The Heat Equation

Abstract

The temperature distribution u in a homogeneous and isotropic heat conducting medium with conductivity k, heat capacity c and mass density ρ satisfies the partial differential equation

$$\frac{\delta u}{\delta t} = \kappa\,\Delta\,u$$

where κ = k/cρ. This equation is called the equation of heat conduction or, shortly, the heat equation and was first derived by Fourier. Simultaneously, the heat equation also occurs in the description of diffusion processes. The heat equation is the standard example for a parabolic differential equation. In this chapter we want to indicate the application of Volterra type integral equations for the solution of initial boundary value problems for the heat equation. Without loss of generality we assume the constant κ = 1.