A stable meshless numerical scheme using hybrid kernels to solve linear Fredholm integral equations of the second kind and its applications

The main challenge in kernel-based approximation theory and its applications is the conflict between accuracy and stability. Hybrid kernels can be used as one of the simplest and most effective tools to manage this challenge. This article uses a meshless scheme based on hybrid radial kernels (HRKs) to solve second kind Fredholm integral equations (FIEs). The method estimates the solution via the discrete collocation procedure based on a hybrid kernel that combines appropriate kernels with suitable weight parameters. The optimal parameters in the hybrid kernels can be calculated using the particle swarm optimization (PSO) algorithm based on the root mean square (RMS) error. This technique converts the problem under investigation into a system of linear equations. Moreover, the convergence of the suggested hybrid approach is studied. Lastly, some numerical experiments are included to reveal the accuracy and stability of the hybrid kernels approach. The numerical results demonstrate that the presented hybrid procedure meaningfully reduces the ill-conditioning of the operational matrices, at the same time, it maintains the accuracy and stability for all values of shape parameters.