Spectral-Collocation Method for Fractional Fredholm Integro-Differential Equations

Yang, Yin; Chen, Yanping; Huang, Yunqing

doi:10.4134/jkms.2014.51.1.203

Cited by 33 publications

(26 citation statements)

References 33 publications

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“…On most of the interval, the solution of Equation (34) behaves like the solution of Equations (1) and (2). However, there is small interval around x = 0 in which the solution of problem (1) and (2) does not agree with the solution of problem (1) and (2).…”

Section: Solution Methodsmentioning

confidence: 85%

“…However, there is small interval around x = 0 in which the solution of problem (1) and (2) does not agree with the solution of problem (1) and (2). To handle this situation, the boundary layer correction subproblem is introduced in step 2.…”

Section: Solution Methodsmentioning

confidence: 99%

“…Using the change of variable x = 2 s, we get −D Using the Pade' approximation of order [2,2], we have θ = 0.0927388622769557. In Table 3, we present the error for x = 0, 0.1, ..., 1 for = 0.001, 0.0001, and 0.00001.…”

Section: Examplementioning

confidence: 99%

“…subject to y(0) = y 0 , y(1) = y 1 (2) where > 0 is a small positive parameter, y 0 and y 1 are constants, and K(x, t) and f (x) are smooth functions. The derivative which we use in this paper is in the Caputo sense.…”

mentioning

confidence: 99%

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A Numerical Method for Solving a Class of Nonlinear Second Order Fractional Volterra Integro-Differntial Type of Singularly Perturbed Problems

Syam

Omar

2018

Mathematics

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Abstract:In this paper, we study a class of fractional nonlinear second order Volterra integro-differential type of singularly perturbed problems with fractional order. We divide the problem into two subproblems. The first subproblems is the reduced problem when = 0. The second subproblems is fractional Volterra integro-differential problem. We use the finite difference method to solve the first problem and the reproducing kernel method to solve the second problem. In addition, we use the pade' approximation. The results show that the proposed analytical method can achieve excellent results in predicting the solutions of such problems. Theoretical results are presented. Numerical results are presented to show the efficiency of the proposed method.

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Section: Solution Methodsmentioning

confidence: 85%

Section: Solution Methodsmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

mentioning

confidence: 99%

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A Numerical Method for Solving a Class of Nonlinear Second Order Fractional Volterra Integro-Differntial Type of Singularly Perturbed Problems

Syam

Omar

2018

Mathematics

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“…Numerical methods are very power tools for solving the complicated problems in many fields (Bianca et al, 2009;Bhrawy and Alghamdi, 2012;Yang et al, 2014;Bhrawy and Aloi, 2013;Doha et al, 2011;Saha Ray, 2009;Mittal and Nigam, 2008;Saeedi and Samimi, 2012;Saeed and Sdeq, 2010;Ahmed and Salh, 2011). Newly, few numerical methods for solving the Fractional Differential Equations (FDEs) and Fractional IntegroDifferential Equations (FIDEs) have been presented.…”

Section: Introductionmentioning

confidence: 99%

Study of Fractional Order Integro-Differential Equations by using Chebyshev Neural Network

Chaharborj¹,

Mahmoudi²

2017

Journal of Mathematics and Statistics

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Abstract:Recently fractional integro-differential equations have been proposed for the modeling of many complex systems. Simulation of these equations requires the development of suitable numerical methods. This paper is devoted to finding the numerical solution of some classes of fractional integro-differential equations by employing the Chebyshev Neural Network (ChNN). The accuracy of proposed method is shown by comparing the numerical results computed by using Chebyshev neural network method with the analytical solution.

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Space‐time finite element adaptive AMG for multi‐term time fractional advection diffusion equations

Yue

Liu

Shu

et al. 2019

Math Methods in App Sciences

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In this study we construct a time-space finite element (FE) scheme and furnish cost-efficient approximations for one-dimensional multi-term time fractional advection diffusion equations on a bounded domain Ω. Firstly, a fully discrete scheme is obtained by the linear FE method in both temporal and spatial directions, and many characterizations on the resulting matrix are established. Secondly, the condition number estimation is proved, an adaptive algebraic multigrid (AMG) method is further developed to lessen computational cost and analyzed in the classical framework. Finally, some numerical experiments are implemented to reach the saturation error order in the L 2 (Ω) norm sense, and present theoretical confirmations and predictable behaviors of the proposed algorithm.Another important note is that no matter which fully discrete scheme is utilized, there always exists the computational challenge in nonlocality caused by fractional differential operators [27]. 15 Quite many scholars are working to identify algorithms most appropriate to overcome the challenge and utilize computer resources. Multigrid exploits a hierarchy of grids or multiscale representations, and reduces the error of the approximation at a number of frequencies (from global smooth to local oscillation) simultaneously [28-31], which makes multigrid as an extremely superior solver or preconditioner of particular interest. Pang and Sun presented an efficient V-cycle geometric multi-20 grid (GMG) with fast Fourier transform (FFT) to solve one-dimensional space-fractional diffusion equations (SFDEs) discretized by an implicit FD scheme [32]. Bu et al. extended and analyzed the V-cycle GMG for one-dimensional MTFADEs via the FD in temporal and FE in spatial directions [26]. Zhou and Wu discussed the FE F-cycle GMG to linear stationary FADEs in Riemann-Liouvlle

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Spectral-Collocation Method for Fractional Fredholm Integro-Differential Equations

Cited by 33 publications

References 33 publications

A Numerical Method for Solving a Class of Nonlinear Second Order Fractional Volterra Integro-Differntial Type of Singularly Perturbed Problems

A Numerical Method for Solving a Class of Nonlinear Second Order Fractional Volterra Integro-Differntial Type of Singularly Perturbed Problems

Study of Fractional Order Integro-Differential Equations by using Chebyshev Neural Network

Space‐time finite element adaptive AMG for multi‐term time fractional advection diffusion equations

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