Analytical Solution of Transient Scalar Wave and Diffusion Problems using RBF Wavelet Series

Title: Analytical Solution of Transient Scalar Wave and Diffusion Problems using RBF Wavelet Series

Abstract: This study applies the RBF (Radial Basis Function) wavelet series to evaluate analytical solutions of linear time-dependent wave and diffusion problem for any dimensionality and geometry. The RBF wavelets can be understood as an alternative for multidimensional problem to the standard Fourier series via fundamental and general solutions of partial differential equations. The present RBF wavelets are infinitely differentiable, compactly supported, orthogonal over different scales, and very simple. The rigorous mathematical proof of completeness and convergence is still missing in this study. This work may open a new window to numerical solution and theoretical analysis of many other high-dimensional, time-dependent PDE (Partial Differential Equation) problems under arbitrary geometry.

Keywords: RBF wavelet series, Helmholtz equation, Fourier series, Gibbs phenomenon, time-dependent problem, high-dimensional problem, geometric complexity.

Introduction: The Fourier series, a historic retrospect: Many important concepts of analysis and computation have their origins in the study of physical problems leading to the partial differential equation (PDE) system. The currently ubiquitous Fourier series and transform came from Fourier's original exploration of the solution of a bar heat transmission problem in the early 1800s. Despite a lack of rigorous proof, Fourier was quite confident of the basic truth of his assertion for obvious physical and geometric grounds. The implications of this discovery go well beyond Fourier's wildest imaginations, leading to numerous important mathematical concepts and techniques, such as Riemann integration, Sturm-Liouville eigenvalue problem, set theory, Laplace transform, Lebesgue integration, Green's function, and distribution theory, functional analysis, and countless applications in numerous fields of science and engineering.

Despite the widespread applicability, the Fourier analysis approach suffers some drawbacks. Most notably, for more than one-dimensional problem

Link to Article: https://arxiv.org/abs/0110055v1 Authors: arXiv ID: 0110055v1