Remarks on fractional derivatives

By Changpin Li, Weihua Deng. Applied Mathematics and Computation, 2007

754 citations recorded by the source corpus.

In this paper, we further discuss the properties of three kinds of fractional derivatives: the Grunwald-Letnikov derivative, the Riemann-Liouville derivative and the Caputo derivative. Especially, we compare the Riemann-Liouville derivative with the Caputo derivative. And sequential property of the Caputo derivative is also derived, which is helpful in translating the higher fractional-order differential systems into lower ones. Besides, we also compare the Riemann-Liouville derivative and the Caputo derivative with the classical derivative.

Subfields: Geometry and Topology, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/aa9d1a481898854d00c573c986598f2ba376dd1c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. N. Kosmatov
2. H. Zhan
3. Quanzhong Huang
4. Can Li
5. Lijing Zhao
6. Dumitru Băleanu
7. Weihua Deng

In Modeling and Simulation

1. N. Kosmatov
2. Lijing Zhao
3. H. Zhan
4. Quanzhong Huang
5. Can Li
6. Dumitru Băleanu
7. G. Diaz
8. Yier Yan

In Statistical and Nonlinear Physics

1. N. Kosmatov
2. Dumitru Băleanu
3. H. Zhan
4. Quanzhong Huang
5. Can Li
6. Lijing Zhao
7. G. Diaz
8. Carlos F. M. Coimbra

Citations in the corpus

• Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation (2009, background): “It is well known that there are different ways of defining a fractional differential operator [22, 35], and this means that there are even more options for defining a variable order differential operator (for example, placing the Gamma function term inside or outside of the integral sign in the equa”
• Well-posedness and numerical algorithm for the tempered fractional differential equations (2019, methodology): “Furthermore, using the composite properties of Riemann-Liouville fractional integral and derivative [36, 26, 38] aD α t [ aI α t (v(t)) ] = v(t), (17) we get aD α,λ t [aI α,λ t u(t)] = e −λt aD α t [ aI α t (e u(t)) ] = u(t), by taking v(t) = eu(t) in Eq.”
• Smoothness and stability of the solutions for nonlinear fractional differential equations (2010, background): “, aD q t (q ∈ (−∞, +∞)) bridges all the gaps among the integer derivatives and the integer integrals, but the Caputo derivative cannot do this [18].”
• Existence of solutions of initial value problems for nonlinear fractional differential equations on (2011, background): “[16], Ouahab [17], El-Sayed [18–20], Nieto [21], Li and Deng [22], Kosmatov [23], Muslim [24], Stojanović [25], Delbosco and Rodino [26] and Babakhani and Gejji [27].”
• A finite element solution for the fractional advection–dispersion equation (2008, methodology): “(2) and (3) can be verified by the method of integration by parts [37,21].”
• Stability analysis of Caputo fractional-order nonlinear systems revisited (2011, background): “The necessary and sufficient stability conditions for linear fractional differential equations (FDEs) and linear time-delayed FDEs have already been obtained in [11–13].”
• Local discontinuous Galerkin methods for fractional diffusion equations (2013, neutral): “Situations for which is true are understood through [15], aD α xv(x) = C a D α xv(x) if v (a) = 0, k = 0, 1, .”