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Asymptotology by homotopy perturbation method
Applied Mathematics and Computation 156 (2004) 591–596 www.elsevier.com/locate/amc
Asymptotology by homotopy perturbation method J.-H. He College of Science, Shanghai Donghua University, 1882 Yan’an Xilu Road, P.O. Box 471, Shanghai 20051, China
Abstract In the paper, an heuristical example is given to illustrate the basic idea of the homotopy perturbation method, so that homotopy perturbation method has made all that is necessary simple, and all that is complex unnecessary. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Perturbation method; Asymptotic method; Homotopy
1. Introduction As V.M. Alexandrov wrote in the foreword of a popular science book ‘‘Asymtotology: ideas, methods, and applications’’, asymptotic methods belong to the, perhaps, most romantic area of modern mathematics [2]. Though computer science is growing very fast, and numerical simulation is applied everywhere, non-numerical issues will still play a large role. There exists now much literature on asymptotic methods, hence, our reference to the literature will not be exhaustive. Rather, our purpose in this paper is to present a new approach called the homotopy perturbation method through an illustrating example, so that everyone can catch the basic idea of the homotopy perturbation method, and apply the method to all branches of human knowledge.
E-mail address: [email protected] (J.-H. He). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.011
592
J.-H. He / Appl. Math. Comput. 156 (2004) 591–596
2. Choice of small parameters Perturbation methods provide the most versatile tools available in nonlinear analysis of engineering problems, and they are constantly being developed and applied to ever more complex problems. But, like other non-linear asymptotic techniques, perturbation methods have their own limitations: 1. Almost all perturbation methods are based on such an assumption that a small parameter must exist in an equation. This so-called small parameter assumption greatly restricts applications of perturbation techniques, as is well known, an overwhelming majority of non-linear problems, especially those having strong non-linearity, have no small parameters at all. 2. It is even more difficult how to determine the so-called small parameter, which seems to be a special art requiring special techniques. An appropriate choice of small parameter may lead to ideal results, however, an unsuitable choice of small parameter results in badly effects, sometimes seriously. 3. Even if there exists a suitable small parameter, the approximate solutions solved by the perturbation methods are valid, in most cases, only for the small values of the parameter. There exist some alternative analytical asymptotic approaches, such as the non-perturbative method [4], modified Lindstedt–Poincare method [10], variational iteration method [5], d-method [1–3], homotopy perturbation method [6,7,14], bookkeeping artificial parameter perturbation method [9]. A recent study [11] also reveals that the numerical technique can also be powerfully applied to perturbation method. A review of recently developed analytical techniques can be found in Refs. [8,13]. There also exist a wide body of literature dealing with the problem of approximate solutions to non-linear equations with various different methodologies.
3. An illustrative example ‘‘In learning science examples are useful than rules’’ (Isaac Newton). To illustrate the non-uniqueness of choice of a parameter for an asymptotic expansion, let us consider a model example––the algebraic equation [2,3] x5 þ x 1 ¼ 0:
ð1Þ
Using NewtonÕs method, we compute that x ¼ 0:75487767; . . . :
ð2Þ
J.-H. He / Appl. Math. Comput. 156 (2004) 591–596
593
Due to no explicit or obvious small parameter in the equation, therefore there is a freedom of choice for a formal small parameter e. Let us consider different possibilities of introducing such a small parameter. 3.1. d-Method The basic idea of the d-method [3] is to introduce an artificial parameter d in the exponent of the non-linear term in (1), x1þd þ x 1 ¼ 0:
ð3Þ
The solution is assumed to be expanded in a power series in d x ¼ c0 þ dc1 þ d2 c2 þ
ð4Þ
The coefficients of this series may be computed easily. The first few coefficients are [2,3] c0 ¼ 0:5; c1 ¼ 0:17328; c2 ¼ 0:08664; c3 ¼ 0:05139; c4 ¼ 0:03377; c5 ¼ 0:02377; c6 ¼ 0:01758: For d ¼ 4 the series surely diverges, and very fast too. The sum of the first six terms equals )54.3224. Comments on the method can be found in Ref. [12]. In engineering, we always interest in the first few terms, it requires cumbersome hard work for longer terms. 3.2. Weak coupling approximation [2,3] Consider the equation ex5 þ x 1 ¼ 0:
ð5Þ
Representing x by a series in powers of e x ¼ x0 þ ex1 þ e2 x2 þ
ð6Þ
By standard steps as illustrated in each textbook of perturbation method, we have x0 ¼ 1, x1 ¼ 1, x2 ¼ 5; . . .. The radius of convergence of the series is R ¼ 0:08192. Hence, for e ¼ 1 the series (6) also diverges, and very fast too. 3.3. Strong coupling approximation [2,3] Now consider the equation x5 þ ex 1 ¼ 0:
ð7Þ
Supposing the solution of Eq. (7) can be expressed in the form of a power series in e as illustrated in (6).
594
J.-H. He / Appl. Math. Comput. 156 (2004) 591–596
Substituting (6) into (7), and equating the coefficients of equal powers of e, the coefficients of the series can be easily determined: x0 ¼ 1, x1 ¼ 1, x2 ¼ 1=25; . . .. Though the present series converges to the exact solution, but it is very slow.
4. Homotopy perturbation method Unlike the previous methods, we will use the homotopy perturbation method [6,7,14] to find the solution. We construct the following homotopy ð1 pÞ½ð1 þ bÞx 1 þ pðx5 þ x 1Þ ¼ 0;
p 2 ½0; 1;
ð8Þ
or ð1 þ bÞx 1 ¼ pðbx x5 Þ;
p 2 ½0; 1;
ð9Þ
where b is an unknown constant, if x ¼ 1=ð1 þ bÞ is an exact solution, then bx x5 vanishes completely. So the constant b can be determined by the following relation Z 1 2 ðbx x5 Þ dx ! min; ð10Þ 0
which leads to the result b ¼ 3=7. The embedding parameter p monotonically increases from zero to unit as the trivial problem, ð1 þ bÞx ¼ 1, is continuously deformed to the original problem, Eq. (1). So if we can construct an iteration formula for the Eq. (9), the series of approximations comes along the solution path, by incrementing the imbedding parameter from zero to one; this continuously maps the initial solution into the solution of the original Eq. (1). Due to the fact that 0 6 p 6 1, so the embedding parameter can be considered as a ‘‘small parameter’’, and the Eq. (9) is called perturbation equation with an embedding parameter. Applying the perturbation technique, we can assume that the solution of the Eq. (9) can be expressed as x ¼ c0 þ pc1 þ p2 c2 þ
ð11Þ
The coefficients of this series can be computed very easily. We write the first few coefficients c0 ¼
1 ¼ 0:7; 1þb
J.-H. He / Appl. Math. Comput. 156 (2004) 591–596
c1 ¼
bc0 c50 ¼ 0:09235; 1þb
c2 ¼
bc1 5c40 c1 ¼ 0:032355: 1þb
595
So we obtain the approximate solutions to order 2 x0 ¼ c0 ¼ 0:7; x1 ¼ c0 þ c1 ¼ 0:79235; x2 ¼ c0 þ c1 þ c2 ¼ 0:75999: The second-order approximate solution differs from the exact answer by only 0.6%. By simple parallel operation, higher-order approximation solutions with higher accuracy can be readily obtained. It is a simple way, and we will not list hereby. There exists alternative approaches to the construction of homotopy. Now we re-write Eq. (1) in the form x ¼ 1 x5 ¼ ð1 xÞð1 þ x þ x2 þ x3 þ x4 Þ:
ð12Þ
We can embed an homotopy parameter in the following forms x ¼ ð1 xÞb1 þ pðx þ x2 þ x3 þ x4 Þc; 2
3
4
x ¼ ð1 xÞb1 þ x þ pðx þ x þ x Þc;
p 2 ½0; 1;
ð13Þ
p 2 ½0; 1:
ð14Þ
Few iteration steps lead to high accuracy.
5. Conclusions We propose a new perturbation technique which is advantageous over the d-method. We hope that the present theory can be applied to most non-linear problems in engineering.
References [1] I. Andrianov, J. Awrejcewicz, Construction of periodic solution to partial differential equations with nonlinear boundary conditions, International Journal of Nonlinear Sciences and Numerical Simulation 1 (4) (2000) 327–332. [2] I. Andrianov, L. Manevitch, Asymptotology: Ideas, Methods, and Applications, Kluwer Academic Publishers, 2003. [3] C.M. Bender, K.S. Pinsky, L.M. Simmons, A new perturbative approach to nonlinear problems, Journal of Mathematical Physics 30 (7) (1989) 1447–1455. [4] B. Delamotte, Nonperturbative method for solving differential equations and finding limit cycles, Physical Review Letters 70 (1993) 3361–3364.
596
J.-H. He / Appl. Math. Comput. 156 (2004) 591–596
[5] J.H. He, Variational iteration method: a kind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics 34 (4) (1999) 699–708. [6] J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178 (1999) 257–262. [7] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Nonlinear Mechanics 35 (1) (2000) 37–43. [8] J.H. He, A review on some new recently developed nonlinear analytical techniques, International Journal of Nonlinear Sciences and Numerical Simulation 1 (1) (2000) 51–70. [9] J.H. He, Bookkeeping parameter in perturbation methods, International Journal of Nonlinear Science and Numerical Simulation 2 (3) (2001) 257–264. [10] J.H. He, Modified Lindstedt–Poincare methods for some strongly nonlinear oscillations. Part III: Double series expansion, International Journal of Nonlinear Science and Numerical Simulation 2 (4) (2001) 317–320. [11] J.H. He, Iteration perturbation method for strongly nonlinear oscillations, Journal of Vibration and Control 7 (5) (2001) 631–642. [12] J.H. He, A note on delta-perturbation expansion method, Applied Mathematics and Mechanics 23 (6) (2002) 634–638. [13] J.H. He, Recent developments in asymptotic methods for nonlinear ordinary equations, in: Invited Lecture Delivered at 10th International Colloquium on Numerical Analysis and Computer Science with Applications, Plovidiv, Bulgaria, 12–17 August 2001International Journal of Computational and Numerical Analysis and Applications 2 (2) (2002) 127–190. [14] J.H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135 (2003) 73–79.
Asymptotology by homotopy perturbation method J.-H. He College of Science, Shanghai Donghua University, 1882 Yan’an Xilu Road, P.O. Box 471, Shanghai 20051, China
Abstract In the paper, an heuristical example is given to illustrate the basic idea of the homotopy perturbation method, so that homotopy perturbation method has made all that is necessary simple, and all that is complex unnecessary. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Perturbation method; Asymptotic method; Homotopy
1. Introduction As V.M. Alexandrov wrote in the foreword of a popular science book ‘‘Asymtotology: ideas, methods, and applications’’, asymptotic methods belong to the, perhaps, most romantic area of modern mathematics [2]. Though computer science is growing very fast, and numerical simulation is applied everywhere, non-numerical issues will still play a large role. There exists now much literature on asymptotic methods, hence, our reference to the literature will not be exhaustive. Rather, our purpose in this paper is to present a new approach called the homotopy perturbation method through an illustrating example, so that everyone can catch the basic idea of the homotopy perturbation method, and apply the method to all branches of human knowledge.
E-mail address: [email protected] (J.-H. He). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.011
592
J.-H. He / Appl. Math. Comput. 156 (2004) 591–596
2. Choice of small parameters Perturbation methods provide the most versatile tools available in nonlinear analysis of engineering problems, and they are constantly being developed and applied to ever more complex problems. But, like other non-linear asymptotic techniques, perturbation methods have their own limitations: 1. Almost all perturbation methods are based on such an assumption that a small parameter must exist in an equation. This so-called small parameter assumption greatly restricts applications of perturbation techniques, as is well known, an overwhelming majority of non-linear problems, especially those having strong non-linearity, have no small parameters at all. 2. It is even more difficult how to determine the so-called small parameter, which seems to be a special art requiring special techniques. An appropriate choice of small parameter may lead to ideal results, however, an unsuitable choice of small parameter results in badly effects, sometimes seriously. 3. Even if there exists a suitable small parameter, the approximate solutions solved by the perturbation methods are valid, in most cases, only for the small values of the parameter. There exist some alternative analytical asymptotic approaches, such as the non-perturbative method [4], modified Lindstedt–Poincare method [10], variational iteration method [5], d-method [1–3], homotopy perturbation method [6,7,14], bookkeeping artificial parameter perturbation method [9]. A recent study [11] also reveals that the numerical technique can also be powerfully applied to perturbation method. A review of recently developed analytical techniques can be found in Refs. [8,13]. There also exist a wide body of literature dealing with the problem of approximate solutions to non-linear equations with various different methodologies.
3. An illustrative example ‘‘In learning science examples are useful than rules’’ (Isaac Newton). To illustrate the non-uniqueness of choice of a parameter for an asymptotic expansion, let us consider a model example––the algebraic equation [2,3] x5 þ x 1 ¼ 0:
ð1Þ
Using NewtonÕs method, we compute that x ¼ 0:75487767; . . . :
ð2Þ
J.-H. He / Appl. Math. Comput. 156 (2004) 591–596
593
Due to no explicit or obvious small parameter in the equation, therefore there is a freedom of choice for a formal small parameter e. Let us consider different possibilities of introducing such a small parameter. 3.1. d-Method The basic idea of the d-method [3] is to introduce an artificial parameter d in the exponent of the non-linear term in (1), x1þd þ x 1 ¼ 0:
ð3Þ
The solution is assumed to be expanded in a power series in d x ¼ c0 þ dc1 þ d2 c2 þ
ð4Þ
The coefficients of this series may be computed easily. The first few coefficients are [2,3] c0 ¼ 0:5; c1 ¼ 0:17328; c2 ¼ 0:08664; c3 ¼ 0:05139; c4 ¼ 0:03377; c5 ¼ 0:02377; c6 ¼ 0:01758: For d ¼ 4 the series surely diverges, and very fast too. The sum of the first six terms equals )54.3224. Comments on the method can be found in Ref. [12]. In engineering, we always interest in the first few terms, it requires cumbersome hard work for longer terms. 3.2. Weak coupling approximation [2,3] Consider the equation ex5 þ x 1 ¼ 0:
ð5Þ
Representing x by a series in powers of e x ¼ x0 þ ex1 þ e2 x2 þ
ð6Þ
By standard steps as illustrated in each textbook of perturbation method, we have x0 ¼ 1, x1 ¼ 1, x2 ¼ 5; . . .. The radius of convergence of the series is R ¼ 0:08192. Hence, for e ¼ 1 the series (6) also diverges, and very fast too. 3.3. Strong coupling approximation [2,3] Now consider the equation x5 þ ex 1 ¼ 0:
ð7Þ
Supposing the solution of Eq. (7) can be expressed in the form of a power series in e as illustrated in (6).
594
J.-H. He / Appl. Math. Comput. 156 (2004) 591–596
Substituting (6) into (7), and equating the coefficients of equal powers of e, the coefficients of the series can be easily determined: x0 ¼ 1, x1 ¼ 1, x2 ¼ 1=25; . . .. Though the present series converges to the exact solution, but it is very slow.
4. Homotopy perturbation method Unlike the previous methods, we will use the homotopy perturbation method [6,7,14] to find the solution. We construct the following homotopy ð1 pÞ½ð1 þ bÞx 1 þ pðx5 þ x 1Þ ¼ 0;
p 2 ½0; 1;
ð8Þ
or ð1 þ bÞx 1 ¼ pðbx x5 Þ;
p 2 ½0; 1;
ð9Þ
where b is an unknown constant, if x ¼ 1=ð1 þ bÞ is an exact solution, then bx x5 vanishes completely. So the constant b can be determined by the following relation Z 1 2 ðbx x5 Þ dx ! min; ð10Þ 0
which leads to the result b ¼ 3=7. The embedding parameter p monotonically increases from zero to unit as the trivial problem, ð1 þ bÞx ¼ 1, is continuously deformed to the original problem, Eq. (1). So if we can construct an iteration formula for the Eq. (9), the series of approximations comes along the solution path, by incrementing the imbedding parameter from zero to one; this continuously maps the initial solution into the solution of the original Eq. (1). Due to the fact that 0 6 p 6 1, so the embedding parameter can be considered as a ‘‘small parameter’’, and the Eq. (9) is called perturbation equation with an embedding parameter. Applying the perturbation technique, we can assume that the solution of the Eq. (9) can be expressed as x ¼ c0 þ pc1 þ p2 c2 þ
ð11Þ
The coefficients of this series can be computed very easily. We write the first few coefficients c0 ¼
1 ¼ 0:7; 1þb
J.-H. He / Appl. Math. Comput. 156 (2004) 591–596
c1 ¼
bc0 c50 ¼ 0:09235; 1þb
c2 ¼
bc1 5c40 c1 ¼ 0:032355: 1þb
595
So we obtain the approximate solutions to order 2 x0 ¼ c0 ¼ 0:7; x1 ¼ c0 þ c1 ¼ 0:79235; x2 ¼ c0 þ c1 þ c2 ¼ 0:75999: The second-order approximate solution differs from the exact answer by only 0.6%. By simple parallel operation, higher-order approximation solutions with higher accuracy can be readily obtained. It is a simple way, and we will not list hereby. There exists alternative approaches to the construction of homotopy. Now we re-write Eq. (1) in the form x ¼ 1 x5 ¼ ð1 xÞð1 þ x þ x2 þ x3 þ x4 Þ:
ð12Þ
We can embed an homotopy parameter in the following forms x ¼ ð1 xÞb1 þ pðx þ x2 þ x3 þ x4 Þc; 2
3
4
x ¼ ð1 xÞb1 þ x þ pðx þ x þ x Þc;
p 2 ½0; 1;
ð13Þ
p 2 ½0; 1:
ð14Þ
Few iteration steps lead to high accuracy.
5. Conclusions We propose a new perturbation technique which is advantageous over the d-method. We hope that the present theory can be applied to most non-linear problems in engineering.
References [1] I. Andrianov, J. Awrejcewicz, Construction of periodic solution to partial differential equations with nonlinear boundary conditions, International Journal of Nonlinear Sciences and Numerical Simulation 1 (4) (2000) 327–332. [2] I. Andrianov, L. Manevitch, Asymptotology: Ideas, Methods, and Applications, Kluwer Academic Publishers, 2003. [3] C.M. Bender, K.S. Pinsky, L.M. Simmons, A new perturbative approach to nonlinear problems, Journal of Mathematical Physics 30 (7) (1989) 1447–1455. [4] B. Delamotte, Nonperturbative method for solving differential equations and finding limit cycles, Physical Review Letters 70 (1993) 3361–3364.
596
J.-H. He / Appl. Math. Comput. 156 (2004) 591–596
[5] J.H. He, Variational iteration method: a kind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics 34 (4) (1999) 699–708. [6] J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178 (1999) 257–262. [7] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Nonlinear Mechanics 35 (1) (2000) 37–43. [8] J.H. He, A review on some new recently developed nonlinear analytical techniques, International Journal of Nonlinear Sciences and Numerical Simulation 1 (1) (2000) 51–70. [9] J.H. He, Bookkeeping parameter in perturbation methods, International Journal of Nonlinear Science and Numerical Simulation 2 (3) (2001) 257–264. [10] J.H. He, Modified Lindstedt–Poincare methods for some strongly nonlinear oscillations. Part III: Double series expansion, International Journal of Nonlinear Science and Numerical Simulation 2 (4) (2001) 317–320. [11] J.H. He, Iteration perturbation method for strongly nonlinear oscillations, Journal of Vibration and Control 7 (5) (2001) 631–642. [12] J.H. He, A note on delta-perturbation expansion method, Applied Mathematics and Mechanics 23 (6) (2002) 634–638. [13] J.H. He, Recent developments in asymptotic methods for nonlinear ordinary equations, in: Invited Lecture Delivered at 10th International Colloquium on Numerical Analysis and Computer Science with Applications, Plovidiv, Bulgaria, 12–17 August 2001International Journal of Computational and Numerical Analysis and Applications 2 (2) (2002) 127–190. [14] J.H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135 (2003) 73–79.