Reference request for certain PDE appearing in a probability application In working on a stochastic processes application, I derived a PDE for the joint transform of some process. It takes the form $$\frac{\partial \zeta(t,z,s)}{\partial t}-(\beta s+1)\frac{\partial\zeta(t,z,s)}{\partial s}=\frac{z}{s}\left(\zeta(t,z,0)-\zeta(t,z,s)\right),$$ where we have boundary conditions $\zeta(0,z,s)=e^{-s}$, $\zeta(t,0,s)=0$ and $\zeta(t,1,0)=1$. Here $\beta>0$ is just some scalar. If convenient, I'm fine with setting it to unity.
I don't have much experience with PDE, and I'm a bit stuck in this problem because I either want to solve this PDE or I'd like to calculate expressions like $\frac{\partial \zeta(t,z,s)}{\partial z}$ evaluated in $(t,1,0)$ for arbitrary $t>0$. In this context, it may be helpful to use the expression $\zeta(t,z,s)=\mathbb Ez^{N(t)}e^{-s\Lambda(t)}$ from my probability application. Rewriting the PDE and using this expression gives us ODEs for moments of $N(t),\Lambda(t)$, but only expressed in higher moments, making a recursive solution inpossible. Therefore I'm looking for something different.
I was hoping this PDE belongs to some well-known class, or for some analytical method to solve this PDE; I'm looking for some reference. Any help is much appreciated.
 A: I'll write $f(t,s)$ rather than $\zeta$, and set $\beta=1$. The equation is
$$\tag{1}
s\partial_tf(t,s)-s(s+1)\partial_s f(t,s)=zf(t,0)-zf(t,s)
$$
This is a non-local differential equation due to the first term on the RHS. It can be recast as an integro-differential equation by writing the RHS as $\int\limits_0^s ds' \frac{\partial f}{\partial s'}$. We will find a class of formal solutions to (1) as integrals. Take a partial Fourier transform in $t$ to get
$$\tag{2}
-i\omega s F(\omega,s)-s(s+1)\partial_s F(\omega,s)=z F(\omega,0)-zF(\omega,s)
$$
Where $F(\omega,s)=\int dt \ f(t,s)e^{i\omega t}$. If we differentiate (2) wrt $s$ we eliminate the non-local term and are left with a linear ODE for $F(s)$ (temporarily suppressing the $\omega$ dependence)
$$\tag{3}
s(s+1)F''+(2s+1+i\omega s-z)F'+i\omega F=0
$$
Using a CAS, we find the general solution to (3) is
$$\tag{4}
(1+s)^{z+i\omega}F(s)=A(\omega)s^z+B(\omega)(-s)^z \beta(-s,-z,i\omega+z)
$$
Where $\beta$ is the incomplete beta function. $A(\omega)$ and $B(\omega)$ are the integration 'constants'. Substituting (4) into (2) we find that (4) satisfies (2) for any $A$ and $B$. Taking the inverse Fourier transform of the first term in (4) yields
$$\tag{5}
f_A(t,s)=\left(\frac{s}{s+1}\right)^z\int\frac{d \omega}{2\pi} \ A(\omega) \exp\left[-i \omega(t+\ln(1+s)) \right]
$$
If the initial condition is given as $f(0,s)=\psi(s)$, then we can; in principle, determine $A$. Let $\eta=\ln(1+s)$, so that at $t=0$ we have from (5)
$$\tag{6}
\frac{1}{(1-e^{-\eta})^z}\psi(e^\eta-1)=\int\frac{d\omega}{2\pi} \ e^{-i\omega \eta} A(\omega)
$$
So that $A(\omega)$ is determined by the initial condition as a Fourier transform in $\eta$.
$$\tag{7}
A(\omega)=\int d\eta \  \frac{e^{i\omega \eta}\psi(e^\eta-1)}{(1-e^{-\eta})^z}
$$
Then (5) reads
$$\tag{8}
f_A(t,s)= \int\frac{d \omega}{2\pi}\int d\eta' \ \left[\frac{1-e^{-\eta}}{1-e^{-\eta'}}\right]^z e^{-i \omega(t+\eta-\eta') } \psi(e^{\eta'}-1) 
$$
The RHS is implicitly a function of $s$ since $\eta=\eta(s)$. All integrals are $\int\limits_{-\infty}^\infty$. Actually performing the integrals analytically seems unfeasible, but (8) already tells you about the full $z$ dependence of $f$. You may differentiate (8) wrt $z$ to find things like $\frac{d}{dz}f|_{z=1}$.
Notes:

*

*You may wonder about taking a partial Fourier transform wrt $s$ instead of $t$. In this case (using the usual rules of Fourier transforms), we find a linear PDE with delta source term

$$\tag{9}
k\partial_{kk}G+(ik+2)\partial_k G + \partial_{kt}G +(1+z)G=2\pi i z  \delta(k)
$$
Where $G=G(t,k)$ is the Fourier transform of $f(t,s)$. (9) says that solutions to (1) are the Fourier transforms of the Green's functions of the linear operator
$$\tag{10}
\hat{L}=k\partial_{kk} +(ik+2)\partial_k + \partial_{kt} +(1+z)
$$
Which is presumably, after a co-ordinate change, equal to the Helmholtz operator. (9) may be useful for generating the solutions numerically.


*Taking a full Fourier transform yields a linear ODE

$$\tag{11}
k F'' +(ik-i\omega+2)F'+i(z+1)F=2\pi i z \delta(k)
$$
Where $F'=\partial_k F$. The solutions of (11) are
$$\tag{12}
e^{ik}k^{1-i\omega}F(\omega,k)=A(\omega)U(-z,i\omega,ik)+B(\omega)L(z,i\omega-1,ik)
$$
Where $U$ is a confluent hypergeometric function and $L$ is a generalized Laguerre polynomial.


*If the nonlocal term were absent, the solution would follow directly from the method of characteristics. The solution in that case is

$$\tag{13}
f(t,s)=z^{1+(1-e^{-t})/s}\psi(e^t(s+1)-1)
$$
