Editing Heat equation (section)

=== Heat flow in non-homogeneous anisotropic media ===
In general, the study of heat conduction is based on several principles. Heat flow is a form of [[energy]] flow, and as such it is meaningful to speak of the time rate of flow of heat into a region of space.
* The time rate of heat flow into a region ''V'' is given by a time-dependent quantity ''q''<sub>''t''</sub>(''V''). We assume ''q'' has a [[Radon-Nikodym Derivative|density]] ''Q'', so that <math display="block"> q_t(V) = \int_V Q(x,t)\,d x \quad </math>
* Heat flow is a time-dependent vector function '''H'''(''x'') characterized as follows: the time rate of heat flowing through an infinitesimal surface element with area ''dS'' and with unit normal vector '''n''' is <math display="block"> \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS .</math> Thus the rate of heat flow into ''V'' is also given by the surface integral <math display="block">  q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS </math> where '''n'''(''x'') is the outward pointing normal vector at ''x''.
* The [[law of heat conduction|Fourier law]] states that heat energy flow has the following linear dependence on the temperature gradient <math display="block"> \mathbf{H}(x) = -\mathbf{A}(x) \cdot \nabla u (x) </math> where '''A'''(''x'') is a 3&nbsp;×&nbsp;3 real [[matrix (mathematics)|matrix]] that is [[symmetric]] and [[positive-definite matrix|positive definite]].
* By the [[divergence theorem]], the previous surface integral for heat flow into ''V'' can be transformed into the volume integral <math display="block">\begin{align}
q_t(V) &= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS \\
&= \int_{\partial V} \mathbf{A}(x) \cdot \nabla u (x) \cdot \mathbf{n}(x) \, dS \\
&= \int_V \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t) \bigr)\,dx 
\end{align}</math>
* The time rate of temperature change at ''x'' is proportional to the heat flowing into an infinitesimal volume element, where the constant of proportionality is dependent on a constant ''κ'' <math display="block"> \partial_t u(x,t) = \kappa(x) Q(x,t)</math>

Putting these equations together gives the general equation of heat flow:
: <math> \partial_t u(x,t) = \kappa(x) \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t)\bigr) </math>

'''Remarks'''
* The coefficient ''κ''(''x'') is the inverse of [[specific heat]] of the substance at ''x'' × [[density]] of the substance at ''x'': <math>\kappa = 1/(\rho c_p)</math>.
* In the case of an isotropic medium, the matrix '''A''' is a scalar matrix equal to [[thermal conductivity]] ''k''.
* In the anisotropic case where the coefficient matrix '''A''' is not scalar and/or if it depends on ''x'', then an explicit formula for the solution of the heat equation can seldom be written down, though it is usually possible to consider the associated abstract [[Cauchy problem]] and show that it is a [[well-posed problem]] and/or to show some qualitative properties (like preservation of positive initial data, infinite speed of propagation, convergence toward an equilibrium, smoothing properties). This is usually done by [[one-parameter semigroup]]s theory: for instance, if ''A'' is a symmetric matrix, then the [[elliptic operator]] defined by <math display="block">Au(x):=\sum_{i, j} \partial_{x_i} a_{i j}(x) \partial_{x_j} u (x)</math> is [[self-adjoint]] and dissipative, thus by the [[spectral theorem]] it generates a [[one-parameter semigroup]].