scholarly journals Numerical differentiation on scattered data through multivariate polynomial interpolation

2021 ◽  
Author(s):  
Francesco Dell’Accio ◽  
Filomena Di Tommaso ◽  
Najoua Siar ◽  
Marco Vianello
Keyword(s):  
Polynomial Interpolation ◽  
Numerical Differentiation ◽  
Scattered Data ◽  
Monomial Basis ◽  
Multivariate Polynomial Interpolation ◽  
Differentiation Formula ◽  
Local Polynomial ◽  
Leja Points ◽  
Derivatives Of ◽  
Numerical Differentiation Formula

AbstractWe discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

scholarly journals Application of Non-polynomial Splines to Solving Differential Equations

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Heat Conduction Problem ◽  
Numerical Differentiation ◽  
Exponential Polynomial ◽  
Order Of Approximation ◽  
Polynomial Splines ◽  
Implicit Difference Scheme ◽  
Local Polynomial ◽  
Second Derivatives ◽  
Implicit Difference Schemes ◽  
Derivatives Of

The application of the local polynomial and non-polynomial to the construction of methods for numerically solving the heat conduction problem is discussed. The non-polynomial splines are used here to approximate the partial derivatives. Formulas for numerical differentiation based on the application of the nonpolynomial splines of the fourth order of approximation are constructed. Particular attention is paid to polynomial, trigonometric, exponential, polynomial-trigonometric and polynomial-exponential splines. This approach allows us to construct explicit and implicit difference schemes. The main focus of the paper is on implicit difference scheme. New approximations with splines of the Lagrange and Hermite type with new properties are obtained. These approximations take into account the first and second derivatives of the function being approximated. Numerical examples are given.

Sign in / Sign up

Export Citation Format

Share Document