scholarly journals Orthonormal Bernstein polynomials for Volterra integral equations of the second kind

2020 ◽  
Vol 9 (1) ◽  
pp. 9
Author(s):  
Jafar Biazar ◽  
Hamed Ebrahimi
Keyword(s):  
Mathematical Modeling ◽  
Integral Equation ◽  
Numerical Method ◽  
Integral Equations ◽  
Method Of Moments ◽  
Volterra Integral Equation ◽  
Bernstein Polynomials ◽  
Volterra Integral Equations ◽  
Linear Volterra Integral Equations ◽  
Relative Errors

The purpose of this research is to provide an effective numerical method for solving linear Volterra integral equations of the second kind. The mathematical modeling of many phenomena in various branches of sciences lead into an integral equation. The proposed approach is based on the method of moments (Galerkin- Ritz) using orthonormal Bernstein polynomials. To solve a Volterra integral equation, the ap-proximation for a solution is considered as an expansion in terms of Bernstein orthonormal polynomials. Ultimately, the usefulness and extraordinary accuracy of the proposed approach will be verified by a few examples where the results are plotted in diagrams, Also the re-sults and relative errors are presented in some Tables.  

scholarly journals Application of the Bernstein polynomials for solving Volterra integral equations with convolution kernels

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 1045-1052 ◽  
Author(s):  
Ahmet Altürk
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Bernstein Polynomials ◽  
Simple Algorithm ◽  
Volterra Integral Equations ◽  
Convolution Product ◽  
Algebraic Equations ◽  
Convolution Kernels ◽  
The Difference ◽  
Linear Volterra Integral Equations

In this article, we consider the second-type linear Volterra integral equations whose kernels based upon the difference of the arguments. The aim is to convert the integral equation to an algebraic one. This is achieved by approximating functions appearing in the integral equation with the Bernstein polynomials. Since the kernel is of convolution type, the integral is represented as a convolution product. Taylor expansion of kernel along with the properties of convolution are used to represent the integral in terms of the Bernstein polynomials so that a set of algebraic equations is obtained. This set of algebraic equations is solved and approximate solution is obtained. We also provide a simple algorithm which depends both on the degree of the Bernstein polynomials and that of monomials. Illustrative examples are provided to show the validity and applicability of the method.

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

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.

scholarly journals Modified Block Pulse Functions for Numerical Solution of Stochastic Volterra Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
K. Maleknejad ◽  
M. Khodabin ◽  
F. Hosseini Shekarabi
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Rate Of Convergence ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Volterra Integral Equations ◽  
Numerical Example ◽  
Block Pulse Functions ◽  
A New Technique ◽  
Modified Block

We present a new technique for solving numerically stochastic Volterra integral equation based on modified block pulse functions. It declares that the rate of convergence of the presented method is faster than the method based on block pulse functions. Efficiency of this method and good degree of accuracy are confirmed by a numerical example.

scholarly journals Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

2021 ◽  
Vol 54 (1) ◽  
pp. 11-24
Author(s):  
Atanaska Georgieva
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Homotopy Analysis Method ◽  
Volterra Integral Equations ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Fuzzy Solution

Abstract The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

scholarly journals Deriving the Composite Simpson Rule by Using Bernstein Polynomials for Solving Volterra Integral Equations

2014 ◽  
Vol 11 (3) ◽  
pp. 1274-1281 ◽  
Author(s):  
Baghdad Science Journal
Keyword(s):  
Exact Solution ◽  
Integral Equations ◽  
Bernstein Polynomials ◽  
Volterra Integral Equations ◽  
One Dimensional ◽  
Linear Volterra Integral Equations

In this paper we use Bernstein polynomials for deriving the modified Simpson's 3/8 , and the composite modified Simpson's 3/8 to solve one dimensional linear Volterra integral equations of the second kind , and we find that the solution computed by this procedure is very close to exact solution.

Uniqueness of solutions for a class of non-linear Volterra integral equations with convolution kernel

1989 ◽  
Vol 106 (3) ◽  
pp. 547-552 ◽  
Author(s):  
P. J. Bushell ◽  
W. Okrasinski
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Convolution Kernel ◽  
Qualitative Behaviour ◽  
Uniqueness Of Solutions ◽  
Linear Diffusion ◽  
Non Linear ◽  
Linear Volterra Integral Equations ◽  
Image Position

The non-linear Volterra integral equationhas been studied recently in connection with non-linear diffusion and percolation problems [4, 6, 10]. The existence, uniqueness and qualitative behaviour of non-negative, non-trivial solutions are the questions of physical interest.

Easy Numerical Method to Solution a System of Linear Volterra Integral Equations

2011 ◽  
Vol 14 (3) ◽  
pp. 144-149
Author(s):  
Ahlam J. Kaleel ◽  
◽  
Huda H. Omran ◽  
Salam J. Majeed ◽  
◽  
...  
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Linear Volterra Integral Equations

scholarly journals EXISTENCE OF RESOLVENT FOR VOLTERRA INTEGRAL EQUATIONS ON TIME SCALES

2010 ◽  
Vol 82 (1) ◽  
pp. 139-155 ◽  
Author(s):  
MURAT ADıVAR ◽  
YOUSSEF N. RAFFOUL
Keyword(s):  
Integral Equation ◽  
Qualitative Analysis ◽  
Integral Equations ◽  
Time Scales ◽  
Volterra Integral Equation ◽  
Sufficient Conditions ◽  
Volterra Integral Equations ◽  
Future Research ◽  
Shift Operators ◽  
Image Position

AbstractWe introduce the concept of ‘shift operators’ in order to establish sufficient conditions for the existence of the resolvent for the Volterra integral equation on time scales. The paper will serve as the foundation for future research on the qualitative analysis of solutions of Volterra integral equations on time scales, using the notion of the resolvent.

scholarly journals Numerical Solution of System of Three Nonlinear Volterra Integral Equations Using Implicit Trapezoidal

2017 ◽  
Vol 10 (1) ◽  
pp. 44
Author(s):  
Dalal Adnan Maturi ◽  
Honaida Mohammed Malaikah
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Numerical Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Volterra Integral Equations

In this project, we will be find numerical solution of Volterra Integral Equation of Second kind through using Implicit trapezoidal and that by using Maple 17 program, then we found that numerical solution was highly accurate when it was compared with exact solution.

Sign in / Sign up

Export Citation Format

Share Document