The Use of Symbolic Computation for Solving Nonlinear and Integro-Differential Equations

这项工作的目的是发展一种精确的数值方法，利用符号计算来求解数学物理中的线性和非线性多变量积分和积分-微分方程组。在可能的情况下，寻求后验(和先验)误差估计，以评估用于获得数值解的投影方法的有效性和准确性。发展了投影方法，例如在全局或局部基础上使用不同的基函数组的配置，以获得准确的和结果。符号计算允许执行以前无法克服的分析操作，目的是(1)开发基于展开的求解方法，以及(2)建立误差估计和收敛速度。这项研究的主要贡献有两个：(1)利用符号处理的发展来增加解析、数值和图形计算，以支持解决和分析数学物理中的非线性积分和积分-微分方程组；(2)研究、推广和发展Kumar和Sloan最近提出的用于多变量方程科学计算的公式。

The goal of this work is to develop an accurate numerical method which employs symbolic computation for solving linear and nonlinear multivariable integral and integro-differential equations of mathematical physics. When possible, a posteriori (and a priori) error estimates are sought to evaluate the effectiveness and accuracy of the projection methods used for obtaining the numeric solution. Projection methods such as collocation using different sets of basis functions on either a global or local basis are developed to obtain accurate and results. Symbolic computation permits previously insurmountable analytic manipulations to be performed for purposes of (1) developing expansion based solution methods, and (2) establishing error estimates and convergence rates. The major contribution of the investigation are twofold; (1) to exploit the development of symbolic manipulation for augmenting analytic, numeric, and graphic computation in support of solving and analyzing nonlinear integral and integro-differential equations of mathematical physics, and (2) to investigate, extend, and develop the recent formulation of Kumar and Sloan for scientific computation of multivariable equations.