Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations

Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S.

doi:10.1016/j.apm.2011.05.011

Cited by 228 publications

(147 citation statements)

References 22 publications

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“…The similar work can be found in [21], where the operational matrix D (α) based on the explicit form of the Legendre polynomials was obtained, which takes O(N 3 ) arithmetic operations. The similar matrix D (α) based on the Chebyshev polynomials can be found in [5,6], where the computational complexity is O(N 3 ).…”

Section: Approximation To the Caputo Derivativementioning

confidence: 99%

Spectral approximations to the fractional integral and derivative

Zeng

Liu

2012

fcaa

153

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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: Approximation To the Caputo Derivativementioning

confidence: 99%

Spectral approximations to the fractional integral and derivative

Zeng

Liu

2012

fcaa

153

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“…We apply the two methods, namely, TFMM, CFMM for different values of α and n. In Table 1, we display a comparison between the results obtained if the two methods TFMM, CFMM are applied with the results obtained by applying Chebyshev spectral method (CSM) which developed in [38]. The displayed results in this table show that our algorithm gives a lesser error in almost all cases.…”

Section: Example 2 (Doha Et Al [38]) Consider the Following Inhomogmentioning

confidence: 93%

A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations

Abd-Elhameed

Youssri

2016

Entropy

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Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. The convergence and error analysis of the suggested Fibonacci expansion were carefully investigated. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. Two accurate semi-analytic polynomial solutions for linear and nonlinear fractional differential equations are the result.

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“…Many researchers have used those methods for the numerical solution of nonlinear PDEs [25], fractional ODEs [26], high-order boundary value problems [27], systems of Volterra integral equations [28], optimal control problems governed by Volterra integral equations [29], Quasi Bang-Bang optimal control problems [30], and ODEs of degenerate types [31]. In relation to many other methods, spectral methods give highly accurate results.…”

Section: The Chebyshev Pseudo-spectral Methodsmentioning

confidence: 99%

A parameterized multi-step Newton method for solving systems of nonlinear equations

Ahmad

Tohidi²,

Carrasco³

2015

Numer Algor

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We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter θ to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of θ, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and, therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spacial and temporal dimensions such as, for instance, the Chebyshev pseudospectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radii of convergence than the multi-step Newton method.Keywords: Multi-step iterative methods; Multi-step Newton method; systems of nonlinear equations; partial differential equations; discretization methods for partial differential equations.

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Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations

Cited by 228 publications

References 22 publications

Spectral approximations to the fractional integral and derivative

Spectral approximations to the fractional integral and derivative

A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations

A parameterized multi-step Newton method for solving systems of nonlinear equations

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