Heat equation

From Academic Kids

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

<math>u_t = k ( u_{xx} + u_{yy} + u_{zz} ) \quad <math>


  • 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;
  • <math>u_{xx}<math>, <math>u_{yy}<math>, and <math>u_{zz}<math> 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

<math>u_t = k \Delta u \quad <math>

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
<math> q_t(V) = \int_V Q(t,x)\,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 d S and with unit normal vector n is
<math> \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> 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 Fourier law states that heat energy flow has the following linear dependence on the temperature gradient
<math> \mathbf{H}(x) = -\mathbf{A}(x) \cdot [\operatorname{grad}(u)] (x) <math>
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

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

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

<math> \partial_t u(t,x) = \kappa(x) \sum_{i, j} \partial_{x_i} a_{i j} \partial_{x_j} u (t,x) <math>


  • 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

de:Wrmeleitungsgleichung ru:Уравнение диффузии sl:difuzijska enačba


Academic Kids Menu

  • Art and Cultures
    • Art (http://www.academickids.com/encyclopedia/index.php/Art)
    • Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
    • Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
    • Music (http://www.academickids.com/encyclopedia/index.php/Music)
    • Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
  • Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
  • Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
  • Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
    • Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
    • Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
    • Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
    • Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
  • History (http://www.academickids.com/encyclopedia/index.php/History)
    • Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
    • Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
    • Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
    • Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
    • Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
    • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
    • United States (http://www.academickids.com/encyclopedia/index.php/United_States)
    • Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
    • World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
  • Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
  • Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
  • Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
  • Science (http://www.academickids.com/encyclopedia/index.php/Science)
    • Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
    • Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
    • Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
    • Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
    • Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
    • Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
    • Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
    • Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
  • Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
    • Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
    • Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
    • Government (http://www.academickids.com/encyclopedia/index.php/Government)
    • Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
    • Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
  • Space and Astronomy
    • Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
    • Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
  • Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
  • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
  • Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
  • US States (http://www.academickids.com/encyclopedia/index.php/US_States)


  • Home Page (http://academickids.com/encyclopedia/index.php)
  • Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

  • Clip Art (http://classroomclipart.com)
Personal tools