The numerical treatment of Love's integral equation having very small parameter

In this paper, we describe a procedure to find the numerical solution of Love's integral equation f(y)+1π∫−11c(x−y)2+c2f(x)dx=1,|y|≤1,0≤c∈R. A crucial role is played by the parameter c. In fact, the complex poles of the kernel function get closer to the real axis as c decreases, and numerical difficulties appear in approximating the solution f. We reduce the above integral equation to an equivalent system of Fredholm integral equations that we solve using a stable and convergent method. Moreover, since the matrices of coefficients of the derived linear systems are structured and can become very “large”, we focus our attention also on the computational aspects related to the numerical solution of such linear systems. Finally, we give some numerical tests considering different values of the parameter c, and compare our results with the ones obtained with other methods in the literature.