An Efficient RBF-Based Collocation Approach for Second-Order Fractional Partial Integro-Differential Problems with Singular Kernel

This paper presents an efficient numerical method for solving second-order time-fractional partial integro-differential equations (PIDEs) with weakly singular kernels. The Caputo fractional derivative is used for temporal discretization, and the spatial derivatives are approximated using a meshless collocation method based on radial basis functions (RBFs), specifically multiquadric (MQ) and Gaussian (GA) types. The method is tested on benchmark problems with both linear and nonlinear source terms. Results show that GA RBFs perform better on coarse grids, while MQ RBFs achieve comparable or superior accuracy on finer grids. Compared to cubic B-spline collocation and Crandall’s methods, the proposed approach yields lower errors and stable convergence rates. Its meshless nature makes it suitable for complex geometries, and it can be extended to higher-dimensional problems.