In this study, we explore linear and nonlinear parabolic integro-differential equations in one and two dimensions. We employ a semi-implicit scheme to discretize the temporal variable and discretize the spatial variable using an integrated radial basis function based on the moving Kriging interpolation (MKI) method. Unlike the global integrated radial basis function (IRBF) method, our proposed approach is particularly well-suited for addressing large-scale problems. Moreover, issues such as the selection of shape parameters leading to matrices with high condition numbers can be mitigated by devising an algorithm to determine optimal shape parameters, resulting in lower condition numbers. To assess the accuracy of the proposed method, we utilize two criteria, namely, and . We compare the computed results obtained from the IRBF-MKI method with those from two grid and finite element methods. To demonstrate the accuracy and efficiency of our approach, we consider irregular areas and scattered Halton points in the two-dimensional case.

中文翻译：

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线性和非线性抛物型积分微分方程的集成径向基函数克里金插值的开发


在本研究中，我们探索一维和二维的线性和非线性抛物型积分微分方程。我们采用半隐式方案来离散化时间变量，并使用基于移动克里金插值（MKI）方法的集成径向基函数来离散化空间变量。与全局积分径向基函数（IRBF）方法不同，我们提出的方法特别适合解决大规模问题。此外，通过设计一种算法来确定最佳形状参数，可以缓解诸如形状参数的选择导致条件数较高的矩阵等问题，从而得到较低的条件数。为了评估所提出方法的准确性，我们使用两个标准，即 和 。我们将 IRBF-MKI 方法获得的计算结果与两种网格和有限元方法获得的计算结果进行比较。为了证明我们方法的准确性和效率，我们考虑二维情况下的不规则区域和分散的霍尔顿点。