# How can the following initial and boundary value problem be solved?

I have derived the following problem and I have to find an analytical solution to it. The following equations describe the temperature variation of an injected fluid (T1) into an underground layer while the heat is transferred to the surroundings by conduction. The second PDE describes the temperature of the surroundings (T2).

$$\frac{∂T_1}{∂t}+A*\frac{∂T_1}{∂r}=B*\frac{∂^2 T_1}{∂Z^2}$$

$$C*\frac{∂T_2}{∂t}=B*\frac{∂^2 T_2}{∂z^2}$$

Where:

$$T_1(r, z, 0)=1$$ $$T_1(0, z, t) = 0$$ $$T_1(r, 1, t) = T_2$$ $$\frac{∂T_1}{∂z}= D*\frac{∂T_2}{∂z}$$

and

$$T_2(r,z,0)=1$$ $$T_2(r,1,t) = T_1$$

• Are you using the heat equation? If so, it should be something like $$\frac{1}{c}\frac{\partial T}{\partial t} = \nabla^2 T = \frac{\partial^2 T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial z^2}$$ Jul 5, 2018 at 7:45
• You are right, but I have introduced the equations in dimensionless form. So, you can find from the initial and boundary conditions values that include 0 and 1. Jul 5, 2018 at 13:52
• That doesn't make sense. Rescaling the variables would not affect the order of derivatives. What were the original PDEs? Jul 5, 2018 at 14:11
• Ok, I'll accept that. What about the 4th boundary condition, where is it evaluated at? Jul 5, 2018 at 15:19
• The equations are as follow: ρ_r * C_r * (∂T/∂t) + ((Q*ρ_w *C_w)/2πrh) *(∂T/∂r) - K_t * (∂^2 T/∂z^2) =0 && ρ_m * C_m * (∂T_m/∂t)=K_m * (∂^2 T_m/∂z^2) && T=T_m=T_o ⟹t=0 && T=T_in⟹r=0 && T=T_m⟹z=b && K_t * (∂T/∂z)=K_m * (∂T_m/∂z). I hope these equations are clear. I need to confirm that, the first equation is heat balance equation for flowing fluid, so it includes local and convection tesms for temperature changes. Jul 5, 2018 at 15:32