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.

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 Fredholm and Volterra Integrals by Newton–Kantorovich and Haar Wavelets Methods

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2034
Author(s):  
Mohammad Hasan Abdul Sathar ◽  
Ahmad Fadly Nurullah Rasedee ◽  
Anvarjon A. Ahmedov ◽  
Norfifah Bachok
Keyword(s):  
Error Analysis ◽  
Numerical Solution ◽  
Numerical Method ◽  
Current Method ◽  
Haar Wavelet ◽  
Numerical Results ◽  
Haar Wavelets ◽  
Numerical Examples ◽  
Hölder Classes ◽  
Holder Classes

The current study proposes a numerical method which solves nonlinear Fredholm and Volterra integral of the second kind using a combination of a Newton–Kantorovich and Haar wavelet. Error analysis for the Holder classes was established to ensure convergence of the Haar wavelets. Numerical examples will illustrate the accuracy and simplicity of Newton–Kantorovich and Haar wavelets. Numerical results of the current method were then compared with previous well-established methods.

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.

Numerical Solution of Stochastic Ito-Volterra Integral Equations using Haar Wavelets

2016 ◽  
Vol 9 (3) ◽  
pp. 416-431 ◽  
Author(s):  
Fakhrodin Mohammadi
Keyword(s):  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
General Procedure ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Computational Method ◽  
Haar Wavelets ◽  
Stochastic Operational Matrix

AbstractThis paper presents a computational method for solving stochastic Ito-Volterra integral equations. First, Haar wavelets and their properties are employed to derive a general procedure for forming the stochastic operational matrix of Haar wavelets. Then, application of this stochastic operational matrix for solving stochastic Ito-Volterra integral equations is explained. The convergence and error analysis of the proposed method are investigated. Finally, the efficiency of the presented method is confirmed by some examples.

scholarly journals Approximate Solution of Urysohn Integral Equations Using the Adomian Decomposition Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Randhir Singh ◽  
Gnaneshwar Nelakanti ◽  
Jitendra Kumar
Keyword(s):  
Approximate Solution ◽  
Error Analysis ◽  
Integral Equations ◽  
Decomposition Method ◽  
Series Solution ◽  
Adomian Decomposition Method ◽  
Numerical Examples ◽  
Recursive Scheme ◽  
Urysohn Integral Equations ◽  
Adomian Decomposition

We apply Adomian decomposition method (ADM) for obtaining approximate series solution of Urysohn integral equations. The ADM provides a direct recursive scheme for solving such problems approximately. The approximations of the solution are obtained in the form of series with easily calculable components. Furthermore, we also discuss the convergence and error analysis of the ADM. Moreover, three numerical examples are included to demonstrate the accuracy and applicability of the method.

Jacobi Collocation Approximation for Solving Multi-dimensional Volterra Integral Equations

2017 ◽  
Vol 18 (5) ◽  
pp. 411-425 ◽  
Author(s):  
Mohamed A. Abdelkawy ◽  
Ahmed Z. M. Amin ◽  
Ali H. Bhrawy ◽  
José A. Tenreiro Machado ◽  
António M. Lopes
Keyword(s):  
Error Analysis ◽  
Integral Equations ◽  
Collocation Method ◽  
Jacobi Polynomials ◽  
Volterra Integral Equations ◽  
The Novel ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Novel Technique ◽  
Shifted Jacobi Polynomials

AbstractThis paper addresses the solution of one- and two-dimensional Volterra integral equations (VIEs) by means of the spectral collocation method. The novel technique takes advantage of the properties of shifted Jacobi polynomials and is applied for solving multi-dimensional VIEs. Several numerical examples demonstrate the efficiency of the method and an error analysis verifies the correctness and feasibility of the proposed method when solving VIE.

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.

scholarly journals An efficient method for the numerical solution of the nonlinear Hammerstein fractional integral equations

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 419-429
Author(s):  
Ahmadabadi Nili ◽  
M.R. Velayati
Keyword(s):  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
Efficient Method ◽  
Fractional Integral ◽  
Singular Integrals ◽  
Computational Cost ◽  
Picard Iteration ◽  
Numerical Examples ◽  
Weakly Singular

In this paper, we present a numerical method for solving nonlinear Hammerstein fractional integral equations. The method approximates the solution by Picard iteration together with a numerical integration designed for weakly singular integrals. Error analysis of the proposed method is also investigated. Numerical examples approve its efficiency in terms of accuracy and computational cost.

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