非线性Volterra型二次积分方程的Sinc配置方法

Sinc Collocation Method for Nonlinear Volterra Integral Equations of Quadratic Type

摘要: 本文聚焦非线性Volterra 型二次积分方程，开展基于修正型Sinc 配置方法的数值分析研究。 针 对该类方程难以直接求解解析解的痛点，本文首次将修正型Sinc 配置算法应用于该类方程数值求 解，依托修正型Sinc 插值公式选取等距配置点完成方程离散，把原积分方程等价转化为可计算的 非线性方程组；分别采用单指数(SE)、双指数(DE)两种变换构造离散格式，并深入对比两类变换 下算法的数值性能。 同时，文中系统推导解的存在唯一性相关理论，清晰梳理本部分理论结论与 现有相关研究成果的关联；完成严格误差界理论分析后，通过多组数值算例仿真验证，结果证实 本文所提算法拥有优异的稳定性、可靠性与计算有效性。

Abstract: This paper focuses on nonlinear Volterra quadratic integral equations and conducts numerical analysis based on the modified Sinc collocation method. Given the difficulty in directly deriving analytical exact solutions for such integral equations, this work is the first to apply the modified Sinc collocation algorithm to numerically solve this class of equations. Relying on the modified Sinc interpolation formula, equidistant collocation points are selected to discretize the original integral equation, which is equivalently transformed into a tractable nonlinear system of equations. Two types of transformations, single exponential (SE) and double exponential (DE), are adopted to construct discrete schemes, and the numerical performance of the algorithm under the two transformations is thoroughly compared. Meanwhile, the theoretical results regarding the existence and uniqueness of solutions are systematically derived, and the relationships between these theoretical conclusions and existing research findings are clearly clarified. After rigorous theoretical analysis of error bounds, multiple numerical examples are carried out for simulation verification, and the results demonstrate that the proposed algorithm possesses favorable stability, reliability and computational effectiveness.