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An effective modification of Ji-Huan He’s variational approach to nonlinear singular oscillator:

An effective modification of Ji-Huan He’s variational approach to nonlinear singular oscillator: In the review article “Asymptotic methods for solitary solutions and compactons” (Ji-Huan He, Abstract and Applied Analysis 2012: 916793), the variational approach and the Hamiltonian approach to nonlinear oscillators are systematically discussed. This paper gives an extension of the methods to singular nonlinear oscillator by the iteration perturbation method. Keywords Nonlinear oscillator, polar coordinate, iteration perturbation method Introduction Chinese mathematician, Prof. Ji-Huan He, systematically discussed the variational approach and the Hamiltonian approach to nonlinear oscillators in his review article, where the history and main development of both methods were elucidated. Although He’s methods have been proved to be effective to nonlinear oscillators, extension of the methods still exists. Consider a nonlinear oscillator in the form x € þfxðÞ ¼ 0 (1) with initial conditions ðÞ ðÞ x 0 ¼ A and x_ 0 ¼ 0 (2) where a dot denotes the differentiation with respect to time t, A is a given constant, fðxÞ is a conservative force. 2–5 There are many approaches to nonlinear oscillators, for example, the homotopy perturbation method, 6–8 9–14 He’s frequency–amplitude formulation and other frequency–amplitude formulae, Akbari-Ganji 15,16 17–19 method, and reviews on various methods are available in the literature. In this paper, we will give an extension of He’s variational approach and Hamiltonian approach. 20–24 A variational principle for equation (1) can be easily established, which reads JxðÞ ¼ x_ FxðÞ dt (3) where F is the potential, defined as @F/@x ¼ f. School of Mathematics & Statistics, Nanjing University of Information Science and Technology, Nanjing, PR China School of Management and Engineering, Nanjing University, Nanjing, PR China Corresponding author: Zhaoling Tao, 219, Ningliu Road, Nanjing 210044, PR China. Email: nj_zaolingt@126.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www. creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 1024 Journal of Low Frequency Noise, Vibration and Active Control 38(3–4) The variational approach and the Hamiltonian approach discussed by He can be applied to equation (3) to obtain the frequency of the nonlinear oscillator. This comment shows that we can couple the iteration perturba- tion method with the variational approach or the Hamiltonian approach to search for a periodic solution. Basic process The oscillator (1)–(2) can be rewritten in a system form ðÞ ðÞ y ¼ x_ ; y_ ¼fx ; x 0 ¼ A; y 0 ¼ 0 (4) ðÞ Introducing polar representations for x; y , i.e. ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ xt ¼ rt cos h t ; yt ¼ rt sin h t (5) with initial conditions ðÞ ðÞ r 0 ¼ A and h 0 ¼ 0 (6) Substituting equation (5) into equation (4), we have r_cos h ¼ 1 þ h rsin h (7) ðÞ r_sin h þ rhcos h ¼frcos h (8) It is easy to arrive at the following results ðÞ cos h frcos h h ¼1 þ cos h  (9) ðÞ rsin 2h ðÞ r_ ¼  sin h frcos h (10) According to the iteration perturbation method, we can construct the following iteration scheme _ ðÞ ½ ðÞ ðÞ ðÞ ½ ðÞ ðÞ r t ¼ Fr t ; h t ; h t ¼ Gr t ; h t (11) kþ1 k k kþ1 k k where k ¼ 0; 1; 2 ... ; and FrðÞ ; h and GrðÞ ; h are, respectively, the right-hand sides of equations (9) and (10) based ðÞ ðÞ on having prior functional forms for r t and h t . By averaging over h and some simple operations, we can first 0 0 obtain functions r_ ðÞ t and h ðÞ t , and then r ðÞ t and h ðÞ t . Therefore, approximation to the solution xtðÞ ¼ r ðÞ t 0 0 0 0 1 ðÞ cos h t can be obtained. Example An important and interesting nonlinear differential equation occurs in the modeling of certain phenomena in 26–29 plasma physics. This example corresponds to the odd nonlinear singular oscillator with e a positive constant ðÞ ðÞ x € ¼ ; x 0 ¼ A; x_ 0 ¼ 0 (12) ex 20–24 The variational principle for equation (12) reads 1 1 JxðÞ ¼ x_  lnx dt (13) 2 e Tao and Chen 1025 The corresponding equation system is ðÞ ðÞ y ¼ x_ ; y_ ¼ ; x 0 ¼ A; y 0 ¼ 0 (14) ex With the polar expressions (5) and (6), we have 2 2 er cos h  1 h ¼1 þ (15) er 2 2 ðÞ er cos h  1 sin h r_ ¼ (16) ercos h Integral averaging the right-hand side of equations (15) and (16) over h with one period, between 0 and 2p, we have ðÞ ðÞ r_ t ¼ 0; h t ¼  1= er (17) 0 0 Combing with the initial conditions in equation (6) yields ðÞ r ðÞ t ¼ A; h ðÞ t ¼  t= eA (18) 0 0 The iteration formula with k ¼ 0; 1; 2; ... based on equation (18) becomes 2 2 er cos h  1 h ðÞ t ¼1 þ (19) kþ1 er 2 2 er cos h  1 sin h k k r_ ðÞ t ¼ (20) kþ1 er cos h k k when k ¼ 0, 2 2 er cos h  1 ðÞ h t ¼1 þ (21) er 2 2 er cos h  1 sin h 0 0 ðÞ r_ t ¼ (22) er cos h 0 0 Approximation to the solution xtðÞ ¼ r ðÞ t  cos h ðÞ t can be obtained just after finding r ðÞ t and h ðÞ t from 1 1 1 1 equations (21) and (22) based on equation (18) ðÞ 2  3eA sin 2Wt h ðÞ t ¼ t  (23) 2eA ðÞ 22 þ eA A 1 A 1 ðÞ ðÞ ðÞ r t ¼ A  þ cos Wt  lncos Wt (24) 2W W 2 eA ðÞ with the amplitude-dependent function, WA ,is 2 þ eA W ¼ (25) 2eA 1026 Journal of Low Frequency Noise, Vibration and Active Control 38(3–4) Conclusions Usually, frequency, period or frequency–amplitude relationship are the main factors for nonlinear oscillators. He gave an elementary introduction to nonlinear vibration systems. This paper gives for the first time an alternative approach to singular oscillators, and the results are much better than those in literature. Hereby, we adopt the iteration method into the variational approach under the polar coordinate system, the modification is also an effective way for more oscillators, and it is extremely effective for singular oscillators as illustrated in this paper. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) received no financial support for the research, authorship, and/or publication of this article. ORCID iD Zhao-Ling Tao http://orcid.org/0000-0001-8674-1843 References 1. He J-H. Asymptotic methods for solitary solutions and compactons. Abstr Appl Anal 2012, http://dx.doi.org/10.1155/ 2012/916793. 2. He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004; 151: 287–292. 3. He JH. Homotopy perturbation method with an auxiliary term. Abstr Appl Anal 2012, http://dx.doi.org/10.1155/ 2012/857612. 4. He JH. Homotopy perturbation method with two expanding parameters. Indian J Phys 2014; 88: 193–196. 5. Liu ZJ, Adamu MY, Suleiman E, et al. Hybridization of homotopy perturbation method and Laplace transformation for the partial differential equations. Therm Sci 2017; 21: 1843–1846. 6. He JH. An improved amplitude-frequency formulation for nonlinear oscillators. Int J Nonlinear Sci Numer Simul 2008; 9: 211–212. 7. He JH. An approximate amplitude-frequency relationship for a nonlinear oscillator with discontinuity. Nonlinear Sci Lett A 2016; 7: 77–85. 8. He JH. Amplitude-frequency relationship for conservative nonlinear oscillators with odd nonlinearities. Int J Appl Comput Math 2017; 3: 1557–1560. 9. Tao ZL. Frequency-amplitude relationship of nonlinear oscillators by He’s parameter-expanding method. Chaos Solitons Fractals 2009; 41: 642–645. 10. Ren Z. He’s frequency-amplitude formulation for nonlinear oscillators. Int J Mod Phys B 2011; 25: 2379–2382. 11. Ren Z. Theoretical basis of He frequency-amplitude formulation for nonlinear oscillators. Nonlinear Sci Lett A 2018; 9: 86–90. 12. Dan T and Zhi L. Period/frequency estimation of a nonlinear oscillator. J Low Frequency Noise Vib Active Control 2018, https://doi.org/10.1177/1461348418756013. 13. Garcıa A. An amplitude-period formula for a second order nonlinear oscillator. Nonlinear Sci Lett A 2017; 8: 340–347. 14. Sua´ rez Antola R. Remarks on an approximate formula for the period of conservative oscillations in nonlinear second order ordinary differential equations. Nonlinear Sci Lett A 2017; 8: 348–351. 15. Balazadeh N and Ganji DD. Akbari-Ganji method for thermal analysis of longitudinal porous fins. Nonlinear Sci Lett A 2018; 9: 1–16. 16. Nimafar M, Akbari MR, Ganji DD, et al. Akbari-Ganji method for vibration under external harmonic load. Nonlinear Sci Lett A 2017; 8: 416–437. 17. He JH. A tutorial review on fractal spacetime and fractional calculus. Int J Theor Phys 2014; 53: 3698–3718. 18. He JH. Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 2006; 20: 1141–1199. 19. He JH. An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int J Mod Phys B 2008; 22: 3487–3578. 20. He JH. An alternative approach to establishment of a variational principle for the torsional problem of piezoelastic beams. Appl Math Lett 2016; 52: 1–3. 21. He JH. Hamilton’s principle for dynamical elasticity. Appl Math Lett 2017; 72: 65–69. 22. He JH. Generalized equilibrium equations for shell derived from a generalized variational principle. Appl Math Lett 2017; 64: 94–100. Tao and Chen 1027 23. Wu Y and He JH. A remark on Samuelson’s variational principle in economics. Appl Math Lett 2018; 84: 143–147. 24. Li XW, Li Y and He JH. On the semi-inverse method and variational principle. Therm Sci 2013; 17: 1565–1568. 25. He JH. Iteration perturbation method for strongly nonlinear oscillations. J Vib Control 2001; 7: 631–642. 26. Depassier MC and Haikala V. Analytic upper and lower bounds for the period of nonlinear oscillators. J Sound Vib 2009; 328: 338–344. 27. Belendez A, Gimeno E, Fernandez E, et al. Accurate approximate solution to nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable. Phys Scr 2008; 77: 065004. 28. Ren Z-F and He J-H. A simple approach to nonlinear oscillators. Phys Lett A 2009; 373: 3749–3752. 29. Mickens RE. Harmonic balance and iteration calculations of periodic solutions to y € þ y ¼ 0. J Sound Vib 2007; 306: 968–972. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE

An effective modification of Ji-Huan He’s variational approach to nonlinear singular oscillator:

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Abstract

In the review article “Asymptotic methods for solitary solutions and compactons” (Ji-Huan He, Abstract and Applied Analysis 2012: 916793), the variational approach and the Hamiltonian approach to nonlinear oscillators are systematically discussed. This paper gives an extension of the methods to singular nonlinear oscillator by the iteration perturbation method. Keywords Nonlinear oscillator, polar coordinate, iteration perturbation method Introduction Chinese mathematician, Prof. Ji-Huan He, systematically discussed the variational approach and the Hamiltonian approach to nonlinear oscillators in his review article, where the history and main development of both methods were elucidated. Although He’s methods have been proved to be effective to nonlinear oscillators, extension of the methods still exists. Consider a nonlinear oscillator in the form x € þfxðÞ ¼ 0 (1) with initial conditions ðÞ ðÞ x 0 ¼ A and x_ 0 ¼ 0 (2) where a dot denotes the differentiation with respect to time t, A is a given constant, fðxÞ is a conservative force. 2–5 There are many approaches to nonlinear oscillators, for example, the homotopy perturbation method, 6–8 9–14 He’s frequency–amplitude formulation and other frequency–amplitude formulae, Akbari-Ganji 15,16 17–19 method, and reviews on various methods are available in the literature. In this paper, we will give an extension of He’s variational approach and Hamiltonian approach. 20–24 A variational principle for equation (1) can be easily established, which reads JxðÞ ¼ x_ FxðÞ dt (3) where F is the potential, defined as @F/@x ¼ f. School of Mathematics & Statistics, Nanjing University of Information Science and Technology, Nanjing, PR China School of Management and Engineering, Nanjing University, Nanjing, PR China Corresponding author: Zhaoling Tao, 219, Ningliu Road, Nanjing 210044, PR China. Email: nj_zaolingt@126.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www. creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 1024 Journal of Low Frequency Noise, Vibration and Active Control 38(3–4) The variational approach and the Hamiltonian approach discussed by He can be applied to equation (3) to obtain the frequency of the nonlinear oscillator. This comment shows that we can couple the iteration perturba- tion method with the variational approach or the Hamiltonian approach to search for a periodic solution. Basic process The oscillator (1)–(2) can be rewritten in a system form ðÞ ðÞ y ¼ x_ ; y_ ¼fx ; x 0 ¼ A; y 0 ¼ 0 (4) ðÞ Introducing polar representations for x; y , i.e. ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ xt ¼ rt cos h t ; yt ¼ rt sin h t (5) with initial conditions ðÞ ðÞ r 0 ¼ A and h 0 ¼ 0 (6) Substituting equation (5) into equation (4), we have r_cos h ¼ 1 þ h rsin h (7) ðÞ r_sin h þ rhcos h ¼frcos h (8) It is easy to arrive at the following results ðÞ cos h frcos h h ¼1 þ cos h  (9) ðÞ rsin 2h ðÞ r_ ¼  sin h frcos h (10) According to the iteration perturbation method, we can construct the following iteration scheme _ ðÞ ½ ðÞ ðÞ ðÞ ½ ðÞ ðÞ r t ¼ Fr t ; h t ; h t ¼ Gr t ; h t (11) kþ1 k k kþ1 k k where k ¼ 0; 1; 2 ... ; and FrðÞ ; h and GrðÞ ; h are, respectively, the right-hand sides of equations (9) and (10) based ðÞ ðÞ on having prior functional forms for r t and h t . By averaging over h and some simple operations, we can first 0 0 obtain functions r_ ðÞ t and h ðÞ t , and then r ðÞ t and h ðÞ t . Therefore, approximation to the solution xtðÞ ¼ r ðÞ t 0 0 0 0 1 ðÞ cos h t can be obtained. Example An important and interesting nonlinear differential equation occurs in the modeling of certain phenomena in 26–29 plasma physics. This example corresponds to the odd nonlinear singular oscillator with e a positive constant ðÞ ðÞ x € ¼ ; x 0 ¼ A; x_ 0 ¼ 0 (12) ex 20–24 The variational principle for equation (12) reads 1 1 JxðÞ ¼ x_  lnx dt (13) 2 e Tao and Chen 1025 The corresponding equation system is ðÞ ðÞ y ¼ x_ ; y_ ¼ ; x 0 ¼ A; y 0 ¼ 0 (14) ex With the polar expressions (5) and (6), we have 2 2 er cos h  1 h ¼1 þ (15) er 2 2 ðÞ er cos h  1 sin h r_ ¼ (16) ercos h Integral averaging the right-hand side of equations (15) and (16) over h with one period, between 0 and 2p, we have ðÞ ðÞ r_ t ¼ 0; h t ¼  1= er (17) 0 0 Combing with the initial conditions in equation (6) yields ðÞ r ðÞ t ¼ A; h ðÞ t ¼  t= eA (18) 0 0 The iteration formula with k ¼ 0; 1; 2; ... based on equation (18) becomes 2 2 er cos h  1 h ðÞ t ¼1 þ (19) kþ1 er 2 2 er cos h  1 sin h k k r_ ðÞ t ¼ (20) kþ1 er cos h k k when k ¼ 0, 2 2 er cos h  1 ðÞ h t ¼1 þ (21) er 2 2 er cos h  1 sin h 0 0 ðÞ r_ t ¼ (22) er cos h 0 0 Approximation to the solution xtðÞ ¼ r ðÞ t  cos h ðÞ t can be obtained just after finding r ðÞ t and h ðÞ t from 1 1 1 1 equations (21) and (22) based on equation (18) ðÞ 2  3eA sin 2Wt h ðÞ t ¼ t  (23) 2eA ðÞ 22 þ eA A 1 A 1 ðÞ ðÞ ðÞ r t ¼ A  þ cos Wt  lncos Wt (24) 2W W 2 eA ðÞ with the amplitude-dependent function, WA ,is 2 þ eA W ¼ (25) 2eA 1026 Journal of Low Frequency Noise, Vibration and Active Control 38(3–4) Conclusions Usually, frequency, period or frequency–amplitude relationship are the main factors for nonlinear oscillators. He gave an elementary introduction to nonlinear vibration systems. This paper gives for the first time an alternative approach to singular oscillators, and the results are much better than those in literature. Hereby, we adopt the iteration method into the variational approach under the polar coordinate system, the modification is also an effective way for more oscillators, and it is extremely effective for singular oscillators as illustrated in this paper. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) received no financial support for the research, authorship, and/or publication of this article. ORCID iD Zhao-Ling Tao http://orcid.org/0000-0001-8674-1843 References 1. He J-H. Asymptotic methods for solitary solutions and compactons. Abstr Appl Anal 2012, http://dx.doi.org/10.1155/ 2012/916793. 2. He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004; 151: 287–292. 3. He JH. Homotopy perturbation method with an auxiliary term. Abstr Appl Anal 2012, http://dx.doi.org/10.1155/ 2012/857612. 4. He JH. Homotopy perturbation method with two expanding parameters. Indian J Phys 2014; 88: 193–196. 5. Liu ZJ, Adamu MY, Suleiman E, et al. Hybridization of homotopy perturbation method and Laplace transformation for the partial differential equations. Therm Sci 2017; 21: 1843–1846. 6. He JH. An improved amplitude-frequency formulation for nonlinear oscillators. Int J Nonlinear Sci Numer Simul 2008; 9: 211–212. 7. He JH. An approximate amplitude-frequency relationship for a nonlinear oscillator with discontinuity. Nonlinear Sci Lett A 2016; 7: 77–85. 8. He JH. Amplitude-frequency relationship for conservative nonlinear oscillators with odd nonlinearities. Int J Appl Comput Math 2017; 3: 1557–1560. 9. Tao ZL. Frequency-amplitude relationship of nonlinear oscillators by He’s parameter-expanding method. Chaos Solitons Fractals 2009; 41: 642–645. 10. Ren Z. He’s frequency-amplitude formulation for nonlinear oscillators. Int J Mod Phys B 2011; 25: 2379–2382. 11. Ren Z. Theoretical basis of He frequency-amplitude formulation for nonlinear oscillators. Nonlinear Sci Lett A 2018; 9: 86–90. 12. Dan T and Zhi L. Period/frequency estimation of a nonlinear oscillator. J Low Frequency Noise Vib Active Control 2018, https://doi.org/10.1177/1461348418756013. 13. Garcıa A. An amplitude-period formula for a second order nonlinear oscillator. Nonlinear Sci Lett A 2017; 8: 340–347. 14. Sua´ rez Antola R. Remarks on an approximate formula for the period of conservative oscillations in nonlinear second order ordinary differential equations. Nonlinear Sci Lett A 2017; 8: 348–351. 15. Balazadeh N and Ganji DD. Akbari-Ganji method for thermal analysis of longitudinal porous fins. Nonlinear Sci Lett A 2018; 9: 1–16. 16. Nimafar M, Akbari MR, Ganji DD, et al. Akbari-Ganji method for vibration under external harmonic load. Nonlinear Sci Lett A 2017; 8: 416–437. 17. He JH. A tutorial review on fractal spacetime and fractional calculus. Int J Theor Phys 2014; 53: 3698–3718. 18. He JH. Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 2006; 20: 1141–1199. 19. He JH. An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int J Mod Phys B 2008; 22: 3487–3578. 20. He JH. An alternative approach to establishment of a variational principle for the torsional problem of piezoelastic beams. Appl Math Lett 2016; 52: 1–3. 21. He JH. Hamilton’s principle for dynamical elasticity. Appl Math Lett 2017; 72: 65–69. 22. He JH. Generalized equilibrium equations for shell derived from a generalized variational principle. Appl Math Lett 2017; 64: 94–100. Tao and Chen 1027 23. Wu Y and He JH. A remark on Samuelson’s variational principle in economics. Appl Math Lett 2018; 84: 143–147. 24. Li XW, Li Y and He JH. On the semi-inverse method and variational principle. Therm Sci 2013; 17: 1565–1568. 25. He JH. Iteration perturbation method for strongly nonlinear oscillations. J Vib Control 2001; 7: 631–642. 26. Depassier MC and Haikala V. Analytic upper and lower bounds for the period of nonlinear oscillators. J Sound Vib 2009; 328: 338–344. 27. Belendez A, Gimeno E, Fernandez E, et al. Accurate approximate solution to nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable. Phys Scr 2008; 77: 065004. 28. Ren Z-F and He J-H. A simple approach to nonlinear oscillators. Phys Lett A 2009; 373: 3749–3752. 29. Mickens RE. Harmonic balance and iteration calculations of periodic solutions to y € þ y ¼ 0. J Sound Vib 2007; 306: 968–972.

Journal

"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: Dec 10, 2018

Keywords: Nonlinear oscillator; polar coordinate; iteration perturbation method

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