Pointwise Approximation by Bernstein Polynomials

Tachev, Gancho

doi:10.1017/s0004972711002838

Cited by 13 publications

(3 citation statements)

References 6 publications

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“…Also, Bernstein polynomials play a prominent role in various areas of mathematics. Many authors have used these polynomials in the solution of integral equations, differential equations, and approximation theory; see for instance [14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials

Behiry

2013

Abstract and Applied Analysis

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A numerical method for solving nonlinear Fredholm integrodifferential equations is proposed. The method is based on hybrid functions approximate. The properties of hybrid of block pulse functions and orthonormal Bernstein polynomials are presented and utilized to reduce the problem to the solution of nonlinear algebraic equations. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method.

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Section: Introductionmentioning

confidence: 99%

Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials

Behiry

2013

Abstract and Applied Analysis

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“…In Equation (),

{scriptB}_{m, M - 1} false(t false), m = 0,1, \dots, M - 1

are the Bernstein polynomials of the ( M − 1)th degree in the interval [0, 1], which is defined as 43

{scriptB}_{m, M - 1} false(t false) = ((), \begin{matrix} center \\ array M - 1 \\ array m \end{matrix}) t^{m} {false(1 - t false)}^{M - 1 - m}, m = 0,1, \dots, M - 1,

where

((), \begin{matrix} center \\ array M - 1 \\ array m \end{matrix}) = false(M - 1 false)! false/ false(m! false(M - 1 - m false)! false) .

…”

Section: Properties Of Hbbpsmentioning

confidence: 99%

Solving fractional differential equation using block‐pulse functions and Bernstein polynomials

Zhang

Tang

Liu

et al. 2021

Math Methods in App Sciences

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The method based on block pulse functions (BPFs) has been proposed to solve different kinds of fractional differential equations (FDEs). However, high accuracy requires considerable BPFs because they are piecewise constant and not so smooth. As a result, it increases the dimension of operational matrix and computational burden. To overcome this deficiency, a novel numerical method is developed to solve fractional differential equations. The method is based upon hybrid of BPFs and Bernstein polynomials (HBBPs), which are piecewise smooth. The HBBPs operational matrix of fractional‐order integral is derived to reduce the FDEs to a system of algebraic equations. Then the numerical solution of the FDEs is obtained through solving the system of algebraic equations. The convergence analysis is conducted for the suggested scheme, and the upper bound of error of the solution is given. Finally, illustrative examples are presented to demonstrate the validity, applicability, and efficiency of the proposed technique in contrast with other approaches.

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“…Definition 9 (See [16,22]). The Bernstein polynomials of degree n are defined on the interval [0, 1] as:…”

Section: Hybrid Bernstein Polynomials and Block Pulse Functionsmentioning

confidence: 99%

Numerical solution of two dimensional nonlinear fuzzy Fredholm integral equations of second kind using hybrid of block-pulse functions and Bernstein polynomials

Samadpour¹,

Ezzati²

2018

Filomat

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In this paper, first, we introduce a successive approximation method in terms of a combination of Bernstein polynomials and block-pulse functions. The proposed method is given for solving two dimensional nonlinear fuzzy Fredholm integral equations of the second kind. Then, we present the convergence of the proposed method. Also we investigate the numerical stability of the method with respect to the choice of the first iteration. Finally, two numerical examples are presented to show the accuracy of the method.

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Pointwise Approximation by Bernstein Polynomials

Cited by 13 publications

References 6 publications

Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials

Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials

Solving fractional differential equation using block‐pulse functions and Bernstein polynomials

Numerical solution of two dimensional nonlinear fuzzy Fredholm integral equations of second kind using hybrid of block-pulse functions and Bernstein polynomials

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