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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Poincaré–Lindstedt method ： ウィキペディア英語版 In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method removes secular terms—terms growing without bound—arising in the straightforward application of perturbation theory to weakly nonlinear problems with finite oscillatory solutions., pp. 181–186.The method is named after Henri Poincaré,, §123–§128. and Anders Lindstedt.A. Lindstedt, Abh. K. Akad. Wiss. St. Petersburg 31, No. 4 (1882)==Example: the Duffing equation==The undamped, unforced Duffing equation is given by:\ddot + x + \varepsilon\, x^3 = 0\,for ''t'' > 0, with 0 J. David Logan. ''Applied Mathematics'', Second Edition, John Wiley & Sons, 1997. ISBN 0-471-16513-1. Consider initial conditions:x(0) = 1,\, \dot x(0) = 0.\,A perturbation-series solution of the form ''x''(''t'') = ''x''0(''t'') + ''ε'' ''x''1(''t'') + …  is sought. The first two terms of the series are:x(t) = \cos(t) + \varepsilon \left(\tfrac\, \left( \cos(3t) - \cos(t) \right) - \tfrac\, t\, \sin(t) \right ) + \cdots.\,This approximation grows without bound in time, which is inconsistent with the physical system that the equation models.The Duffing equation has an invariant energy \scriptstyle E=\tfrac12\, \dot^2 + \tfrac12\, x^2 + \tfrac14\, \varepsilon\, x^4 = constant, as can be seen by multiplying the Duffing equation with \scriptstyle \dot and integrating with respect to time ''t''. For the example considered, from its initial conditions, is found: ''E'' = ½ + ¼ ''ε''. The term responsible for this unbounded growth, called the secular termsecular term redirects here-->, is ''t'' sin t. The Poincaré–Lindstedt method allows for the creation of an approximation that is accurate for all time, as follows.In addition to expressing the solution itself as an asymptotic series, form another series with which to scale time ''t''::\tau = \omega t,\, where \omega = \omega_0 + \varepsilon \omega_1 + \cdots.\,For convenience, take ''ω''0 = 1 because the leading order of the solution's angular frequency is 1. Then the original problem becomes:\omega^2\, x''(\tau) + x(\tau) + \varepsilon\, x^3(\tau) = 0\,with the same initial conditions. Now search for a solution of the form ''x''(''τ'') = ''x''0(''τ'') + ''ε'' ''x''1(''τ'') + … . The following solutions for the zeroth and first order problem in ''ε'' are obtained: : \begin x_0 &= \cos(\tau) \\ \text x_1 &= \tfrac\, \left(\cos(3\tau)-\cos(\tau)\right) + \left( \omega_1 - \tfrac \right)\, \tau\, \sin(\tau). \endSo the secular term can be removed through the choice: ''ω''1 = . Higher orders of accuracy can be obtained by continuing the perturbation analysis along this way. As of now, the approximation—correct up to first order in ''ε''—is: x(t) \approx \cos\Bigl(\left(1 + \tfrac\, \varepsilon \right)\, t \Bigr) + \tfrac\, \varepsilon\, \left(3 \left(1 + \tfrac\,\varepsilon\, \right)\, t \Bigr)-\cos\Bigl(\left(1 + \tfrac\,\varepsilon\, \right)\, t \Bigr)\right ). \, In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method removes secular terms—terms growing without bound—arising in the straightforward application of perturbation theory to weakly nonlinear problems with finite oscillatory solutions.〔, pp. 181–186.〕 The method is named after Henri Poincaré,〔, §123–§128.〕 and Anders Lindstedt.〔A. Lindstedt, Abh. K. Akad. Wiss. St. Petersburg 31, No. 4 (1882)〕 ==Example: the Duffing equation== The undamped, unforced Duffing equation is given by :$\backslash ddot\; +\; x\; +\; \backslash varepsilon\backslash ,\; x^3\; =\; 0\backslash ,$ for ''t'' > 0, with 0 < ''ε'' (unicode:≪) 1.〔J. David Logan. ''Applied Mathematics'', Second Edition, John Wiley & Sons, 1997. ISBN 0-471-16513-1.〕 Consider initial conditions :$x(0)\; =\; 1,\backslash ,$ $\backslash dot\; x(0)\; =\; 0.\backslash ,$ A perturbation-series solution of the form ''x''(''t'') = ''x''0(''t'') + ''ε'' ''x''1(''t'') + …  is sought. The first two terms of the series are :$x(t)\; =\; \backslash cos(t)\; +\; \backslash varepsilon\; \backslash left(\backslash tfrac\backslash ,\; \backslash left(\; \backslash cos(3t)\; -\; \backslash cos(t)\; \backslash right)\; -\; \backslash tfrac\backslash ,\; t\backslash ,\; \backslash sin(t)\; \backslash right)\; +\; \backslash cdots.\backslash ,$ This approximation grows without bound in time, which is inconsistent with the physical system that the equation models.〔The Duffing equation has an invariant energy $\backslash scriptstyle\; E=\backslash tfrac12\backslash ,\; \backslash dot^2\; +\; \backslash tfrac12\backslash ,\; x^2\; +\; \backslash tfrac14\backslash ,\; \backslash varepsilon\backslash ,\; x^4$ = constant, as can be seen by multiplying the Duffing equation with $\backslash scriptstyle\; \backslash dot$ and integrating with respect to time ''t''. For the example considered, from its initial conditions, is found: ''E'' = ½ + ¼ ''ε''.〕 The term responsible for this unbounded growth, called the secular term, is ''t'' sin t. The Poincaré–Lindstedt method allows for the creation of an approximation that is accurate for all time, as follows. In addition to expressing the solution itself as an asymptotic series, form another series with which to scale time ''t'': :$\backslash tau\; =\; \backslash omega\; t,\backslash ,$ where $\backslash omega\; =\; \backslash omega\_0\; +\; \backslash varepsilon\; \backslash omega\_1\; +\; \backslash cdots.\backslash ,$ For convenience, take ''ω''0 = 1 because the leading order of the solution's angular frequency is 1. Then the original problem becomes :$\backslash omega^2\backslash ,\; x\text{'}\text{'}(\backslash tau)\; +\; x(\backslash tau)\; +\; \backslash varepsilon\backslash ,\; x^3(\backslash tau)\; =\; 0\backslash ,$ with the same initial conditions. Now search for a solution of the form ''x''(''τ'') = ''x''0(''τ'') + ''ε'' ''x''1(''τ'') + … . The following solutions for the zeroth and first order problem in ''ε'' are obtained: :$$ \begin x_0 &= \cos(\tau) \\ \text x_1 &= \tfrac\, \left(\cos(3\tau)-\cos(\tau)\right) + \left( \omega_1 - \tfrac \right)\, \tau\, \sin(\tau). \end So the secular term can be removed through the choice: ''ω''1 = . Higher orders of accuracy can be obtained by continuing the perturbation analysis along this way. As of now, the approximation—correct up to first order in ''ε''—is :$$ x(t) \approx \cos\Bigl(\left(1 + \tfrac\, \varepsilon \right)\, t \Bigr) + \tfrac\, \varepsilon\, \left(3 \left(1 + \tfrac\,\varepsilon\, \right)\, t \Bigr)-\cos\Bigl(\left(1 + \tfrac\,\varepsilon\, \right)\, t \Bigr)\right ). \, 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method removes secular terms—terms growing without bound—arising in the straightforward application of perturbation theory to weakly nonlinear problems with finite oscillatory solutions., pp. 181–186.The method is named after Henri Poincaré,, §123–§128. and Anders Lindstedt.A. Lindstedt, Abh. K. Akad. Wiss. St. Petersburg 31, No. 4 (1882)==Example: the Duffing equation==The undamped, unforced Duffing equation is given by:\ddot + x + \varepsilon\, x^3 = 0\,for ''t'' > 0, with 0 J. David Logan. ''Applied Mathematics'', Second Edition, John Wiley & Sons, 1997. ISBN 0-471-16513-1. Consider initial conditions:x(0) = 1,\, \dot x(0) = 0.\,A perturbation-series solution of the form ''x''(''t'') = ''x''0(''t'') + ''ε'' ''x''1(''t'') + …  is sought. The first two terms of the series are:x(t) = \cos(t) + \varepsilon \left(\tfrac\, \left( \cos(3t) - \cos(t) \right) - \tfrac\, t\, \sin(t) \right ) + \cdots.\,This approximation grows without bound in time, which is inconsistent with the physical system that the equation models.The Duffing equation has an invariant energy \scriptstyle E=\tfrac12\, \dot^2 + \tfrac12\, x^2 + \tfrac14\, \varepsilon\, x^4 = constant, as can be seen by multiplying the Duffing equation with \scriptstyle \dot and integrating with respect to time ''t''. For the example considered, from its initial conditions, is found: ''E'' = ½ + ¼ ''ε''. The term responsible for this unbounded growth, called the secular termsecular term redirects here-->, is ''t'' sin t. The Poincaré–Lindstedt method allows for the creation of an approximation that is accurate for all time, as follows.In addition to expressing the solution itself as an asymptotic series, form another series with which to scale time ''t''::\tau = \omega t,\, where \omega = \omega_0 + \varepsilon \omega_1 + \cdots.\,For convenience, take ''ω''0 = 1 because the leading order of the solution's angular frequency is 1. Then the original problem becomes:\omega^2\, x''(\tau) + x(\tau) + \varepsilon\, x^3(\tau) = 0\,with the same initial conditions. Now search for a solution of the form ''x''(''τ'') = ''x''0(''τ'') + ''ε'' ''x''1(''τ'') + … . The following solutions for the zeroth and first order problem in ''ε'' are obtained: : \begin x_0 &= \cos(\tau) \\ \text x_1 &= \tfrac\, \left(\cos(3\tau)-\cos(\tau)\right) + \left( \omega_1 - \tfrac \right)\, \tau\, \sin(\tau). \endSo the secular term can be removed through the choice: ''ω''1 = . Higher orders of accuracy can be obtained by continuing the perturbation analysis along this way. As of now, the approximation—correct up to first order in ''ε''—is: x(t) \approx \cos\Bigl(\left(1 + \tfrac\, \varepsilon \right)\, t \Bigr) + \tfrac\, \varepsilon\, \left(3 \left(1 + \tfrac\,\varepsilon\, \right)\, t \Bigr)-\cos\Bigl(\left(1 + \tfrac\,\varepsilon\, \right)\, t \Bigr)\right ). \,」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.