# Thermal equation

Model of a heating pipe that is cooled by a metal strut at different times

The heat conduction equation or diffusion equation is a partial differential equation used to describe heat conduction . It is the typical example of a parabolic differential equation , describes the relationship between the temporal and spatial change in temperature at a location in a body and is suitable for calculating transient temperature fields . In the one-dimensional case (without heat sources) it means that the (temporal) derivative of the temperature is the product of the second spatial derivative and the thermal diffusivity. This has a clear meaning: If the second spatial derivative is not equal to zero at a location, the first derivatives differ shortly before and after this location. According to Fourier's law , the heat flow that flows to this location differs from that that flows away from it. So the temperature in this place must change over time. Mathematically, the thermal conduction equation and the diffusion equation are identical, instead of temperature and thermal diffusivity , concentration and diffusion coefficient occur here . The heat conduction equation can be derived from the law of energy conservation and Fourier's law of heat conduction. The fundamental solution to the heat conduction equation is called the heat conduction core.

## formulation

### Homogeneous equation

In homogeneous media the equation of heat conduction reads

${\ displaystyle {\ frac {\ partial} {\ partial t}} u ({\ vec {x}}, t) -a \ Delta u ({\ vec {x}}, t) = 0,}$

wherein the temperature at the site at the time , the Laplacian with respect to and the constant , the thermal diffusivity is of the medium. ${\ displaystyle u ({\ vec {x}}, t)}$${\ displaystyle {\ vec {x}}}$${\ displaystyle t}$${\ displaystyle \ Delta}$${\ displaystyle {\ vec {x}}}$${\ displaystyle a> 0}$

In the stationary case, i.e. when the time derivative is zero, the equation changes into the Laplace equation . ${\ displaystyle {\ tfrac {\ partial u} {\ partial t}}}$ ${\ displaystyle \ Delta u = 0}$

A frequently used simplification only takes one dimension of space into account and describes, for example, the change in temperature over time in a thin, relatively long rod made of solid material. This makes the Laplace operator a simple second derivative:

${\ displaystyle {\ frac {\ partial} {\ partial t}} u (x, t) -a {\ frac {\ partial ^ {2}} {\ partial x ^ {2}}} {u (x, t)} = 0}$

### Inhomogeneous equation

In media with additional heat sources (e.g. by Joule heat or a chemical reaction ) the inhomogeneous heat conduction equation is then

${\ displaystyle {\ frac {\ partial} {\ partial t}} u ({\ vec {x}}, t) -a \ Delta u ({\ vec {x}}, t) = f ({\ vec {x}}, t),}$

where the right-hand side is the quotient of the volume-related heat source density (the amount of heat produced per volume and time) and the volume-related heat capacity (the product of density and mass-related heat capacity ). In the stationary case, i.e. when the time derivative is zero, the equation changes into the Poisson equation . ${\ displaystyle f}$

## Derivation

The heat balance on a small volume element (volume ) is considered. In a closed system that does not do volume work, the energy present in the system is preserved according to the first law of thermodynamics and it applies . The continuity equation for the internal energy can thus be written as: ${\ displaystyle V}$${\ displaystyle dU = \ delta Q}$

${\ displaystyle {\ frac {\ partial q} {\ partial t}} + {\ vec {\ nabla}} \ cdot {\ vec {q}} = 0}$,

where the change in heat density denotes and with the thermal conductivity is the heat flux density . ${\ displaystyle \ delta q = {\ tfrac {\ delta Q} {V}}}$${\ displaystyle {\ vec {q}} = - \ lambda {\ vec {\ nabla}} T}$ ${\ displaystyle \ lambda}$

With the connection to the heat capacity or the specific heat capacity over ${\ displaystyle C}$ ${\ displaystyle c}$

${\ displaystyle Q = CT = cmT}$

with the mass and accordingly with the volume-related size ${\ displaystyle m}$

${\ displaystyle q = c \ rho T}$

with the density results under the assumption that there is no mass transport or heat radiation losses, as well as the homogeneity of the material: ${\ displaystyle \ rho}$

${\ displaystyle c \ rho {\ frac {\ partial T} {\ partial t}} - {\ vec {\ nabla}} \ cdot (\ lambda {\ vec {\ nabla}} T) = c \ rho {\ frac {\ partial T} {\ partial t}} - \ lambda \ Delta T = 0}$.

The equation above follows with the thermal diffusivity ${\ displaystyle a = {\ tfrac {\ lambda} {\ rho c}}}$

${\ displaystyle {\ frac {\ partial T} {\ partial t}} - a \ Delta T = 0}$.

## Classic solutions

### Fundamental solution

A special solution to the heat conduction equation is the so-called fundamental solution of the heat conduction equation. This is for a one-dimensional problem

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

and with a -dimensional problem ${\ displaystyle n}$

${\ displaystyle H ({\ vec {x}}; t) = {\ frac {1} {(4 \ pi at) ^ {n / 2}}} \ exp \ left (- {\ frac {\ | { \ vec {x}} \ | ^ {2}} {4at}} \ right),}$

where is the square of the Euclidean norm of . ${\ displaystyle \ textstyle \ | {\ vec {x}} \ | ^ {2} = \ sum _ {k = 1} ^ {n} x_ {k} ^ {2}}$${\ displaystyle {\ vec {x}}}$

${\ displaystyle H}$is also used as heat kernel (or engl. heat kernel ), respectively. The functional form corresponds to that of a Gaussian normal distribution with . ${\ displaystyle \ sigma ^ {2} = 2at}$

### Solution formula for the homogeneous Cauchy problem

With the help of the fundamental solution of the heat conduction equation given above, one can give a general solution formula for the homogeneous Cauchy problem of the heat conduction equation. In continuum theory for given initial data at the time the additional initial condition${\ displaystyle u_ {0}}$${\ displaystyle t = 0}$

${\ displaystyle \ forall \ {\ vec {x}} \ in \ mathbb {R} ^ {n}: u ({\ vec {x}}, t = 0) = u_ {0} ({\ vec {x }})}$

in the form of a delta distribution . The solution of the homogeneous initial value problem is obtained by convolving the fundamental solution with the given initial data : ${\ displaystyle u ({\ vec {x}}, t)}$${\ displaystyle t> 0}$${\ displaystyle H}$${\ displaystyle u_ {0}}$

${\ displaystyle u ({\ vec {x}}, t) = (H * u_ {0}) ({\ vec {x}}, t) = \ int _ {\ mathbb {R} ^ {n}} H ({\ vec {x}} - {\ vec {y}}, t) u_ {0} ({\ vec {y}}) \, d {\ vec {y}}}$

### Solution formula for the inhomogeneous Cauchy problem with zero initial data

For the inhomogeneous initial value problem with zero initial data , we get, analogously to the homogeneous case, by folding the fundamental solution with the given right hand side of the differential equation as the solution formula: ${\ displaystyle u_ {0} ({\ vec {x}}) = 0}$${\ displaystyle H}$${\ displaystyle f}$

${\ displaystyle u ({\ vec {x}}, t) = (H * f) ({\ vec {x}}, t) = \ int _ {0} ^ {t} \ int _ {\ mathbb { R} ^ {n}} H ({\ vec {x}} - {\ vec {y}}, ts) f ({\ vec {y}}, s) \, d {\ vec {y}} \ , ds}$

### General solution formula

The solution formula for the inhomogeneous Cauchy problem with any initial data is obtained due to the linearity of the heat conduction equation by adding the solution of the homogeneous Cauchy problem with the solution of the inhomogeneous Cauchy problem with zero initial data, so in total:

${\ displaystyle u ({\ vec {x}}, t) = \ int _ {\ mathbb {R} ^ {n}} H ({\ vec {x}} - {\ vec {y}}, t) u_ {0} ({\ vec {y}}) \, d {\ vec {y}} + \ int _ {0} ^ {t} \ int _ {\ mathbb {R} ^ {n}} H ( {\ vec {x}} - {\ vec {y}}, ts) f ({\ vec {y}}, s) \, d {\ vec {y}} \, ds}$

### More solutions

In some cases, one can find solutions to the equation using the symmetry approach:

${\ displaystyle u (x, t) = f \ left ({\ frac {x} {\ sqrt {at}}} \ right)}$

This leads to the following ordinary differential equation for : ${\ displaystyle f}$

${\ displaystyle \ xi f ^ {\ prime} (\ xi) = - 2f ^ {\ prime \ prime} (\ xi)}$

Another one-dimensional solution is

${\ displaystyle u (x, t) = \ sin \ left (2c ^ {2} at-xc \ right) \ exp (-cx),}$

where is a constant. It can be used to model the heat storage behavior when an object (with a temporally sinusoidal temperature) is heated. ${\ displaystyle c}$

## Properties of classic solutions

### Maximum principle

Solution of a two-dimensional heat equation

Let be a function that gives the temperature of a solid as a function of location and time, so . is time-dependent because the thermal energy spreads over the material over time. The physical self-evident fact that heat does not arise from nowhere is mathematically reflected in the maximum principle : The maximum value (over time and space) of the temperature is assumed either at the beginning of the considered time interval or at the edge of the considered spatial area. This property is common to parabolic partial differential equations and can be easily proven. ${\ displaystyle u}$${\ displaystyle u = u (x_ {1}, x_ {2}, x_ {3}, t)}$${\ displaystyle u}$

### Smoothing property

Another interesting feature is that even if at the time a discontinuity has the function at any point is continuous in space. So if two pieces of metal at different temperatures are firmly connected, the mean temperature will suddenly set at the connection point (according to this modeling) and the temperature curve will run steadily through both workpieces. ${\ displaystyle u}$${\ displaystyle t = t_ {0}}$${\ displaystyle u}$${\ displaystyle t> t_ {0}}$${\ displaystyle t = t_ {0}}$

## literature

• Gerhard Dziuk: Theory and Numerics of Partial Differential Equations. de Gruyter, Berlin 2010, ISBN 978-3-11-014843-5 , pp. 183-253.
• Lawrence C. Evans : Partial Differential Equations. Reprinted with corrections. American Mathematical Society, Providence RI 2008, ISBN 978-0-8218-0772-9 ( Graduate studies in mathematics 19).
• John Rozier Cannon: The One-Dimensional Heat Equation. Addison-Wesley Publishing Company / Cambridge University Press, 1984, ISBN 978-0-521-30243-2 .