# How to solve two partial differential equations that are coupled through the boundary conditions

My problem is a heat transfer problem between two systems that are connected through an interface. The heat balance equation for each system is a second order PDE and the systems have equal temperature and heat flux at the common interface between them. Actually, I have the final solution, but I don't know what are the steps that should be followed to solve this system of PDEs as the solution of each one depends on the solution of the second system.

PDE for the first system is:

$$\rho_r C_r \frac{\partial T}{\partial t} + \left(\frac{Q \rho_f C_f}{2πrh} \right) \frac{\partial T}{\partial r} - K_t \frac{\partial^{2} T}{\partial z^{2}}=0$$

The initial and boundary conditions:

\begin{align} T(r,z,0) &= T_{i} \\ T(0,z,t) &= T_{0} \end{align}

The PDE of the surrounding:

$$\rho_m C_m \frac{\partial T_m}{\partial t} = K_m \frac{\partial^{2} T_m}{\partial z^{2}}$$

The initial and boundary conditions for the surrounding:

$$T_m(z,0)=T_i$$

The common boundary conditions ($z = h$):

\begin{align} T_m(z,t) &= T(z,t) \\ k_m \frac{\partial T_m}{\partial z} &= K_t \frac{\partial T}{\partial z} \end{align}

• Why don't you post the problem and your solution? As it stands, what you have written means very little. Jul 8, 2018 at 23:42