A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations

Esmaeili, Shahrokh; Shamsi, M.

doi:10.1016/j.cnsns.2010.12.008

Cited by 127 publications

(76 citation statements)

References 21 publications

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“…An automatic quadrature method based on the Chebyshev polynomials was presented for approximating the Caputo derivative in [26]. Some other computational schemes, such as the L1, L2 and L2C schemes, etc., are also introduced [8,11,13,14,16,18,19,21,23,25,28,29,31].…”

Section: Introductionmentioning

confidence: 99%

Spectral approximations to the fractional integral and derivative

Zeng

Liu

2012

fcaa

151

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AbstractIn this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

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Section: Introductionmentioning

confidence: 99%

Spectral approximations to the fractional integral and derivative

Zeng

Liu

2012

fcaa

151

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“…. , m − 1, and c kj are given explicitly as (24) and (25), respectively. The matrix D corresponding to the treatment of multi-point boundary conditions (21), its elements can be written as…”

Section: A Shifted Legendre Tau Methodsmentioning

confidence: 99%

“…In [18,22,23], the authors have presented spectral tau method for numerical solution of some FDEs. Recently, Esmaeili and Shamsi [24] introduced a direct solution technique for obtaining the spectral solution of a special family of fractional initial value problems using a pseudo-spectral method, and Pedas and Tamme [25] developed the spline collocation method for solving FDEs subject to initial conditions. Multi-point boundary value problems appear in wave propagation and in elastic stability.…”

Section: Introductionmentioning

confidence: 99%

A shifted Legendre spectral method for fractional-order multi-point boundary value problems

Bhrawy

Al-Shomrani

2012

Adv Differ Equ

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In this article, a shifted Legendre tau method is introduced to get a direct solution technique for solving multi-order fractional differential equations (FDEs) with constant coefficients subject to multi-point boundary conditions. The fractional derivative is described in the Caputo sense. Also, this article reports a systematic quadrature tau method for numerically solving multi-point boundary value problems of fractional-order with variable coefficients. Here the approximation is based on shifted Legendre polynomials and the quadrature rule is treated on shifted Legendre Gauss-Lobatto points. We also present a Gauss-Lobatto shifted Legendre collocation method for solving nonlinear multi-order FDEs with multi-point boundary conditions. The main characteristic behind this approach is that it reduces such problem to those of solving a system of algebraic equations. Thus we can find directly the spectral solution of the proposed problem. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms.

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“…In general, extrapolation method is applied in order to obtain a high accuracy [10,16]. It is well-known that spectral methods are superior to finite different methods in many instances for partial differential equations [20][21][22][23][24]. In the recent paper [25], the authors presented a spectral method to calculate the fractional derivative and integral, and studied the numerical solution of differential equations by the spectral method.…”

Section: Introductionmentioning

confidence: 99%

Pseudo-Spectral Method for Space Fractional Diffusion Equation

Huang¹,

Zheng²

2013

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This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogonal polynomials or interpolation polynomials. Then, by using pseudo-spectral method, the SFDE is reduced to a system of ordinary differential equations for time variable t. The high order Runge-Kutta scheme can be used to solve the system. So, a high order numerical scheme is derived. Numerical examples illustrate that the results obtained by this method agree well with the analytical solutions.

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A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations

Cited by 127 publications

References 21 publications

Spectral approximations to the fractional integral and derivative

Spectral approximations to the fractional integral and derivative

A shifted Legendre spectral method for fractional-order multi-point boundary value problems

Pseudo-Spectral Method for Space Fractional Diffusion Equation

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