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Iterated He’s homotopy perturbation method for quadratic Riccati differential equation
Applied Mathematics and Computation 175 (2006) 581–589 www.elsevier.com/locate/amc
Iterated HeÕs homotopy perturbation method for quadratic Riccati differential equation S. Abbasbandy Department of Mathematics, Imam Khomeini International University, P.O. Box 288, Ghazvin 34194, Iran
Abstract In this paper, iterated HeÕs homotopy perturbation method is proposed to solving quadratic Riccati differential equation. Comparisons are made between AdomianÕs decomposition method (ADM) and the exact solution and the proposed method. The results reveal that the method is very effective and simple. 2005 Elsevier Inc. All rights reserved. Keywords: Riccati equation; Homotopy perturbation method; Adomian decomposition method
1. Introduction Riccati differential equation is an important equation in the optimal control literature. Solution of this equation can be obtained using classical numerical method as Runge–Kutta method or the forward Euler method. We can use AdomianÕs decomposition method (ADM) [1,2] to solve the nonlinear Riccati E-mail address: [email protected] 0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.07.035
582
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
differential equation in an analytic form. ADM gives the solution as an infinite series usually converging to an accurate solution. El-Tawil et al. [4] applied the multistage AdomianÕs decomposition method to solving Riccati differential equation and compared the results with standard ADM. The application of the homotopy perturbation method [3,20,21] in nonlinear problems has been devoted by scientists and engineers, because this method is to continuously deform a simple problem easy to solve into the difficult problem under study. The most perturbation methods are assumed a small parameter exists, but most nonlinear problems have no small parameter at all. Many new methods, such as variational iterations method [6,9,7,13,22], various modified Lindstedt–Poincare method [15–17,23], are proposed to eliminate the shortcomings arising in the small parameter assumption. A review of recently developed nonlinear analysis methods can be found in [12]. Recently, the applications of homotopy theory among scientists were appeared [5,8,10,11], and the homotopy theory becomes a powerful mathematical tool, when it is successfully coupled with perturbation theory, [18,19]. In this paper, we iterate HeÕs homotopy perturbation method to solve quadratic Riccati differential equation. Comparisons are made between standard ADM and the exact solution and the iterated homotopy perturbation method. The results reveal that the proposed method is very effective and simple. 2. Iterated HeÕs homotopy perturbation method To illustrate the iterated homotopy perturbation method (IHPM), He [11] considered the following nonlinear differential equation: AðuÞ ¼ f ðrÞ;
r 2 Xj ;
ð1Þ
with boundary conditions Bðu; ou=onÞ ¼ 0;
r 2 Cj ;
ð2Þ
where A is a general differential operator, B is a boundary operator, f(r) is a known analytic function, Cj is the boundary of the domain Xj and X = ¨jXj and C = ¨jCj. The operator A can be divided into two parts L and N, where L is linear and N is nonlinear. Therefore (1) can be rewritten as follows: LðuÞ þ N ðuÞ ¼ f ðrÞ.
ð3Þ
He [14] constructed a homotopy vj ðr; pÞ : Xj ½0; 1 ! R which satisfies Hðvj ; pÞ ¼ ð1 pÞ½Lðvj Þ Lðy j;0 Þ þ p½Aðvj Þ f ðrÞ ¼ 0
ð4Þ
Hðvj ; pÞ ¼ Lðvj Þ Lðy j;0 Þ þ pLðy j;0 Þ þ p½N ðvj Þ f ðrÞ ¼ 0;
ð5Þ
or
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
583
where r 2 Xj and p 2 [0, 1] is an imbedding parameter, yj,0 is an initial approximation of (1). Hence, it is obvious that Hðvj ; 0Þ ¼ Lðvj Þ Lðy j;0 Þ ¼ 0; Hðvj ; 1Þ ¼ Aðvj Þ f ðrÞ ¼ 0; and the changing process of p from 0 to 1, is just that of Hðvj ; pÞ from L(vj) L(yj,0) to A(vj) f(r). In topology, this is called deformation, L(vj) L(yj,0) and A(vj) f(r) are called homotopic. Applying the perturbation technique [24], due to the fact that 0 6 p 6 1 can be considered as a small parameter, we can assume that the solution of (4) or (5) can be expressed as a series in p, as follows: vj ¼ vj;0 þ pvj;1 þ p2 vj;2 þ p3 vj;3 þ ;
ð6Þ
when p ! 1, (4) or (5) corresponds to (3) and (6) becomes the approximate solution of (3), i.e., u ¼ lim vj ¼ vj;0 þ vj;1 þ vj;2 þ . p!1
ð7Þ
The series (7) is convergent for most cases, and also the rate of convergence depends on A(vj), [10]. 3. Numerical implementation We will consider the quadratic Riccati equation [4,26,25], dY ðtÞ ¼ 2Y ðtÞ Y 2 ðtÞ þ 1; dt
ð8Þ
with initial condition Y(0) = 0. The exact solution of (8), Fig. 1, was found to be !! pffiffiffi pffiffiffi pffiffiffi 1 21 Y ðtÞ ¼ 1 þ 2 tanh 2t þ log pffiffiffi . 2 2þ1
Fig. 1. The exact solution.
584
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
Expanding Y(t) using Taylor expansion about t = 0 gives 1 1 7 7 53 7 71 8 t þ t þ . Y ðtÞ ¼ t þ t2 þ t3 t4 t5 t6 þ 3 3 15 45 315 315 Applying ADM [1] to solve (8), starting from t = 0, we get Y 0 ðtÞ ¼ t; 1 Y 1 ðtÞ ¼ t2 t3 ; 3 2 3 2 4 2 Y 2 ðtÞ ¼ t t þ t5 ; 3 3 15 and hence 1 2 2 Y ðtÞ ffi t þ t2 þ t3 t4 þ t5 ; 3 3 15
8t 2 ½0; T ;
where T = 4. Fig. 2 illustrates the solution obtained compared with the exact solution. Unfortunately, this solution gives a good approximation only in the neighborhood of the initial time. But, in the optimal control problems, the required solution is the steady state one, which cannot be reached form this scheme [4]. In what follows, we introduce iterated homotopy perturbation method. Applying this method gives the complete solution for Riccati equation over any time horizon and consequently the steady state solution can be reached. Consider (8) defined on the time horizon t 2 [0, T]. Divide [0, T] into n equal subinterval DT = Tj+1 Tj, j = 0, 1, . . . , n 1 with T0 = 0 and Tn = T. Suppose L(u) = du/dt 2u and N(u) = u2 and we assume the initial approximation of (8) for t 2 [Tj, Tj+1] has the form y j;0 ðtÞ ¼ t tj þ uðtj Þ. Substituting (6) into (5), and equating the terms with the identical powers of p, we have
6 4 2 0 0
1
2
3
4
Fig. 2. The exact solution (dashed line), versus the ADM solution (solid line).
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
2 1.5 1 0.5 0 0
1
2
3
4
Fig. 3. Result for DT = 0.5.
Fig. 4. Result for DT = 0.25.
Fig. 5. Result for DT = 0.2.
p0 : Lðvj;0 Þ Lðy j;0 Þ ¼ 0; p1 : Lðvj;1 Þ þ Lðy j;0 Þ þ v2j;0 1 ¼ 0; p2 : Lðvj;2 Þ þ 2vj;0 vj;1 ¼ 0;
vj;1 ðtj Þ ¼ 0;
vj;2 ðtj Þ ¼ 0.
For simplicity we always set vj,0(t) = yj,0(t). Accordingly we have
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S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
Fig. 6. Result for DT = 0.1.
Fig. 7. Result for DT = 0.4.
Fig. 8. Result for DT = 0.25.
1 v0;1 ðtÞ ¼ ð1 þ e2t 2t þ 2t2 Þ; 4 1 v0;2 ðtÞ ¼ ðt2 e2t t2 þ 2t3 Þ. 4 Applying IHPM, the results obtained are illustrated in Figs. 3–6 for various DT. So we can see clearly that the obtained solution is of high accuracy.
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
587
Fig. 9. Result for DT = 0.2.
Fig. 10. Result for DT = 0.1.
Let we assume the initial approximation of (8) for t 2 [Tj, Tj+1] has the form y j;0 ðtÞ ¼
uðtj Þ t. tj
Applying IHPM, the results obtained are illustrated in Figs. 7–10 for various DT. 4. Conclusion In this work, we successfully apply the iterated HeÕs homotopy perturbation method and compared with standard AdomianÕs decomposition method to solving quadratic Riccati differential equation. The main advantage of iterated homotopy perturbation method compared to ADM is the capability to achieve the solution for the whole time horizon with only two terms.
Acknowledgement Many thanks are due to financial support from the Imam Khomeini International University of Iran.
588
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
References [1] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, 1994. [2] G. Adomian, R. Rach, On the solution of algebraic equations by the decomposition method, Math. Anal. Appl. 105 (1985) 141–166. [3] M. El-Shahed, Application of HeÕs homotopy perturbation method to VolterraÕs integrodifferential equation, Int. J. Nonlinear Sci. Numer. Simul. 6 (2) (2005) 163–168. [4] M.A. El-Tawil, A.A. Bahnasawi, A. Abdel-Naby, Solving Riccati differential equation using AdomianÕs decomposition method, Appl. Math. Comput. 157 (2004) 503–514. [5] C. Hillermeier, Generalized homotopy approach to multiobjective optimization, Int. J. Optim. Theory Appl. 110 (3) (2001) 557–583. [6] J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng. 167 (1–2) (1998) 57–68. [7] J.H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Eng. 167 (1–2) (1998) 69–73. [8] J.H. He, An approximate solution technique depending upon an artificial parameter, Commun. Nonlinear Sci. Simulat. 3 (2) (1998) 92–97. [9] J.H. He, Variational iteration method: a kind of nonlinear analytical technique: some examples, Int. J. Non-Linear Mech. 34 (4) (1999) 699–708. [10] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178 (3/4) (1999) 257–262. [11] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech. 35 (1) (2000) 37–43. [12] J.H. He, A review on some new recently developed nonlinear analytical techniques, Int. J. Nonlinear Sci. Numer. Simul. 1 (1) (2000) 51–70. [13] J.H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput. 114 (2–3) (2000) 115–123. [14] J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlinear Sci. Numer. Simul. 2 (3) (2001) 257–264. [15] J.H. He, Modified Lindsted–Poincare methods for some strongly nonlinear oscillations, Part III: Double series expansion, Int. J. Nonlinear Sci. Numer. Simul. 2 (4) (2001) 317– 320. [16] J.H. He, Modified Lindstedt–Poincare methods for some strongly non-linear oscillations, Part I: expansion of a constant, Int. J. Nonlinear Mech. 37 (2) (2002) 309–314. [17] J.H. He, Modified Lindstedt–Poincare methods for some strongly non-linear oscillations, Part II: a new transformation, Int. J. Nonlinear Mech. 37 (2) (2002) 315–320. [18] J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput. 151 (2004) 287–292. [19] J.H. He, Comparison of homotopy perturbtion method and homotopy analysis method, Appl. Math. Comput. 156 (2004) 527–539. [20] J.H. He, Asymptotology by homotopy perturbtion method, Appl. Math. Comput. 156 (2004) 591–596. [21] J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul. 6 (2) (2005) 207–208. [22] J.H. He, Y.Q. Wan, Q. Guo, An iteration formulation for normalized diode characteristics, Int. J. Circ. Theor. Appl. 32 (6) (2004) 629–632. [23] H.M. Liu, Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt–Poincare method, Chaos, Solitons & Fractals 23 (2) (2005) 577–579. [24] A.H. Nayfeh, Problems in Perturbation, J. Wiley, New York, 1985.
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
589
[25] Q. Wang, Y. Chen, H.Q. Zhang, A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation, Chaos, Solitons & Fractals 25 (5) (2005) 1019–1028. [26] F. Xie, X. Gao, Exact travelling wave solutions for a class of nonlinear partial differential equations, Chaos, Solitons & Fractals 19 (5) (2004) 1113–1117.
Iterated HeÕs homotopy perturbation method for quadratic Riccati differential equation S. Abbasbandy Department of Mathematics, Imam Khomeini International University, P.O. Box 288, Ghazvin 34194, Iran
Abstract In this paper, iterated HeÕs homotopy perturbation method is proposed to solving quadratic Riccati differential equation. Comparisons are made between AdomianÕs decomposition method (ADM) and the exact solution and the proposed method. The results reveal that the method is very effective and simple. 2005 Elsevier Inc. All rights reserved. Keywords: Riccati equation; Homotopy perturbation method; Adomian decomposition method
1. Introduction Riccati differential equation is an important equation in the optimal control literature. Solution of this equation can be obtained using classical numerical method as Runge–Kutta method or the forward Euler method. We can use AdomianÕs decomposition method (ADM) [1,2] to solve the nonlinear Riccati E-mail address: [email protected] 0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.07.035
582
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
differential equation in an analytic form. ADM gives the solution as an infinite series usually converging to an accurate solution. El-Tawil et al. [4] applied the multistage AdomianÕs decomposition method to solving Riccati differential equation and compared the results with standard ADM. The application of the homotopy perturbation method [3,20,21] in nonlinear problems has been devoted by scientists and engineers, because this method is to continuously deform a simple problem easy to solve into the difficult problem under study. The most perturbation methods are assumed a small parameter exists, but most nonlinear problems have no small parameter at all. Many new methods, such as variational iterations method [6,9,7,13,22], various modified Lindstedt–Poincare method [15–17,23], are proposed to eliminate the shortcomings arising in the small parameter assumption. A review of recently developed nonlinear analysis methods can be found in [12]. Recently, the applications of homotopy theory among scientists were appeared [5,8,10,11], and the homotopy theory becomes a powerful mathematical tool, when it is successfully coupled with perturbation theory, [18,19]. In this paper, we iterate HeÕs homotopy perturbation method to solve quadratic Riccati differential equation. Comparisons are made between standard ADM and the exact solution and the iterated homotopy perturbation method. The results reveal that the proposed method is very effective and simple. 2. Iterated HeÕs homotopy perturbation method To illustrate the iterated homotopy perturbation method (IHPM), He [11] considered the following nonlinear differential equation: AðuÞ ¼ f ðrÞ;
r 2 Xj ;
ð1Þ
with boundary conditions Bðu; ou=onÞ ¼ 0;
r 2 Cj ;
ð2Þ
where A is a general differential operator, B is a boundary operator, f(r) is a known analytic function, Cj is the boundary of the domain Xj and X = ¨jXj and C = ¨jCj. The operator A can be divided into two parts L and N, where L is linear and N is nonlinear. Therefore (1) can be rewritten as follows: LðuÞ þ N ðuÞ ¼ f ðrÞ.
ð3Þ
He [14] constructed a homotopy vj ðr; pÞ : Xj ½0; 1 ! R which satisfies Hðvj ; pÞ ¼ ð1 pÞ½Lðvj Þ Lðy j;0 Þ þ p½Aðvj Þ f ðrÞ ¼ 0
ð4Þ
Hðvj ; pÞ ¼ Lðvj Þ Lðy j;0 Þ þ pLðy j;0 Þ þ p½N ðvj Þ f ðrÞ ¼ 0;
ð5Þ
or
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
583
where r 2 Xj and p 2 [0, 1] is an imbedding parameter, yj,0 is an initial approximation of (1). Hence, it is obvious that Hðvj ; 0Þ ¼ Lðvj Þ Lðy j;0 Þ ¼ 0; Hðvj ; 1Þ ¼ Aðvj Þ f ðrÞ ¼ 0; and the changing process of p from 0 to 1, is just that of Hðvj ; pÞ from L(vj) L(yj,0) to A(vj) f(r). In topology, this is called deformation, L(vj) L(yj,0) and A(vj) f(r) are called homotopic. Applying the perturbation technique [24], due to the fact that 0 6 p 6 1 can be considered as a small parameter, we can assume that the solution of (4) or (5) can be expressed as a series in p, as follows: vj ¼ vj;0 þ pvj;1 þ p2 vj;2 þ p3 vj;3 þ ;
ð6Þ
when p ! 1, (4) or (5) corresponds to (3) and (6) becomes the approximate solution of (3), i.e., u ¼ lim vj ¼ vj;0 þ vj;1 þ vj;2 þ . p!1
ð7Þ
The series (7) is convergent for most cases, and also the rate of convergence depends on A(vj), [10]. 3. Numerical implementation We will consider the quadratic Riccati equation [4,26,25], dY ðtÞ ¼ 2Y ðtÞ Y 2 ðtÞ þ 1; dt
ð8Þ
with initial condition Y(0) = 0. The exact solution of (8), Fig. 1, was found to be !! pffiffiffi pffiffiffi pffiffiffi 1 21 Y ðtÞ ¼ 1 þ 2 tanh 2t þ log pffiffiffi . 2 2þ1
Fig. 1. The exact solution.
584
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
Expanding Y(t) using Taylor expansion about t = 0 gives 1 1 7 7 53 7 71 8 t þ t þ . Y ðtÞ ¼ t þ t2 þ t3 t4 t5 t6 þ 3 3 15 45 315 315 Applying ADM [1] to solve (8), starting from t = 0, we get Y 0 ðtÞ ¼ t; 1 Y 1 ðtÞ ¼ t2 t3 ; 3 2 3 2 4 2 Y 2 ðtÞ ¼ t t þ t5 ; 3 3 15 and hence 1 2 2 Y ðtÞ ffi t þ t2 þ t3 t4 þ t5 ; 3 3 15
8t 2 ½0; T ;
where T = 4. Fig. 2 illustrates the solution obtained compared with the exact solution. Unfortunately, this solution gives a good approximation only in the neighborhood of the initial time. But, in the optimal control problems, the required solution is the steady state one, which cannot be reached form this scheme [4]. In what follows, we introduce iterated homotopy perturbation method. Applying this method gives the complete solution for Riccati equation over any time horizon and consequently the steady state solution can be reached. Consider (8) defined on the time horizon t 2 [0, T]. Divide [0, T] into n equal subinterval DT = Tj+1 Tj, j = 0, 1, . . . , n 1 with T0 = 0 and Tn = T. Suppose L(u) = du/dt 2u and N(u) = u2 and we assume the initial approximation of (8) for t 2 [Tj, Tj+1] has the form y j;0 ðtÞ ¼ t tj þ uðtj Þ. Substituting (6) into (5), and equating the terms with the identical powers of p, we have
6 4 2 0 0
1
2
3
4
Fig. 2. The exact solution (dashed line), versus the ADM solution (solid line).
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
2 1.5 1 0.5 0 0
1
2
3
4
Fig. 3. Result for DT = 0.5.
Fig. 4. Result for DT = 0.25.
Fig. 5. Result for DT = 0.2.
p0 : Lðvj;0 Þ Lðy j;0 Þ ¼ 0; p1 : Lðvj;1 Þ þ Lðy j;0 Þ þ v2j;0 1 ¼ 0; p2 : Lðvj;2 Þ þ 2vj;0 vj;1 ¼ 0;
vj;1 ðtj Þ ¼ 0;
vj;2 ðtj Þ ¼ 0.
For simplicity we always set vj,0(t) = yj,0(t). Accordingly we have
585
586
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
Fig. 6. Result for DT = 0.1.
Fig. 7. Result for DT = 0.4.
Fig. 8. Result for DT = 0.25.
1 v0;1 ðtÞ ¼ ð1 þ e2t 2t þ 2t2 Þ; 4 1 v0;2 ðtÞ ¼ ðt2 e2t t2 þ 2t3 Þ. 4 Applying IHPM, the results obtained are illustrated in Figs. 3–6 for various DT. So we can see clearly that the obtained solution is of high accuracy.
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
587
Fig. 9. Result for DT = 0.2.
Fig. 10. Result for DT = 0.1.
Let we assume the initial approximation of (8) for t 2 [Tj, Tj+1] has the form y j;0 ðtÞ ¼
uðtj Þ t. tj
Applying IHPM, the results obtained are illustrated in Figs. 7–10 for various DT. 4. Conclusion In this work, we successfully apply the iterated HeÕs homotopy perturbation method and compared with standard AdomianÕs decomposition method to solving quadratic Riccati differential equation. The main advantage of iterated homotopy perturbation method compared to ADM is the capability to achieve the solution for the whole time horizon with only two terms.
Acknowledgement Many thanks are due to financial support from the Imam Khomeini International University of Iran.
588
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
References [1] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, 1994. [2] G. Adomian, R. Rach, On the solution of algebraic equations by the decomposition method, Math. Anal. Appl. 105 (1985) 141–166. [3] M. El-Shahed, Application of HeÕs homotopy perturbation method to VolterraÕs integrodifferential equation, Int. J. Nonlinear Sci. Numer. Simul. 6 (2) (2005) 163–168. [4] M.A. El-Tawil, A.A. Bahnasawi, A. Abdel-Naby, Solving Riccati differential equation using AdomianÕs decomposition method, Appl. Math. Comput. 157 (2004) 503–514. [5] C. Hillermeier, Generalized homotopy approach to multiobjective optimization, Int. J. Optim. Theory Appl. 110 (3) (2001) 557–583. [6] J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng. 167 (1–2) (1998) 57–68. [7] J.H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Eng. 167 (1–2) (1998) 69–73. [8] J.H. He, An approximate solution technique depending upon an artificial parameter, Commun. Nonlinear Sci. Simulat. 3 (2) (1998) 92–97. [9] J.H. He, Variational iteration method: a kind of nonlinear analytical technique: some examples, Int. J. Non-Linear Mech. 34 (4) (1999) 699–708. [10] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178 (3/4) (1999) 257–262. [11] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech. 35 (1) (2000) 37–43. [12] J.H. He, A review on some new recently developed nonlinear analytical techniques, Int. J. Nonlinear Sci. Numer. Simul. 1 (1) (2000) 51–70. [13] J.H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput. 114 (2–3) (2000) 115–123. [14] J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlinear Sci. Numer. Simul. 2 (3) (2001) 257–264. [15] J.H. He, Modified Lindsted–Poincare methods for some strongly nonlinear oscillations, Part III: Double series expansion, Int. J. Nonlinear Sci. Numer. Simul. 2 (4) (2001) 317– 320. [16] J.H. He, Modified Lindstedt–Poincare methods for some strongly non-linear oscillations, Part I: expansion of a constant, Int. J. Nonlinear Mech. 37 (2) (2002) 309–314. [17] J.H. He, Modified Lindstedt–Poincare methods for some strongly non-linear oscillations, Part II: a new transformation, Int. J. Nonlinear Mech. 37 (2) (2002) 315–320. [18] J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput. 151 (2004) 287–292. [19] J.H. He, Comparison of homotopy perturbtion method and homotopy analysis method, Appl. Math. Comput. 156 (2004) 527–539. [20] J.H. He, Asymptotology by homotopy perturbtion method, Appl. Math. Comput. 156 (2004) 591–596. [21] J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul. 6 (2) (2005) 207–208. [22] J.H. He, Y.Q. Wan, Q. Guo, An iteration formulation for normalized diode characteristics, Int. J. Circ. Theor. Appl. 32 (6) (2004) 629–632. [23] H.M. Liu, Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt–Poincare method, Chaos, Solitons & Fractals 23 (2) (2005) 577–579. [24] A.H. Nayfeh, Problems in Perturbation, J. Wiley, New York, 1985.
S. Abbasbandy / Appl. Math. Comput. 175 (2006) 581–589
589
[25] Q. Wang, Y. Chen, H.Q. Zhang, A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation, Chaos, Solitons & Fractals 25 (5) (2005) 1019–1028. [26] F. Xie, X. Gao, Exact travelling wave solutions for a class of nonlinear partial differential equations, Chaos, Solitons & Fractals 19 (5) (2004) 1113–1117.