I have an expression such as $\int_0^{x+l}y(z)g(x-z) dz$ and I want to evaluate its Laplace transform w.r.t $x$ in terms of the Laplace transform of $y(x)$. I know that I can substitute $t=x+l$, and coerce it into the standard from to get the Laplace transform w.r.t $t$, but I need its transform w.r.t $x$.


I would like to solve an integral equation: $y(x) = f(x) + \int_0^{x+l} y(z) g(x-z) dz$.

If the integral limit had been to $x$, we would have had $y(x) = f(x) + \int_0^{x} y(z) g(x-z) dz$. This leads to $Y(s) = \frac{F(s)}{1+K(s)}$.

  • $\begingroup$ May I ask what made this equation to you? $\endgroup$ – mrs Jun 28 '12 at 7:48
  • $\begingroup$ @BabakSorouh, as you may guess, I have a certain process where I am trying to calculate some quantity along an axis, the $f(x)$ is an external component and $y(z)g(x-z)$ are local interactions. The applications are related to my research, but it could occur anywhere from EM fields, heat conduction, or chemical and biological processes. $\endgroup$ – highBandWidth Jun 28 '12 at 17:00
  • $\begingroup$ eqworld.ipmnet.ru/en/solutions/ie/ie0217.pdf might be helpful. $\endgroup$ – doraemonpaul Jul 11 '12 at 1:15







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