THE DIFFUSION EQUATION IN SPACES OF DIFFERENTIAL K-FORMS DEFINED ON A RIEMANNIAN MANIFOLD WITHOUT BOUNDARY, USING THE EXAMPLE OF A SPHERICAL APPROXIMATION OF THE EARTH'S SURFACE

Abstract

This article substantiates a new approach to systems studies of the propagation of temperatures, fluid flows, and electromagnetic waves on manifolds without boundary (this applies to the heat equation, the Navier-Stokes system of equations, and the Maxwell system of equations) using, as a generalization, the invariant form of the diffusion equation in spaces of differential forms of different ranks and  pseudodifferential operators on them (one of these is the Laplace-Beltrami operator). The spherical surface of the Earth is given as an example of a manifold without boundary, and the need for such studies is raised. The relationship between the solutions of the heat conduction equations, the Navier--Stokes system of equations, and the Maxwell system of equations is noted, through the spectrum of the Laplace-Beltrami operator, on a manifold without boundary (in our case, on a spherical surface of Earth's radius).