The direct RBF-based partition of unity method for solving nonlinear fractional parabolic equations

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.