1
$\begingroup$

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 $x$.

Motivation:

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 $Y(s) = \frac{F(s)}{1+K(s)}$.

  • 0
    http://eqworld.ipmnet.ru/en/solutions/ie/ie0217.pdf might be helpful.2012-07-11

1 Answers 1

0

Hint:

$y(x)=f(x)+\int_0^{x+l}y(z)g(x-z)~dz$

$y(x)=f(x)+\int_x^{-l}y(x-t)g(t)~d(x-t)$

$y(x)=f(x)-\int_x^{-l}y(x-t)g(t)~dt$

$y(x)=f(x)+\int_{-l}^xy(x-t)g(t)~dt$