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

摘要

In this paper, we propose a simple and efficient method for numerically solving the following Love's integral equation u(x)+∫~1_(-1) d/π(d~2+(x-t)~2) u(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) - 1/2, using the property that the solution u(x) of Love's integral equation satisfies u(x) → 1/2 for x ∈ (-1, 1), when the parameter d → 0~+. Numerical results are presented to illustrate the efficiency of the proposed method.