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== | In [[perturbation theory]], the '''Poincaré–Lindstedt method''' or '''Lindstedt–Poincaré method''' is a technique for uniformly approximating [[periodic function|periodic]] solutions to [[ordinary differential equation]]s, 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.<ref>{{citation | first=P.G. | last=Drazin | title=Nonlinear systems | publisher=Cambridge University Press | year=1992 | isbn=0-521-40668-4 }}, pp. 181–186.</ref> | ||
The method is named after [[Henri Poincaré]],<ref>{{citation | first=H. | last=Poincaré | authorlink=Henri Poincaré | title=Les Méthodes Nouvelles de la Mécanique Célèste | volume=II | publisher=Dover Publ. | location=New York | year=1957 | origyear=1893 }}, §123–§128.</ref> and [[Anders Lindstedt]].<ref>A. Lindstedt, Abh. K. Akad. Wiss. St. Petersburg 31, No. 4 (1882)</ref> | |||
== | ==Example: the Duffing equation== | ||
The undamped, unforced [[Duffing equation]] is given by | |||
:<math>\ddot{x} + x + \varepsilon\, x^3 = 0\,</math> | |||
for ''t'' > 0, with 0 < ''ε'' {{Unicode|≪}} 1.<ref name=logan>J. David Logan. ''Applied Mathematics'', Second Edition, John Wiley & Sons, 1997. ISBN 0-471-16513-1.</ref> | |||
Consider initial conditions | |||
:<math>x(0) = 1,\,</math> {{pad|5em}} <math> \dot x(0) = 0.\,</math> | |||
A [[perturbation theory|perturbation-series]] solution of the form ''x''(''t'') = ''x''<sub>0</sub>(''t'') + ''ε'' ''x''<sub>1</sub>(''t'') + … is sought. The first two terms of the series are | |||
:<math>x(t) = \cos(t) + \varepsilon \left[ \tfrac{1}{32}\, \left( \cos(3t) - \cos(t) \right) - \tfrac{3}{8}\, t\, \sin(t) \right] + \cdots.\,</math> | |||
This approximation grows without bound in time, which is inconsistent with the physical system that [[Duffing equation|the equation]] models.<ref>The Duffing equation has an invariant energy <math>\scriptstyle E=\tfrac12\, \dot{x}^2 + \tfrac12\, x^2 + \tfrac14\, \varepsilon\, x^4</math> = constant, as can be seen by multiplying the Duffing equation with <math>\scriptstyle \dot{x}</math> and integrating with respect to time ''t''. For the example considered, from its initial conditions, is found: ''E'' = ½ + ¼ ''ε''.</ref> The term responsible for this unbounded growth, called the '''secular term'''<!-- [[secular 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'': | |||
:<math>\tau = \omega t,\,</math> {{pad|4em}} where {{pad|4em}} <math>\omega = \omega_0 + \varepsilon \omega_1 + \cdots.\,</math> | |||
For convenience, take ''ω''<sub>0</sub> = 1 because the [[leading-order|leading order]] of the solution's [[angular frequency]] is 1. Then the original problem becomes | |||
:<math>\omega^2\, x''(\tau) + x(\tau) + \varepsilon\, x^3(\tau) = 0\,</math> | |||
with the same initial conditions. Now search for a solution of the form ''x''(''τ'') = ''x''<sub>0</sub>(''τ'') + ''ε'' ''x''<sub>1</sub>(''τ'') + … . The following solutions for the zeroth and first order problem in ''ε'' are obtained: | |||
:<math> | |||
\begin{align} | |||
x_0 &= \cos(\tau) \\ | |||
\text{and } | |||
x_1 &= \tfrac{1}{32}\, \cos(3\tau) + \left( \omega_1 - \tfrac{3}{8} \right)\, \tau\, \sin(\tau). | |||
\end{align} | |||
</math> | |||
So the secular term can be removed through the choice: ''ω''<sub>1</sub> = <small>{{frac|3|8}}</small>. 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 | |||
:<math> | |||
x(t) \approx \cos\Bigl(\left(1 + \tfrac{3}{8}\, \varepsilon \right)\, t \Bigr) | |||
+ \tfrac{1}{32}\, \varepsilon\, \cos\Bigl( 3 \left(1 + \tfrac{3}{8}\,\varepsilon\, \right)\, t \Bigr). \, | |||
</math> | |||
== References and notes == | |||
<references/> | |||
{{DEFAULTSORT:Poincare-Lindstedt method}} | |||
[[Category:Perturbation theory]] | |||
Revision as of 04:53, 31 October 2013
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.[1]
The method is named after Henri Poincaré,[2] and Anders Lindstedt.[3]
Example: the Duffing equation
The undamped, unforced Duffing equation is given by
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Consider initial conditions
A perturbation-series solution of the form x(t) = x0(t) + ε x1(t) + … is sought. The first two terms of the series are
This approximation grows without bound in time, which is inconsistent with the physical system that the equation models.[5] 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:
- Template:Pad where Template:Pad
For convenience, take ω0 = 1 because the leading order of the solution's angular frequency is 1. Then the original problem becomes
with the same initial conditions. Now search for a solution of the form x(τ) = x0(τ) + ε x1(τ) + … . The following solutions for the zeroth and first order problem in ε are obtained:
So the secular term can be removed through the choice: ω1 = Template:Frac. 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
References and notes
- ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, pp. 181–186. - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, §123–§128. - ↑ A. Lindstedt, Abh. K. Akad. Wiss. St. Petersburg 31, No. 4 (1882)
- ↑ J. David Logan. Applied Mathematics, Second Edition, John Wiley & Sons, 1997. ISBN 0-471-16513-1.
- ↑ The Duffing equation has an invariant energy = constant, as can be seen by multiplying the Duffing equation with and integrating with respect to time t. For the example considered, from its initial conditions, is found: E = ½ + ¼ ε.