Variational formulation of the problem
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Here we will limit ourselves to presenting the problem with only the conditions at the limits of imposed temperature [:ref:`§2.1 <§2.1>`], of imposed normal flow [:ref:`§2.3 <§2.3>`] or of exchange [:ref:`§2.4 <§2.4>`]. The boundary conditions of wall exchange [:ref:`§2.5 <§2.5>`] are treated in [:ref:`§4 <§4>`] and those with linear relationships [:ref:`§2.2 <§2.2>`] are easily reduced to that of [:ref:`§2.1 <§2.1>`].

Let :math:`\Omega` be an open of :math:`{ℝ}^{3}`, of border :math:`\Gamma ={\Gamma }_{1}\cup {\Gamma }_{2}\cup {\Gamma }_{3}`.

The weak formulation of the heat equation is:

:math:`\underset{\Omega }{\int }\rho {C}_{p}\frac{\partial T}{\partial t}\text{.}vd\Omega +\underset{\Omega }{\int }\lambda \nabla T\text{.}\nabla vd\Omega -\underset{\Gamma }{\int }\lambda \frac{\partial T}{\partial n}\text{.}vd\Gamma =\underset{\Omega }{\int }s\text{.}vd\Omega`

where :math:`v` is a sufficiently regular function cancelling uniformly over :math:`{\Gamma }_{1}`. With the following boundary conditions:

:math:`\{\begin{array}{ccc}T={T}_{1}(r,t)& & \text{sur}{\Gamma }_{1}\\ \lambda \frac{\partial T}{\partial n}=q(r,t)& & \text{sur}{\Gamma }_{2}\\ \lambda \frac{\partial T}{\partial n}=h(r,t)({T}_{\text{ext}}(r,t)-T)& & \text{sur}{\Gamma }_{3}\end{array}`

The variational formulation of the problem is:

:math:`\underset{\Omega }{\int }\rho {C}_{p}\frac{\partial T}{\partial t}\text{.}vd\Omega +\underset{\Omega }{\int }\lambda \nabla T\text{.}\nabla vd\Omega +\underset{{\Gamma }_{3}}{\int }hT\text{.}vd\Gamma =\underset{\Omega }{\int }s\text{.}vd\Omega +\underset{{\Gamma }_{2}}{\int }q\text{.}vd\Gamma +\underset{{\Gamma }_{3}}{\int }h{T}_{\text{ext}}\text{.}vd\Gamma`