paper [25]. For the application of modified HPM to (1) we can write (3) as follows by expanding f ðxÞ into a Taylor series around x0 : 00 f 0 ðx0 Þ 2 f ðx0 Þ þ ðx x0 Þ þ . . . f ðx0 Þ þ pf ðx0 Þ þ pð1 pÞa ¼ 0: f ðx0 Þ þ ðx x0 Þ ð8Þ 1! 2! Substitution of (6) into (8) yields 00 f 0 ðx0 Þ 2 f ðx0 Þ þ ðx0 þ px1 þ p2 x2 þ . . . x0 Þ þ ... f ðx0 Þ þ ðx0 þ px1 þ p2 x2 þ . . . x0 Þ 1! 2! f ðx0 Þ þ pf ðx0 Þ þ pð1 pÞa ¼ 0:
ð9Þ
By equating the terms with identical powers of p, we have p0 : f ðx0 Þ f ðx0 Þ ¼ 0; 1
0
p : x1 f ðx0 Þ þ f ðx0 Þ þ a ¼ 0; 1 p2 : x2 f 0 ðx0 Þ þ x21 f 00 ðx0 Þ a ¼ 0; 2 1 3 0 p : x3 f ðx0 Þ þ x1 x2 f 00 ðx0 Þ þ x31 f 000 ðx0 Þ ¼ 0: 6
ð10Þ ð11Þ ð12Þ ð13Þ
We try to find parameter a, such that x2 ¼ 0:
ð14Þ
124
A. Golbabai, M. Javidi / Applied Mathematics and Computation 191 (2007) 122–127
0 Þþa Hence, by substituting x1 ¼ f fðx0 ðx from (11) into (12), we have 0Þ 2 1 f ðx0 Þ þ a f 00 ðx0 Þ a ¼ 0: x2 f 0 ðx0 Þ þ 2 f 0 ðx0 Þ
ð15Þ
We can rewrite (15) as follows: x2 f 0 ðx0 Þ þ Aa2 þ ð2Af ðx0 Þ 1Þa þ Af 2 ðx0 Þ ¼ 0;
ð16Þ
00 1 f ðx0 Þ . 2 f 02 ðx0 Þ
Setting x2 ¼ 0 into (16) and solve it, we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 02 ðx0 Þ f ðx0 Þf 00 ðx0 Þ f 0 ðx0 Þ f 02 ðx0 Þ 2f ðx0 Þf 00 ðx0 Þ a¼ f 00 ðx0 Þ
where A ¼
By substituting (17) into (11), we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 00 0 x1 ¼
f ðx0 Þf ðx0 Þf ðx0 Þf ðx0 Þ f 02 ðx0 Þ2f ðx0 Þf 00 ðx0 Þ f 00 ðx0 Þ f 0 ðx0 Þ
ð17Þ
þ f ðx0 Þ
ð18Þ
By substituting (18) into (7), we can obtain the zero of Eq. (1) as follows pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 02 ðx0 Þf ðx0 Þf 00 ðx0 Þf 0 ðx0 Þ f 02 ðx0 Þ2f ðx0 Þf 00 ðx0 Þ þ f ðx0 Þ f 00 ðx0 Þ þ : x ¼ x0 þ x1 þ x2 þ ¼ x0 0 f ðx0 Þ
ð19Þ
This formulations allows us to suggest the following iterative method for solving nonlinear Eq. (1). Algorithm 2.1. For a given z0, calculate the approximation solution znþ1 by the iterative scheme pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 0 ðzn Þ þ f 02 ðzn Þ 2f ðzn Þf 00 ðzn Þ : znþ1 ¼ zn f 00 ðzn Þ
ð20Þ
Algorithm 2.2. For a given z0, calculate the approximation solution znþ1 by the iterative scheme znþ1 ¼ zn
f 0 ðzn Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 02 ðzn Þ 2f ðzn Þf 00 ðzn Þ : f 00 ðzn Þ
ð21Þ
We consider the convergence of Algorithm 2.2. Definition 2.1. Let en ¼ zn r be the truncation error in the nth iterate. If there exists a number k P 1 and a constant c 6¼ 0 such that lim
n!1
jenþ1 j jen jk
¼ c;
ð22Þ
then k is called the order of convergence of the method. Theorem 2.1. Consider the nonlinear equation f ðxÞ ¼ 0. Suppose f is sufficiently differentiable. Then for the iterative method defined by Eq. (21), the convergence is at least of order 3. Proof. Let r be a simple zero of f. Since, f is sufficiently differentiable, by expanding f ðzn Þ; f 0 ðzn Þ and f 00 ðzn Þ around r, we get f ðzn Þ ¼ f 0 ðrÞ½en þ c2 e2n þ c3 e3n þ c4 e4n þ c5 e5n . . .; f 0 ðzn Þ ¼ f 0 ðrÞ½1 þ 2c2 en þ 3c3 e2n þ 4c4 e3n þ 5c5 e4n þ 6c6 e5n . . .; 00
0
f ðzn Þ ¼ f ðrÞ½2c2 þ 6c3 en þ where cn ¼
ðnÞ 1 f ðrÞ ; n! f 0 ðrÞ
12c4 e2n
þ
20c5 e3n
þ
n ¼ 1; 2; 3; . . . and en ¼ zn r.
30c6 e4n
þ
42c7 e5n
ð23Þ . . .;
A. Golbabai, M. Javidi / Applied Mathematics and Computation 191 (2007) 122–127
Now from (23) and
125
pffiffiffiffiffiffiffiffiffiffiffi 1 x ’ 1 12 x 18 x2 , we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 02 ðzn Þ 2f ðzn Þf 00 ðzn Þ ¼ f 0 ðrÞ 1 ½6c3 e2n þ ð20c4 þ 4c2 c3 Þe3n þ ;
ð24Þ
1 1 2 ’ 1 f6c3 e2n þ ð20c4 þ 4c2 c3 Þe3n þ g f6c3 e2n þ ð20c4 þ 4c2 c3 Þe3n þ g : 2 8 Substituting (23), (24) into (21) yields: enþ1 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 02 ðzn Þ 2f ðzn Þf 00 ðzn ÞÞ 2f ðzn Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2c3 e3n þ Oðe4n Þ f 0 ðzn Þ þ f 02 ðzn Þ 2f ðzn Þf 00 ðzn Þ
en ðf 0 ðzn Þ þ
From the above term, we have lim
n!1
jenþ1 j 3
jen j
¼ 2c3 ¼
1 f 000 ðrÞ 3 f 0 ðrÞ
ð25Þ
which shows that Algorithm 2.2 is at least a third order convergent method, the required result. 3. Applications We present some examples to illustrate the efficiency of the developed method in this paper. We apply the Algorithm 2.1 (A1) and Algorithm 2.2 (A2) for the following test problems.
Table 1 Numerical results for Example 1 Method
x0
n
xn
jf ðxn Þj
f1 A1 A1 A2 A2
10 1,000,000 10 1,000,000
1 1 1 1
0 1.00000000000000i 0 1.00000000000000i 0 + 1.00000000000000i 0 + 1.00000000000000i
0 0 0 0
f2 A1 A2
10 10
1 1
0.62500000000000 0.78062474979980i 0.62500000000000 + 0.78062474979980i
0 0
f3 A1 A1 A2 A2 A1
1 1 1 1 10
5 5 2 4 6
0.23673291566846 2.04159920739809i 0.23673290386456 2.04159922690925i 0.47347871922668 0.47346580772913 0.23673290386456 + 2.04159922690925i
2.0127e7 1.8310e15 6.0329e5 1.9984e15 1.8310e15
f4 A1 A1 A2 A2 A2 A2
10 10 10 10 5 5
3 4 3 4 4 5
0.90999663733283 0.91000757248871 0.45896226625028 0.45896226753695 3.73307902854733 0.00000000002014i 3.73307902863281
3.2539e5 4.4408e16 4.3562e9 2.2204e16 1.7046e9 7.1054e15
f5 A1 A1 A2 A2
10 10 10 10
5 6 5 7
2.05892687903733 2.06061115234403i 2.05892548616325 2.06061121597737i 0.11791163218484 + 0.00006005627555i 0.11785097232650
1.7224e5 3.5527e15 7.6693e4 1.1102e15
126
A. Golbabai, M. Javidi / Applied Mathematics and Computation 191 (2007) 122–127
Table 2 Numerical results for Example 2 Method
x0
n
xn
jf ðxn Þj
f1 A1 A1 A1 A2 A2 A2
0 0 0 0 0 0
1 2 3 8 9 10
0.25834261322606 0.25753028540925 0.25753028543986 2.08395653730930 2.73333363329271i 2.08438140711048 2.73300672534965i 2.08438140713089 2.73300672536096i
0.0030 1.1565e10 4.4408e16 0.0047 2.0636e10 5.0242e15
f2 A1 A1 A2 A2
0 0 0 0
6 7 5 6
1.33997399226311 0.75857399599429i 1.33996949441870 0.75857208933040i 1.20764782713135 + 0.00000000000093i 1.20764782713092
1.1282e4 4.8849e15 2.0881e11 3.5527e15
f3 A1 A1 A1 A1 A1 A1 A1 A2 A2 A2
3 3 3 8 8 8 8 8 8 8
2 3 4 5 6 7 8 8 9 10
3.14158789502121 3.14159265358979 3.14159265358979 9.02169881646105 + 0.04130522314755i 9.00077636954597 + 0.00002556098149i 8.99999999610981 0.00000000038504i 9.00000000000000 + 0.00000000000000i 9.42473099126943 0.00015790982573i 9.42477796077744 + 0.00000000000685i 9.42477796076938
0.1518 1.0265e11 3.9088e12 3.2753e14 5.9949e16 3.0280e21 1.2013e37 4.7879e17 3.0743e24 1.0677e28
Example 1. Consider the following nonlinear equations: f1 ¼ x2 þ 1 ¼ 0; f2 ¼ 4x2 5x þ 4 ¼ 0; f3 ¼ x3 þ 4x þ 2 ¼ 0;
ð26Þ
2
f4 ¼ expðxÞ 3x ¼ 0; f5 ¼ x3 þ 4x2 þ 8x 1 ¼ 0: In Table 1, we list the results obtained by modified homotopy perturbation method. Example 2. Consider the following nonlinear equations: f 1 ¼ x2 exp ðxÞ 3x þ 2 ¼ 0; 2
f2 ¼ x exp ðx2 Þ sin ðxÞ þ 3 cos ðxÞ þ 5 ¼ 0;
ð27Þ
f 3 ¼ log ðx þ 10Þexp3 ðxÞ sinðxÞ: In Table 2, we list the results obtained by modified homotopy perturbation method. 4. Conclusions A numerical method based modified homotopy perturbation method is proposed for solving nonlinear algebric equations. To check the numerical method, it is applied to solve different test problems. Convergence is also observed in the numerical solutions when the calculation is refined by increasing the iteration number used. The numerical results confirm the validity of the numerical method and suggest that it is an interesting and viable alternative to existing numerical methods for solving the problem under consideration. This method also calculate the complex and real zero’s of nonlinear algebric equations.
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127
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