Numerical Investigation of the Fredholm Integral Equations with Oscillatory Kernels Based on Compactly Supported Radial Basis Functions

Abstract

The integral equations with oscillatory kernels are of great concern in applied sciences and computational engineering, particularly for large-scale data points and high frequencies. Therefore, the interest of this work is to develop an accurate, efficient, and stable algorithm for the computation of the Fredholm integral equations (FIEs) with the oscillatory kernel. The oscillatory part of the FIEs is evaluated by the Levin quadrature coupled with a compactly supported radial basis function (CS-RBF). The algorithm exhibits sparse and well-conditioned matrix even for large-scale data points, as compared to its counterpart, multi-quadric radial basis function (MQ-RBF) coupled with the Levin quadrature. Usually, the RBFs behave with spherical symmetry about the centers, known as radial. The comparison of convergence and stability analysis of both types of RBFs are performed and numerically verified. The proposed algorithm is tested with benchmark problems and compared with the counterpart methods in the literature. It is concluded that the algorithm in this work is accurate, robust, and stable than the existing methods in the literature based on MQ-RBF and the Chebyshev interpolation matrix.

Keywords

• compactly supported radial basis functions
• fredholm integral equations
• levin method
• stability analysis
• highly oscillatory kernels
• high frequency