The numerical solution of weakly singular integral equations based on the meshless product integration (MPI) method with error analysis

This article investigates a numerical scheme based on the radial basis functions (RBFs) for solving weakly singular Fredholm integral equations by combining the product integration and collocation methods. A set of scattered points over the domain of integration is utilized to approximate the unknown function by using the RBFs. Since the proposed scheme does not require any background mesh for its approximations and numerical integrations unlike other product integration methods, it is called the meshless product integration (MPI) method. The method can be easily implemented and its algorithm is simple and effective to solve weakly singular integral equations. This approach reduces the solution of linear weakly singular integral equations to the solution of linear systems of algebraic equations. The error analysis of the proposed method is provided. The validity and efficiency of the new technique are demonstrated through several tests.