Numerical approach for nonlinear system of Fredholm-Volterra integral equations

2021 ◽  
Author(s):  
Zainidin Eshkuvatov ◽  
Husnida Mamatova ◽  
Shahrina Ismail ◽  
Ilyani Abdullah ◽  
Rakhmatillo Aloev
Keyword(s):  
Nonlinear System ◽  
Integral Equations ◽  
Numerical Approach ◽  
Volterra Integral Equations

scholarly journals Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein–Kantorovich Operators

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1193
Author(s):  
Suzan Cival Buranay ◽  
Mehmet Ali Özarslan ◽  
Sara Safarzadeh Falahhesar
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Unknown Function ◽  
Input Data ◽  
Linear Equations ◽  
Numerical Approach ◽  
Volterra Integral Equations ◽  
Test Problems ◽  
The Stability ◽  
Kantorovich Operators

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations.

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 Non-standard Volterra integral equations: a mean-value theorem numerical approach

2020 ◽  
Vol 14 (9) ◽  
pp. 423-432
Author(s):  
Paolo De Angelis ◽  
Roberto De Marchis ◽  
Antonio Luciano Martire ◽  
Stefano Patri
Keyword(s):  
Integral Equations ◽  
Numerical Approach ◽  
Mean Value ◽  
Volterra Integral Equations ◽  
Mean Value Theorem

scholarly journals Solutions of Linear and Nonlinear Volterra Integral Equations Using Hermite and Chebyshev Polynomials

2013 ◽  
Vol 11 (8) ◽  
pp. 2910-2920
Author(s):  
Md. Shafiqul Islam ◽  
Md. Azizur Rahman
Keyword(s):  
Integral Equations ◽  
Good Accuracy ◽  
Numerical Approach ◽  
Volterra Integral Equations ◽  
Basis Functions ◽  
Matrix Formulation ◽  
Numerical Examples ◽  
Singular Kernels ◽  
Piecewise Continuous ◽  
Linear And Nonlinear

The purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on Galerkin weighted residual approximation. In this method Hermite and Chebyshev piecewise, continuous and differentiable polynomials are exploited as basis functions. A rigorous effective matrix formulation is proposed to solve the linear and nonlinear Volterra integral equations of the first and second kind with regular and singular kernels. The algorithm is simple and can be coded easily. The efficiency of the proposed method is tested on several numerical examples to get the desired and reliable good accuracy.

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.

Sign in / Sign up

Export Citation Format

Share Document