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

本工作的目的是发展一种精确的数值方法，利用符号计算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。在可能的情况下，寻求后验（和先验）误差估计来评估用于获得数值解的投影方法的有效性和准确性。投影方法，如在全局或局部基础上使用不同的基函数集进行配置，以获得准确的结果。符号计算允许执行以前无法克服的分析操作，以实现以下目的：(1)开发基于扩展的解决方法，以及(2)建立误差估计和收敛率。调查的主要贡献有两方面；(1)利用符号操作的发展来扩展解析、数值和图形计算，以支持求解和分析数学物理中的非线性积分和积分-微分方程；(2)研究、扩展和发展库马尔和斯隆最近的公式，用于多变量方程的科学计算。

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.