Integral Equations curiosity question

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Discussion Overview

The discussion centers around the applications and insights provided by integral equations, exploring their relevance in various fields and their relationship to differential equations. Participants seek examples and clarification on how integral equations are utilized in practice.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant expresses curiosity about integral equations and requests examples of their applications.
  • Another participant explains that integral equations are used similarly to differential equations, modeling processes with integrals instead of derivatives, and cites fields such as potential theory, electrostatics, electrodynamics, and fluid mechanics.
  • A third participant notes that integral equations often arise as reformulations of differential equations, specifically mentioning Volterra and Fredholm integral equations, and suggests that integral operators may offer better analytical and numerical behavior than differential operators.
  • This participant also indicates an intention to provide a simple example illustrating the connection between periodic solutions of a nonlinear differential equation and a corresponding nonlinear integral equation.

Areas of Agreement / Disagreement

Participants generally agree on the relevance of integral equations in various applications, but the discussion remains open-ended with no consensus on specific examples or the best way to illustrate their use.

Contextual Notes

The discussion does not resolve the specific applications of integral equations or provide detailed examples, leaving some assumptions and definitions implicit.

Andreol263
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Hello, I'm getting interest in the name integral equations and what it can give to me of insights, but i don't see any application of this, can anyone give me a example where it is used??
 
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Andreol263 said:
Hello, I'm getting interest in the name integral equations and what it can give to me of insights, but i don't see any application of this, can anyone give me a example where it is used??
Just like differential equations, where the change in a certain quantity w.r.t. time for example, is used to model certain phenomena, integral equations are used where the processes involved are expressed with integrals rather than derivatives.

https://en.wikipedia.org/wiki/Integral_equation

Integral equations are found commonly in potential theory, electrostatics and electrodynamics, fluid mechanics, etc.
 
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This is a nice question.

In addition to being used directly for modeling purposes, integral equations (IEs) also arise quite frequently in a more indirect and "hidden" way, namely as reformulations of problems involving differential equations (DEs). In this context, linear initial value problems for DEs lead to so-called Volterra IEs (variable limits of integration) while linear boundary value problems give rise to Fredholm IEs (fixed limits of integration). One advantage of such a reformulation is that the behavior of integral operators is usually much better from an analytical and numerical point of view than the behavior of the original differential operators.

Shortly, I hope to post elsewhere on this site a simple example that will hopefully illustrate the relationship between periodic solutions of a nonlinear DE, a boundary value problem and a certain type of nonlinear integral equation.
 
Thanks for the answers!, and Krylov i will wait for your post about it :)
 

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