# Heat equation

The heat equation or diffusion equation is a partial differential equation. It is the typical example of a parabolic differential equation and describes the relationship between the temporal change ( derivative) and the spatial variation of the temperature at a location in a body and is suitable for the calculation of unsteady temperature fields. In the one-dimensional case (without a heat source) it says that the time derivative of the temperature is the product of the second spatial derivative of the temperature and conductivity. This has an intuitive meaning: When the second spatial derivative in one place is not zero, then the first derivatives differ just before and behind this place. The heat current flowing to this place, therefore, differs according to the Fourier 's law of the flowing away from him. It needs to change with time that is the temperature at this location. Mathematically, the heat equation and diffusion equation identical, instead of temperature and thermal diffusivity, enter here the concentration and diffusion coefficient on. The heat conduction equation can be derived from the energy conservation law and Fourier's law of heat conduction. The fundamental solution of the heat equation is called the heat kernel.

- 2.1 Fundamental solution
- 2.2 Solution formula for the homogeneous Cauchy problem
- 2.3 solution formula for the inhomogeneous Cauchy problem with zero initial data
- 2.4 General solution formula
- 2.5 Other Solutions

- 3.1 Maximum Principle
- 3.2 smoothing property

## Formulation

### Homogeneous equation

In homogeneous media, the heat conduction equation is

Wherein the temperature at the site at the time of the Laplacian with respect to the constant and the thermal conductivity of the medium.

In the stationary case, when the time derivative is zero, the equation becomes Laplace's equation.

Simplification commonly used considers only a space dimension, for example, describes the variation of the temperature in a thin, relatively to the long bar of solid material. Therefore, the Laplacian operator is to a simple second derivative:

### Non-homogeneous equation

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

With the right side of the quotient of volume-related heat flux density and the ( volume-based ) is heat capacity. In the stationary case, when the time derivative is zero, the equation becomes the Poisson equation.

## Classical solutions

### Fundamental solution

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

And in one -dimensional problem

Where the square of the Euclidean norm of being.

Is also referred to as a heat pipe or heat core kernel. The functional form corresponds to that of a Gaussian normal distribution.

### Solution formula for the homogeneous Cauchy problem

With the help of the above fundamental solution of the heat equation one can specify a general solution formula for the homogeneous Cauchy problem of the heat equation. To do this additionally provides for given initial data at the time the initial condition

The solution of the homogeneous initial value problem is obtained by the convolution of the fundamental solution with the given initial data:

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

For the inhomogeneous initial value problem with zero initial data, we obtain analogously to the homogeneous case by the convolution of the fundamental solution with the given right-hand side as a solution formula:

### General solution formula

The solution formula for the inhomogeneous Cauchy problem with arbitrary initial data, we obtain due to the linearity of the heat equation by adding the solution of the homogeneous Cauchyproblems with the solution of the inhomogeneous Cauchyproblems with zero initial data, a total of:

### More Solutions

In some cases it is possible to find solutions to the equation by means of the approach of symmetry:

This leads to the following ordinary differential equation:

Another one-dimensional solution is

Where c is a constant. It can be used to model the heat storage behavior when an object is heated ( with a time- sinusoidal temperature).

## Properties of classical solutions

### Maximum principle

Let be a function that indicates the temperature of a solid as a function of location and time, ie. is time-dependent, because the thermal energy propagates with time over the material. The physical self-evident, that heat is not created out of nothing, reflected mathematically in the maximum principle down: The maximum value (over time and space ) the temperature is assumed at the beginning of the considered time interval or at the edge of the observed spatial region either. This property is generally in parabolic partial differential equations and can be easily proved.

### Smoothing property

Another interesting feature is that even when the time has a discontinuity, the function at any point is continuous in space. So if two pieces of metal of different temperature are connected with fixed, the average temperature will ( according to this modeling ) at the junction abruptly adjust and extend the temperature curve steadily through both workpieces.

- Infinite speed of propagation of information
- Weak solutions
- Numerical Methods
- Applications