Interpolation Formulas for Functions with Large Gradients in the Boundary Layer and their Application

Abstract

Interpolation of functions on the basis of Lagrange’s polynomials is widely used. However in the case when the function has areas of large gradients, application of polynomials of Lagrange leads to essential errors. It is supposed that the function of one variable has the representation as a sum of regular and boundary layer components. It is supposed that derivatives of a regular component are bounded to a certain order, and the boundary layer component is a function, known within a multiplier; its derivatives are not uniformly bounded. A solution of a singularly perturbed boundary value problem has such a representation. Interpolation formulas, which are exact on a boundary layer component, are constructed. Interpolation error estimates, uniform in a boundary layer component and its derivatives are obtained. Application of the constructed interpolation formulas to creation of formulas of the numerical differentiation and integration of such functions is investigated.

Keywords

• function of one variable
• boundary layer component
• nonpolynomial interpolation
• quadrature formulas
• formulas of numerical differentiation
• error estimate.