基于径向基函数插值的非线性积分方程求解
Nonlinear Integral Equation Solution Based on Radial Basis Function Interpolation
DOI: 10.12677/AAM.2019.811209, PDF,    国家自然科学基金支持
作者: 王斌斌, 曾 光*, 雷 莉, 熊 晗：东华理工大学理学院，江西 南昌
关键词: 径向基函数；MQ函数；插值；形状参数α；数值积分；Radial Basis Function； MQ Function； Interpolation； Shape Parameter； Numerical Integration
摘要: 将径向基函数(radialbasisfunction, RBF)插值应用于非线性积分方程的求解中，本文选用的是RBF中插值性能优异的多重二次曲面(multiquadric, MQ)函数，首先将待求函数表示为RBF的线性组合，再通过配置法将方程离散为非线性方程组，求得权系数后给出待求函数的近似表示。基于MQ函数插值求解非线性积分方程可以在较少的节点下得到更髙的精度，且可扩展到髙维的积分方程中。本文证明了MQ基函数插值问题的存在唯一性，并且通过数值算例对MQ函数构造非线性函数的插值基函数进行了分析，得到了理想的逼近效果。
Abstract: The radial basis function (RBF) interpolation is applied to the solution of the nonlinear integral equation. In this paper, the multiquadric (MQ) function with excellent interpolation performance in RBF is selected. First, the function to be solved is expressed as a linear combination of RBF, and then the equation is discretized into a non-linear square by collocation method. The approximate expression of the function to be solved is given after the weight coefficients obtained. Solving nonlinear integral equation based on MQ function interpolation can obtain higher accuracy with fewer nodes, which can be extended to high-dimensional integral equation. In this paper, we prove the existence and uniqueness of MQ basis function interpolation problem, and analyse the interpolation basis function of constructing non-linear function of MQ function by numerical examples, and obtain the ideal approximation effect and result.
文章引用：王斌斌, 曾光, 雷莉, 熊晗. 基于径向基函数插值的非线性积分方程求解[J]. 应用数学进展, 2019, 8(11): 1795-1801. https://doi.org/10.12677/AAM.2019.811209

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