第二类 Volterra 型积分方程的数值算法研究

摘要/Abstract

摘要：

该文将最小二乘法与再生核法相结合, 提出了求解第二类 Volterra 型积分方程的新算法. 通过构造再生核空间的多尺度正交基, 得到了模型的解的表达式. 为了减少计算量, 简化计算过程, 文章利用最小二乘法将模型转化为线性代数方程进而得到 $\varepsilon$ 近似解. 此外, 为了验证算法的严谨性, 文章详细证明了新算法的一致收敛性和稳定性, 并对误差估计进行了讨论分析. 通过算例验证了该算法的可行性和适用性, 并与一些已知的方法相比, 所得结果更精准.

关键词: 最小二乘法, 再生核空间, Volterra 积分方程

Abstract:

In this paper, a new algorithm for solving the second Volterra type integral equation is proposed by combining the least square method with the reproducing kernel method. By constructing the multi-scale orthogonal basis of the reproducing kernel space, the solution expression of the model is obtained. To reduce the amount of computation and simplify the calculation process, this paper transforms the model into linear algebraic equation by using least square method and obtains the approximate solution of $\varepsilon$. In addition, to verify the rigor of the algorithm, the uniform convergence and stability of the algorithm are proved in detail, and the error estimation is discussed and analyzed. The feasibility and applicability of the proposed algorithm are verified by numerical examples. Compared with some known methods, the results obtained in this paper are more accurate.

Key words: Least square method, Reproducing kernel space, Volterra integral equation

中图分类号:

O241.8

代雪飞, 于一康, 牛晶. 第二类 Volterra 型积分方程的数值算法研究[J]. 数学物理学报, 2024, 44(5): 1283-1301.

Dai Xuefei, Yu Yikang, Niu Jing. Numerical Algorithm for Volterra Type Integral Equation of the Second Kind[J]. Acta mathematica scientia,Series A, 2024, 44(5): 1283-1301.

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