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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
- 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;
- uxx, uyy, and uzz are the second spatial derivatives (thermal conductions) of temperature in the x, y, and z directions, respectively
- k is a material-specific constant (thermal diffusivity)
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 .
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
- 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
Thus the rate of heat flow into V is also given by the surface integral
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
By Green's theorem, the previous surface integral for heat flow into V can be transformed into the volume integral
- 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
Putting these equations together gives the general equation of heat flow:
- 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.
- Derivation of the heat equation
- Linear heat equations: Particular solutions and boundary value problems - from EqWorld
- Nonlinear heat equations: Exact solutions - from EqWorld
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