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**System of PDEs--Heat Equation For Two Objects**

Hello everyone,

Before is a system of partial differential equations; to be specific, it is this system:

[itex]\frac{\partial U_A }{\partial t} = - \frac{k_B}{k_A} \alpha_A \left( \frac{\partial^2 U_B}{\partial x^2} + \frac{\partial^2 U_B}{\partial y^2} + \frac{\partial^2 U_B}{\partial z^2} \right)[/itex]

and

[itex]\frac{\partial U_B }{\partial t} = \alpha_B \left( \frac{\partial^2 U_B}{\partial x^2} + \frac{\partial^2 U_B}{\partial y^2} + \frac{\partial^2 U_B}{\partial z^2} \right)[/itex]

I am not very certain as to how to solve this--as a matter of fact, I do not even know if it is possible to solve this. So, does this system have a solution [itex]U_A(x,y,z,t)[/itex] and [itex]U_B(x,y,z,t)[/itex]? And if it does, could someone help me with solving it, such as providing hints or suitable reading materials? I would certainly appreciate it.

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