# Help solving a 2nd Order Parabolic Linear PDE with One non-constant Coefficient

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: $$\large A,J,r_1,\theta_r \implies \text{Constants}$$

• Does your initial condition satisfy the boundary conditions? (Is your initial condition space-dependent?) If not, then I think this is not be a well-posed problem. – nukeguy Mar 8 '17 at 20:31
• Yes, i think it does. My initial condition is space-dependent. I solved it numerically and it does plot a good result, but i need the analytical solution. – shewlong Mar 8 '17 at 20:54
• @nukeguy and i can also change the second Boundary Condition to the 1rst derivative of "Theta(L,t)" = 0. Do you think it will help solving it? "r" is a space parameter. The Boundary Conditions and the Initial Conditions that i modelled were based on the experiment. I need the Analytical Solution to adjust the parameters with the flow-experiment. – shewlong Mar 8 '17 at 21:04
• I'm not sure. It's hard enough obtaining nice analytical solutions to ODEs with non-constant coefficients, let alone a PDE. Even if you could find some way to write down an explicit solution, it will likely be in terms of obscure mathematical functions. Have you tried plugging it into Mathematica? Also, is there a reason your numerical simulation is insufficient? I'm still not sure I understand why you need an analytical solution. – nukeguy Mar 8 '17 at 21:06
• @nukeguy Yes, i used mathematica to solve it numerically and it gave me the results. This solution is probably going to have an Hermite Polynomial, but it doesn't matter how odd are the functions, i just need to have the equation to adjust the parameters experimentally. I can do it numerically, but it takes me a looong time, sometimes doesn't fit correctly. Can i e-mail you, so that we can talk about it properly to discuss it? – shewlong Mar 8 '17 at 23:42