scholarly journals Numerical Solution of Nonlinear Volterra Integral Equations System Using Simpson’s 3/8 Rule

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Adem Kılıçman ◽  
L. Kargaran Dehkordi ◽  
M. Tavassoli Kajani
Keyword(s):  
Approximate Solution ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Numerical Solution ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Block System

The Simpson’s 3/8 rule is used to solve the nonlinear Volterra integral equations system. Using this rule the system is converted to a nonlinear block system and then by solving this nonlinear system we find approximate solution of nonlinear Volterra integral equations system. One of the advantages of the proposed method is its simplicity in application. Further, we investigate the convergence of the proposed method and it is shown that its convergence is of orderO(h4). Numerical examples are given to show abilities of the proposed method for solving linear as well as nonlinear systems. Our results show that the proposed method is simple and effective.

Solving systems of fractional two-dimensional nonlinear partial Volterra integral equations by using Haar wavelets

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Amir Ahmad Khajehnasiri ◽  
R. Ezzati ◽  
M. Afshar Kermani
Keyword(s):  
Nonlinear System ◽  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
Fractional Integration ◽  
Volterra Integral Equations ◽  
Haar Wavelets ◽  
Operational Matrices ◽  
Two Dimensional ◽  
Numerical Examples

Abstract The main aim of this paper is to use the operational matrices of fractional integration of Haar wavelets to find the numerical solution for a nonlinear system of two-dimensional fractional partial Volterra integral equations. To do this, first we present the operational matrices of fractional integration of Haar wavelets. Then we apply these matrices to solve systems of two-dimensional fractional partial Volterra integral equations (2DFPVIE). Also, we present the error analysis and convergence as well. At the end, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

scholarly journals Numerical solution of nonlinear stochastic Itô–Volterra integral equations based on Haar wavelets

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jieheng Wu ◽  
Guo Jiang ◽  
Xiaoyan Sang
Keyword(s):  
Approximate Solution ◽  
Error Analysis ◽  
Numerical Solution ◽  
Numerical Method ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Haar Wavelets ◽  
Stochastic Integration ◽  
Numerical Examples

AbstractIn this paper, an efficient numerical method is presented for solving nonlinear stochastic Itô–Volterra integral equations based on Haar wavelets. By the properties of Haar wavelets and stochastic integration operational matrixes, the approximate solution of nonlinear stochastic Itô–Volterra integral equations can be found. At the same time, the error analysis is established. Finally, two numerical examples are offered to testify the validity and precision of the presented method.

scholarly journals Convergence Analysis of Legendre Pseudospectral Scheme for Solving Nonlinear Systems of Volterra Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Emran Tohidi ◽  
O. R. Navid Samadi ◽  
S. Shateyi
Keyword(s):  
Nonlinear Systems ◽  
Theoretical Prediction ◽  
Numerical Solution ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Volterra Equations ◽  
Exponential Rate ◽  
Exponential Rate Of Convergence ◽  
Legendre Spectral Method ◽  
Pseudospectral Scheme

We are concerned with the extension of a Legendre spectral method to the numerical solution of nonlinear systems of Volterra integral equations of the second kind. It is proved theoretically that the proposed method converges exponentially provided that the solution is sufficiently smooth. Also, three biological systems which are known as the systems of Lotka-Volterra equations are approximately solved by the presented method. Numerical results confirm the theoretical prediction of the exponential rate of convergence.

scholarly journals Numerical solution of linear stochastic Volterra integral equations via new basis functions

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 5959-5966
Author(s):  
Tofigh Cheraghi ◽  
Morteza Khodabin ◽  
Reza Ezzati
Keyword(s):  
Linear System ◽  
Numerical Solution ◽  
Integral Equations ◽  
Orthogonal Basis ◽  
Volterra Integral Equations ◽  
Basis Functions ◽  
New Method ◽  
Algebraic Equations ◽  
Numerical Examples

In this article, we use a new method based on orthogonal basis functions for the numerical solution of stochastic Volterra integral equations of the second kind (SVIE). By using this method, a SVIE can be reduced to a linear system of algebraic equations. Finally, to show the efficiency of the proposed method, we give two numerical examples.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

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