Hybrid Radial Kernels for Solving Weakly Singular Fredholm Integral Equations: Balancing Accuracy and Stability in Meshless Methods
Davoud Moazami, Mohsen Esmaeilbeigi, Tahereh Akbari
TL;DR
This work develops a meshless framework based on Hybrid Radial Kernels (HRKs) to solve second-kind weakly singular Fredholm integral equations. By blending an infinitely smooth kernel with a piecewise-smooth kernel, HRKs balance high accuracy with numerical stability, and are optimized using an Enhanced Particle Swarm Optimization (PSO) to minimize the maximum absolute error. The method relies on dispersed nodes and a nonuniform Gauss–Legendre quadrature for weakly singular integrals, enabling efficient linear-system formulations across 1D, 2D, and high-dimensional problems, with a rigorous convergence analysis. Numerical experiments across multiple dimensions demonstrate superior accuracy and conditioning of HRKs compared to pure kernels, and confirm the effectiveness of the PSO-based parameter selection. The approach is mesh-free, scalable to irregular domains, and adaptable to complex geometries, highlighting its practical impact for solving WSIEs in science and engineering.
Abstract
Over the past few decades, kernel-based approximation methods had achieved astonishing success in solving different problems in the field of science and engineering. However, when employing the direct or standard method of performing computations using infinitely smooth kernels, a conflict arises between the accuracy that can be theoretically attained and the numerical stability. In other words, when the shape parameter tends to zero, the operational matrix for the standard bases with infinitely smooth kernels become severely ill-conditioned. This conflict can be managed applying hybrid kernels. The hybrid kernels extend the approximation space and provide high flexibility to strike the best possible balance between accuracy and stability. In the current study, an innovative approach using hybrid radial kernels (HRKs) is provided to solve weakly singular Fredholm integral equations (WSFIEs) of the second kind in a meshless scheme. The approach employs hybrid kernels built on dispersed nodes as a basis within the discrete collocation technique. This method transforms the problem being studied into a linear system of algebraic equations. Also, the particle swarm optimization (PSO) algorithm is utilized to calculate the optimal parameters for the hybrid kernels, which is based on minimizing the maximum absolute error (MAE). We also study the error estimate of the suggested scheme. Lastly, we assess the accuracy and validity of the hybrid technique by carrying out various numerical experiments. The numerical findings show that the estimates obtained from hybrid kernels are significantly more accurate in solving WSFIEs compared to pure kernels. Additionally, it was revealed that the hybrid bases remain stable across various values of the shape parameters.