Heat Equation

$u\_t\; =\; k\; (\; u\_\{xx\}\; +\; u\_\{yy\}\; +\; u\_\{zz\}\; )\; \backslash quad$

• u(t, x, y, z) is temperature as a function of time and space;

• u_t is the rate of change of temperature at a point over time;

• $u\_\{xx\}$, $u\_\{yy\}$, and $u\_\{zz\}$ 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)

Heat conduction in non-homogeneous anisotropic media

• The time rate of heat flow into a region V is given by a time-dependent quantity q_t(V). We assume q has a density, so that

$q\_t(V)\; =\; \backslash int\_V\; Q(t,x)\backslash ,d\; x\; \backslash quad$

• 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

$\backslash mathbf\{H\}(x)\; \backslash cdot\; \backslash mathbf\{n\}(x)\; \backslash ,\; dS$

$q\_t(V)=\; -\; \backslash int\_\{\backslash partial\; V\}\; \backslash mathbf\{H\}(x)\; \backslash cdot\; \backslash mathbf\{n\}(x)\; \backslash ,\; dS$

• The Fourier law states that heat energy flow has the following linear dependence on the temperature gradient

$\backslash mathbf\{H\}(x)\; =\; -\backslash mathbf\{A\}(x)\; \backslash cdot\backslash operatorname\{grad\}(u)(x)$

where A(x) is a 3 × 3 real matrix, which in fact is symmetric and non-negative.

$q\_t(V)\; =\; -\; \backslash int\_\{\backslash partial\; V\}\; \backslash mathbf\{H\}(x)\; \backslash cdot\; \backslash mathbf\{n\}(x)\; \backslash ,\; dS$

$=\; \backslash int\_\{\backslash partial\; V\}\; \backslash mathbf\{A\}(x)\; \backslash cdot\backslash operatorname\{grad\}(u)(x)\; \backslash cdot\; \backslash mathbf\{n\}(x)\; \backslash ,\; dS$

$=\; \backslash int\_V\; \backslash sum\_\{i,\; j\}\; \backslash partial\_\{x\_i\}\; a\_\{i\; j\}\; \backslash partial\_\{x\_j\}\; u\; (t,x)\backslash ,dx$

• 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

$\backslash partial\_t\; u(t,x)\; =\; \backslash kappa(x)\; Q(t,x)\backslash ,\; dx$

$\backslash partial\_t\; u(t,x)\; =\; \backslash kappa(x)\; \backslash sum\_\{i,\; j\}\; \backslash partial\_\{x\_i\}\; a\_\{i\; j\}\; \backslash partial\_\{x\_j\}\; u\; (t,x)$

• 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.

External links

• Derivation of the heat equation
• Linear heat equations: Particular solutions and boundary value problems - from EqWorld
• Nonlinear heat equations: Exact solutions - from EqWorld
• Systems of reaction-diffusion equations - from EqWorld

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