- #1
Bashyboy
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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.
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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