Help with solving this equation I'm very new to differential functional equations and I'm trying to solve the following:
$$\frac{\partial f(t,x)}{\partial t} + a f(t,x)+\frac{1}{x} f(t,x+1)+b = 0.$$
So what we know is that $t \in [0, T]$ and $x \geqslant 2 $ and integer for all$ t \in [0, T]$. We also know that $f(T, x) = 0$.
I'm very confused about this type of PDEs and don't know how to solve them.
I studied the method of characteristics and The Myshkys Method of Steps provided in this link for ordinary differential functional equations but I could not adapt their method to my problem. I also tried to use methods in this link, but it was very heavy for me and I didn't understand the material properly.

You can try to do the following thing. First, let me replace the time $t=T-\tau$, so that new time $\tau$ changes from 0 to 2, and the initial conditions for the new functions
$$
u(\tau,x)=u(T-t,x)=f(t,x)=f(T-\tau)
$$
are set at $\tau=0$. I will also be using index notation
$$
u_n(\tau)=u(\tau,x), \quad n=2,3,\ldots
$$
Now your equation reads
$$
\dot u_n = au_n+b+\frac 1n u_{n+1},\quad n=2,3\ldots,\quad u_n(0)=0,\quad \tau\in[0,2]. 
$$
Assume that for sufficiently large $n+1$ we can disregard the term
$$
\frac{1}{n+1}u_{n+1},
$$
then for $u_{n+1}$ we have the simple ODE
$$
\dot u_{n+1}=au_{n+1}+b,\quad u_{n+1}(0)=0\implies u_{n+1}(\tau)=\frac{b}{a}e^{a\tau}-\frac{b}{a}\,.
$$
Now you can sequentially solve all the equations for $n,n-1,\ldots,2$. The question is which $n+1$ to pick.
It seems (but I do not have rigorous arguments) you can take $n+1=3$. Then (if I did not make any mistakes),
$$
u_2(\tau)=\frac{b \left(3 a e^{a \tau}+a \tau e^{a \tau}-e^{a \tau}-3 a+1\right)}{3 a^2}\,.
$$
Why such $n+1$? The only argument I have is that I experimented a little, and it seems that taking $n+1$ bigger does not really change the resulting $u_2$. Yes, the analytical expressions become more and more cumbersome, but the graph for $u_2$ remains the same.
Of course it all also depends on which $f_n$ (and hence $u_n$) you need.
P.S. Your equation is the so-called mixed functional differential equation. This is the last big project by Anatoly Myshkis, who worked on it at the end of the last century. You can find the English translation of his work here.
