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

这项工作的目标是开发一种精确的数值方法，采用符号计算， 求解线性和非线性多变量积分和积分微分 数学物理方程 如果可能的话，后验（和 先验）误差估计寻求评估的有效性， 用于获得数字的投影方法的准确性 溶液 投影方法，例如使用不同基集的配置 开发了全局或局部函数，以获得准确的 结果 符号计算允许以前 难以克服的分析操作进行的目的（1）发展 基于扩展的求解方法，以及（2）建立误差估计， 收敛速度 研究的主要贡献有两个方面：（1）利用符号操纵的发展， 增强分析，数值和图形计算，以支持解决和 分析非线性积分微分方程 物理，（2）调查，扩展，并发展最近的配方 Kumar和Sloan的多变量科学计算 方程

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.