Reformulation of elliptic equations for heat transfer and diffusion in solids with space-time algebra

Abstract

In this paper, we demonstrate the application of non-commutative space-time algebra of sedeons to generalize the system of equations describing heat transfer and impurity diffusion in solids at finite velocity. It is shown that by analogy with electrodynamics, these transfer processes can be described using a compact second-order sedeonic equation for generalized scalar and vector potentials. On the one hand, this equation is reduced to the system of first-order differential equations for vortex-less mass and heat flows, and on the other hand, it can be transformed to the second-order elliptical equations for the profiles of temperature and impurity concentration. The comparison of peculiarities in transfer within the frames of parabolic and elliptic equations is discussed.