  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.
