On Riemann‐Liouville and Caputo Derivatives

Li, Changpin; Qian, Deliang; Chen, YangQuan

doi:10.1155/2011/562494

Cited by 241 publications

(141 citation statements)

References 23 publications

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“…For a given function, its classical Riemann-Liouville derivative or Caputo derivative ( [15], [16] and [10]) may not exist in general ( [17], [18] and [19]). Even if they do, the Riemann-Liouville derivative and the Caputo derivative are not necessarily the same.…”

Section: Distributions In D (R + )mentioning

confidence: 99%

The Abel Integral Equations in Distribution

Li¹,

Li²,

Kacsmar³

et al. 2017

AAN

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Section: Distributions In D (R + )mentioning

confidence: 99%

The Abel Integral Equations in Distribution

Li¹,

Li²,

Kacsmar³

et al. 2017

AAN

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“…, n − 1 < q < n then the Grunwald-Letnikov fractional derivative is given by (see [11], Definition 1.3)…”

Section: (T − S)mentioning

confidence: 99%

Stability of Caputo fractional differential equations by Lyapunov functions

2015

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“…Unlike classical order derivative, fractional derivative has several kinds of definitions such as Caputo, Riemann-Liouville, Grünwald-Letnikov and Riesz. Detailed discussion about various definitions of fractional derivative can be found in [5] [6].…”

Section: Introductionmentioning

confidence: 99%

Lagrange’s Spectral Collocation Method for Numerical Approximations of Two-Dimensional Space Fractional Diffusion Equation

Molla¹,

Nova²

2018

AJCM

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Due to the ability to model various complex phenomena where classical calculus failed, fractional calculus is getting enormous attention recently. There are several approaches available for numerical approximations of various types of fractional differential equations. For fractional diffusion equations spectral collocation is one of the efficient and most popular approximation techniques. In this research, we introduce spectral collocation method based on Lagrange's basis polynomials for numerical approximations of two-dimensional (2D) space fractional diffusion equations where spatial fractional derivative is described in Riemann-Liouville sense. We consider four different types of nodes to generate Lagrange's basis polynomials and as collocation points in the proposed spectral collocation technique. Spectral collocation method converts the diffusion equation into a system of ordinary differential equations (ODE) for time variable and we use 4 th order Runge-Kutta method to solve the resulting system of ODE. Two examples are considered to verify the efficiency of different types of nodes in the proposed method. We compare approximated solution with exact solution and find that Lagrange's spectral collocation method gives very high accuracy approximation. Among the four types of nodes, nodes from Jacobi polynomial give highest accuracy and nodes from Chebyshev polynomials of 1 st kind give lowest accuracy in the proposed method.

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On Riemann‐Liouville and Caputo Derivatives

Cited by 241 publications

References 23 publications

The Abel Integral Equations in Distribution

The Abel Integral Equations in Distribution

Stability of Caputo fractional differential equations by Lyapunov functions

Lagrange’s Spectral Collocation Method for Numerical Approximations of Two-Dimensional Space Fractional Diffusion Equation

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