Investigations in Wavelet Analysis and Numerical Solutions of Partial Differential Equations

本课题的目的是在最近发现的紧支持小波基函数的基础上，发展微分方程初值和边值问题的求解方法。另一个目的是继续研究小波的理论和数值性质。第一阶段将研究线性和非线性一维和二维标量方程的问题，以确定使用小波求解微分方程的可行性。之后的阶段将包括在流体力学、场论和有限弹性中出现的矢量值问题。第三阶段将开发商业可用的计算机程序，用于实际应用的基于小波的偏微分方程解。

The purpose of this project is to develop solution methods for initial and/or boundary value problems for differential equations based on the recently discovered compactly supported wavelet basis functions. An additional purpose is to continue the investigation of the theoretical and numerical properties of wavelets. The first phase will investigate the problem of linear and nonlinear one and two-dimensional scalar equations to determine the feasibility of using wavelets to solve differential equations. Later phases will include vector-valued problems arising in fluid dynamics, field theory and finite elasticity. Phase III will develop commercially usable computer programs for wavelet-based solutions to partial differential equations for practical applications.