Discrete Heat Equation Research Articles

Graphs Between the Elliptic and Parabolic Harnack Inequalities

We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L2 Poincare inequality. The first example of graph satisfying the elliptic but not the parabolic Harnack inequality is due to M. Barlow and R. Bass. It satisfies the volume regularity and not the Poincare inequality. We construct another example that does not satisfy the volume regularity.

Discrete derivatives and symmetries of difference equations

We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra, defined in terms of shift operators, isomorphic to that of the continuous heat equation.

The stationary state and the heat equation for a variant of Davies' model of heat conduction

We study a variant of Davies' model of heat conduction, consisting of a chain of (classical or quantum) harmonic oscillators, whose ends are coupled to thermal reservoirs at different temperatures, and where neighboring oscillators interact via intermediate reservoirs. In the weak coupling limit, we show that a unique stationary state exists, and that a discretized heat equation holds. We give an explicit expression of the stationary state in the case of two classical oscillators. The heat equation is obtained in the hydrodynamic limit, and it is proved that it completely describes the macroscopic behavior of the model.

Two-Dimensional Quasi-Stationary Temperature Rise in Multicable Configurations Under Arbitrary Cyclic Load

The quasi-stationary temperature field is computed for a buried multicable system under an arbitrary cyclic load. The technique is based upon the development of a discretized complex heat equation, relying on finite elements approach, solved by direct frontal solution for a sinusoidal load pattern. The response to an arbitrary pattern is then found by using Fourier analysis. This versatile simulation makes it possible to calculate with great accuracy cyclic temperature fields for nearly all cable configurations. It removes many of the usual limiting assumptions and is used to ascertain their numerical influence.