Distance function wavelets - Part I: Helmholtz and convection-diffusion transforms and series

TL;DR

This paper introduces distance function wavelets based on fundamental and general solutions of Helmholtz and convection-diffusion equations, including transforms and series, with applications to diffusion problems and handling discontinuous data.

Contribution

It presents new distance function wavelet transforms and series, including isotropic and anisotropic types, with novel notation and applications to diffusion problems, addressing edge effects and alternative existence conditions.

Findings

Introduces Helmholtz-Fourier and Helmholtz-Laplace transforms and series.

Addresses edge effects in Helmholtz-Fourier series.

Proposes a new notation for simplifying transform expressions.

Abstract

This report aims to present my research updates on distance function wavelets (DFW) based on the fundamental solutions and the general solutions of the Helmholtz, modified Helmholtz, and convection-diffusion equations, which include the isotropic Helmholtz-Fourier (HF) transform and series, the Helmholtz-Laplace (HL) transform, and the anisotropic convection-diffusion wavelets and ridgelets. The latter is set to handle discontinuous and track data problems. The edge effect of the HF series is addressed. Alternative existence conditions for the DFW transforms are proposed and discussed. To simplify and streamline the expression of the HF and HL transforms, a new dimension-dependent function notation is introduced. The HF series is also used to evaluate the analytical solutions of linear diffusion problems of arbitrary dimensionality and geometry. The weakness of this report is lacking of rigorous mathematical analysis due to the author's limited mathematical knowledge.