Volterra Equations: Applications in Physics

[alecrimi]

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

 

[Dickfore]

A differential equation:

$$y' = f(x, y)$$

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

$$y(x) = y_{0} + \int_{x_{0}}^{x}{f(t, y(t)) \, dt}$$

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).

 

[alecrimi]

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 ?

 

[Dickfore]

Is what I typed a definition?

 

[sardar]

Hello Alex,

Volterra equations, also known as integral equations, have many applications in physics. They are used to model a wide range of physical phenomena, including heat transfer, diffusion, wave propagation, and transport processes. These equations are particularly useful in situations where the system being studied is nonlinear or has memory effects.

One example of their application is in the study of heat transfer in materials. Volterra equations can be used to model the temperature distribution in a material as it changes over time, taking into account factors such as thermal conductivity, heat generation, and boundary conditions.

In the field of fluid dynamics, Volterra equations are used to model the transport of pollutants in a river or ocean. By incorporating the effects of advection, diffusion, and source terms, these equations can provide insights into the dispersion of contaminants and their impact on the environment.

Volterra equations have also been applied in the study of wave propagation, such as in the analysis of seismic waves in geophysics. They have also been used to model the behavior of electromagnetic fields in materials, with applications in optics and photonics.

In summary, Volterra equations have a wide range of applications in physics, particularly in systems that exhibit nonlinear behavior or memory effects. They provide a valuable tool for understanding and predicting the behavior of complex physical systems. I hope this helps answer your question.