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Thread: Solving an integral equation

  1. #1
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    Solving an integral equation

    Hello everyone,

    I am trying to solve a seemingly simple integral equation:
    $\displaystyle g(z)=\int_{z-1}^{\infty}\frac{g(t)}{t}dt$
    where $\displaystyle g(z)$ is known to be zero for $\displaystyle z\le 1$.

    $\displaystyle g(z)$ is a probability distribution function of a random process
    $\displaystyle z(t+1)=a(t)z(t)+1$ where $\displaystyle a(t)=U(0,1)$ are iid random variables from a uniform distribution. I ran simulations of this process and got that $\displaystyle g(z)$ is constant for $\displaystyle 1>z\ge 2$ and decays for $\displaystyle z>2$. The decay does not seem to be exponential or 1/z. I did not manage to find an analytical solution or some transform (Laplace/Mellin) that can help.

    Any suggestions would be welcome!
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  2. #2
    A Plied Mathematician
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    I can't pretend I'm an expert, but one of my first lines of attack on an integral equation is to see if I can't convert it to a differential equation and then solve using standard techniques. In your case, if I differentiate with respect to z, I get

    $\displaystyle g'(z)=-\frac{g(z-1)}{z-1},$

    assuming that $\displaystyle g(z)$ decays to zero at least as fast as $\displaystyle 1/z.$

    This result is a delay differential equation. My guess is that unless you are a really good guesser, you won't be able to find a solution analytically. However, there are numerical solvers available for delay differential equations. Take a look here.

    And we have now reached the limits of my knowledge on the subject.
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