Heat Equation · aiwiki.page

The heat equation is a linear partial differential equation that describes how temperature changes through heat conduction, and more generally how a quantity spreads by diffusion. Its standard form is $u_t=\alpha\Delta u$ , where $u$ depends on position and time, $\alpha>0$ is a diffusivity, and $\Delta$ is the spatial Laplacian. It is a prototype of a parabolic partial differential equation, characterized by smoothing and dissipative evolution. (web.stanford.edu)

Mathematical form and physical derivation

For position $x$ in $n$ -dimensional Euclidean space and time $t>0$ , the equation is

\frac{\partial u}{\partial t} =\alpha\Delta u, \qquad \Delta u=\sum_{j=1}^{n} \frac{\partial^2u}{\partial x_j^2}.

The Laplace operator combines second spatial partial derivatives. In one dimension, this becomes $u_t=\alpha u_{xx}$ . Its linearity permits the superposition principle: linear combinations of solutions remain solutions, with correspondingly combined initial and boundary data. (web.stanford.edu)

The physical derivation combines conservation of energy with Fourier’s law, which states that conductive heat flux is proportional to the negative temperature gradient. For a stationary, isotropic material,

\mathbf q=-k\nabla u, \qquad \rho c\,u_t=\nabla\cdot(k\nabla u)+Q.

Here $k$ is thermal conductivity, $\rho$ is density, $c$ is specific heat capacity, and $Q$ is heat production per unit volume per unit time. With constant coefficients and no source, this reduces to the standard equation with thermal diffusivity $\alpha=k/(\rho c)$ , measured in length squared per unit time. The negative flux sign expresses heat transport toward lower temperatures. (ocw.mit.edu)

Initial and boundary conditions

A heat problem requires an initial distribution,

u(x,0)=f(x).

On a bounded spatial domain $\Omega$ , boundary conditions specify interaction with the surroundings, forming an initial–boundary value problem. Dirichlet conditions prescribe temperature; Neumann conditions prescribe its normal derivative and thus, through Fourier’s law, conductive flux; Robin conditions relate temperature to that derivative. An insulated boundary has zero normal heat flux. Boundary conditions determine whether heat is retained, supplied, or removed. (ocw.mit.edu)

For sufficiently regular compatible data, classical solutions satisfy the equation pointwise. Less regular data can instead be treated using weak solutions, which express the equation through integration against test functions. On all of space, uniqueness also requires a suitable solution class, such as bounded solutions: unrestricted growth at infinity can invalidate uniqueness. (web.stanford.edu)

Heat kernel and solution formulas

On $\mathbb R^n$ , the fundamental solution, or heat kernel, is

G(x,t)=\frac{1}{(4\pi\alpha t)^{n/2}} \exp\left(-\frac{|x|^2}{4\alpha t}\right).

For suitable initial data, the solution is the convolution

u(x,t)=\int_{\mathbb R^n}G(x-y,t)f(y)\,dy.

The kernel is positive and has integral one. It is a Gaussian distribution with coordinate variance $2\alpha t$ , so its characteristic spreading distance grows as $\sqrt{\alpha t}$ . Although it approaches a point concentration as $t$ tends to zero, it is smooth for every positive time. (web.stanford.edu)

The Fourier transform gives an equivalent description:

\widehat u(\xi,t) =e^{-\alpha|\xi|^2t}\widehat f(\xi).

Each spatial frequency decays exponentially, and higher frequencies decay faster. This explains why fine-scale variations disappear more rapidly than broad ones. (ocw.mit.edu)

On an interval $0<x<L$ with zero endpoint temperatures, separation of variables yields a Fourier sine series:

u(x,t)=\sum_{m=1}^{\infty} b_m e^{-\alpha(m\pi/L)^2t} \sin(m\pi x/L).

The coefficients represent the initial distribution. On more general domains, analogous expansions use Laplacian eigenvalues and eigenfunctions adapted to the boundary conditions. (web.stanford.edu)

Qualitative properties

The maximum principle states that, for the source-free equation on a bounded space–time region, a classical solution cannot exceed the largest temperature prescribed initially or on the spatial boundary. Applying the same statement to differences of solutions gives uniqueness and stability with respect to data. Nonnegative initial and boundary values remain nonnegative. (web.stanford.edu)

For constant material properties, insulated boundaries conserve the spatial integral of temperature. Meanwhile, on a bounded domain with homogeneous Dirichlet or Neumann conditions,

\frac{d}{dt}\frac12\int_\Omega u^2\,dx =-\alpha\int_\Omega|\nabla u|^2\,dx\leq0.

Thus spatial variation dissipates. On a connected bounded insulated domain, solutions approach their initial spatial average; with zero-temperature Dirichlet boundaries, they approach zero. (web.stanford.edu)

Probability and computation

In probability theory, the heat kernel is the transition density of scaled Brownian motion. Standard Brownian motion corresponds to $\alpha=\tfrac12$ ; scaling it by $\sqrt{2\alpha}$ gives diffusivity $\alpha$ . The same equation therefore describes both deterministic diffusion and the evolution of a random particle’s distribution. (stat.berkeley.edu)

A basic finite difference method uses

U_j^{m+1}=U_j^m+ r(U_{j+1}^m-2U_j^m+U_{j-1}^m), \qquad r=\frac{\alpha\Delta t}{(\Delta x)^2}.

For the standard one-dimensional explicit scheme, numerical stability requires $r\leq\tfrac12$ . Implicit schemes avoid this particular restriction but require solving algebraic systems at each step. (ocw.mit.edu)

Recovering an earlier temperature field from later measurements is a backward inverse problem. Forward evolution suppresses high frequencies, so reversal amplifies measurement errors exponentially. This makes the backward problem ill-posed in usual function-space norms, even when a solution exists. (math.nyu.edu)