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

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.

Welcome any comments to [email protected]

References in corpus (4)

• Relationship between boundary integral equation and radial basis function
• Analytical solution of transient scalar wave and diffusion problems of arbitrary dimensionality and geometry by RBF wavelet series
• Errata and supplements to: Orthonormal RBF Wavelet and Ridgelet-like Series and Transforms for High-Dimensional Problems
• Distance function wavelets - Part II: Extended results and conjectures