Heat equation

In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. Since then, the heat equation and its variants have been found to be fundamental in many parts of both pure and applied mathematics.

Definition Given an open subset U of R n {\displaystyle \mathbb {R} ^{n}} and a subinterval I of R {\displaystyle \mathbb {R} } , one says that a function u : U × I → R {\displaystyle u:U\times I\to \mathbb {R} } is a solution of the heat equation if

∂ u ∂ t = ∂ 2 u ∂ x 1 2 + ⋯ + ∂ 2 u ∂ x n 2 , {\displaystyle {\frac {\partial u}{\partial t}}={\frac {\partial ^{2}u}{\partial x_{1}^{2}}}+\cdots +{\frac {\partial ^{2}u}{\partial x_{n}^{2}}},}

where ( x 1 , x 2 , ⋯ , x n , t ) {\displaystyle (x_{1},x_{2},\cdots ,x_{n},t)} denotes a general point of the domain. It is typical to refer to t {\displaystyle t} as time and ( x 1 , x 2 , ⋯ , x n ) {\displaystyle (x_{1},x_{2},\cdots ,x_{n})} as spatial variables, even in abstract contexts where these phrases fail to have their intuitive meaning. The collection of spatial variables is often referred to simply as x. For any given value of t, the right-hand side of the equation is the Laplacian of the function u ( ⋅ , t ) : U → R {\displaystyle u(\cdot ,t):U\to \mathbb {R} } . As such, the heat equation is often written more compactly as

In physics and engineering contexts, especially in the context of diffusion through a medium, it is more common to fix a Cartesian coordinate system and then to consider the specific case of a function u ( x , y , z , t ) {\displaystyle u(x,y,z,t)} of three spatial variables ( x , y , z ) {\displaystyle (x,y,z)} and time variable t {\displaystyle t} . One then says that u is a solution of the heat equation if

∂ u ∂ t = α ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 ) {\displaystyle {\frac {\partial u}{\partial t}}=\alpha \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\right)}

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