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.

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

\(\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x_1^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2},\)

where \((x_1, x_2, \cdots, x_n,t)\) denotes a general point of the domain. It is typical to refer to \(t\) as time and \((x_1, x_2,\cdots, x_n)\) 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 \(u(\cdot,t): U \to \mathbb{R}\). As such, the heat equation is often written more compactly as

\(\frac{\partial u}{\partial t}=\nabla^2 u\)

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

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

in which \(\alpha\) 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 \(\alpha\) would not be a fixed coefficient, and would instead depend on \((x,y,z)\); the equation would also have a slightly different form. In some physics and engineering literature, it is common to use \(\Delta\) to denote the Laplacian, rather than \(\nabla^2\). But the "nabla-squared" \((\nabla^2)\) notation is more modern and recommended, while \(\Delta\) may denote a simple change in some cases.

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

\(\dot u=\nabla^2 u\)

Note also that the ability to use either \(\Delta\) or \(\nabla^2\) 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 \(\alpha\) 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

\(\frac{\partial u}{\partial t}=\alpha\nabla^2 u.\)

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

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

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

The nonhomogeneous heat equation is

\(\frac{\partial u}{\partial t} = \nabla^2 u + f\)

for a given function \(f = f(x,t)\) which is allowed to depend on both \(x\) and \(t\). The inhomogeneous heat equation models thermal problems in which a heat source modeled by f is switched on. For example, it can be used to model the temperature throughout a room with a heater switched on. If \(S \subset U\) is the region of the room where the heater is and the heater is constantly generating \(q\) units of heat per unit of volume, then \(f\) would be given by \(f(x,t) = q 1_S(x)\).

A solution to the heat equation \(\frac{\partial u}{\partial t} = \nabla^2 u\) is said to be a steady-state solution if it does not vary with respect to time:

\(0 = \frac{\partial u}{\partial t} = \nabla^2 u.\)

Flowing u via. the heat equation causes it to become closer and closer as time increases to a steady-state solution. For very large time, u is closely approximated by a steady-state solution. A steady state solution of the heat equation is equivalently a solution of Laplace's equation.

Similarly, a solution to the nonhomogeneous heat equation \(\frac{\partial u}{\partial t} = \nabla^2 u + f\) is said to be a steady-state solution if it does not vary with respect to time:

\(0 = \frac{\partial u}{\partial t} = \nabla^2 u + f.\)

This is equivalently a solution of Poisson's equation.

In the steady-state case, a nonzero spatial thermal gradient \(\nabla u\) may (or may not) be present, but if it is, it does not change in time. The steady-state equation 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 steady-state equations are simpler and can help to understand better the physics of the materials without focusing on the dynamics of heat transport. It is widely used for simple engineering problems assuming there is equilibrium of the temperature fields and heat transport, with time.

Informally, the Laplacian operator \(\nabla^2\) gives the difference between the average value of a function in the neighborhood of a point, and its value at that point. Thus, if \(u\) is the temperature, \(\nabla^2 u\) conveys if (and by how much) the material surrounding each point is hotter or colder, on the average, than the material at that point.

By the second law of thermodynamics, heat will flow from hotter bodies to adjacent colder bodies, in proportion to the difference of temperature and of the thermal conductivity of the material between them. When heat flows into (respectively, out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount (mass) of material, with a proportionality factor called the specific heat capacity of the material.

By the combination of these observations, the heat equation says the rate \(\dot u\) at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The coefficient α in the equation takes into account the thermal conductivity, specific heat, and density of the material.

The first half of the above physical thinking can be put into a mathematical form. The key is that, for any fixed \(x\), one has

\(\begin{aligned} u_{(x)}(0)&=u(x)\\ u_{(x)}'(0)&=0\\ u_{(x)}''(0)&=\frac{1}{n}\nabla^2 u(x) \end{aligned}\)

where u_(x)(r) is the single-variable function denoting the average value of \(u\) over the surface of the sphere of radius r centered at x; it can be defined by

\(u_{(x)}(r)=\frac{1}{\omega_{n-1}r^{n-1}}\int_{\{y:|x-y|=r\}}u\,d\mathcal{H}^{n-1},\)

in which \(\omega_{n-1}\) denotes the surface area of the unit ball in \(n\)-dimensional Euclidean space. This formalizes the above statement that the value of \(\nabla^2 u\) at a point \(x\) measures the difference between the value of \(u(x)\) and the value of u at points nearby to \(x\), in the sense that the latter is encoded by the values of \(u_{(x)}(r)\) for small positive values of \(r\).

Following this observation, one may interpret the heat equation as imposing an infinitesimal averaging of a function. Given a solution of the heat equation, the value of \(u(x,t+\tau)\) for a small positive value of \(\tau\) may be approximated as \(\frac{1}{2n}\) times the average value of the function \(u(\cdot,t)\) over a sphere of very small radius centered at \(x\).

The heat equation implies that peaks (local maxima) of \(u\) 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 \(A x + B y + C z + D\), then the value at the center of that neighborhood will not be changing at that time (that is, the derivative \(\dot u\) will be zero).

A more subtle consequence is the maximum principle, that says that the maximum value of \(u\) in any region \(R\) of the medium will not exceed the maximum value that previously occurred in \(R\), unless it is on the boundary of \(R\). That is, the maximum temperature in a region \(R\) can increase only if heat comes in from outside \(R\). 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 \(u\) 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 \(u_0\) and \(u_1\), 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 \(u\) will gradually vary between \(u_0\) and \(u_1\).

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.

For heat flow, the heat equation follows from the physical laws of conduction of heat and conservation of energy (Cannon 1984).

By Fourier's law for an isotropic medium, the rate of flow of heat energy per unit area through a surface is proportional to the negative temperature gradient across it:

\(\mathbf{q} = - k \, \nabla u\)

where \(k\) is the thermal conductivity of the material, \(u=u(\mathbf{x},t)\) is the temperature, and \(\mathbf{q} = \mathbf{q}(\mathbf{x},t)\) is a vector field that represents the magnitude and direction of the heat flow at the point \(\mathbf{x}\) of space and time \(t\).

If the medium is a thin rod of uniform section and material, the position x is a single coordinate and the heat flow \(q = q(t,x)\) towards \(x\) is a scalar field. The equation becomes

\(q = -k \,\frac{\partial u}{\partial x}\)

Let \(Q=Q(x,t)\) be the internal energy (heat) per unit volume of the bar at each point and time. The rate of change in heat per unit volume in the material, \(\partial Q/\partial t\), is proportional to the rate of change of its temperature, \(\partial u/\partial t\). That is,

\(\frac{\partial Q}{\partial t} = c \, \rho \, \frac{\partial u}{\partial t}\)

where \(c\) is the specific heat capacity (at constant pressure, in case of a gas) and \(\rho\) is the density (mass per unit volume) of the material. This derivation assumes that the material has constant mass density and heat capacity through space as well as time.

Applying the law of conservation of energy to a small element of the medium centred at \(x\), one concludes that the rate at which heat changes at a given point \(x\) is equal to the derivative of the heat flow at that point (the difference between the heat flows either side of the particle). That is,

\(\frac{\partial Q}{\partial t} = - \frac{\partial q}{\partial x}\)

\(\frac{\partial u}{\partial t} \;=\; - \frac{1}{c \rho} \frac{\partial q}{\partial x} \;=\; - \frac{1}{c \rho} \frac{\partial}{\partial x} \left(-k \,\frac{\partial u}{\partial x} \right) \;=\; \frac{k}{c \rho} \frac{\partial^2 u}{\partial x^2}\)

\(\alpha = \frac{k}{c\rho}\)

Condensed: the full section is in Wikipedia.

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 \[q_t(V) = \int_V Q(x,t)\,d x \quad\]
• 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 \[\mathbf{H}(x) \cdot \mathbf{n}(x) \, dS .\] Thus the rate of heat flow into V is also given by the surface integral \[q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS\] 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 \[\mathbf{H}(x) = -\mathbf{A}(x) \cdot \nabla u (x)\] 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 \[\begin{aligned} q_t(V) &= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS \\ &= \int_{\partial V} \mathbf{A}(x) \cdot \nabla u (x) \cdot \mathbf{n}(x) \, dS \\ &= \int_V \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t) \bigr)\,dx \end{aligned}\]
• 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 \[\partial_t u(x,t) = \kappa(x) Q(x,t)\]

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

\(\partial_t u(x,t) = \kappa(x) \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t)\bigr)\)

Remarks

• The coefficient κ(x) is the inverse of specific heat of the substance at x × density of the substance at x: \(\kappa = 1/(\rho c_p)\).
• 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 \[Au(x):=\sum_{i, j} \partial_{x_i} a_{i j}(x) \partial_{x_j} u (x)\] is self-adjoint and dissipative, thus by the spectral theorem it generates a one-parameter semigroup.

In the special cases of propagation of heat in an isotropic and homogeneous medium in a 3-dimensional space, this equation is

\(\frac{\partial u}{\partial t} = \alpha \nabla^2 u = \alpha \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2 }\right)\) \(= \alpha \left( u_{xx} + u_{yy} + u_{zz} \right)\)

where:

• \(u = u(x, y, z, t)\) is temperature as a function of space and time;
• \(\tfrac{\partial u}{\partial t}\) 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;
• \(\alpha \equiv \tfrac{k}{c_p\rho}\) is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity \(k\), the specific heat capacity \(c_p\), and the mass density \(\rho\).

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

If the medium is not the whole space, in order to solve the heat equation uniquely we also need to specify boundary conditions for u. To determine uniqueness of solutions in the whole space it is necessary to assume additional conditions, for example an exponential bound on the growth of solutions or a sign condition (nonnegative solutions are unique by a result of David Widder).

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. Generally, many different states and starting conditions will tend toward the same stable equilibrium. As a consequence, to reverse the solution and conclude something about earlier times or initial conditions from the present heat distribution is very inaccurate except over the shortest of time periods.

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

Using the Laplace operator, the heat equation can be simplified, and generalized to similar equations over spaces of arbitrary number of dimensions, as

\(u_t = \alpha \nabla^2 u = \alpha \Delta u,\)

Condensed: the full section is in Wikipedia.

The function u above represents temperature of a body. Alternatively, it is sometimes convenient to change units and represent u as the heat density of a medium. Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units.

Suppose that a body obeys the heat equation and, in addition, generates its own heat per unit volume (e.g., in watts/litre - W/L) at a rate given by a known function q varying in space and time. Then the heat per unit volume u satisfies an equation

\(\frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q.\)

For example, a tungsten light bulb filament generates heat, so it would have a positive nonzero value for q when turned on. While the light is turned off, the value of q for the tungsten filament would be zero.

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

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

where the function f is given, and the boundary conditions

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:

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

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

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:

1. Suppose that λ < 0. Then there exist real numbers B, C such that \[X(x) = B e^{\sqrt{-\lambda} \, x} + C e^{-\sqrt{-\lambda} \, x}.\] 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 \[T(t) = A e^{-\lambda \alpha t}\] and \[X(x) = B \sin\left(\sqrt{\lambda} \, x\right) + C \cos\left(\sqrt{\lambda} \, x\right).\] From (3) we get C = 0 and that for some positive integer n, \[\sqrt{\lambda} = n \frac{\pi}{L}.\]

\(u(x,t) = \sum_{n = 1}^{\infty} D_n \sin \left(\frac{n\pi x}{L}\right) e^{-\frac{n^2 \pi^2 \alpha t}{L^2}}\)

\(D_n = \frac{2}{L} \int_0^L f(x) \sin \left(\frac{n\pi x}{L}\right ) \, dx.\)

Condensed: the full section is in Wikipedia.

The solution technique used above can be greatly extended to many other types of equations. The idea is that the operator u_xx with the zero boundary conditions can be represented in terms of its eigenfunctions. This leads naturally to one of the basic ideas of the spectral theory of linear self-adjoint operators.

Consider the linear operator Δu = u_xx. The infinite sequence of functions

\(e_n(x) = \sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right)\)

for n ≥ 1 are eigenfunctions of Δ. Indeed,

\(\Delta e_n = -\frac{n^2 \pi^2}{L^2} e_n.\)

Moreover, any eigenfunction f of Δ with the boundary conditions f(0) = f(L) = 0 is of the form e_n for some n ≥ 1. The functions e_n for n ≥ 1 form an orthonormal sequence with respect to a certain inner product on the space of real-valued functions on [0, L]. This means

\(\langle e_n, e_m \rangle = \int_0^L e_n(x) e^*_m(x) dx = \delta_{mn}\)

Finally, the sequence {e_n}_{n ∈ N} spans a dense linear subspace of L((0, L)). This shows that in effect we have diagonalized the operator Δ.