The Legendre wavelet method for solving fractional differential equations

Rehman, Mujeeb ur; Khan, Rahmat Ali

doi:10.1016/j.cnsns.2011.01.014

Cited by 200 publications

(95 citation statements)

References 32 publications

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“…The existence and uniqueness of solution of the initial value problem (13)- (14) have been studied in [10] in noise-free case. We are going to numerically solve this problem by using the fractional Jacobi differentiator and the modulation functions method in the noisy case.…”

Section: Fractional Linear Systemsmentioning

confidence: 99%

“…Then, different methods were considered to solve this problem by solving a linear system of algebraic equations. These methods include operational matrix method [14], collocation method [15], and spectral tau method [13]. Unlike the spectral tau method used in [13], the operational matrix method in general uses polynomials to approximate the solutions of fractional differential equations, and the fractional derivatives or the fractional integrals of these solutions.…”

Section: Introductionmentioning

confidence: 99%

“…Solutions were approximated by truncated series expansions involving different bases, such as orthogonal polynomials [13] and wavelets [14,15], with unknown coefficients. Thus, the problem of solving a fractional differential equation was transformed into a problem of identifying the coefficients of an equation.…”

Section: Introductionmentioning

confidence: 99%

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Fractional order differentiation by integration: an application to fractional linear systems

Liu

Laleg‐Kirati

Gibaru

2013

IFAC Proceedings Volumes

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In this article, we propose a robust method to compute the output of a fractional linear system defined through a linear fractional differential equation (FDE) with timevarying coefficients, where the input can be noisy. We firstly introduce an estimator of the fractional derivative of an unknown signal, which is defined by an integral formula obtained by calculating the fractional derivative of a truncated Jacobi polynomial series expansion. We then approximate the FDE by applying to each fractional derivative this formal algebraic integral estimator. Consequently, the fractional derivatives of the solution are applied on the used Jacobi polynomials and then we need to identify the unknown coefficients of the truncated series expansion of the solution. Modulating functions method is used to estimate these coefficients by solving a linear system issued from the approximated FDE and some initial conditions. A numerical result is given to confirm the reliability of the proposed method.

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Section: Fractional Linear Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fractional order differentiation by integration: an application to fractional linear systems

Liu

Laleg‐Kirati

Gibaru

2013

IFAC Proceedings Volumes

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“…Spectral methods have been proposed to solve fractional differential equations, such as the Legendre collocation method [20,36], Legendre wavelets method [32,34], homotopy perturbation method [40] and Jacobi-Gauss-Lobatto collocation method [4]. The authors in [12,13,39] constructed an efficient spectral method for the numerical approximation of fractional integro-differential equations based on tau and pseudo-spectral methods.…”

Section: Introductionmentioning

confidence: 99%

Spectral-Collocation Method for Fractional Fredholm Integro-Differential Equations

Yang¹,

Chen²,

Huang³

2014

Journal of the Korean Mathematical Society

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Abstract. We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of FredholmVolterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L ∞ norm and weighted L 2 -norm. The numerical examples are given to illustrate the theoretical results.

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“…Recently the operational matrices of fractional order integration for the Haar wavelets, the Chebyshev wavelets and the Legendre wavelet have been developed in [14], [17], [18] and [19] to solve the fractional order differential equations.…”

Section: Introductionmentioning

confidence: 99%

Chebyshev Wavelets Method for Solving Partial Differential Equations of Fractional Order

Mohammed

Ameen

2017

JNUS

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In this paper, we use the second kind Chebyshev wavelet operational matrix of fractional integration for solving fractional order linear partial differential equations.By using this method the fractional order partial differential equation is translated into Lyapunov type matrix equation and the computation effort become convenient. Two illustrative examples are provided to demonstrate the validity, simplicity and applicability of the numerical scheme based on the Chebyshev set of functions.

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The Legendre wavelet method for solving fractional differential equations

Cited by 200 publications

References 32 publications

Fractional order differentiation by integration: an application to fractional linear systems

Fractional order differentiation by integration: an application to fractional linear systems

Spectral-Collocation Method for Fractional Fredholm Integro-Differential Equations

Chebyshev Wavelets Method for Solving Partial Differential Equations of Fractional Order

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