scholarly journals An algebraic exposition of umbral calculus with application to general linear interpolation problem: A survey

2014 ◽  
Vol 96 (110) ◽  
pp. 67-83 ◽  
Author(s):  
Francesco Costabile ◽  
Elisabetta Longo
Keyword(s):  
Interpolation Problem ◽  
Linear Interpolation ◽  
Umbral Calculus ◽  
General Linear ◽  
Numerical Examples ◽  
Polynomial Sequences ◽  
Sheffer Sequences ◽  
Determinantal Form ◽  
Systematic Exposition ◽  
Sheffer Polynomial

A systematic exposition of Sheffer polynomial sequences via determinantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper.

scholarly journals Umbral interpolation

2016 ◽  
Vol 99 (113) ◽  
pp. 165-175 ◽  
Author(s):  
Francesco Costabile ◽  
Elisabetta Longo
Keyword(s):  
Interpolation Problem ◽  
Truncation Error ◽  
Linear Interpolation ◽  
General Linear ◽  
Numerical Examples ◽  
Sheffer Polynomials

A general linear interpolation problem is posed and solved. This problem is called umbral interpolation problem because its solution can be expressed by a basis of Sheffer polynomials. The truncation error and its bounds are considered. Some examples are discussed, in particular generalizations of Abel-Gontscharoff and central interpolation are studied. Numerical examples are given too.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

Research on Contour Interpolation in Reverse Engineering

2012 ◽  
Vol 170-173 ◽  
pp. 3304-3307
Author(s):  
Hong Liang Wang ◽  
Hai Fei Ding ◽  
Jin Qi Wang
Keyword(s):  
Rapid Prototyping ◽  
Reverse Engineering ◽  
Interpolation Problem ◽  
Minimum Distance ◽  
Linear Interpolation ◽  
Contour Interpolation ◽  
Vehicle Engine

In reverse engineering and rapid prototyping, several intermediate contours interpolated between adjacent two-layer ICT slice are needed to meet the requirement of rapid prototyping. A method of linear interpolation based on minimum distance is adopted in this paper. On the basis of the research on interpolation problem between single contours, some researches on the interpolation problem of multiply contours was done in this paper. Satisfactory results are attained by the experiments of a vehicle engine.

An algebraic approach to Sheffer polynomial sequences

2013 ◽  
Vol 25 (4) ◽  
pp. 295-311 ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Elisabetta Longo
Keyword(s):  
Algebraic Approach ◽  
Polynomial Sequences ◽  
Sheffer Polynomial

scholarly journals Degenerate Catalan-Daehee numbers and polynomials of order $ r $ arising from degenerate umbral calculus

2021 ◽  
Vol 7 (3) ◽  
pp. 3845-3865
Author(s):  
Hye Kyung Kim ◽  
◽  
Dmitry V. Dolgy ◽  
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
Bell Polynomials ◽  
Psychological Burden ◽  
Special Polynomials ◽  
Sheffer Sequences ◽  
Special Numbers And Polynomials ◽  
Bernoulli Polynomials And Numbers ◽  
Euler Polynomials And Numbers ◽  
Degenerate Version

<abstract><p>Many mathematicians have studied degenerate versions of some special polynomials and numbers that can take into account the surrounding environment or a person's psychological burden in recent years, and they've discovered some interesting results. Furthermore, one of the most important approaches for finding the combinatorial identities for the degenerate version of special numbers and polynomials is the umbral calculus. The Catalan numbers and the Daehee numbers play important role in connecting relationship between special numbers.</p> <p>In this paper, we first define the degenerate Catalan-Daehee numbers and polynomials and aim to study the relation between well-known special polynomials and degenerate Catalan-Daehee polynomials of order $ r $ as one of the generalizations of the degenerate Catalan-Daehee polynomials by using the degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bernoulli polynomials and numbers of order $ r $, the degenerate Euler polynomials and numbers of order $ r $, the degenerate Daehee polynomials of order $ r $, the degenerate Bell polynomials, and so on.</p></abstract>

scholarly journals A Class of Sheffer Sequences of Some Complex Polynomials and Their Degenerate Types

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1064 ◽  
Author(s):  
Dojin Kim
Keyword(s):  
Trigonometric Polynomial ◽  
Complex Polynomials ◽  
Polynomial Sequences ◽  
Special Polynomials ◽  
Sheffer Sequences

We study some properties of Sheffer sequences for some special polynomials with complex Changhee and Daehee polynomials introducing their complex versions of the polynomials and splitting them into real and imaginary parts using trigonometric polynomial sequences. Moreover, considering their degenerate types of Sheffer sequences based on umbral composition, we present some useful expressions, properties, and examples about complex versions of the degenerate polynomials.

scholarly journals On a linear interpolation problem for n-dimensional vector polynomials

2015 ◽  
Vol 199 ◽  
pp. 45-62 ◽  
Author(s):  
Mikhail Kudryavtsev ◽  
Sergio Palafox ◽  
Luis O. Silva
Keyword(s):  
Interpolation Problem ◽  
Linear Interpolation ◽  
Dimensional Vector

scholarly journals A Finite Difference Scheme for Compressible Miscible Displacement Flow in Porous Media on Grids with Local Refinement in Time

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Wei Liu
Keyword(s):  
Porous Media ◽  
Finite Difference ◽  
Linear Interpolation ◽  
Miscible Displacement ◽  
Flow In Porous Media ◽  
Finite Difference Schemes ◽  
Two Dimensional ◽  
Local Refinement ◽  
Numerical Examples ◽  
Compressible Miscible Displacement

Considering two-dimensional compressible miscible displacement flow in porous media, finite difference schemes on grids with local refinement in time are constructed and studied. The construction utilizes a modified upwind approximation and linear interpolation at the slave nodes. Error analysis is presented in the maximum norm and numerical examples illustrating the theory are given.

scholarly journals Degenerate Lah–Bell polynomials arising from degenerate Sheffer sequences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hye Kyung Kim
Keyword(s):  
Differential Operators ◽  
Umbral Calculus ◽  
Bell Polynomials ◽  
Linear Functionals ◽  
Inversion Formulas ◽  
Sheffer Sequence ◽  
Sheffer Sequences ◽  
Special Numbers And Polynomials ◽  
Symmetric Identities ◽  
Degenerate Version

AbstractUmbral calculus is one of the important methods for obtaining the symmetric identities for the degenerate version of special numbers and polynomials. Recently, Kim–Kim (J. Math. Anal. Appl. 493(1):124521, 2021) introduced the λ-Sheffer sequence and the degenerate Sheffer sequence. They defined the λ-linear functionals and λ-differential operators, respectively, instead of the linear functionals and the differential operators of umbral calculus established by Rota. In this paper, the author gives various interesting identities related to the degenerate Lah–Bell polynomials and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derives the inversion formulas of these identities.

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