The purpose of this note is to present the different fractional order derivatives definition that are commonly used in the literature on one hand and to present a table of fractional order derivatives of some functions in Riemann-Liouville sense On other the hand. We present some advantages and disadvantages of these fractional derivatives. And finally we propose alternative fractional derivative definition.
Fractional calculus has been used to model physical and engineering processes, which are found to be best described by fractional differential equations. It is worth nothing that the standard mathematical models of integer-order derivatives, including nonlinear models, do not work adequately in many cases. In the recent years, fractional calculus has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, notably control theory, and signal and image processing. Major topics include anomalous diffusion, vibration and control, continuous time random walk, Levy statistics, fractional Brownian motion, fractional neutron point kinetic model, power law, Riesz potential, fractional derivative and fractals, computational fractional derivative equations, nonlocal phenomena, history-dependent process, porous media, fractional filters, biomedical engineering, fractional phase-locked loops, fractional variational principles, fractional transforms, fractional wavelet, fractional predator-prey system, soft matter mechanics, fractional signal and image processing; singularities analysis and integral representations for fractional differential systems; special functions related to fractional calculus, non-Fourier heat conduction, acoustic dissipation, geophysics, relaxation, creep, viscoelasticity, rheology, fluid dynamics, chaos and groundwater problems. An excellent literature of this can be found in [
There exists a vast literature on different definitions of fractional derivatives. The most popular ones are the Riemann-Liouville and the Caputo derivatives. For Caputo we have
In this section we present the fractional of some special functions. The fractional derivatives in Table
Fractional order derivatives for some functions.
Functions | L-fractional derivatives |
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Cos |
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Sin |
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ln |
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Sinh |
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Cosh |
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Arcsin |
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Arccos |
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Arctan |
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Zeta[ |
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Erf |
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In Table
It is very important to point out that all these fractional derivative order definitions have their advantages and disadvantages; here we will include Caputo, variational order, Riemann-Liouville Jumarie and Weyl. We will examine first the Variational order differential operator. Anomalous diffusion phenomena are extensively observed in physics, chemistry, and biology fields [
Although these fractional derivative display great advantages, they are not applicable in all the situations. We shall begin with the Liouville-Riemann type. The Riemann-Liouville derivative has certain disadvantages when trying to model real-world phenomena with fractional differential equations. The Riemann-Liouville derivative of a constant is not zero. In addition, if an arbitrary function is a constant at the origin, its fractional derivation has a singularity at the origin for instant exponential and Mittag-Leffler functions. Theses disadvantages reduce the field of application of the Riemann-Liouville fractional derivative. Caputo’s derivative demands higher conditions of regularity for differentiability: to compute the fractional derivative of a function in the Caputo sense, we must first calculate its derivative. Caputo derivatives are defined only for differentiable functions while functions that have no first-order derivative might have fractional derivatives of all orders less than one in the Riemann-Liouville sense. With the Jumarie fractional derivative, if the function is not continuous at the origin, the fractional derivative will not exist, for instance what will be the fractional derivative of
Let
Let
Recently, Guy Jumarie proposed a simple alternative definition to the Riemann-Liouville derivative. His modified Riemann-Liouville derivative has the advantage of both standard Riemann-Liouville and Caputo fractional derivatives: it is defined for arbitrary continuous (nondifferentiable) functions and the fractional derivative of a constant is equal to zero. However if the function is not defined at the origin, the fractional derivative will not exist, therefore in order to circumvent this defeat we propose the following definition. Let
The above fractional order derivative definition can be used in many field for instance in the field of groundwater. Because this definition does not produce a fractional derivative with any kind of singularity as in the case of Jumarie and the traditional Riemann-Liouville fractional order derivative. This concept was introduced by Hadamard [
We presented the definitions of the commonly used fractional derivatives operators which are ranging from Riemann-Liouville to Guy Jumarie. We presented the disadvantages and advantages of each definition. No definition has fulfilled the entire requirement needed; for example, the Jumarie definition fulfills some interesting requirements including the derivative of a constant is zero, and a nondifferentiable function may have a fractional derivative. However, if the function is not defined at the origin, it may not have a fractional derivative in Jumarie sense. With the Riemann-Liouville fractional derivative, the function needs not to be continuous at the origin and needs not to be differentiable; however, the derivative of a constant is not zero; in addition, his has certain disadvantages when trying to model real-world phenomena with fractional differential equations. Also if an arbitrary function is a zero at the origin, its fractional derivation has a singularity at the origin, for instance exponential and Mittag-Leffler functions. Theses disadvantages reduce the field of application of the Riemann-Liouville fractional derivative. The Caputo derivative is very useful when dealing with real-world problem because, it allows traditional initial and boundary conditions to be included in the formulation of the problem and in addition the derivative of a constant is zero; however, functions that are not differentiable do not have fractional derivative, which reduces the field of application of Caputo derivative. It is in addition important to notice that, to characterize anomalous diffusion phenomena, constant-order fractional diffusion equations have been introduced and have received tremendous success. However, it has been found that the constant order fractional diffusion equations are not capable of characterizing some complex diffusion processes. To solve the above problems, the variable-order (VO) fractional diffusion equation models have been suggested for use; however, the calculations involved in these definitions are very difficult to handle analytically; therefore, numerical attentions are needed for these cases. To solve the problem found in Jumarie definition, we proposed an alternative fractional derivative and we extended the definition to the case of variational differential operator. We provided a table of Liouville fractional derivative of some special functions. Now we can conclude here by observing, that all fractional derivatives examined here are all useful, and they have to be used according to the support of the function.
The authors declare that they have no conflict of interests.
Abdon Atangana wrote the first draft and Aydin Secer corrected final version. All authors read and approved the final draft
The authors would like to thank the referee for some valuable comments and helpful suggestions. Special thanks go to the editor for his valuable time spent to evaluate this paper.