Can you help me solving analytically this PDE? This is a flow problem that i am studying. I was able to do it numerically for several boundary conditions, but i want to find an Analytical Solution to this problem for these simple boundary conditions. I've already tried a bunch of solutions, but in none of them i was able to figure out how to deal with the non-constant coefficient. The time is going from $0$ to $\infty$.
Thank you very much for the attention!
$$\large A\cdot \frac{\partial^2 \theta}{\partial r^2}-J\cdot r\cdot \frac{\partial \theta}{\partial r}-J\cdot r_1\cdot \frac{\partial \theta}{\partial r}-J\cdot \theta+J\cdot \theta_r=\frac{\partial \theta}{\partial t}$$ Boundary conditions: $$\large \theta(0,t)=\theta_1$$ $$\large \theta(L,t)=\theta_2$$ Initial Conditions: $$\large \theta(r,0)=\theta_0\implies 0<r<L$$ Where: $$\large A,J,r_1,\theta_r \implies \text{Constants}$$