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Equation

Given a domain \$\Omega \subset \mathbb{R}^d, d=1,2,3\$, \$\Omega\$ is partitioned into \$N_r\$ regions \$\Omega_i,i=1,\ldots,N_r\$ corresponding to different materials (solid or fluid).

\$\rho C_p \frac{\partial T}{\partial t} + \mathb{u} \cdot \nabla T - \nabla \cdot \left\( k \nabla T \right\) = Q\$

which is completed with initial value and boundary conditions

Initial value
\$\text{at } t=0, \quad T(x,0) = T_0(x)\$
Dirichlet Boundary condition
\$\\$

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