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Heat equation

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Partial differential equation describing the evolution of temperature in a region

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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.

Inmathematics andphysics (more specificallythermodynamics), theheat equation is aparabolic partial differential equation. The theory of the heat equation was first developed byJoseph Fourier in 1822 for the purpose of modeling how a quantity such asheat 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 subsetU ofRⁿ and a subintervalI ofR, one says that a functionu :U ×I →R is a solution of theheat 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(x₁, ...,x_n,t) denotes a general point of the domain.^[1] It is typical to refer tot as time andx₁, ...,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 asx. For any given value oft, the right-hand side of the equation is theLaplacian of the functionu(⋅,t) :U →R. As such, the heat equation is often written more compactly as

${\frac {\partial u}{\partial t}}=\Delta u$

In physics and engineering contexts, especially in the context of diffusion through a medium, it is more common to fix aCartesian coordinate system and then to consider the specific case of afunctionu(x,y,z,t) of three spatial variables(x,y,z) andtime variablet. One then says thatu is a solution of the heat equation if

{\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 whichα is a positivecoefficient called thethermal diffusivity of the medium. In addition to other physical phenomena, this equation describes the flow of heat in a homogeneous and isotropic medium, withu(x,y,z,t) being the temperature at the point(x,y,z) and timet. If the medium is not homogeneous and isotropic, thenα 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 useNewton's notation for time derivatives, so that ${\dot {u}}$ is used to denote⁠∂u/∂t⁠, so the equation can be written

${\dot {u}}=\Delta u$

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.

Diffusivity constant

The diffusivity constantα 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. Letu be a function with

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

Define a new function $v(t,x)=u(t/\alpha ,x)$ . Then, according to thechain rule, one has

{\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 ofα and solutions of the heat equation withα = 1. As such, for the sake of mathematical analysis, it is often sufficient to only consider the caseα = 1.

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

Nonhomogeneous heat equation

The nonhomogeneous heat equation is

{\frac {\partial u}{\partial t}}=\Delta u+f

for a given function $f=f(x,t)$ which is allowed to depend on bothx andt.^[1] The inhomogeneous heat equation models thermal problems in which a heat source modeled byf 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 generatingq units of heat per unit of volume, thenf would be given by $f(x,t)=q1_{S}(x)$ .

Steady-state equation

A solution to the heat equation $\partial u/\partial t=\Delta u$ is said to be a steady-state solution if it does not vary with respect to time:

0={\frac {\partial u}{\partial t}}=\Delta u.

Flowingu 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 ofLaplace's equation.

Similarly, a solution to the nonhomogeneous heat equation $\partial u/\partial t=\Delta u+f$ is said to be a steady-state solution if it does not vary with respect to time:

0={\frac {\partial u}{\partial t}}=\Delta u+f.

This is equivalently a solution ofPoisson'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.

Interpretation

Informally, the Laplacian operator∆ gives the difference between the average value of a function in the neighborhood of a point, and its value at that point. Thus, ifu is the temperature,∆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 thesecond law of thermodynamics, heat will flow from hotter bodies to adjacent colder bodies, in proportion to the difference of temperature and of thethermal 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 aproportionality factor called thespecific 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 coefficientα in the equation takes into account the thermal conductivity, specific heat, anddensity of the material.

Interpretation of the equation

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

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

whereu_(x)(r) is the single-variable function denoting theaverage value ofu over the surface of the sphere of radiusr centered atx; 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 whichω_{n − 1} denotes thesurface area of the unit ball inn-dimensional Euclidean space. This formalizes the above statement that the value of∆u at a pointx measures the difference between the value ofu(x) and the value ofu at points nearby tox, in the sense that the latter is encoded by the values ofu_(x)(r) for small positive values ofr.

Following this observation, one may interpret the heat equation as imposing aninfinitesimal averaging of a function. Given a solution of the heat equation, the value ofu(x,t + τ) for a small positive value ofτ may be approximated as⁠1/2n⁠ times the average value of the functionu(⋅,t) over a sphere of very small radius centered atx.

Character of the solutions

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

{\displaystyle u} — Solution of a 1D heat partial differential equation. The temperature ( $u {\displaystyle u}$ ) 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 $u {\displaystyle u}$ 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 $Ax+By+Cz+D$ , 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 themaximum principle, that says that the maximum value of $u {\displaystyle u}$ in any region $R {\displaystyle R}$ of the medium will not exceed the maximum value that previously occurred in $R {\displaystyle R}$ , unless it is on the boundary of $R {\displaystyle R}$ . That is, the maximum temperature in a region $R {\displaystyle R}$ can increase only if heat comes in from outside $R {\displaystyle R}$ . This is a property ofparabolic partial differential equations and is not difficult to prove mathematically (see below).

Another interesting property is that even if $u {\displaystyle u}$ 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 $u_{0}$ 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 $u {\displaystyle u}$ 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 adiffusion wave. Unlike theelastic andelectromagnetic 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

Heat flow in a uniform rod

For heat flow, the heat equation follows from the physical laws ofconduction of heat andconservation of energy (Cannon 1984).

ByFourier'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 $k {\displaystyle k}$ is thethermal conductivity of the material, $u=u(\mathbf {x} ,t)$ is the temperature, and $\mathbf {q} =\mathbf {q} (\mathbf {x} ,t)$ is avector field that represents the magnitude and direction of the heat flow at the point $\mathbf {x}$ of space and time $t {\displaystyle t}$ .

If the medium is a thin rod of uniform section and material, the positionx is a single coordinate and the heat flow $q=q(t,x)$ towards $x {\displaystyle x}$ is ascalar field. The equation becomes

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

Let $Q=Q(x,t)$ be theinternal 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 $c {\displaystyle c}$ 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 $x {\displaystyle x}$ , one concludes that the rate at which heat changes at a given point $x {\displaystyle x}$ 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}}

From the above equations it follows that

{\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}}}

which is the heat equation in one dimension, with diffusivity coefficient

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

This quantity is called thethermal diffusivity of the medium.

Accounting for radiative loss

An additional term may be introduced into the equation to account for radiative loss of heat. According to theStefan–Boltzmann law, this term is $\mu \left(u^{4}-v^{4}\right)$ , where $v=v(x,t)$ is the temperature of the surroundings, and $\mu$ is a coefficient that depends on theStefan-Boltzmann constant, theemissivity of the material, and the geometry. The rate of change in internal energy becomes

{\frac {\partial Q}{\partial t}}=-{\frac {\partial q}{\partial x}}-\mu \left(u^{4}-v^{4}\right)

and the equation for the evolution of $u {\displaystyle u}$ becomes

{\frac {\partial u}{\partial t}}={\frac {k}{c\rho }}{\frac {\partial ^{2}u}{\partial x^{2}}}-{\frac {\mu }{c\rho }}\left(u^{4}-v^{4}\right).

Non-uniform isotropic medium

Note that the state equation, given by thefirst law of thermodynamics (i.e. conservation of energy), is written in the following form (assuming no mass transfer or radiation). This form is more general and particularly useful to recognize which property (e.g.c_p or $\rho$ ) influences which term.

\rho c_{p}{\frac {\partial T}{\partial t}}-\nabla \cdot \left(k\nabla T\right)={\dot {q}}_{V}

where ${\dot {q}}_{V}$ is the volumetric heat source.

Heat flow in non-homogeneous anisotropic media

In general, the study of heat conduction is based on several principles. Heat flow is a form ofenergy 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 regionV is given by a time-dependent quantityq_t(V). We assumeq has adensityQ, so that $q_{t}(V)=\int _{V}Q(x,t)\,dx\quad$
Heat flow is a time-dependent vector functionH(x) characterized as follows: the time rate of heat flowing through an infinitesimal surface element with areadS and with unit normal vectorn is $\mathbf {H} (x)\cdot \mathbf {n} (x)\,dS.$ Thus the rate of heat flow intoV is also given by the surface integral $q_{t}(V)=-\int _{\partial V}\mathbf {H} (x)\cdot \mathbf {n} (x)\,dS$ wheren(x) is the outward pointing normal vector atx.
TheFourier law states that heat energy flow has the following linear dependence on the temperature gradient $\mathbf {H} (x)=-\mathbf {A} (x)\cdot \nabla u(x)$ whereA(x) is a 3 × 3 realmatrix that issymmetric andpositive definite.
By thedivergence theorem, the previoussurface integral for heat flow intoV can be transformed into the volume integral ${\begin{aligned}q_{t}(V)&=-\int _{\partial V}\mathbf {H} (x)\cdot \mathbf {n} (x)\,dS\\&=\int _{\partial V}\mathbf {A} (x)\cdot \nabla u(x)\cdot \mathbf {n} (x)\,dS\\&=\int _{V}\sum _{i,j}\partial _{x_{i}}{\bigl (}a_{ij}(x)\partial _{x_{j}}u(x,t){\bigr )}\,dx\end{aligned}}$
The time rate of temperature change atx is proportional to the heat flowing into an infinitesimal volume element, where the constant of proportionality is dependent on a constantκ $\partial _{t}u(x,t)=\kappa (x)Q(x,t)$

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_{ij}(x)\partial _{x_{j}}u(x,t){\bigr )}

Remarks

The coefficientκ(x) is the inverse ofspecific heat of the substance atx ×density of the substance atx: $\kappa =1/(\rho c_{p})$ .
In the case of an isotropic medium, the matrixA is a scalar matrix equal tothermal conductivityk.
In the anisotropic case where the coefficient matrixA is not scalar and/or if it depends onx, 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 abstractCauchy problem and show that it is awell-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 byone-parameter semigroups theory: for instance, ifA is asymmetric matrix, then theelliptic operator defined by $Au(x):=\sum _{i,j}\partial _{x_{i}}a_{ij}(x)\partial _{x_{j}}u(x)$ isself-adjoint and dissipative, thus by thespectral theorem it generates aone-parameter semigroup.

Three-dimensional problem

In the special cases of propagation of heat in anisotropic andhomogeneous 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 spatialderivatives (thermal conductions) of temperature in the $x {\displaystyle x}$ , $y {\displaystyle y}$ , and $z {\displaystyle z}$ directions, respectively;
$\alpha \equiv {\tfrac {k}{c_{p}\rho }}$ is thethermal diffusivity, a material-specific quantity depending on thethermal conductivity $k {\displaystyle k}$ , thespecific heat capacity $c_{p}$ , and themass density $\rho$ .

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

If the medium is not the whole space, in order to solve the heat equation uniquely we also need to specifyboundary conditions foru. 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^[2] or a sign condition (nonnegative solutions are unique by a result ofDavid Widder).^[3]

Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow ofheat from warmer to colder areas of an object. Generally, many different states and starting conditions will tend toward the same stableequilibrium. 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 aparabolic partial differential equation.

Using theLaplace 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,

where the Laplace operator, denoted as either Δ or as ∇² (the divergence of the gradient), is taken in the spatial variables.

The heat equation governs heat diffusion, as well as other diffusive processes, such asparticle diffusion or the propagation ofaction potential in nerve cells. Although they are not diffusive in nature, some quantum mechanics problems are also governed by a mathematical analog of the heat equation (see below). It also can be used to model some phenomena arising infinance, like theBlack–Scholes or theOrnstein-Uhlenbeck processes. The equation, and various non-linear analogues, has also been used in image analysis.

The heat equation is, technically, in violation ofspecial relativity, because its solutions involve instantaneous propagation of a disturbance. The part of the disturbance outside the forwardlight cone can usually be safely neglected, but if it is necessary to develop a reasonable speed for the transmission of heat, ahyperbolic problem should be considered instead – like a partial differential equation involving a second-ordertime derivative. Some models of nonlinear heat conduction (which are also parabolic equations) have solutions with finite heat transmission speed.^[4]^[5]

Internal heat generation

The functionu above represents temperature of a body. Alternatively, it is sometimes convenient to change units and representu 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 functionq varying in space and time.^[6] Then the heat per unit volumeu 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 forq when turned on. While the light is turned off, the value ofq for the tungsten filament would be zero.

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 byJoseph Fourier in his treatiseThéorie 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 u_{t}=\alpha u_{xx}

1

whereu =u(x,t) is a function of two variablesx andt. Here

x is the space variable, sox ∈ [0,L], whereL is the length of the rod.
t is the time variable, sot ≥ 0.

We assume the initial condition

u(x,0)=f(x)\quad \forall x\in [0,L]

2

where the functionf is given, and the boundary conditions

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 ofu onx,t is separated, that is:

u(x,t)=X(x)T(t).

4

This solution technique is calledseparation of variables. Substitutingu back into equation (1),

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

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

T'(t)=-\lambda \alpha T(t)

5

and

X''(x)=-\lambda X(x).

6

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

Suppose thatλ < 0. Then there exist real numbersB,C such that $X(x)=Be^{{\sqrt {-\lambda }}\,x}+Ce^{-{\sqrt {-\lambda }}\,x}.$ From (3) we getX(0) = 0 =X(L) and thereforeB = 0 =C which impliesu is identically 0.
Suppose thatλ = 0. Then there exist real numbersB,C such thatX(x) =Bx +C. From equation (3) we conclude in the same manner as in 1 thatu is identically 0.
Therefore, it must be the case thatλ > 0. Then there exist real numbersA,B,C such that $T(t)=Ae^{-\lambda \alpha t}$ and $X(x)=B\sin \left({\sqrt {\lambda }}\,x\right)+C\cos \left({\sqrt {\lambda }}\,x\right).$ From (3) we getC = 0 and that for some positive integern, ${\sqrt {\lambda }}=n{\frac {\pi }{L}}.$

This solves the heat equation in the special case that the dependence ofu 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

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

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

Generalizing the solution technique

The solution technique used above can be greatly extended to many other types of equations. The idea is that the operatoru_xx with the zero boundary conditions can be represented in terms of itseigenfunctions. This leads naturally to one of the basic ideas of thespectral theory of linearself-adjoint operators.

Consider thelinear operator Δu =u_xx. The infinite sequence of functions

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

forn ≥ 1 are eigenfunctions of Δ. Indeed,

\Delta e_{n}=-{\frac {n^{2}\pi ^{2}}{L^{2}}}e_{n}.

Moreover, any eigenfunctionf of Δ with the boundary conditionsf(0) =f(L) = 0 is of the forme_n for somen ≥ 1. The functionse_n forn ≥ 1 form anorthonormal sequence with respect to a certaininner 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 ofL²((0,L)). This shows that in effect we havediagonalized the operator Δ.

Mean-value property

Solutions of the heat equations

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

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

\Delta u=0,

though a bit more complicated. Precisely, ifu solves

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

and

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

then

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

where $E_{\lambda }$ is aheat-ball, that is a super-level set of the fundamental solution of the heat equation:

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

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

Notice that

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

as $\lambda \to \infty$ so the above formula holds for any $(x,t)$ in the (open) set $\mathrm {dom} (u)$ for $\lambda$ large enough.^[7]

Fundamental solutions

See also:Weierstrass transform

Afundamental solution of the heat equation is a solution that corresponds 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).

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

{\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 $\delta$ is theDirac delta function. The fundamental solution to this problem is given by theheat kernel

\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 conditionu(x, 0) =g(x) for −∞ <x < ∞ and 0 <t < ∞ by applying aconvolution:

u(x,t)=\int \Phi (x-y,t)g(y)dy.

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

{\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}}

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

\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 onRⁿ is then obtained by a convolution, so that to solve the initial value problem withu(x, 0) =g(x), one has

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 inRⁿ is

{\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 eitherDirichlet orNeumann boundary data. AGreen'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 themethod of images,separation of variables, andLaplace 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.^[8] In some of these, the spatial domain is (−∞,∞). In others, it is the semi-infinite interval (0,∞) with eitherNeumann orDirichlet boundary conditions. One further variation is that some of these solve the inhomogeneous equation

u_{t}=ku_{xx}+f.

wheref is some given function ofx andt.

Homogeneous heat equation

Initial value problem on (−∞,∞): ${\begin{cases}u_{t}=ku_{xx}&(x,t)\in \mathbb {R} \times (0,\infty )\\u(x,0)=g(x)&{\text{Initial condition}}\end{cases}}$; $u(x,t)={\frac {1}{\sqrt {4\pi kt}}}\int _{-\infty }^{\infty }\exp \left(-{\frac {(x-y)^{2}}{4kt}}\right)g(y)\,dy$

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

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

Comment. This solution is theconvolution with respect to the variablex of the fundamental solution

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

and the functiong(x). (TheGreen'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

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

Moreover,

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

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

so that, by general facts aboutapproximation to the identity, Φ(⋅,t) ∗g →g ast → 0 in various senses, according to the specificg. For instance, ifg is assumed bounded and continuous onR thenΦ(⋅,t) ∗g converges uniformly tog ast → 0, meaning thatu(x,t) is continuous onR × [0, ∞) withu(x, 0) =g(x).

Initial value problem on (0,∞) with homogeneous Dirichlet boundary conditions: ${\begin{cases}u_{t}=ku_{xx}&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=g(x)&{\text{IC}}\\u(0,t)=0&{\text{BC}}\end{cases}}$; $u(x,t)={\frac {1}{\sqrt {4\pi kt}}}\int _{0}^{\infty }\left[\exp \left(-{\frac {(x-y)^{2}}{4kt}}\right)-\exp \left(-{\frac {(x+y)^{2}}{4kt}}\right)\right]g(y)\,dy$

Comment. This solution is obtained from the preceding formula as applied to the datag(x) suitably extended toR, so as to be anodd function, that is, lettingg(−x) := −g(x) for allx. Correspondingly, the solution of the initial value problem on (−∞,∞) is an odd function with respect to the variablex for all values oft, and in particular it satisfies the homogeneous Dirichlet boundary conditionsu(0,t) = 0.TheGreen's function number of this solution is X10.

Initial value problem on (0,∞) with homogeneous Neumann boundary conditions: ${\begin{cases}u_{t}=ku_{xx}&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=g(x)&{\text{IC}}\\u_{x}(0,t)=0&{\text{BC}}\end{cases}}$; $u(x,t)={\frac {1}{\sqrt {4\pi kt}}}\int _{0}^{\infty }\left[\exp \left(-{\frac {(x-y)^{2}}{4kt}}\right)+\exp \left(-{\frac {(x+y)^{2}}{4kt}}\right)\right]g(y)\,dy$

Comment. This solution is obtained from the first solution formula as applied to the datag(x) suitably extended toR so as to be aneven function, that is, lettingg(−x) :=g(x) for allx. Correspondingly, the solution of the initial value problem onR is an even function with respect to the variablex for all values oft > 0, and in particular, being smooth, it satisfies the homogeneous Neumann boundary conditionsu_x(0,t) = 0. TheGreen's function number of this solution is X20.

Problem on (0,∞) with homogeneous initial conditions and non-homogeneous Dirichlet boundary conditions: ${\begin{cases}u_{t}=ku_{xx}&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=0&{\text{IC}}\\u(0,t)=h(t)&{\text{BC}}\end{cases}}$; $u(x,t)=\int _{0}^{t}{\frac {x}{\sqrt {4\pi k(t-s)^{3}}}}\exp \left(-{\frac {x^{2}}{4k(t-s)}}\right)h(s)\,ds,\qquad \forall x>0$

Comment. This solution is theconvolution with respect to the variablet of

\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 functionh(t). Since Φ(x,t) is the fundamental solution of

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

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

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

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

so that, by general facts aboutapproximation to the identity,ψ(x, ⋅) ∗h →h asx → 0 in various senses, according to the specifich. For instance, ifh is assumed continuous onR with support in [0, ∞) thenψ(x, ⋅) ∗h converges uniformly on compacta toh asx → 0, meaning thatu(x,t) is continuous on[0, ∞) × [0, ∞) withu(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 inR², that is with respect to both the variablesx andt, of the fundamental solution

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

and the functionf(x,t), both meant as defined on the wholeR² and identically 0 for allt → 0. One verifies that

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

which expressed in the language of distributions becomes

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

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

Problem on (0,∞) with homogeneous Dirichlet boundary conditions and initial conditions: ${\begin{cases}u_{t}=ku_{xx}+f(x,t)&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=0&{\text{IC}}\\u(0,t)=0&{\text{BC}}\end{cases}}$; $u(x,t)=\int _{0}^{t}\int _{0}^{\infty }{\frac {1}{\sqrt {4\pi k(t-s)}}}\left(\exp \left(-{\frac {(x-y)^{2}}{4k(t-s)}}\right)-\exp \left(-{\frac {(x+y)^{2}}{4k(t-s)}}\right)\right)f(y,s)\,dy\,ds$

Comment. This solution is obtained from the preceding formula as applied to the dataf(x,t) suitably extended toR × [0,∞), so as to be an odd function of the variablex, that is, lettingf(−x,t) := −f(x,t) for allx andt. Correspondingly, the solution of the inhomogeneous problem on (−∞,∞) is an odd function with respect to the variablex for all values oft, and in particular it satisfies the homogeneous Dirichlet boundary conditionsu(0,t) = 0.

Problem on (0,∞) with homogeneous Neumann boundary conditions and initial conditions: ${\begin{cases}u_{t}=ku_{xx}+f(x,t)&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=0&{\text{IC}}\\u_{x}(0,t)=0&{\text{BC}}\end{cases}}$; $u(x,t)=\int _{0}^{t}\int _{0}^{\infty }{\frac {1}{\sqrt {4\pi k(t-s)}}}\left(\exp \left(-{\frac {(x-y)^{2}}{4k(t-s)}}\right)+\exp \left(-{\frac {(x+y)^{2}}{4k(t-s)}}\right)\right)f(y,s)\,dy\,ds$

Comment. This solution is obtained from the first formula as applied to the dataf(x,t) suitably extended toR × [0,∞), so as to be an even function of the variablex, that is, lettingf(−x,t) :=f(x,t) for allx andt. Correspondingly, the solution of the inhomogeneous problem on (−∞,∞) is an even function with respect to the variablex for all values oft, and in particular, being a smooth function, it satisfies the homogeneous Neumann boundary conditionsu_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 appropriatelinear combination of the above Green's function solutions.

For example, to solve

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

letu =w +v wherew andv solve the problems

{\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

{\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}}

letu =w +v +r wherew,v, andr solve the problems

{\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}}

Applications

As the prototypicalparabolic partial differential equation, the heat equation is among the most widely studied topics inpure mathematics, and its analysis is regarded as fundamental to the broader field ofpartial differential equations. The heat equation can also be considered onRiemannian manifolds, leading to many geometric applications. Following work ofSubbaramiah Minakshisundaram andÅke Pleijel, the heat equation is closely related withspectral geometry. A seminalnonlinear variant of the heat equation was introduced todifferential geometry byJames Eells andJoseph Sampson in 1964, inspiring the introduction of theRicci flow byRichard Hamilton in 1982 and culminating in the proof of thePoincaré conjecture byGrigori Perelman in 2003. Certain solutions of the heat equation known asheat kernels provide subtle information about the region on which they are defined, as exemplified through their application to theAtiyah–Singer index theorem.^[9]

The heat equation, along with variants thereof, is also important in many fields of science andapplied mathematics. Inprobability theory, the heat equation is connected with the study ofrandom walks andBrownian motion via theFokker–Planck equation. TheBlack–Scholes equation offinancial mathematics is a small variant of the heat equation, and theSchrödinger equation ofquantum mechanics can be regarded as a heat equation inimaginary time. Inimage analysis, the heat equation is sometimes used to resolve pixelation and toidentify edges. FollowingRobert Richtmyer andJohn von Neumann's introduction of artificial viscosity methods, solutions of heat equations have been useful in the mathematical formulation ofhydrodynamical shocks. Solutions of the heat equation have also been given much attention in thenumerical analysis literature, beginning in the 1950s with work of Jim Douglas, D.W. Peaceman, and Henry Rachford Jr.

Particle diffusion

Main article:Diffusion equation

One can model particlediffusion by an equation involving either:

the volumetricconcentration of particles, denotedc, in the case ofcollective diffusion of a large number of particles, or
theprobability density function associated with the position of a single particle, denotedP.

In either case, one uses the heat equation

c_{t}=D\Delta c,

or

P_{t}=D\Delta P.

Bothc andP are functions of position and time.D is the diffusion coefficient that controls the speed of the diffusive process, and is typically expressed in meters squared over second. If the diffusion coefficientD is not constant, but depends on the concentrationc (orP in the second case), then one gets thenonlinear diffusion equation.

Brownian motion

Let thestochastic process $X {\displaystyle X}$ be the solution to thestochastic differential equation

{\begin{cases}\mathrm {d} X_{t}={\sqrt {2k}}\;\mathrm {d} B_{t}\\X_{0}=0\end{cases}}

where $B {\displaystyle B}$ is theWiener process (standard Brownian motion). Theprobability density function of $X {\displaystyle X}$ is given at any time $t {\displaystyle t}$ by

{\frac {1}{\sqrt {4\pi kt}}}\exp \left(-{\frac {x^{2}}{4kt}}\right)

which is the solution to the initial value problem

{\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 $\delta$ is theDirac delta function.

Schrödinger equation for a free particle

Main article:Schrödinger equation

With a simple division, theSchrödinger equation for a single particle ofmassm in the absence of any applied force field can be rewritten in the following way:

\psi _{t}={\frac {i\hbar }{2m}}\Delta \psi

,

wherei is theimaginary unit,ħ is thereduced Planck constant, andψ is thewave function of the particle.

This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation:

{\begin{aligned}c(\mathbf {R} ,t)&\to \psi (\mathbf {R} ,t)\\D&\to {\frac {i\hbar }{2m}}\end{aligned}}

Applying this transformation to the expressions of the Green functions determined in the case of particle diffusion yields the Green functions of theSchrödinger equation, which in turn can be used to obtain thewave function at any time through an integral on thewave function att = 0:

\psi (\mathbf {R} ,t)=\int \psi \left(\mathbf {R} ^{0},t=0\right)G\left(\mathbf {R} -\mathbf {R} ^{0},t\right)dR_{x}^{0}\,dR_{y}^{0}\,dR_{z}^{0},

with

G(\mathbf {R} ,t)=\left({\frac {m}{2\pi i\hbar t}}\right)^{3/2}e^{-{\frac {\mathbf {R} ^{2}m}{2i\hbar t}}}.

Remark: this analogy between quantum mechanics and diffusion is a purely formal one. Physically, the evolution of thewave function satisfying theSchrödinger equation might have an origin other than diffusion^{[citation needed]}.

Thermal diffusivity in polymers

A direct practical application of the heat equation, in conjunction withFourier theory, in spherical coordinates, is the prediction of thermal transfer profiles and the measurement of thethermal diffusivity inpolymers (Unsworth andDuarte). This dual theoretical-experimental method is applicable to rubber, various other polymeric materials of practical interest, and microfluids. These authors derived an expression for the temperature at the center of a sphereT_C

{\frac {T_{C}-T_{S}}{T_{0}-T_{S}}}=2\sum _{n=1}^{\infty }(-1)^{n+1}\exp \left({-{\frac {n^{2}\pi ^{2}\alpha t}{L^{2}}}}\right)

whereT₀ is the initial temperature of the sphere andT_S the temperature at the surface of the sphere, of radiusL. This equation has also found applications in protein energy transfer and thermal modeling in biophysics.

Financial Mathematics

The heat equation arises in anumber of phenomena and is often used infinancial mathematics in themodeling ofoptions. TheBlack–Scholes option pricing model'sdifferential equation can be transformed into the heat equation allowing relatively easy solutions from a familiar body of mathematics. Many of the extensions to the simple option models do not have closed form solutions and thus must be solved numerically to obtain a modeled option price. The equation describing pressure diffusion in a porous medium is identical in form with the heat equation.Diffusion problems dealing withDirichlet,Neumann andRobin boundary conditions have closed form analytic solutions (Thambynayagam 2011).

Image Analysis

The heat equation is also widely used in image analysis (Perona & Malik 1990) and inmachine learning as the driving theory behindscale-space orgraph Laplacian methods. The heat equation can be efficiently solved numerically using the implicitCrank–Nicolson method of (Crank & Nicolson 1947). This method can be extended to many of the models with no closed form solution, see for instance (Wilmott, Howison & Dewynne 1995).

Riemannian geometry

An abstract form of heat equation onmanifolds provides a major approach to theAtiyah–Singer index theorem, and has led to much further work on heat equations inRiemannian geometry.

See also

Notes

^^a ^bEvans 2010, p. 44.
^Stojanovic, Srdjan (2003), "3.3.1.3 Uniqueness for heat PDE with exponential growth at infinity",Computational Financial Mathematics using MATHEMATICA: Optimal Trading in Stocks and Options, Springer, pp. 112–114,ISBN 9780817641979
^John, Fritz (1991-11-20).Partial Differential Equations. Springer Science & Business Media. p. 222.ISBN 978-0-387-90609-6.
^TheMathworld: Porous Medium Equation and the other related models have solutions with finite wave propagation speed.
^Juan Luis Vazquez (2006-12-28),The Porous Medium Equation: Mathematical Theory, Oxford University Press, USA,ISBN 978-0-19-856903-9
^Note that the units ofu must be selected in a manner compatible with those ofq. Thus instead of being for thermodynamic temperature (Kelvin - K), units ofu should be J/L.
^Conversely, any functionu satisfying the above mean-value property on an open domain ofRⁿ ×R is a solution of the heat equation
^TheGreen's Function Library contains a variety of fundamental solutions to the heat equation.
^Berline, Nicole; Getzler, Ezra; Vergne, Michèle. Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften, 298. Springer-Verlag, Berlin, 1992. viii+369 pp.ISBN 3-540-53340-0

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"(PDF),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
Jili, Latif M. (2009),Heat Conduction, Springer (3rd ed.), Berlin-Heidelberg: Springer-Verlag,ISBN 978-3-642-01266-2
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 aboutHeat equation

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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 – viaYouTube.

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