Volterra Equations: Applications in Physics

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Volterra equations are utilized in various physical phenomena, particularly in modeling systems where memory effects or history dependence are significant. They serve as integral representations of differential equations, allowing for the analysis of problems where traditional methods may fall short. Applications include inverse problems, where the goal is to deduce system characteristics from observed data. These equations are particularly relevant in fields like control theory, fluid dynamics, and biological systems. Understanding when to use Volterra equations enhances the analysis of complex dynamic systems.
alecrimi
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Hi Guys!
I have a (stupid) question. In which physical phenomena do you use Volterra equations (or similar equations) ?
I mean if we go back to traditional heat,diffusion,wave, transport... and so on we know more or less when to use them. Are integral equation just a dual representation or is there a specific reason to use them ?
Thanx
Alex
 
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A differential equation:

<br /> y&#039; = f(x, y)<br />

with the initial condition y(x_{0}) = y_{0} is equivalent to the integral equation:

<br /> y(x) = y_{0} + \int_{x_{0}}^{x}{f(t, y(t)) \, dt}<br />

This is a Volterra (since the upper bound of the integral is variable) integral equation of the second kind (since the unknown function y(x) is both under the integral and outside).
 
Probably my question was not clear. I didn't ask for a definition (everybody can look up wikipedia), I asked when do you need to use them ?
some inverse problem... for example ? I am asking when did you meet them, in which phenomena ?
 
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Is what I typed a definition?
 
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