scholarly journals Numerical Techniques for Solving Linear Volterra Fractional Integral Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Safa’ Hamdan ◽  
Naji Qatanani ◽  
Adnan Daraghmeh
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Fractional Integral ◽  
Integration Method ◽  
Haar Wavelet ◽  
Numerical Techniques ◽  
Product Integration ◽  
Fractional Integral Equation ◽  
Product Integration Method ◽  
Good Agreement

Two numerical techniques, namely, Haar Wavelet and the product integration methods, have been employed to give an approximate solution of the fractional Volterra integral equation of the second kind. To test the applicability and efficiency of the numerical method, two illustrative examples with known exact solution are presented. Numerical results show clearly that the accuracy of these methods are in a good agreement with the exact solution. A comparison between these methods shows that the product integration method provides more accurate results than its counterpart.

scholarly journals AN EXTENSION OF THE PRODUCT INTEGRATION METHOD TO L1 WITH APPLICATIONS IN ASTROPHYSICS

2016 ◽  
Vol 21 (6) ◽  
pp. 774-793 ◽  
Author(s):  
Laurence Grammont ◽  
Mario Ahues ◽  
Hanane Kaboul
Keyword(s):  
Integral Equation ◽  
Existence And Uniqueness ◽  
Integration Method ◽  
Sufficient Conditions ◽  
Iterative Refinement ◽  
Weakly Singular Kernel ◽  
Product Integration ◽  
Weakly Singular ◽  
Uniqueness Of The Solution ◽  
Product Integration Method

A Fredholm integral equation of the second kind in L1([a, b], C) with a weakly singular kernel is considered. Sufficient conditions are given for the existence and uniqueness of the solution. We adapt the product integration method proposed in C0 ([a, b], C) to apply it in L1 ([a, b], C), and discretize the equation. To improve the accuracy of the approximate solution, we use different iterative refinement schemes which we compare one to each other. Numerical evidence is given with an application in Astrophysics.

An adapted integration method for Volterra integral equation of the second kind with weakly singular kernel

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ahlem Nemer ◽  
Hanane Kaboul ◽  
Zouhir Mokhtari
Keyword(s):  
Integral Equation ◽  
Integration Method ◽  
New Product ◽  
Linear Interpolation ◽  
Weakly Singular Kernel ◽  
Product Integration ◽  
Weakly Singular ◽  
Product Integration Method ◽  
Singular Kernels ◽  
Linear Volterra Integral Equations

Abstract In this paper, we consider general cases of linear Volterra integral equations under minimal assumptions on their weakly singular kernels and introduce a new product integration method in which we involve the linear interpolation to get a better approximate solution, figure out its effect and also we provide a convergence proof. Furthermore, we apply our method to a numerical example and conclude this paper by adding a conclusion

A Spectral Numerical Method for Solving Distributed-Order Fractional Initial Value Problems

2018 ◽  
Vol 13 (10) ◽  
Author(s):  
M. A. Zaky ◽  
E. H. Doha ◽  
J. A. Tenreiro Machado
Keyword(s):  
Integral Equation ◽  
Initial Value Problems ◽  
Fractional Integral ◽  
Recurrence Relations ◽  
Fractional Integrals ◽  
Initial Value ◽  
Fractional Integral Equation ◽  
Collocation Scheme ◽  
Gauss Quadrature Formula ◽  
Distributed Order

In this paper, we construct and analyze a Legendre spectral-collocation method for the numerical solution of distributed-order fractional initial value problems. We first introduce three-term recurrence relations for the fractional integrals of the Legendre polynomial. We then use the properties of the Caputo fractional derivative to reduce the problem into a distributed-order fractional integral equation. We apply the Legendre–Gauss quadrature formula to compute the distributed-order fractional integral and construct the collocation scheme. The convergence of the proposed method is discussed. Numerical results are provided to give insights into the convergence behavior of our method.

scholarly journals A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation

2016 ◽  
Vol 14 (1) ◽  
pp. 237-246 ◽  
Author(s):  
Nasrin Eghbali ◽  
Vida Kalvandi ◽  
John M. Rassias
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fractional Order ◽  
Fractional Integral ◽  
Compact Interval ◽  
Ulam Stability ◽  
Fixed Point Approach ◽  
Delay Integral Equation ◽  
Fractional Integral Equation

AbstractIn this paper, we have presented and studied two types of the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation. We prove that the fractional order delay integral equation is Mittag-Leffler-Hyers-Ulam stable on a compact interval with respect to the Chebyshev and Bielecki norms by two notions.

Controllability of Abstract Systems of Fractional Order

2015 ◽  
Vol 18 (6) ◽  
Author(s):  
Therese Mur ◽  
Hernán R. Henríquez
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Integral Equation ◽  
Control Systems ◽  
Banach Spaces ◽  
Fractional Differential Equation ◽  
Fractional Integral ◽  
First Order ◽  
Fractional Integral Equation ◽  
Fractional Differential

AbstractIn this paper we are concerned with the controllability of control systems governed by a fractional differential equation in Banach spaces. Using the properties of the Mittag-Leffler function we generalize to these systems a result of Korobov and Rabakh, which was established for first order systems. We apply our results to study the controllability of a system modeled by a fractional integral equation in a Hilbert space.

scholarly journals Computational Methods for Solving Linear Fuzzy Volterra Integral Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jihan Hamaydi ◽  
Naji Qatanani
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Methods ◽  
Volterra Integral Equation ◽  
Taylor Expansion ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Iteration Methods ◽  
Good Agreement

Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results.

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