Superconvergent cubic trigonometric quasi-interpolant splines for solving Love's integral equation by approximating the right kernel

Abstract

In this paper, we propose a numerical method for solving Love's integral equation, based on a degenerate approximation of the kernel. The method consists of approximating the right section of the integral kernel by a cubic trigonometric quasi-interpolant spline. This approximation leads to a degenerate representation of the kernel, transforming the Fredholm integral equation of the second kind into a linear system whose coefficients are expressed in the form of integrals. We present in detail the explicit construction of the quasi-interpolant, including the calculation of local functionals, the associated quasi-Lagrange functions, and the properties of stability and accuracy. The integrals involved in the linear system are evaluated using a high-order numerical quadrature formula. A convergence analysis is established, showing that the approximate solution converges to the exact solution with an overall order of four. Numerical results illustrate the efficiency and accuracy of the proposed method.