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Integral equations in Wolfram Language

Posted 8 years ago
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Hello, Could someone point me to approaches at solving the following type of integral equation in Mathematica (p and q are variables, " f " is a function of either p, or q , and \omega is a function of p, q or both p and q, as well as \mu and m. A, \mu and m are constants): enter image description here

POSTED BY: Yaj Bhattacharya
8 Replies

Thank you Sean and Henrick. I too am not a mathematician, and this is my first encounter with an Integral equation! There was no expectation or intent to punch in this sort of an equation in a Notebook and press "Shift-Enter" for a solution. I don't want to convert it to a differential equation.

POSTED BY: Yaj Bhattacharya

Hennrik, On a completely separate note, how do you typset your post to include such nice math symbols?

POSTED BY: Yaj Bhattacharya

Dear Yaj,

this typesetting is a nice feature, isn't it! To use it one just hast to apply the LaTeX syntax, see How to Post ...

Concerning integral equations: I find very helpful

Andrei D. Polyanin, Alexander V. Manzhirov; Handbook of Integral Equations

Henrik

POSTED BY: Henrik Schachner

Hi Yaj,

I hope my remark is not too silly (because unfortunately I am not a mathematician): Are you sure that your equation has a solution anyway? The general form is:

$f(q) = \int f(p) K(q,p)\mbox{d}p$

and this is true for $K(q,p) = \delta(p-q)$, but in your case $K$ is not the Dirac-Deltafunction ...

Henrik

POSTED BY: Henrik Schachner

Ok, here comes a final note:

If the above equation is discretized, giving:

$ f(q_m) = \sum_n f(p_n) K(q_m, p_n) $

one can see that $ f(q) $ needs to be an eigenvector for the matrix $ K $ with the eigenvalue 1. So maybe there are solutions. This actually could be explored with Mathematica. (This is always true when $ K $ were the Kronecker-Delta - in analogy to the above.)

In principal I want to support what Sean was indicating: Before applying any tool or algorithm one has to switch off the computer, take paper and pencil and think about true nature of the problem.

Henrik

POSTED BY: Henrik Schachner

Don't use special characters. Just use regular english keyboard characters.

Other than that, I would see if you can work out how to turn this into a differential equation on a chalkboard.

POSTED BY: Sean Clarke

When I try to post the code, I get this

[q] (2 \[Omega][q, \[Mu]] - A) = \[Pi] \[Integral](
   f[p] \[DifferentialD]p)/(\[Omega][
     q, \[Mu]] (\[Omega][p - q, m])^2 \[Omega][p, \[Mu]])
POSTED BY: Yaj Bhattacharya

The first step in most cases is to write out the equation in Wolfram Language Code.

That said, there isn't a way to just simply solve an integral equation. Can it be expressed as a differential equation of some kind?

POSTED BY: Sean Clarke
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