Perturbation Theory, Again

In summary, the conversation discusses the use of perturbation coefficients and the roots of an equation. The conversation also touches on using a Taylor series for ln(x) and the limitations of its convergence. The participants also discuss the use of numerical solutions and the difficulty of finding a simple function to represent the solutions.
  • #1
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From the following attachments I understand how the roots of the equation and the perturbation coefficients were found. What I don't get is the solid line in the graph that is allegedly the plot of two of the three roots versus epsilon. Can somebody clear this up for me? Also, how would I proceed w/the following question where (x2 – 4) = ε ln(x)? What series would I use for ln(x)?
 

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  • #2
jbowers9 said:
Also, how would I proceed w/the following question where (x2 – 4) = ε ln(x)? What series would I use for ln(x)?

The well known Taylor series around x=1 for the logarithm is

[tex]
\textrm{log}(x) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}(x-1)^k.
[/tex]

The nasty thing with this is, that it converges only for [tex]0<x<2[/tex], so you need to be careful when using it. After attempt [tex]x=x_0 + \epsilon x_1 + \epsilon^2 x_2 + O(\epsilon^3)[/tex], it would be smart to do this:

[tex]
\textrm{log}(x_0 \;+\; \epsilon x_1 \;+\; \epsilon^2 x_2 \;+\; O(\epsilon^3))\; =\; \textrm{log}(x_0) \;+\; \textrm{log}\big(1 \;+\; \epsilon\frac{x_1}{x_0} \;+\; \epsilon^2\frac{x_2}{x_0} \;+\; O(\epsilon^3)\big)
[/tex]

Then use the Taylor series only to the second logarithm on the right side. As long as [tex]x_1[/tex] and [tex]x_2[/tex] are not going to be significantly larger than [tex]x_0[/tex], the approximation should be working.
 
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  • #3
Thank you for the heads up on the ln(x) series

But does anyone know what the solid line in the previous attachment for the solution is referring too? The author refers to it as being 2 of the 3 solutions and I follow that because 2 of them are positive. But what equation is he using in the plot?
 
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  • #4
The whole point is that you can't find a solution in terms of any simple function. I would presume that the solid line is generated through numerical solutions for a variety of values of [itex]\epsilon[/b].
 

What is perturbation theory?

Perturbation theory is a mathematical method used to approximate the behavior of a system when it is subjected to small changes or disturbances.

Why is perturbation theory important?

Perturbation theory is important because it allows us to study complex systems that would otherwise be too difficult to analyze. It also provides a way to make predictions about how a system will behave in the presence of small changes.

What are the applications of perturbation theory?

Perturbation theory has many applications in physics, engineering, and other fields. It is commonly used in quantum mechanics, fluid dynamics, and celestial mechanics, among others.

What are the limitations of perturbation theory?

One limitation of perturbation theory is that it only works for small changes or disturbances. If the changes are too large, the approximations made by perturbation theory may become inaccurate. Additionally, perturbation theory may not work for highly nonlinear systems.

How is perturbation theory used in practice?

In practice, perturbation theory involves breaking down a complex system into simpler parts and then analyzing how those parts are affected by small changes. This is often done using mathematical techniques such as series expansions and integrals.

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