A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations
A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations
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DOI：
10.1016/j.amc.2019.01.013
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发表时间：
2019-06
期刊：
影响因子：
--
通讯作者：
P. Assari;M. Dehghan
中科院分区：
文献类型：
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作者：
P. Assari;M. Dehghan
本文给出了求解二维非线性第二类Volterra积分方程的数值格式。该方法基于移动最小二乘（MLS）方法作为局部加权最小二乘多项式拟合，采用伽辽金方法估计解。离散伽辽金方法是由与该方案相关的所有积分的数值积分得到的。在目前的工作中，我们采用复合高斯-勒让德积分规则来近似该方法中出现的积分。由于该方法是在一组分散的点上构造的，因此不需要任何背景网格，因此我们将其称为无网格局部离散伽辽金方法。该方案的算法在计算上具有吸引力，并且易于在计算机上实现。得到了该方法的误差界和收敛速度。算例清楚地表明了新技术的可靠性和有效性，并证实了理论误差估计。
This article describes a numerical scheme to solve two-dimensional nonlinear Volterra integral equations of the second kind. The method estimates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin method results from the numerical integration of all integrals associated with the scheme. In the current work, we employ the composite Gauss-Legendre integration rule to approximate the integrals appearing in the method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The algorithm of the described scheme is computationally attractive and easy to implement on computers. The error bound and the convergence rate of the presented method are obtained. Illustrative examples clearly show the reliability and efficiency of the new technique and confirm the theoretical error estimates.