Enhanced stability and accuracy in solving nonlinear Fredholm integral equations using hybrid radial kernels and particle swarm optimization

Over the past few decades, kernel-based approximation methods had achieved astonishing success in solving various 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, as the shape parameter goes 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 second kind Fredholm integral equations (FIEs) 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 nonlinear system of algebraic equations. Also, the particle swarm optimization (PSO) algorithm is employed to determine the optimal parameters for the hybrid kernels, which is based on minimizing the root mean square (RMS) error. We also study the error estimate of the suggested method. Lastly, we assess the accuracy and validity of the hybrid approach by carrying out various numerical experiments. The numerical findings show that the estimates obtained from hybrid kernels are significantly more accurate in solving FIEs compared to pure kernels. Additionally, it was revealed that the hybrid bases remain stable across various values of the shape parameters. Also, we compared the numerical result of the HRKs method with Hilbert-Schmidt SVD (HS-SVD) approach in both CPU time and accuracy and concluded that HRKs method has better results.