Volterra equation, asymptotic behaviour

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Discussion Overview

The discussion revolves around finding the asymptotic behavior of the Volterra integral equation of the second kind as the variable "tau" approaches infinity. Participants explore analytical methods and numerical comparisons to derive an accurate expression for the solution.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant, perr, seeks help in deriving an asymptotic expression for "c(tau)" as "tau" approaches infinity, noting discrepancies between their analytical results and numerical solutions.
  • Another participant suggests that perr's approximation may have been derived around zero, prompting clarification from perr regarding the nature of their expansion.
  • Perr clarifies that their expansion was for large values of the argument, not around zero, and questions the suggestion made by the other participant.
  • A subsequent reply acknowledges the validity of perr's expansion for large values of "x" and prompts perr to compare their analytical result with numerical solutions for large "x".
  • Perr confirms that their derived formula is valid only for large "x" and provides a fitting formula that closely matches the numerical solution, expressing concern about missing important factors in their analytical derivation.
  • Another participant introduces the idea of deriving expansions for both large and small "x" and combining them, referencing perturbation methods as a potential approach.
  • Perr expresses uncertainty about deriving an approximate expression with the correct factors, reiterating that both terms in their expression are valid only for large "tau".

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to derive the asymptotic expression, and multiple competing views on the methods and expansions remain present throughout the discussion.

Contextual Notes

Participants express uncertainty regarding the derivation of specific factors in their analytical approximations and the applicability of different expansion methods. There is also a lack of clarity on how to effectively mesh expansions for different ranges of "x".

perr
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Dear all,

I want to solve the Volterra integral equation (of 2nd kind). But I only need to solve it analytically for large times "tau", i.e. I only need the asymptotic behaviour as "tau -> infinity".

By simple algebra, I obtain an approximative analytical expression in this limit. However, this expression is not in agreement with the exact numerical solution of the relevant Volterra equation. Please see PDF-file attached!

Can anyone help me finding a right asymptotic expression for "c(tau)"?

I appreciate any feedback!

Al the best,
perr


PS:
I have also tried to solve this applying Laplace transformation. However, I'm not sure that this method will help me here: Suppose that the solution "c(tau)" scales as "1/tau", I am not sure how to get a "1/tau" term from Laplace transform analysis since the Laplace transform of "1/t" diverges.
 

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You can find an expansions close to 0 with the method you have applied, or you can look for an approximation elsewhere. It looks as if you have taken an approximation around 0.
 
Thank you for your reply!

I have expanded the Cosine- and Sine integral for large values of the argument. (Hence, I have not 'taken an approximation around 0', as I understand it). I don't understand what you mean by 'find an expansion close to 0'.

What do you mean?
 
Oh right, that is effectively what I meant, you were expanding the kernel for large values of x and the formula that you found will be valid for large values of x only, have you tried to compare the approximation you derived and (I presume you have solved it numerically) the solution for large values of x?

Is the approximation that you have stated valid for all x or just large x?
 
The formula I have derived analytically is value for large x only. The result is (see PDF-file):

c(tau) = exp(-tau/2) + exp(i*b*tau)/(Pi*b*tau)This analytical derivation is almost right, but not quite: The "fitting formula"

c(tau) = exp(-tau/2) + exp(i*b*tau)/(2*Pi*i*(b-1)*b*tau)is, on the other hand, very close to the exact numerical solution. (I have solved the Volterra equation exact numerically in C++, see PDF-file). Hence, I know that the right answer should be close to this "fitting formula".

Unfortunately, my analytical approximation is missing some important factors, (e.g. "b-1" as well as a factor "2*i"). (Typically, b=10).

How can I derive this without missing the important factors?
 
One thing I have seen done is that an expansion is derived for large x and an expansion is derived for small x and then the two can be then meshed together to obtain a solution valid for the whole region, see perturbation methods by Hinch and Applied maths by J. David Logan.
 
OK, but as I see it, both the "exp(-tau/2)"-term and the "1/tau"-term are valid for large x only, i.e. for large "tau" only. And for very large "tau", the latter term is dominant. Hence, as I understand it, this is not an issue meshing togehter a "small x" region and "large x" region.

I still don't know how to derive an approximate expression with the right factors. I'm close, but still not close enough.

Any suggestions?
 

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