Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. The height and redness indicate the temperature at each point. The initial state has a uniformly hot hoof-shaped region (red) surrounded by uniformly cold region (yellow). As time passes the heat diffuses into the cold region.

In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. 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.

As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications. Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, inspiring the introduction of the Ricci flow by Richard Hamilton in 1982 and culminating in the proof of the Poincaré conjecture by Grigori Perelman in 2003. Certain solutions of the heat equation known as heat kernels provide subtle information about the region on which they are defined, as exemplified through their application to the Atiyah–Singer index theorem.^[1]

The heat equation, along with variants thereof, is also important in many fields of science and applied mathematics. In probability theory, the heat equation is connected with the study of random walks and Brownian motion via the Fokker–Planck equation. The Black–Scholes equation of financial mathematics is a small variant of the heat equation, and the Schrödinger equation of quantum mechanics can be regarded as a heat equation in imaginary time. In image analysis, the heat equation is sometimes used to resolve pixelation and to identify edges. Following Robert Richtmyer and John von Neumann's introduction of "artificial viscosity" methods, solutions of heat equations have been useful in the mathematical formulation of hydrodynamical shocks. Solutions of the heat equation have also been given much attention in the numerical analysis literature, beginning in the 1950s with work of Jim Douglas, D.W. Peaceman, and Henry Rachford Jr.

Statement of the equation

In mathematics, if given an open subset U of Rⁿ and a subinterval I of R, one says that a function u : U × I → R is a solution of the heat equation if

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

where (x₁, …, x_n, t) denotes a general point of the domain. It is typical to refer to t as "time" and x₁, …, 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. 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) of three spatial variables (x, y, z) and time variable t. One then says that u is a solution of the heat equation if

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

in which α is a positive coefficient called the thermal diffusivity of the medium. In addition to other physical phenomena, this equation describes the flow of heat in a homogeneous and isotropic medium, with u(x, y, z, t) being the temperature at the point (x, y, z) and time t. If the medium is not homogeneous and isotropic, then α would not be a fixed coefficient, and would instead depend on (x, y, z); the equation would also have a slightly different form. In the physics and engineering literature, it is common to use ∇² to denote the Laplacian, rather than ∆.

In mathematics as well as in physics and engineering, it is common to use Newton's notation for time derivatives, so that ${\displaystyle {\dot {u}}}$ is used to denote ∂u/∂t, so the equation can be written

Note also that the ability to use either ∆ or ∇² to denote the Laplacian, without explicit reference to the spatial variables, is a reflection of the fact that the Laplacian is independent of the choice of coordinate system. In mathematical terms, one would say that the Laplacian is "translationally and rotationally invariant." In fact, it is (loosely speaking) the simplest differential operator which has these symmetries. This can be taken as a significant (and purely mathematical) justification of the use of the Laplacian and of the heat equation in modeling any physical phenomena which are homogeneous and isotropic, of which heat diffusion is a principal example.

The "diffusivity constant" α is often not present in mathematical studies of the heat equation, while its value can be very important in engineering. This is not a major difference, for the following reason. Let u be a function with

${\displaystyle {\frac {\partial u}{\partial t}}=\alpha \Delta u.}$

Define a new function ${\displaystyle v(t,x)=u(t/\alpha ,x)}$. Then, according to the chain rule, one has

${\displaystyle {\frac {\partial }{\partial t}}v(t,x)={\frac {\partial }{\partial t}}u(t/\alpha ,x)=\alpha ^{-1}{\frac {\partial u}{\partial t}}(t/\alpha ,x)=\Delta u(t/\alpha ,x)=\Delta v(t,x)}$

(⁎)

Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of α and solutions of the heat equation with α = 1. As such, for the sake of mathematical analysis, it is often sufficient to only consider the case α = 1.

Since ${\displaystyle \alpha >0}$ there is another option to define a ${\displaystyle v}$ satisfying ${\textstyle {\frac {\partial }{\partial t}}v=\Delta v}$ as in (⁎) above by setting ${\displaystyle v(t,x)=u(t,\alpha ^{1/2}x)}$. Note that the two possible means of defining the new function ${\displaystyle v}$ discussed here amount, in physical terms, to changing the unit of measure of time or the unit of measure of length.

Interpretation

Character of the solutions

Solution of a 1D heat partial differential equation. The temperature (${\displaystyle u}$) is initially distributed over a one-dimensional, one-unit-long interval (x = [0,1]) with insulated endpoints. The distribution approaches equilibrium over time.

The behavior of temperature when the sides of a 1D rod are at fixed temperatures (in this case, 0.8 and 0 with initial Gaussian distribution). The temperature approaches a linear function because that is the stable solution of the equation: wherever temperature has a nonzero second spatial derivative, the time derivative is nonzero as well.

The heat equation implies that peaks (local maxima) of ${\displaystyle u}$ will be gradually eroded down, while depressions (local minima) will be filled in. The value at some point will remain stable only as long as it is equal to the average value in its immediate surroundings. In particular, if the values in a neighborhood are very close to a linear function ${\displaystyle Ax+By+Cz+D}$, then the value at the center of that neighborhood will not be changing at that time (that is, the derivative ${\displaystyle {\dot {u}}}$ will be zero).

A more subtle consequence is the maximum principle, that says that the maximum value of ${\displaystyle u}$ in any region ${\displaystyle R}$ of the medium will not exceed the maximum value that previously occurred in ${\displaystyle R}$, unless it is on the boundary of ${\displaystyle R}$. That is, the maximum temperature in a region ${\displaystyle R}$ can increase only if heat comes in from outside ${\displaystyle R}$. This is a property of parabolic partial differential equations and is not difficult to prove mathematically (see below).

Another interesting property is that even if ${\displaystyle u}$ initially has a sharp jump (discontinuity) of value across some surface inside the medium, the jump is immediately smoothed out by a momentary, infinitesimally short but infinitely large rate of flow of heat through that surface. For example, if two isolated bodies, initially at uniform but different temperatures ${\displaystyle u_{0}}$ and ${\displaystyle u_{1}}$, are made to touch each other, the temperature at the point of contact will immediately assume some intermediate value, and a zone will develop around that point where ${\displaystyle u}$ will gradually vary between ${\displaystyle u_{0}}$ and ${\displaystyle u_{1}}$.

If a certain amount of heat is suddenly applied to a point in the medium, it will spread out in all directions in the form of a diffusion wave. Unlike the elastic and electromagnetic waves, the speed of a diffusion wave drops with time: as it spreads over a larger region, the temperature gradient decreases, and therefore the heat flow decreases too.

Specific examples

Solving the heat equation using Fourier series

Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions.

The following solution technique for the heat equation was proposed by Joseph Fourier in his treatise Théorie analytique de la chaleur, published in 1822. Consider the heat equation for one space variable. This could be used to model heat conduction in a rod. The equation is

${\displaystyle \displaystyle u_{t}=\alpha u_{xx}}$

(1)

where u = u(x, t) is a function of two variables x and t. Here

• x is the space variable, so x ∈ [0, L], where L is the length of the rod.
• t is the time variable, so t ≥ 0.

We assume the initial condition

${\displaystyle u(x,0)=f(x)\quad \forall x\in [0,L]}$

(2)

where the function f is given, and the boundary conditions

${\displaystyle u(0,t)=0=u(L,t)\quad \forall t>0}$.

(3)

Let us attempt to find a solution of (1) that is not identically zero satisfying the boundary conditions (3) but with the following property: u is a product in which the dependence of u on x, t is separated, that is:

${\displaystyle u(x,t)=X(x)T(t).}$

(4)

This solution technique is called separation of variables. Substituting u back into equation (1),

${\displaystyle {\frac {T'(t)}{\alpha T(t)}}={\frac {X''(x)}{X(x)}}.}$

Since the right hand side depends only on x and the left hand side only on t, both sides are equal to some constant value −λ. Thus:

${\displaystyle T'(t)=-\lambda \alpha T(t)}$

(5)

and

${\displaystyle X''(x)=-\lambda X(x).}$

(6)

We will now show that nontrivial solutions for (6) for values of λ ≤ 0 cannot occur:

1. Suppose that λ < 0. Then there exist real numbers B, C such that
   $${\displaystyle X(x)=Be^{{\sqrt {-\lambda }}\,x}+Ce^{-{\sqrt {-\lambda }}\,x}.}$$

   
   From (3) we get X(0) = 0 = X(L) and therefore B = 0 = C which implies u is identically 0.
2. Suppose that λ = 0. Then there exist real numbers B, C such that X(x) = Bx + C. From equation (3) we conclude in the same manner as in 1 that u is identically 0.
3. Therefore, it must be the case that λ > 0. Then there exist real numbers A, B, C such that
   $${\displaystyle T(t)=Ae^{-\lambda \alpha t}}$$

   
   and
   $${\displaystyle X(x)=B\sin \left({\sqrt {\lambda }}\,x\right)+C\cos \left({\sqrt {\lambda }}\,x\right).}$$

   
   From (3) we get C = 0 and that for some positive integer n,
   $${\displaystyle {\sqrt {\lambda }}=n{\frac {\pi }{L}}.}$$

   

This solves the heat equation in the special case that the dependence of u has the special form (4).

In general, the sum of solutions to (1) that satisfy the boundary conditions (3) also satisfies (1) and (3). We can show that the solution to (1), (2) and (3) is given by

${\displaystyle u(x,t)=\sum _{n=1}^{\infty }D_{n}\sin \left({\frac {n\pi x}{L}}\right)e^{-{\frac {n^{2}\pi ^{2}\alpha t}{L^{2}}}}}$

where

${\displaystyle D_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n\pi x}{L}}\right)\,dx.}$

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 q_t(V). We assume q has a density Q, so that
  $${\displaystyle q_{t}(V)=\int _{V}Q(x,t)\,dx\quad }$$

  
• 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 dS and with unit normal vector n is
  $${\displaystyle \mathbf {H} (x)\cdot \mathbf {n} (x)\,dS.}$$

  
  Thus the rate of heat flow into V is also given by the surface integral
  $${\displaystyle q_{t}(V)=-\int _{\partial V}\mathbf {H} (x)\cdot \mathbf {n} (x)\,dS}$$

  
  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
  $${\displaystyle \mathbf {H} (x)=-\mathbf {A} (x)\cdot \nabla u(x)}$$

  
  where A(x) is a 3 × 3 real matrix that is symmetric and positive definite.
• By the divergence theorem, the previous surface integral for heat flow into V can be transformed into the volume integral
  $${\displaystyle {\begin{aligned}q_{t}(V)&=-\int _{\partial V}\mathbf {H} (x)\cdot \mathbf {n} (x)\,dS\\&=\int _{\partial V}\mathbf {A} (x)\cdot \nabla u(x)\cdot \mathbf {n} (x)\,dS\\&=\int _{V}\sum _{i,j}\partial _{x_{i}}{\bigl (}a_{ij}(x)\partial _{x_{j}}u(x,t){\bigr )}\,dx\end{aligned}}}$$

  
• 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
  $${\displaystyle \partial _{t}u(x,t)=\kappa (x)Q(x,t)}$$

  

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

${\displaystyle \partial _{t}u(x,t)=\kappa (x)\sum _{i,j}\partial _{x_{i}}{\bigl (}a_{ij}(x)\partial _{x_{j}}u(x,t){\bigr )}}$

Remarks.

• The coefficient κ(x) is the inverse of specific heat of the substance at x × density of the substance at x: ${\displaystyle \kappa =1/(\rho c_{p})}$.
• In the case of an isotropic medium, the matrix A is a scalar matrix equal to thermal conductivity k.
• In the anisotropic case where the coefficient matrix A is not scalar and/or if it depends on x, then an explicit formula for the solution of the heat equation can seldom be written down, though it is usually possible to consider the associated abstract Cauchy problem and show that it is a well-posed problem and/or to show some qualitative properties (like preservation of positive initial data, infinite speed of propagation, convergence toward an equilibrium, smoothing properties). This is usually done by one-parameter semigroups theory: for instance, if A is a symmetric matrix, then the elliptic operator defined by
  $${\displaystyle Au(x):=\sum _{i,j}\partial _{x_{i}}a_{ij}(x)\partial _{x_{j}}u(x)}$$

  
  is self-adjoint and dissipative, thus by the spectral theorem it generates a one-parameter semigroup.

Fundamental solutions

A fundamental solution, also called a heat kernel, is a solution of the heat equation corresponding to the initial condition of an initial point source of heat at a known position. These can be used to find a general solution of the heat equation over certain domains; see, for instance, (Evans 2010) for an introductory treatment.

In one variable, the Green's function is a solution of the initial value problem (by Duhamel's principle, equivalent to the definition of Green's function as one with a delta function as solution to the first equation)

${\displaystyle {\begin{cases}u_{t}(x,t)-ku_{xx}(x,t)=0&(x,t)\in \mathbb {R} \times (0,\infty )\\u(x,0)=\delta (x)&\end{cases}}}$

where ${\displaystyle \delta }$ is the Dirac delta function. The solution to this problem is the fundamental solution (heat kernel)

${\displaystyle \Phi (x,t)={\frac {1}{\sqrt {4\pi kt}}}\exp \left(-{\frac {x^{2}}{4kt}}\right).}$

One can obtain the general solution of the one variable heat equation with initial condition u(x, 0) = g(x) for −∞ < x < ∞ and 0 < t < ∞ by applying a convolution:

${\displaystyle u(x,t)=\int \Phi (x-y,t)g(y)dy.}$

In several spatial variables, the fundamental solution solves the analogous problem

${\displaystyle {\begin{cases}u_{t}(\mathbf {x} ,t)-k\sum _{i=1}^{n}u_{x_{i}x_{i}}(\mathbf {x} ,t)=0&(\mathbf {x} ,t)\in \mathbb {R} ^{n}\times (0,\infty )\\u(\mathbf {x} ,0)=\delta (\mathbf {x} )\end{cases}}}$

The n-variable fundamental solution is the product of the fundamental solutions in each variable; i.e.,

${\displaystyle \Phi (\mathbf {x} ,t)=\Phi (x_{1},t)\Phi (x_{2},t)\cdots \Phi (x_{n},t)={\frac {1}{\sqrt {(4\pi kt)^{n}}}}\exp \left(-{\frac {\mathbf {x} \cdot \mathbf {x} }{4kt}}\right).}$

The general solution of the heat equation on Rⁿ is then obtained by a convolution, so that to solve the initial value problem with u(x, 0) = g(x), one has

${\displaystyle u(\mathbf {x} ,t)=\int _{\mathbb {R} ^{n}}\Phi (\mathbf {x} -\mathbf {y} ,t)g(\mathbf {y} )d\mathbf {y} .}$

The general problem on a domain Ω in Rⁿ is

${\displaystyle {\begin{cases}u_{t}(\mathbf {x} ,t)-k\sum _{i=1}^{n}u_{x_{i}x_{i}}(\mathbf {x} ,t)=0&(\mathbf {x} ,t)\in \Omega \times (0,\infty )\\u(\mathbf {x} ,0)=g(\mathbf {x} )&\mathbf {x} \in \Omega \end{cases}}}$

with either Dirichlet or Neumann boundary data. A Green's function always exists, but unless the domain Ω can be readily decomposed into one-variable problems (see below), it may not be possible to write it down explicitly. Other methods for obtaining Green's functions include the method of images, separation of variables, and Laplace transforms (Cole, 2011).

Some Green's function solutions in 1D

A variety of elementary Green's function solutions in one-dimension are recorded here; many others are available elsewhere.^[7] In some of these, the spatial domain is (−∞,∞). In others, it is the semi-infinite interval (0,∞) with either Neumann or Dirichlet boundary conditions. One further variation is that some of these solve the inhomogeneous equation

${\displaystyle u_{t}=ku_{xx}+f.}$

where f is some given function of x and t.

Homogeneous heat equation

Initial value problem on (−∞,∞)

${\displaystyle {\begin{cases}u_{t}=ku_{xx}&(x,t)\in \mathbb {R} \times (0,\infty )\\u(x,0)=g(x)&{\text{Initial condition}}\end{cases}}}$

${\displaystyle u(x,t)={\frac {1}{\sqrt {4\pi kt}}}\int _{-\infty }^{\infty }\exp \left(-{\frac {(x-y)^{2}}{4kt}}\right)g(y)\,dy}$

Fundamental solution of the one-dimensional heat equation. Red: time course of ${\displaystyle \Phi (x,t)}$. Blue: time courses of ${\displaystyle \Phi (x_{0},t)}$ for two selected points x₀ = 0.2 and x₀ = 1. Note the different rise times/delays and amplitudes.
Interactive version.

Comment. This solution is the convolution with respect to the variable x of the fundamental solution

${\displaystyle \Phi (x,t):={\frac {1}{\sqrt {4\pi kt}}}\exp \left(-{\frac {x^{2}}{4kt}}\right),}$

and the function g(x). (The Green's function number of the fundamental solution is X00.)

Therefore, according to the general properties of the convolution with respect to differentiation, u = g ∗ Φ is a solution of the same heat equation, for

${\displaystyle \left(\partial _{t}-k\partial _{x}^{2}\right)(\Phi *g)=\left[\left(\partial _{t}-k\partial _{x}^{2}\right)\Phi \right]*g=0.}$

Moreover,

${\displaystyle \Phi (x,t)={\frac {1}{\sqrt {t}}}\,\Phi \left({\frac {x}{\sqrt {t}}},1\right)}$

${\displaystyle \int _{-\infty }^{\infty }\Phi (x,t)\,dx=1,}$

so that, by general facts about approximation to the identity, Φ(⋅, t) ∗ g → g as t → 0 in various senses, according to the specific g. For instance, if g is assumed bounded and continuous on R then Φ(⋅, t) ∗ g converges uniformly to g as t → 0, meaning that u(x, t) is continuous on R × [0, ∞) with u(x, 0) = g(x).

Initial value problem on (0,∞) with homogeneous Dirichlet boundary conditions

${\displaystyle {\begin{cases}u_{t}=ku_{xx}&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=g(x)&{\text{IC}}\\u(0,t)=0&{\text{BC}}\end{cases}}}$

${\displaystyle u(x,t)={\frac {1}{\sqrt {4\pi kt}}}\int _{0}^{\infty }\left[\exp \left(-{\frac {(x-y)^{2}}{4kt}}\right)-\exp \left(-{\frac {(x+y)^{2}}{4kt}}\right)\right]g(y)\,dy}$

Comment. This solution is obtained from the preceding formula as applied to the data g(x) suitably extended to R, so as to be an odd function, that is, letting g(−x) := −g(x) for all x. Correspondingly, the solution of the initial value problem on (−∞,∞) is an odd function with respect to the variable x for all values of t, and in particular it satisfies the homogeneous Dirichlet boundary conditions u(0, t) = 0. The Green's function number of this solution is X10.

Initial value problem on (0,∞) with homogeneous Neumann boundary conditions

${\displaystyle {\begin{cases}u_{t}=ku_{xx}&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=g(x)&{\text{IC}}\\u_{x}(0,t)=0&{\text{BC}}\end{cases}}}$

${\displaystyle u(x,t)={\frac {1}{\sqrt {4\pi kt}}}\int _{0}^{\infty }\left[\exp \left(-{\frac {(x-y)^{2}}{4kt}}\right)+\exp \left(-{\frac {(x+y)^{2}}{4kt}}\right)\right]g(y)\,dy}$

Comment. This solution is obtained from the first solution formula as applied to the data g(x) suitably extended to R so as to be an even function, that is, letting g(−x) := g(x) for all x. Correspondingly, the solution of the initial value problem on R is an even function with respect to the variable x for all values of t > 0, and in particular, being smooth, it satisfies the homogeneous Neumann boundary conditions u_x(0, t) = 0. The Green's function number of this solution is X20.

Problem on (0,∞) with homogeneous initial conditions and non-homogeneous Dirichlet boundary conditions

${\displaystyle {\begin{cases}u_{t}=ku_{xx}&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=0&{\text{IC}}\\u(0,t)=h(t)&{\text{BC}}\end{cases}}}$

${\displaystyle u(x,t)=\int _{0}^{t}{\frac {x}{\sqrt {4\pi k(t-s)^{3}}}}\exp \left(-{\frac {x^{2}}{4k(t-s)}}\right)h(s)\,ds,\qquad \forall x>0}$

Comment. This solution is the convolution with respect to the variable t of

${\displaystyle \psi (x,t):=-2k\partial _{x}\Phi (x,t)={\frac {x}{\sqrt {4\pi kt^{3}}}}\exp \left(-{\frac {x^{2}}{4kt}}\right)}$

and the function h(t). Since Φ(x, t) is the fundamental solution of

${\displaystyle \partial _{t}-k\partial _{x}^{2},}$

the function ψ(x, t) is also a solution of the same heat equation, and so is u := ψ ∗ h, thanks to general properties of the convolution with respect to differentiation. Moreover,

${\displaystyle \psi (x,t)={\frac {1}{x^{2}}}\,\psi \left(1,{\frac {t}{x^{2}}}\right)}$

${\displaystyle \int _{0}^{\infty }\psi (x,t)\,dt=1,}$

so that, by general facts about approximation to the identity, ψ(x, ⋅) ∗ h → h as x → 0 in various senses, according to the specific h. For instance, if h is assumed continuous on R with support in [0, ∞) then ψ(x, ⋅) ∗ h converges uniformly on compacta to h as x → 0, meaning that u(x, t) is continuous on [0, ∞) × [0, ∞) with u(0, t) = h(t).

Depicted is a numerical solution of the non-homogeneous heat equation. The equation has been solved with 0 initial and boundary conditions and a source term representing a stove top burner.

Inhomogeneous heat equation

Problem on (-∞,∞) homogeneous initial conditions

Comment. This solution is the convolution in R², that is with respect to both the variables x and t, of the fundamental solution

${\displaystyle \Phi (x,t):={\frac {1}{\sqrt {4\pi kt}}}\exp \left(-{\frac {x^{2}}{4kt}}\right)}$

and the function f(x, t), both meant as defined on the whole R² and identically 0 for all t → 0. One verifies that

${\displaystyle \left(\partial _{t}-k\partial _{x}^{2}\right)(\Phi *f)=f,}$

which expressed in the language of distributions becomes

${\displaystyle \left(\partial _{t}-k\partial _{x}^{2}\right)\Phi =\delta ,}$

where the distribution δ is the Dirac's delta function, that is the evaluation at 0.

Problem on (0,∞) with homogeneous Dirichlet boundary conditions and initial conditions

${\displaystyle {\begin{cases}u_{t}=ku_{xx}+f(x,t)&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=0&{\text{IC}}\\u(0,t)=0&{\text{BC}}\end{cases}}}$

${\displaystyle u(x,t)=\int _{0}^{t}\int _{0}^{\infty }{\frac {1}{\sqrt {4\pi k(t-s)}}}\left(\exp \left(-{\frac {(x-y)^{2}}{4k(t-s)}}\right)-\exp \left(-{\frac {(x+y)^{2}}{4k(t-s)}}\right)\right)f(y,s)\,dy\,ds}$

Comment. This solution is obtained from the preceding formula as applied to the data f(x, t) suitably extended to R × [0,∞), so as to be an odd function of the variable x, that is, letting f(−x, t) := −f(x, t) for all x and t. Correspondingly, the solution of the inhomogeneous problem on (−∞,∞) is an odd function with respect to the variable x for all values of t, and in particular it satisfies the homogeneous Dirichlet boundary conditions u(0, t) = 0.

Problem on (0,∞) with homogeneous Neumann boundary conditions and initial conditions

${\displaystyle {\begin{cases}u_{t}=ku_{xx}+f(x,t)&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=0&{\text{IC}}\\u_{x}(0,t)=0&{\text{BC}}\end{cases}}}$

${\displaystyle u(x,t)=\int _{0}^{t}\int _{0}^{\infty }{\frac {1}{\sqrt {4\pi k(t-s)}}}\left(\exp \left(-{\frac {(x-y)^{2}}{4k(t-s)}}\right)+\exp \left(-{\frac {(x+y)^{2}}{4k(t-s)}}\right)\right)f(y,s)\,dy\,ds}$

Comment. This solution is obtained from the first formula as applied to the data f(x, t) suitably extended to R × [0,∞), so as to be an even function of the variable x, that is, letting f(−x, t) := f(x, t) for all x and t. Correspondingly, the solution of the inhomogeneous problem on (−∞,∞) is an even function with respect to the variable x for all values of t, and in particular, being a smooth function, it satisfies the homogeneous Neumann boundary conditions u_x(0, t) = 0.

Examples

Since the heat equation is linear, solutions of other combinations of boundary conditions, inhomogeneous term, and initial conditions can be found by taking an appropriate linear combination of the above Green's function solutions.

For example, to solve

${\displaystyle {\begin{cases}u_{t}=ku_{xx}+f&(x,t)\in \mathbb {R} \times (0,\infty )\\u(x,0)=g(x)&{\text{IC}}\end{cases}}}$

let u = w + v where w and v solve the problems

${\displaystyle {\begin{cases}v_{t}=kv_{xx}+f,\,w_{t}=kw_{xx}\,&(x,t)\in \mathbb {R} \times (0,\infty )\\v(x,0)=0,\,w(x,0)=g(x)\,&{\text{IC}}\end{cases}}}$

Similarly, to solve

${\displaystyle {\begin{cases}u_{t}=ku_{xx}+f&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=g(x)&{\text{IC}}\\u(0,t)=h(t)&{\text{BC}}\end{cases}}}$

let u = w + v + r where w, v, and r solve the problems

${\displaystyle {\begin{cases}v_{t}=kv_{xx}+f,\,w_{t}=kw_{xx},\,r_{t}=kr_{xx}&(x,t)\in [0,\infty )\times (0,\infty )\\v(x,0)=0,\;w(x,0)=g(x),\;r(x,0)=0&{\text{IC}}\\v(0,t)=0,\;w(0,t)=0,\;r(0,t)=h(t)&{\text{BC}}\end{cases}}}$

Mean-value property for the heat equation

Solutions of the heat equations

${\displaystyle (\partial _{t}-\Delta )u=0}$

satisfy a mean-value property analogous to the mean-value properties of harmonic functions, solutions of

${\displaystyle \Delta u=0,}$

though a bit more complicated. Precisely, if u solves

${\displaystyle (\partial _{t}-\Delta )u=0}$

and

${\displaystyle (x,t)+E_{\lambda }\subset \mathrm {dom} (u)}$

then

${\displaystyle u(x,t)={\frac {\lambda }{4}}\int _{E_{\lambda }}u(x-y,t-s){\frac {|y|^{2}}{s^{2}}}ds\,dy,}$

where E_λ is a "heat-ball", that is a super-level set of the fundamental solution of the heat equation:

${\displaystyle E_{\lambda }:=\{(y,s):\Phi (y,s)>\lambda \},}$

${\displaystyle \Phi (x,t):=(4t\pi )^{-{\frac {n}{2}}}\exp \left(-{\frac {|x|^{2}}{4t}}\right).}$

Notice that

${\displaystyle \mathrm {diam} (E_{\lambda })=o(1)}$

as λ → ∞ so the above formula holds for any (x, t) in the (open) set dom(u) for λ large enough.^[8] This can be shown by an argument similar to the analogous one for harmonic functions.

Steady-state heat equation

The steady-state heat equation is by definition not dependent on time. In other words, it is assumed conditions exist such that:

${\displaystyle {\frac {\partial u}{\partial t}}=0}$

This condition depends on the time constant and the amount of time passed since boundary conditions have been imposed. Thus, the condition is fulfilled in situations in which the time equilibrium constant is fast enough that the more complex time-dependent heat equation can be approximated by the steady-state case. Equivalently, the steady-state condition exists for all cases in which enough time has passed that the thermal field u no longer evolves in time.

In the steady-state case, a spatial thermal gradient may (or may not) exist, but if it does, it does not change in time. This equation therefore describes the end result in all thermal problems in which a source is switched on (for example, an engine started in an automobile), and enough time has passed for all permanent temperature gradients to establish themselves in space, after which these spatial gradients no longer change in time (as again, with an automobile in which the engine has been running for long enough). The other (trivial) solution is for all spatial temperature gradients to disappear as well, in which case the temperature become uniform in space, as well.

The equation is much simpler and can help to understand better the physics of the materials without focusing on the dynamic of the heat transport process. It is widely used for simple engineering problems assuming there is equilibrium of the temperature fields and heat transport, with time.

Steady-state condition:

${\displaystyle {\frac {\partial u}{\partial t}}=0}$

The steady-state heat equation for a volume that contains a heat source (the inhomogeneous case), is the Poisson's equation:

${\displaystyle -k\nabla ^{2}u=q}$

where u is the temperature, k is the thermal conductivity and q is the rate of heat generation per unit volume.

In electrostatics, this is equivalent to the case where the space under consideration contains an electrical charge.

The steady-state heat equation without a heat source within the volume (the homogeneous case) is the equation in electrostatics for a volume of free space that does not contain a charge. It is described by Laplace's equation:

${\displaystyle \nabla ^{2}u=0}$

Applications

See also

• Caloric polynomial
• Curve-shortening flow
• Diffusion equation
• Relativistic heat conduction
• Schrödinger equation
• Weierstrass transform

Notes

References

• Cannon, John Rozier (1984), The one–dimensional heat equation, Encyclopedia of Mathematics and its Applications, vol. 23, Reading, MA: Addison-Wesley Publishing Company, Advanced Book Program, ISBN 0-201-13522-1, MR 0747979, Zbl 0567.35001
• Crank, J.; Nicolson, P. (1947), "A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-Conduction Type", Proceedings of the Cambridge Philosophical Society, 43 (1): 50–67, Bibcode:1947PCPS...43...50C, doi:10.1017/S0305004100023197, S2CID 16676040
• Evans, Lawrence C. (2010), Partial Differential Equations, Graduate Studies in Mathematics, vol. 19 (2nd ed.), Providence, RI: American Mathematical Society, ISBN 978-0-8218-4974-3
• Perona, P; Malik, J. (1990), "Scale-Space and Edge Detection Using Anisotropic Diffusion" (PDF), IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (7): 629–639, doi:10.1109/34.56205, S2CID 14502908
• Thambynayagam, R. K. M. (2011), The Diffusion Handbook: Applied Solutions for Engineers, McGraw-Hill Professional, ISBN 978-0-07-175184-1
• Wilmott, Paul; Howison, Sam; Dewynne, Jeff (1995), The mathematics of financial derivatives. A student introduction, Cambridge: Cambridge University Press, ISBN 0-521-49699-3

Further reading

• Carslaw, H.S.; Jaeger, J.C. (1988), Conduction of heat in solids, Oxford Science Publications (2nd ed.), New York: The Clarendon Press, Oxford University Press, ISBN 978-0-19-853368-9
• Cole, Kevin D.; Beck, James V.; Haji-Sheikh, A.; Litkouhi, Bahan (2011), Heat conduction using Green's functions, Series in Computational and Physical Processes in Mechanics and Thermal Sciences (2nd ed.), Boca Raton, FL: CRC Press, ISBN 978-1-43-981354-6
• Einstein, Albert (1905), "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen", Annalen der Physik, 322 (8): 549–560, Bibcode:1905AnP...322..549E, doi:10.1002/andp.19053220806
• Friedman, Avner (1964), Partial differential equations of parabolic type, Englewood Cliffs, N.J.: Prentice-Hall
• Unsworth, J.; Duarte, F. J. (1979), "Heat diffusion in a solid sphere and Fourier Theory", Am. J. Phys., 47 (11): 891–893, Bibcode:1979AmJPh..47..981U, doi:10.1119/1.11601
• Widder, D.V. (1975), The heat equation, Pure and Applied Mathematics, vol. 67, New York-London: Academic Press [Harcourt Brace Jovanovich, Publishers]

External links

Wikiversity has learning resources about Heat equation

Wikimedia Commons has media related to Heat equation.

• Derivation of the heat equation
• Linear heat equations: Particular solutions and boundary value problems - from EqWorld
• "The Heat Equation". PBS Infinite Series. November 17, 2017. Archived from the original on 2021-12-11 – via YouTube.