A quasi-interpolation product integration based method for solving Love’s integral equation with a very small parameter

In this paper, we propose a simple and efficient method for numerically solving the following Love’s integral equation u(x)+∫−11dπd2+(x−t)2u(t)dt=1,x∈[−1,1],where d>0 is a very small parameter. We apply the product integration method based on discrete spline quadratic quasi-interpolation, by considering a new unknown function v(x)=u(x)−12, using the property that the solution u(x) of Love’s integral equation satisfies u(x)→12 for x∈(−1,1), when the parameter d→0+. Numerical results are presented to illustrate the efficiency of the proposed method.