Heat equation

The heat 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

Related Topics:
Partial differential equation - Isotropic - Homogeneous - Dimension

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:u_t = k ( u_{xx} + u_{yy} + u_{zz} ) quad

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where:

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• 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 called thermal diffusivity.

The heat equation is a consequence of Fourier's law of cooling (see heat conduction).

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

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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.

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

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

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:u_t = k Delta u, quad

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where the Laplace operator is taken in the spatial variables.

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The heat equation also describes other physical processes, such as diffusion. It also can be used to model some processes in finance.

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Heat equation: Solving the heat equation using Fourier series ►