Term information
The heat equation is an important partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time. For a function u(x,y,z,t) of three spatial variables (x,y,z) and the time variable t, the heat equation is \frac{\partial u}{\partial t} -\alpha\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right)=0 also written \frac{\partial u}{\partial t} - \alpha \nabla^2 u=0 or sometimes \frac{\partial u}{\partial t} - \alpha \Delta u=0 where is a positive constant and \Delta\ or \nabla^2\ denotes the Laplacian operator. For the mathematical treatment it is sufficient to consider the case = 1. For the case of variation of temperature u(x,y,z,t) is the temperature and is the thermal diffusivity source: http://en.wikipedia.org/wiki/Heat_equation
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mrow> <mfrac> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>t</mi> </mrow> </mfrac> <mo>-</mo> <mrow> <mi>α</mi> <mo>⁢</mo> <mfenced close=")" open="("> <mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <mo>⁢</mo> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <mo>⁢</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <mo>⁢</mo> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <mo>⁢</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <mo>⁢</mo> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <mo>⁢</mo> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </mfenced> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mrow> <mfrac> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>t</mi> </mrow> </mfrac> <mo>-</mo> <mrow> <mi>α</mi> <mo>⁢</mo> <msup> <mo>∇</mo> <mn>2</mn> </msup> <mo>⁢</mo> <mi>u</mi> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math>
\frac{\partial u}{\partial t} - \alpha \nabla^2 u=0
\frac{\partial u}{\partial t} -\alpha\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right)=0
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mfrac> <mrow> <mo>∂</mo> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>-</mo> <mi>α</mi> <msup> <mo>∇</mo> <mn>2</mn> </msup> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> <annotation encoding="SnuggleTeX">\[ \frac{\partial u}{\partial t} - \alpha \nabla^2 u=0 \]</annotation> </semantics> </math>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mfrac> <mrow> <mo>∂</mo> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>-</mo> <mi>α</mi> <mfenced close=")" open="("> <mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </mfenced> <mo>=</mo> <mn>0</mn> </mrow> <annotation encoding="SnuggleTeX">\[ \frac{\partial u}{\partial t} -\alpha\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right)=0 \]</annotation> </semantics> </math>