An efficient solution of system of generalized Abel integral equations using Bernstein polynomials wavelet bases

This paper deals with a new implementation of the Bernstein polynomials method to the numerical solution of a special kind of singular system. For this aim, first the truncated Bernstein series polynomials of the solution functions are substituted in the given problem. Using some properties of these polynomials, the solution of the problem is reduced to solve a linear system of algebraic equations. In order to confirm the reliability and accuracy of the proposed method, some weakly Abel integral equations systems with comparisons are solved in detail as numerical examples.

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A family of methods for Abel integral equations of the second kind

Journal of Computational and Applied Mathematics ◽

10.1016/0377-0427(91)90043-j ◽

1991 ◽

Vol 34 (2) ◽

pp. 211-219 ◽

Cited By ~ 19

Author(s):

H. Brunner ◽

M.R. Crisci ◽

E. Russo ◽

A. Vecchio

Keyword(s):

Integral Equations ◽

Abel Integral Equations

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Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

Journal of Scientific Research ◽

10.3329/jsr.v2i2.4483 ◽

2010 ◽

Vol 2 (2) ◽

pp. 264-272 ◽

Cited By ~ 7

Author(s):

A. Shirin ◽

M. S. Islam

Keyword(s):

Integral Equation ◽

Galerkin Method ◽

Integral Equations ◽

Fredholm Integral Equation ◽

Numerical Solutions ◽

Bernstein Polynomials ◽

Fredholm Integral Equations ◽

Matrix Formulation ◽

Piecewise Polynomials ◽

The Galerkin Method

In this paper, Bernstein piecewise polynomials are used to solve the integral equations numerically. A matrix formulation is given for a non-singular linear Fredholm Integral Equation by the technique of Galerkin method. In the Galerkin method, the Bernstein polynomials are used as the approximation of basis functions. Examples are considered to verify the effectiveness of the proposed derivations, and the numerical solutions guarantee the desired accuracy. Keywords: Fredholm integral equation; Galerkin method; Bernstein polynomials. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i2.4483 J. Sci. Res. 2 (2), 264-272 (2010)

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Systems of Generalized Abel Integral Equations.

10.21236/ada050001 ◽

1977 ◽

Cited By ~ 1

Author(s):

M. Lowengrub ◽

J. R. Walton

Keyword(s):

Integral Equations ◽

Abel Integral Equations

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Numerical solution of nonlinear stochastic Itô – Volterra integral equation driven by fractional Brownian motion

Engineering Computations ◽

10.1108/ec-01-2020-0039 ◽

2020 ◽

Vol 37 (9) ◽

pp. 3243-3268

Author(s):

S. Saha Ray ◽

S. Singh

Keyword(s):

Integral Equation ◽

Brownian Motion ◽

Fractional Brownian Motion ◽

Integral Equations ◽

Volterra Integral Equation ◽

Stochastic Integral ◽

Bernstein Polynomials ◽

Numerical Results ◽

Content Type ◽

Stochastic Integral Equations

Purpose This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations. Design/methodology/approach Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed. Findings Two illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method. Originality/value To the best of the authors’ knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method.

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