第二类Volterra积分方程的广义多步配置法

Generalized Multistep Collocation Methods for the Second-Kind Volterra Integral Equations

Volterra积分方程作为一种重要的数学模型,被广泛应用于多个科学研究领域。针对第二类Volterra积分方程的数值解,本文在经典多步配置法的基础上,结合边值方法的思想,研究出一种广义多步配置方法。该方法利用Lagrange插值公式,以不同的节点作为插值节点,将原方程离散成为一个线性方程组。通过实验,本文验证了该方法在求解第一类Volterra积分方程的有效性,并且可以达到较高的收敛阶。Volterra积分方程在实际科学领域中有广泛的应用,因此对其数值求解方法的研究具有重要的理论意义和应用价值。本文所提出的广义多步配置方法为求解第二类Volterra积分方程提供了一种高效且可靠的数值逼近方案,为相关领域的研究和应用提供了新的思路和方法。

As an important mathematical model, Volterra integral equation is widely used in many scientific research fields. Aiming at the numerical solution of the second kind of Volterra integral equation, this paper studies a generalized multi-step collocation method based on the classical multi-step collocation method and the idea of boundary value method. This method uses the Lagrange inter-polation formula to discretize the original equation into a linear equation set with different nodes as interpolation nodes. Through experiments, this paper verifies the effectiveness of the method in solving the first kind of Volterra integral equation, and can achieve higher convergence order. The Volterra integral equation has a wide range of applications in the field of practical science, so the research on its numerical solution method has important theoretical significance and application value. The generalized multi-step collocation method proposed in this paper provides an efficient and reliable numerical approximation scheme for solving the second kind of Volterra integral equations, and provides new ideas and methods for research and application in related fields.