I have a PDE: $$\frac{\partial N (x,u)}{\partial x}=\int _0^uN(x,u)f(u-u')du'$$ $$N(0,u) = \delta (u)$$
Here $f(u)$ is a probability density function for $0 \le u \le u_{max}$, $\int _0 ^ {u_{max}} f(u)du = 1 $ Functions $N(x,u)$ and $f(u)$ are positive and real.
If we make a Laplace transformation, that equation becames an ordinary first-order PDE: $$\frac{\partial N (x,p)}{\partial x}=N(x,p)f(p)$$
The solution is: $$N(x,p)=e^{f(p)x}$$
How can I calculate $N(x,u)$ from that if $f(u)$ is an experimental dependance, I mean I have descrete analog as $f_i=f(u_i)$, and I need to numericaly calculate $N_i(x)=N(x,u_i)$.
So the main task is to find an apropriate transformation where linear convolution of nonperiodic finite function with itself transfer into multiplications as it is in Laplace transformation, this function $f(p)$ in new domain has to be calculated numericaly, and also I have to numericaly calculate inverse transformation of result functions. That functions might not be only exponential but also like $$\frac 1 {1-f(p)}$$ or even $$\frac {e^{f(p)x}}{1-f(p)} $$ and others similar ones, expandeble into power series.
Thank you!