Heat equation

The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. In the special case of heat propagation in an isotropic and homogeneous medium in the 3-dimensional space, this equation is

[itex]u_t = k ( u_{xx} + u_{yy} + u_{zz} ) \quad [itex]

where:

• u(t, x, y, z) is temperature as a function of time and space;
• ut is the rate of change of temperature at a point over time;
• [itex]u_{xx}[itex], [itex]u_{yy}[itex], and [itex]u_{zz}[itex] are the second spatial derivatives (thermal conductions) of temperature in the x, y, and z directions, respectively

To solve the heat equation, we also need to specify boundary conditions for u.

Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object.

The heat equation is the prototypical example of a parabolic partial differential equation.

Using the Laplace operator, the heat equation can be generalized to

[itex]u_t = k \Delta u \quad [itex]

Heat conduction 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 qt(V). We assume q has a density, so that
[itex] q_t(V) = \int_V Q(t,x)\,d x \quad [itex]
• 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 d S and with unit normal vector n is
[itex] \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS [itex]

Thus the rate of heat flow into V is also given by the surface integral

[itex] q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS [itex]

where n(x) is the outward pointing normal vector at x.

• The Fourier law states that heat energy flow has the following linear dependence on the temperature gradient
[itex] \mathbf{H}(x) = -\mathbf{A}(x) \cdot [\operatorname{grad}(u)] (x) [itex]
where A(x) is a 3 × 3 real matrix, which in fact is symmetric and non-negative.

By Green's theorem, the previous surface integral for heat flow into V can be transformed into the volume integral

[itex] q_t(V) = - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS [itex]
[itex] = \int_{\partial V} \mathbf{A}(x) \cdot [\operatorname{grad}(u)] (x) \cdot \mathbf{n}(x) \, dS [itex]
[itex] = \int_V \sum_{i, j} \partial_{x_i} a_{i j} \partial_{x_j} u (t,x)\,dx [itex]
• 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
[itex] \partial_t u(t,x) = \kappa(x) Q(t,x)\, dx[itex]

Putting these equations together gives the general equation of heat flow:

[itex] \partial_t u(t,x) = \kappa(x) \sum_{i, j} \partial_{x_i} a_{i j} \partial_{x_j} u (t,x) [itex]

Remarks.

• The constant κ(x) is the inverse of specific heat of the substance at x × density of the substance at x.
• In the case of an isotropic medium, the matrix A is a scalar matrix equal to thermal conductivity.
• The Jacobi theta function is the unique solution to the one-dimensional heat equation with periodic boundary conditions at t = 0.

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