任意维度和几何的瞬态标量波与扩散问题的RBF小波级数解析解
Analytical solution of transient scalar wave and diffusion problems of arbitrary dimensionality and geometry by RBF wavelet series
AI总结：
本研究提出用RBF小波级数求解任意维度和几何的线性瞬态波与扩散问题，首次获得此类解析解，为高维时间相关偏微分方程提供新途径。
摘要：
本研究将RBF小波级数应用于求解任意维度和几何的线性时间相关波与扩散问题的解析解。据作者所知，此类解析解此前从未实现过。RBF小波可被理解为多维问题中标准傅里叶级数的替代方案，其基础是偏微分方程的基本解和通解。本文提出的RBF小波具有无限可微、紧支撑、跨尺度正交且形式非常简单的特点。本研究中尚缺乏关于完备性和收敛性的严格数学证明。当前工作可能为许多其他任意几何下高维时间相关偏微分方程问题的数值求解和理论分析打开新的窗口。
英文摘要：
This study applies the RBF wavelet series to the evaluation of analytical solutions of linear time-dependent wave and diffusion problems of any dimensionality and geometry. To the best of the author's knowledge, such analytical solutions have never been achieved before. The RBF wavelets can be understood an alternative for multidimensional problems to the standard Fourier series via fundamental and general solutions of partial differential equation. The present RBF wavelets are infinitely differential, compactly supported, orthogonal over different scales and very simple. The rigorous mathematical proof of completeness and convergence is still missing in this study. The present work may open a new window to numerical solution and theoretical analysis of many other high-dimensional time-dependent PDE problems under arbitrary geometry.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
