The reproducing kernel method for solving the system of the linear Volterra integral equations with variable coefficients

Yang, Lihong; Shen, Jihong; Wang, Yue

doi:10.1016/j.cam.2011.11.026

Cited by 27 publications

(18 citation statements)

References 13 publications

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“…In this paper, the authors generalize the idea of RKT method to provide a numerical solution for solving both fuzzy and fractional orders as given in (1). The present work is the extension of the past published works (Yang et al, 2012;Javadi et al, 2014;Abu Arqub, 2015;Bushnaq et al, 2013). To the best of the author's knowledge, the said problem has not been discussed before.…”

Section: Introductionmentioning

confidence: 83%

“…The Reproducing Kernel Theory (RKT) has potential applications in integral equations, integrodifferential equations, statistics, numerical analysis (Cattani, 2010;Abu Arqub et al, 2012;Jiang and Chen, 2013) among the other numerical and analytical methods. The RKT method has been successfully employed in the concerned literature to investigate certain scientific applications (Yang et al, 2012;Javadi et al, 2014;Abu Arqub, 2015).…”

Section: Introductionmentioning

confidence: 99%

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New implementation of reproducing kernel Hilbert space method for solving a fuzzy integro-differential equation of integer and fractional orders

Albzeirat

Ahmad

Momani

et al. 2018

Journal of King Saud University - Science

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a b s t r a c t This paper presents a novel technique for solving two new form of equation with fuzzy and integrodifferential equations. The proposed numerical iterative technique is based on the use of the reproducing Kernel theory. Two numerical examples are given to show the effectiveness and performance of the proposed technique. Simulation results are illustrated and comparative studies with past published works to the exact solution from Laplace transform of order integer have been performed to emphasize the simplicity and accuracy of the proposed technique. Moreover, future applications of the proposed technique are also discussed. Numerical experimental results fully support the findings of the proposed analytical approaches. Ó 2017 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

New implementation of reproducing kernel Hilbert space method for solving a fuzzy integro-differential equation of integer and fractional orders

Albzeirat

Ahmad

Momani

et al. 2018

Journal of King Saud University - Science

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“…Hossein and Biazar [13] introduced a new numerical approach for the solution of the systems of VIEs in which a modified homotopy perturbation method was presented for the second kind VIEs. Yang et al [14] described the reproducing kernel approach with variable coefficients to solve the system of the linear VIEs. Sidorov [15] presented the solvability of systems with piecewise continuous kernels of VIEs of the first Kind.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Bernstein Polynomial Approach for the System of Volterra Integral Equations on Arbitrary Interval and Its Convergence Analysis

2022

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In this paper, a new numerical technique is introduced to find the solution of the system of Volterra integral equations based on symmetric Bernstein polynomials. The use of Bernstein polynomials to find the numerical solutions of differential and integral equations increased due to its fast convergence. Here, the numerical solution of the system of Volterra integral equations on any finite interval [m,n] is obtained by replacing the unknown functions with the generalized Bernstein basis functions. The proposed technique converts the given system of equations into the system of algebraic equations which can be solved by using any standard rule. Further, Hyers–Ulam stability criteria are used to check the stability of the given technique. The comparison between exact and numerical solution for the distinct nodes is demonstrated to show its fast convergence.

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“…Wu and Li [11] applied iterative reproducing kernel method to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. Yang et al [12] used this method for solving the system of the linear Volterra integral equations with variable coefficients. A particular singular integral equation was solved by Du and Shen [13].…”

Section: Introductionmentioning

confidence: 99%

Solving Delay Differential Equations by an Accurate Method with Interpolation

Akgül

Kılıçman

2015

Abstract and Applied Analysis

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The reproducing kernel method for solving the system of the linear Volterra integral equations with variable coefficients

Cited by 27 publications

References 13 publications

New implementation of reproducing kernel Hilbert space method for solving a fuzzy integro-differential equation of integer and fractional orders

New implementation of reproducing kernel Hilbert space method for solving a fuzzy integro-differential equation of integer and fractional orders

Symmetric Bernstein Polynomial Approach for the System of Volterra Integral Equations on Arbitrary Interval and Its Convergence Analysis

Solving Delay Differential Equations by an Accurate Method with Interpolation

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