Nonstandard Gauss—Lobatto quadrature approximation to fractional derivatives

Esmaeili, Shahrokh; Milovanović, Gradimir V.

doi:10.2478/s13540-014-0215-z

Cited by 8 publications

(3 citation statements)

References 44 publications

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“…If α = 1, these fractional derivatives coincide with the classical integer‐order derivatives . It is well known that the fractional derivative of Riemann–Liouville and Caputo type are closely linked by the following relationship:

_{0} D_{t}^{α} f (t) = \frac{f (0)}{t^{α} Γ (1 - α)} +_{0}^{C} D_{t}^{α} f (t) .

In order to approximate fractional derivatives, a number of methods have been proposed (e.g., ).…”

Section: Preliminary Considerationsmentioning

confidence: 99%

“…In order to approximate fractional derivatives, a number of methods have been proposed (e.g., [27,29,30]). Unlike integer order derivatives, which are local operators, the singular nature of the kernel in the definition (1) or (2) of non-integer order derivatives makes the problem global; indeed, the value of On the other hand, the presence of a significantly increasing persistent memory, with respect to the integer order case, adds to the complexity of the numerical treatment of related differential problems, especially for long-time integration.…”

Section: Fractional Calculusmentioning

confidence: 99%

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Solving 2D time‐fractional diffusion equations by a pseudospectral method and Mittag‐Leffler function evaluation

Esmaeili

2016

Math Methods in App Sciences

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Two‐dimensional time‐fractional diffusion equations with given initial condition and homogeneous Dirichlet boundary conditions in a bounded domain are considered. A semidiscrete approximation scheme based on the pseudospectral method to the time‐fractional diffusion equation leads to a system of ordinary fractional differential equations. To preserve the high accuracy of the spectral approximation, an approach based on the evaluation of the Mittag‐Leffler function on matrix arguments is used for the integration along the time variable. Some examples along with numerical experiments illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

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_{0} D_{t}^{α} f (t) = \frac{f (0)}{t^{α} Γ (1 - α)} +_{0}^{C} D_{t}^{α} f (t) .

In order to approximate fractional derivatives, a number of methods have been proposed (e.g., ).…”

Section: Preliminary Considerationsmentioning

confidence: 99%

Section: Fractional Calculusmentioning

confidence: 99%

Solving 2D time‐fractional diffusion equations by a pseudospectral method and Mittag‐Leffler function evaluation

Esmaeili

2016

Math Methods in App Sciences

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“…In order to approximate fractional derivatives, a number of methods have been proposed (cf., e.g., [6,31]). …”

Section: Fractional Derivativesmentioning

confidence: 99%

The Numerical Solution of the Bagley-Torvik Equation by Exponential Integrators

Esmaeili

2017

Scientia Iranica

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Abstract. This paper presents a family of computational schemes for the solution to the Bagley-Torvik equation. The schemes are based on the reformulation of the original problem into a system of fractional di erential equations of order 1/2. Then, suitable exponential integrators are devised to solve the resulting system accurately. The attainable order of convergence of exponential integrators for solving the fractional problem is studied. Theoretical ndings are validated by means of some numerical examples. The advantages of the proposed method are illustrated by comparing several existing methods.

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On fractional-Legendre spectral Galerkin method for fractional Sturm–Liouville problems

Al‐Mdallal

2018

Chaos, Solitons & Fractals

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Nonstandard Gauss—Lobatto quadrature approximation to fractional derivatives

Abstract: A family of nonstandard Gauss-Jacobi-Lobatto quadratures for numerical calculating integrals of the form

Cited by 8 publications

References 44 publications

Solving 2D time‐fractional diffusion equations by a pseudospectral method and Mittag‐Leffler function evaluation

Solving 2D time‐fractional diffusion equations by a pseudospectral method and Mittag‐Leffler function evaluation

The Numerical Solution of the Bagley-Torvik Equation by Exponential Integrators

On fractional-Legendre spectral Galerkin method for fractional Sturm–Liouville problems

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