This paper aims to analyze a novel localized radial basis function method known as the ’direct RBF-based partition of unity method’ for solving nonlinear fractional parabolic equations. In the proposed method, the weight functions are not operated on by the differential operators, resulting in a decrease in computational cost and algorithmic complexity. Another advantage of the direct RBF-based partition of unity method compared to the standard RBF-based partition of unity method is that in the direct method, it is possible to use discontinuous weights, which is not applicable in the standard method. Furthermore, we conduct a convergence analysis of the method and derive an error bound for the local approximation. This error bound is determined by considering conditions on the eigenvalues of the Laplacian operator matrix. To ensure the accuracy and reliability of our proposed approach, we conducted a numerical simulation and provided two numerical examples for comparison. We evaluated the consistency between the theoretical and numerical results. Our comparison between the standard RBF-based partition of unity and our proposed framework shows that the direct method is more efficient computationally, but both versions have relatively the same accuracy.

中文翻译：

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求解非线性分数抛物型方程的直接基于RBF的统一划分法

本文旨在分析一种新颖的局部径向基函数方法，称为“直接基于 RBF 的单位划分方法”，用于求解非线性分数抛物线方程。在所提出的方法中，权函数不通过微分算子进行操作，从而降低了计算成本和算法复杂度。与标准的基于RBF的统一划分方法相比，直接基于RBF的统一划分方法的另一个优点是，在直接方法中，可以使用不连续的权重，这在标准方法中不适用。此外，我们对该方法进行了收敛分析，并得出了局部近似的误差界。该误差界限是通过考虑拉普拉斯算子矩阵特征值的条件来确定的。为了确保我们提出的方法的准确性和可靠性，我们进行了数值模拟并提供了两个数值例子进行比较。我们评估了理论结果和数值结果之间的一致性。我们将标准的基于 RBF 的统一划分与我们提出的框架进行比较，结果表明直接方法的计算效率更高，但两个版本的精度相对相同。