Volterra equation, asymptotic behaviour

  • Thread starter perr
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In summary: One thing you could do is to try to find a general expression for c(tau) that is valid for large x and large "tau". This might require some experimentation.Alternatively, you could try to find a more accurate analytical approximation for c(tau) for small x and small "tau", by expanding the kernel for small x and small "tau" and then finding an expression for c(tau) that is valid for large x and large "tau".This approach might be more difficult, but it may be more accurate.
  • #1
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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  • #2
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.
 
  • #3
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?
 
  • #4
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?
 
  • #5
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?
 
  • #6
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.
 
  • #7
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?
 

Related to Volterra equation, asymptotic behaviour

1. What is a Volterra equation?

A Volterra equation is a type of integral equation that involves both an unknown function and its integral. It is used to model systems in which the future behavior of a variable is determined by its past values.

2. What is the difference between a Volterra equation and a Fredholm equation?

The main difference between a Volterra equation and a Fredholm equation is that a Volterra equation involves an integral of the unknown function, while a Fredholm equation involves the unknown function itself. This results in different mathematical properties and solution methods for the two types of equations.

3. What is meant by the asymptotic behavior of a Volterra equation?

The asymptotic behavior of a Volterra equation refers to the long-term behavior of its solutions. This includes how the solutions approach a steady state or equilibrium over time, as well as any periodic or oscillatory behavior that may occur.

4. How does the order of a Volterra equation affect its asymptotic behavior?

The order of a Volterra equation refers to the highest derivative of the unknown function that appears in the equation. Generally, a higher order Volterra equation will have more complex asymptotic behavior, potentially including multiple steady states or chaotic solutions.

5. What are some applications of Volterra equations in science and engineering?

Volterra equations have been used in many fields, including biology, economics, and physics, to model systems that exhibit memory or delayed effects. They are particularly useful for studying population dynamics, chemical reactions, and control systems. Additionally, they have been used in signal processing and image reconstruction problems.

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