Parabolic Equations

Abstract

Suppose a long, thin rod of length l is situated on the interval (0, l) along the x-axis. We shall assume that the material of the rod is homogeneous. Heat may be put into or removed from the rod, and we assume that the temperature u at any point in the rod is a function only of x, the location of a particular cross section, and of t, the time. We write u = u(x, t). Under certain assumptions on the physical properties of the rod, the differential equation governing the flow of heat (in appropriate units) in the rod is given by

$$ \frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}} = f(x,t). $$

. The function f is the rate of heat removal in the bar. The temperature function u(x, t) satisfies a maximum principle somewhat different from the one which was established for elliptic equations and inequalities.