On a Class of Hermite Interpolation Polynomials for Nonlinear Second Order Partial Differential Operators

Leonid A Yanovich,Marina V Ignatenko,Gh Adam,J Buša,M Hnatič,D Podgainy

doi:10.1051/epjconf/201817303023

Abstract

This article is devoted to the problem of construction of Hermite interpolation formulas with knots of the second multiplicity for second order partial differential operators given in the space of continuously differentiable functions of two variables. The obtained formulas contain the Gateaux differentials of a given operator. The construction of operator interpolation formulas is based on interpolation polynomials for scalar functions with respect to an arbitrary Chebyshev system of functions. An explicit representation of the interpolation error has been obtained.

Highlights

The interpolation of the functions provides the basis for the construction and research of approximate and numerical methods for the solution of many classes of problems. It is widely used for the approximate representation and calculation of functions, the numerical integration and differentiation, the construction of approximate methods for solving various classes of linear and nonlinear equations, etc
We consider the Hermite interpolation operator formulas with respect to knots of the second multiplicity
As fundamental interpolation polynomials h1,k(x), q1,k(x) (k = 0, 1) we choose the algebraic polynomials h1,k(x) = l12,k(x) 1 + 2 l1,1−k(x), q1,k(x) = l12,k(x) (x − xk), where l1,k(x) = (x − x1−k)/(xk − x1−k), k = 0, 1, and we take as interpolation nodes xk(t, s), k = 0, 1, the system of functions x0(t, s) ≡ 3, x1 (t, s) = sin t

Summary

Introduction

The interpolation of the functions provides the basis for the construction and research of approximate and numerical methods for the solution of many classes of problems. A number of other interpolation formulas for operators given in the spaces of functions and matrices are given in [1]–[4]. Interpolation operator formulas of the Hermite type We consider the operator polynomials of the form

Results

Conclusion