Application of Local Integrated Radial Basis Function Method for Solving System of Fredholm Integro-Differential Equations

This paper thoroughly examines the Local Integrated Radial Basis Function (LIRBF) method's performance in addressing linear systems and first- to higher-order Fredholm integro-differential problems. Utilizing a meshless approach with Gauss-Lobatto quadrature points for spatial discretization, we rigorously assess the method's accuracy and efficiency across various numerical problems from the existing literature. Evaluation criteria, including maximum absolute errors and rates of convergence, validate the method's effectiveness. To gauge the proposed LIRBF method's efficacy, we benchmark it against well-established numerical techniques like Multi-Scale-Galerkin's, Alpert Multiwavelets, Legendre multi-wavelets collocation, Legendre-Galerkin, Legendre polynomial approximation, and variational iteration methods. A comparative analysis based on criterion norms assesses the numerical results obtained by each method. The findings reveal that the proposed method demonstrates a significant reduction in sensitivity to the shape parameter compared to the RBF method. This observation establishes the robustness and stability of the proposed method, highlighting its ability to maintain accuracy and efficiency across diverse conditions. Results from numerical experiments and comparisons with other established techniques affirm the efficiency and accuracy of the LIRBF method in solving Fredholm integro-differential problems. The outcomes demonstrate promising performance, emphasizing the LIRBF method's potential as a compelling alternative for addressing similar problems with high precision and computational efficiency.