Integral equation with bounded unknown kernel

In summary, the conversation is about solving an integral equation where the unknown kernel, K, needs to be found. The equation must satisfy certain conditions, such as being positive and bounded, and there is a search for a solution other than K(omega,y) = omega. Various attempts have been made, including using orthogonal functions and considering the derivative of K.
  • #1
JohnXYZ
2
0
I need to solve an integral equation of the form

$$\forall \omega \in [0,1], ~ \int_{\mathbb{R}} K(\omega,y)f(y)dy = \omega$$

where

- f is known and positive with $$\int_{\mathbb{R}} f(y)dy = 1$$

- K: [0,1] x R -> [0,1] is the unknown kernel

I am looking for a solution other than K(omega,y) = omega. I do not know if such a solution exists, so I am looking either for a solution, or for a proof that K(omega,y) = \omega is the only solution.

----------------------

Let me tell you more about my attempts :

- looking for K(omega,y) = a(omega)b(y) fails because it leads to a(omega) = \omega and b(y) = 1

- looking for K(omega,y) = a(omega-y) and trying to solve the convolution equation $$\forall \omega \in [0,1], ~ (a \star f)(\omega) = \omega$$ fails (I proved it with Fourier transforms, although I'm not 100% confident about my proof)

- setting K(omega,y) = omega + H(omega,y) and trying to solve $$\int_{\mathbb{R}} H(\omega,y) f(y) = 0$$ with either H(omega,y) = a(omega)b(y) or H(omega,y) = a(y-omega) fails as well

There may not be a solution but I hope you can help me find one or prove that it's impossible ! Thank you
 
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  • #2
One way of thinking about it is to assume we can differentiate inside the integral sign with respect to [itex] \omega [/itex]

[tex] \int_{\mathbb{R}} K'(\omega,y)f(y)dy = 1 [/tex]
[tex] \int_{\mathbb{R}} K"(\omega,y)f(y)dy = 0 [/tex]

We can consider [itex] K" [/itex] to be a member of a family of functions of [itex] y [/itex] parameterized by [itex] \omega [/itex] such that each member of the family is orthogonal to [itex] f(y) [/itex] on [itex] \mathbb{R} [/itex].

Suppose we have an orthogonal basis [itex]\{f_1(y), f_2(y),f_3(y)...\}[/itex] for some space of functions and that [itex] f = f_1 [/itex] Then [itex] \omega f_2 + (1 - \omega) f_3 [/itex] is a family of functions that is orthogonal to [itex] f [/itex].

I'm just making these suggestion off the top of my head. I haven't tried to work this out with any concrete example.
 
  • #3
Thank you for your answer, I will also try your approach. Working from K'' may still be difficult since we have a boundedness condition on K.
 

1. What is an integral equation with a bounded unknown kernel?

An integral equation with a bounded unknown kernel is a mathematical equation that involves an unknown function as both the integrand and the limits of integration. The unknown function is bounded, meaning it has finite values and does not tend to infinity.

2. What are some examples of integral equations with bounded unknown kernels?

Examples of integral equations with bounded unknown kernels include the Fredholm integral equation, the Volterra integral equation, and the Abel integral equation. These equations are commonly used in physics, engineering, and other scientific fields to model real-world phenomena.

3. How do you solve an integral equation with a bounded unknown kernel?

There are various methods for solving integral equations with bounded unknown kernels, such as the method of successive approximations, the method of moments, and the numerical method. The specific method used will depend on the nature of the equation and the desired level of accuracy.

4. What are the applications of integral equations with bounded unknown kernels?

Integral equations with bounded unknown kernels have a wide range of applications in physics, engineering, and other scientific fields. They are particularly useful for modeling problems involving boundary value and initial value conditions, and for solving differential equations.

5. What are the advantages of using integral equations with bounded unknown kernels?

Integral equations with bounded unknown kernels offer several advantages over other methods of solving equations, such as their ability to model complex systems and their flexibility in handling different types of boundary conditions. They also provide a more general solution that can be applied to a variety of problems.

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