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

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.

scholarly journals Numerical approach of riemann-liouville fractional derivative operator

2021 ◽  
Vol 11 (6) ◽  
pp. 5367
Author(s):  
Ramzi B. Albadarneh ◽  
Iqbal M. Batiha ◽  
Ahmad Adwai ◽  
Nedal Tahat ◽  
A. K. Alomari
Keyword(s):  
Fractional Derivative ◽  
Nonlinear Problems ◽  
Numerical Approach ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Weighted Mean ◽  
Derivative Operator ◽  
Liouville Fractional Derivative ◽  
Linear And Nonlinear ◽  
Residual Errors

<p>This article introduces some new straightforward and yet powerful formulas in the form of series solutions together with their residual errors for approximating the Riemann-Liouville fractional derivative operator. These formulas are derived by utilizing some of forthright computations, and by utilizing the so-called weighted mean value theorem (WMVT). Undoubtedly, such formulas will be extremely useful in establishing new approaches for several solutions of both linear and nonlinear fractionalorder differential equations. This assertion is confirmed by addressing several linear and nonlinear problems that illustrate the effectiveness and the practicability of the gained findings.</p>

scholarly journals A Novel Numerical Method of Two-Dimensional Fredholm Integral Equations of the Second Kind

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Yanying Ma ◽  
Jin Huang ◽  
Hu Li
Keyword(s):  
Approximate Solution ◽  
Integral Operator ◽  
Numerical Method ◽  
Integral Equations ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Fredholm Integral Equations ◽  
Two Dimensional ◽  
Error Analyses ◽  
Approximate Integral

A novel numerical method is developed for solving two-dimensional linear Fredholm integral equations of the second kind by integral mean value theorem. In the proposed algorithm, each element of the generated discrete matrix is not required to calculate integrals, and the approximate integral operator is convergent according to collectively compact theory. Convergence and error analyses of the approximate solution are provided. In addition, an algorithm is given. The reliability and efficiency of the proposed method will be illustrated by comparison with some numerical results.

scholarly journals Numerical approach for approximating the Caputo fractional-order derivative operator

2021 ◽  
Vol 6 (11) ◽  
pp. 12743-12756
Author(s):  
Ramzi B. Albadarneh ◽  
◽  
Iqbal Batiha ◽  
A. K. Alomari ◽  
Nedal Tahat ◽  
...  
Keyword(s):  
Fractional Order ◽  
Truncation Error ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Analytic Solutions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Linear And Nonlinear ◽  
Caputo Fractional Order Derivative

<abstract><p>This work aims to propose a new simple robust power series formula with its truncation error to approximate the Caputo fractional-order operator $ D_{a}^{\alpha}y(t) $ of order $ m-1 &lt; \alpha &lt; m $, where $ m\in\mathbb{N} $. The proposed formula, which are derived with the help of the weighted mean value theorem, is expressed ultimately in terms of a fractional-order series and its reminder term. This formula is used successfully to provide approximate solutions of linear and nonlinear fractional-order differential equations in the form of series solution. It can be used to determine the analytic solutions of such equations in some cases. Some illustrative numerical examples, including some linear and nonlinear problems, are provided to validate the established formula.</p></abstract>

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.

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