scholarly journals Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 542
Author(s):  
Frank Filbir ◽  
Donatella Occorsio ◽  
Woula Themistoclakis
Keyword(s):  
Simultaneous Approximation ◽  
Bernstein Polynomials ◽  
Reference Interval ◽  
Pointwise Estimates ◽  
Numerical Tests ◽  
Equidistant Points ◽  
Hadamard Transforms ◽  
Boolean Sums ◽  
Generalized Bernstein Polynomials ◽  
Theoretical Results

In the present paper, we propose a numerical method for the simultaneous approximation of the finite Hilbert and Hadamard transforms of a given function f, supposing to know only the samples of f at equidistant points. As reference interval we consider [ − 1 , 1 ] and as approximation tool we use iterated Boolean sums of Bernstein polynomials, also known as generalized Bernstein polynomials. Pointwise estimates of the errors are proved, and some numerical tests are given to show the performance of the procedures and the theoretical results.

scholarly journals Stable Filtering Procedures for Nodal Discontinuous Galerkin Methods

2021 ◽  
Vol 87 (1) ◽  
Author(s):  
Jan Nordström ◽  
Andrew R. Winters
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Galerkin Methods ◽  
Theoretical Discussion ◽  
Numerical Tests ◽  
Dg Methods ◽  
Theoretical Results ◽  
Filtering Procedure

AbstractWe prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre–Gauss–Lobatto quadrature. The theoretical discussion re-contextualizes stable filtering results for finite difference methods into the DG setting. Numerical tests verify and validate the underlying theoretical results.

An analysis of discretisations of inverse diffusion equations

2006 ◽  
Vol 2 (3) ◽  
pp. 117-129
Author(s):  
M. Breuß
Keyword(s):  
Diffusion Processes ◽  
Diffusion Equations ◽  
Numerical Tests ◽  
Inverse Diffusion ◽  
Theoretical Results

We discuss some important issues arising when approximating numerically stabilised inverse diffusion processes. We prove rigorously the necessity of a minmod-type stabilisation. Furthermore, we give rigorously verified assertions concerning the occurence of undesirable staircasing aka terracing artefacts. The theoretical results are supplemented by numerical tests.

Generalized Bernstein Polynomials

2004 ◽  
Vol 44 (1) ◽  
pp. 63-78 ◽  
Author(s):  
Stanisław Lewanowicz ◽  
Paweł Woźny
Keyword(s):  
Bernstein Polynomials ◽  
Generalized Bernstein Polynomials

scholarly journals An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

2019 ◽  
Vol 17 (1) ◽  
pp. 1567-1598
Author(s):  
Tianbao Liu ◽  
Xiwen Qin ◽  
Qiuyue Li
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Third Order ◽  
Numerical Tests ◽  
The Family ◽  
Theoretical Results ◽  
High Computational Efficiency

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

ABSORBING BOUNDARY CONDITIONS IN BLOCK GAUSS–SEIDEL METHODS FOR CONVECTION PROBLEMS

1996 ◽  
Vol 06 (04) ◽  
pp. 481-502 ◽  
Author(s):  
FREDERIC NATAF
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary ◽  
Convection Diffusion ◽  
Radiation Boundary Conditions ◽  
Convection Diffusion Equation ◽  
Numerical Tests ◽  
Radiation Boundary ◽  
Theoretical Results

In the context of convection-diffusion equation, the use of absorbing boundary conditions (also called radiation boundary conditions) is considered in block Gauss–Seidel algorithms. Theoretical results and numerical tests show that the convergence is thus accelerated.

scholarly journals A New Class of Iterative Processes for Solving Nonlinear Systems by Using One Divided Differences Operator

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 776 ◽  
Author(s):  
Alicia Cordero ◽  
Cristina Jordán ◽  
Esther Sanabria ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Boundary Problem ◽  
Fourth Order ◽  
Difference Operator ◽  
Divided Difference ◽  
Order Convergence ◽  
New Class ◽  
Numerical Tests ◽  
New Family ◽  
Theoretical Results

In this manuscript, a new family of Jacobian-free iterative methods for solving nonlinear systems is presented. The fourth-order convergence for all the elements of the class is established, proving, in addition, that one element of this family has order five. The proposed methods have four steps and, in all of them, the same divided difference operator appears. Numerical problems, including systems of academic interest and the system resulting from the discretization of the boundary problem described by Fisher’s equation, are shown to compare the performance of the proposed schemes with other known ones. The numerical tests are in concordance with the theoretical results.

scholarly journals A generalization of the Bernstein polynomials

1999 ◽  
Vol 42 (2) ◽  
pp. 403-413 ◽  
Author(s):  
Haul Oruç ◽  
George M. Phillips ◽  
Philip J. Davis
Keyword(s):  
Bernstein Polynomials ◽  
Classical Case ◽  
Geometric Progression ◽  
Generalized Bernstein Polynomials

This paper is concerned with a generalization of the classical Bernstein polynomials where the function is evaluated at intervals which are in geometric progression. It is shown that, when the function is convex, the generalized Bernstein polynomials Bn are monotonic in n, as in the classical case.

scholarly journals The Truncatedq-Bernstein Polynomials in the Caseq>1

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Mehmet Turan
Keyword(s):  
Bernstein Polynomials ◽  
Numerical Examples ◽  
Theoretical Results

The truncatedq-Bernstein polynomialsBn,m,qf;x,n∈ℕ, and  m∈ℕ0emerge naturally when theq-Bernstein polynomials of functions vanishing in some neighbourhood of 0 are considered. In this paper, the convergence of the truncatedq-polynomials on0,1is studied. To support the theoretical results, some numerical examples are provided.

scholarly journals Convergence of Generalized Bernstein Polynomials

2002 ◽  
Vol 116 (1) ◽  
pp. 100-112 ◽  
Author(s):  
Alexander Il'inskii ◽  
Sofiya Ostrovska
Keyword(s):  
Bernstein Polynomials ◽  
Generalized Bernstein Polynomials
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