The Heat Equation

Abstract

The temperature distribution uin a homogeneous and isotropic heat conducting medium with conductivity k,heat capacity c, and mass density psatisfies the partial differential

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

equation where K= k/cp.This is called the equation of heat conductionor, shortly, the heat equation;it 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 parabolicdifferential equation. In this chapter we want to indicate the application of Volterra-type integral equations of the second kind for the solution of initial boundary value problems for the heat equation. Without loss of generality we assume the constant K=1.For a more comprehensive study of integral equations of the second kind for the heat equation we refer to Cannon [20], Friedman [45], and Pogorzelski [145].