In an effort to solve a delay partial differential equation $$\partial_t f(t,x)= \Phi(x) f(t,x)+\Psi(x) f(t,x-\alpha),$$ with $$f(0,x)=1,\hspace{0.3cm} f(t,0)=1$$ Where $\alpha$ is the delay ( a real number) and in particular I have $\Phi(x)=e^{-x}+e^{x}-2$ and $\Psi(x)=1-e^{-x}$,

I used Laplace transform $$ L(f)=\int_{0}^{t} e^{-st} f(x,t) dt: =U(s,t) $$ and I got a delay equation with no derivatives involved : $$s\hspace{0.1cm}U(s,x)-1 = \Phi(x) U(s,x)+\Psi(x) U(s,x-\alpha)$$

This is where I stopped. Does anyone know how I can find an explicit formula for $U(s,x)$ from the last equation? Or is there any other way to solve the delay PDE expressed on top? Obviously the separation of variables method doesn't work here because it doesn't satisfy initial and boundary conditions.