Non-uniform UE-spline quasi-interpolants and their application to the numerical solution of integral equations

Abstract

A construction of Marsden's identity for UE-splines is developed and a complete proof is given. With the help of this identity, a new non-uniform quasi-interpolant that reproduces the spaces of polynomial, trigonometric and hyperbolic functions are defined. Efficient quadrature rules based on integrating these quasi-interpolation schemes are derived and analyzed. Then, a quadrature formula associated with non-uniform quasi-interpolation along with Nyström's method is used to numerically solve Hammerstein and Fredholm integral equations. Numerical results that illustrate the effectiveness of these rules are presented.