# Thread: Solving an integral equation

1. ## 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!

2. 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.