Integral Equations curiosity question

In summary, integral equations are used in various fields such as potential theory, electrostatics, and fluid mechanics as an alternative to differential equations. They can also arise as reformulations of problems involving differential equations, leading to more favorable analytical and numerical behavior. A simple example can demonstrate the relationship between periodic solutions of a nonlinear differential equation, a boundary value problem, and a nonlinear integral equation.
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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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  • #2
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.
 
  • #4
Thanks for the answers!, and Krylov i will wait for your post about it :)
 

1. What is an integral equation?

An integral equation is a mathematical equation that involves an unknown function in an integral. It is used to solve problems in various fields of science and engineering.

2. What is the difference between a differential equation and an integral equation?

The main difference is that a differential equation involves derivatives of an unknown function, while an integral equation involves integrals of an unknown function. In other words, a differential equation relates a function to its derivatives, while an integral equation relates a function to its integrals.

3. How are integral equations solved?

Integral equations are solved using various techniques such as separation of variables, Laplace transform, and numerical methods like the Gauss-Seidel method. The specific method used depends on the type and complexity of the integral equation.

4. What are some applications of integral equations?

Integral equations are used in many fields, including physics, engineering, and economics. They are commonly used to model problems involving continuous systems, such as fluid flow, heat transfer, and electromagnetic fields.

5. Are there any real-world problems that can be solved using integral equations?

Yes, integral equations are used to solve many real-world problems. For example, they can be used to calculate the distribution of temperature in a solid object, the flow of water through a pipe, or the electric potential in a circuit.

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