Numerical Solutions for Quadratic Integro-Differential Equations of Fractional Orders

Alhendi, Fatheah Ahmad; Shammakh, Wafa; Al-badrani, Hind

doi:10.4236/ojapps.2017.74014

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…For example, Wazwaz (2010) confirmed that the VIM is very reliable in solving first-and second-kind integral equations of the Volterra type and most calculations can be significantly reduced. Alhendi, Shammakh, and Al-Badrani (2017) found that the VIM and HPM are very effective when applying them to solve quadratic fractional integro-differential equations. Elborai, Abdou, and Youssef (2013) studied the mean square convergence of the series solution for a stochastic integro-differential equation and estimated the truncation error by the ADM. Kurt and Tasbozan (2019) utilized the HAM to solve the modified Burgers equation.…”

Section: Introductionmentioning

confidence: 99%

On an approximate solution of a boundary value problem for a nonlinear integro-differential equation

Abdou

Youssef

2021

Arab Journal of Basic and Applied Sciences

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The aim of this work is to discuss the solvability of a boundary value problem for a nonlinear integro-differential equation. First, we derive an equivalent nonlinear Fredholm integral equation (NFIE) to this problem. Second, we prove the existence of a solution to the NFIE using the Krasnosel'skii fixed point theorem under verifying some sufficient conditions. Third, we solve the NFIE numerically and study the convergence rate via methods based upon applying the modified Adomian decomposition method and Liao's homotopy analysis method. As applications, some examples are illustrated to support our work. The results in this work refer to both methods are efficient and converge rapidly, but the homotopy analysis method may converge faster when we succeed in choosing the optimal homotopy control parameter.

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Section: Introductionmentioning

confidence: 99%

On an approximate solution of a boundary value problem for a nonlinear integro-differential equation

Abdou

Youssef

2021

Arab Journal of Basic and Applied Sciences

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“…The obtained results reveal that this method is very effective, Momani [19] and Qaralleh [20] applied Adomian polynomials to solve fractional integro-differential equations and systems of fractional integro-differential equations, Kadem and Kilicman [15] utilized the HPM and VIM methods for integrodifferential equation of fractional order with initial-boundary conditions, Yang [21] used the hybrid of block pulse function and Chebyshev polynomials to solve nonlinear Fredholm fractional integro-differential equations, Yang and Hou [22] applied the Laplace decomposition method to solve the fractional integro-differential equations, Mittal and Nigam [18] utilized the Adomian decomposition method to approximate solutions of fractional integro-differential equations, and Ma and Huang [17] applied hybrid collocation method to study integro-differential equations of fractional order. Moreover, properties of the fractional integro-differential equations have been studied by several authors [4,12,16].…”

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confidence: 99%

The Approximate Solutions of Fractional Volterra-Fredholm Integro-Differential Equations by using Analytical Techniques

Hamoud¹,

Ghadle²

2018

Issues of Analysis

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This paper demonstrates a study on some significant latest innovations in the approximation techniques to find the approximate solutions of Caputo fractional Volterra-Fredholm integro-differential equations. To apply this, the study uses Adomian decomposition method and modified Laplace Adomian decomposition method. A wider applicability of these techniques is based on their reliability and reduction in the size of the computational work. This study provides analytical approximate to determine the behavior of the solution. It proves the existence and uniqueness results and convergence of the solution. In addition, it brings an example to examine the validity and applicability of the proposed techniques.

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Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

Mirzaee

Alipour

2018

Numerical Methods Partial

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In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h3). The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.

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Numerical Solutions for Quadratic Integro-Differential Equations of Fractional Orders

Cited by 3 publications

References 17 publications

On an approximate solution of a boundary value problem for a nonlinear integro-differential equation

On an approximate solution of a boundary value problem for a nonlinear integro-differential equation

The Approximate Solutions of Fractional Volterra-Fredholm Integro-Differential Equations by using Analytical Techniques

Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

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