Qualitative Properties of Conductive Heat Transfer

In Section 8.1 we overview the classical theory of heat conduction formulated in thermodynamic terms. The dissipative character of pure heat conduction is manifested in the heat conductional inequality, the maximum principle, and other related properties (Section 8.2). These properties can be stated in general, far beyond the linear theory. The classical heat equation yields an infinite velocity of propagation. The hyperbolic heat equation has been proposed to overcome this paradox. However, the maximum principle is not valid for a hyperbolic equation; a satisfactory theory involving the maximum principle, as well as finite propagation, is not yet known (Section 8.3). The required basic properties may be used as postulates in searching for a new theory. An attempt of this kind is outlined. For a homogeneous one-dimensional-medium it is demonstrated in Section 8.4 that a theory for stationary heat conduction can be derived. The solution of the initial-boundary value problem for the classical linear heat conduction equation of parabolic type has several special characteristic properties, like contractivity in time, nonoscillatory behavior, exponential convergence, and others. In Section 8.5 we list some such properties. As is typical, the continuous problem cannot be solved analytically. In Section 8.6 some numerical processes are applied. This means that we define the approximate solution at the discrete points of a mesh. Obviously, the basic question is the convergence, that is, when refining the mesh, the numerical solution should be convergent to the solution of the original continuous problem. It is no less important to require the preservation of the discrete analogues of the basic qualitative properties of the continuous solution mentioned the above. In Section 8.7 the exact conditions for the preservation of some qualitative properties are established. There are damped traveling wave solutions of the sourceless parabolic heat equation. In Section 8.8 we list all the “shape-preserving signal forms,” that is, all the signal forms that can propagate inside the body without distortion, after a transient period. The solutions of this “shape-preserving” type have an attractive property: the asymptotic solutions belonging to different initial conditions tend to solutions of that kind.