Abstract
In this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.
1 Introduction
Fractional calculus is a generalization of ordinary calculus, where derivatives and integrals of arbitrary real or complex order are defined. These fractional operators may model more efficiently certain real world phenomena, especially when the dynamics is affected by constraints inherent to the system. There exist several definitions for fractional derivatives and fractional integrals, like the Riemann-Liouville, Caputo, Hadamard, Riesz, Griinwald-Letnikov, Marchaud, etc. (see e.g., [14, 17] and references therein). Although most of them are already well-studied, some of the usual features concerning the differentiation of functions fail, like the Leibniz rule, the chain rule, the semigroup property, to name a few. As it was mentioned in [6], “These definitions, however, are non-local in nature, which makes them unsuitable for investigating properties related to local scaling or fractional differentiability”. Recently, the concept of local fractional derivatives have gained relevance, namely because they kept some of the properties of ordinary derivatives, although they loss the memory condition inherent to the usual fractional derivatives. For example, in [9], a concept similar to the Caputo fractional derivative is presented, but the first-order derivative ƒ′(t) is replaced by another operator: limϵ→0(f(x + ϵt) — f(x))/ϵα; in [12], the local fractional derivative is given by f(α)(t) = limϵ→0(ƒ(t exp(ϵt–α) — f(t)))/ϵ, and some fundamental properties like the algebraic rules or the mean value theorem are obtained; later, in [3], the same concept of local fractional derivative is considered and an anti-derivative operator is defined, as well some applications to quantum mechanics; in [15, 16], the local fractional derivative is defined by the expression f(α)(t) = limx→t (Dα (f(x)—ƒ(t))), where Dα denotes the Riemann-Liouville fractional derivative, and it is proven that the Weierstrass Function is a-differentiable. One question is what is the best local fractional derivative definition that we should consider, and the answer is not unique. Similarly to what happens to the classical definitions of fractional operators, the best choice depends on the experimental data that fits better in the theoretical model, and because of this we find already a vast number of definitions for local fractional derivatives.
2 Local fractional derivative
We present a definition of local fractional derivative using kernels.
Let k : [a, b] → ℝ be acontinuous nonnegative map such that k(t) ≠ 0, whenevert > a. Given a function ƒ : [a, b] → ℝ and α ∈ (0, 1) a real, we say that ƒ is α-differentiable at t > a, with respect to kernel k, if the limit
exists. The α-derivative at t = a is defined by
if the limit exists.
Consider the limit α → 1–. In this case, for t > a, we obtain the classical definition for derivative of a function, f(α)(t) = ƒ′(t). Our definition is a more general concept, compared to others that we find in the literature. For example, taking k(t) = t and a = 0, we get the definition from [7, 8, 10, 11, 13] (also called conformable fractional derivative); when k(t) = t — a, the one from [1, 2, 18]; for k(t) = t + 1/Γ(α), the definition in [4, 5].
The following result is trivial, and we omit the proof.
Let ƒ : [a, b] → ℝ be a differentiablefunction and t > a. Then, ƒ is a-differentiable at t and
Also, if f′ is continuous at t = a, then
However, there exist α-differentiable functions which are not differentiable in the usual sense. For example, consider the function
If f(α)(t) exists for t > a, then ƒ is differentiable at t and
It follows from
Of course we can not conclude anything at the initial point t = a, as was discussed before.
Combining the two previous results, we have the main result of our paper.
A function ƒ : [a,b] → ℝ is α-differentiable at t > a if and only if it is differentiable at t. In that case, we have the relation
3 Conclusion
In this short note we show that some of the existent notions about local fractional derivative are very close related to the usual derivative function. In fact, the α-derivative of a function is equal to the first-order derivative, multiplied by a continuous function. Also, using formula (2), most of the results concerning α-differentiation can be deduced trivially from the ordinary ones. In the authors opinion, local fractional calculus is an interesting idea and deserves further research, but definitions like (1) are not the best ones and a different path should be followed.
Acknowledgement
Research supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2013 (R. Almeida) and by the Warsaw School of Economics grant KAE/S16/03/16 (T. Odzijewicz). The authors are very grateful to the two anonymous Reviewers for their comments that improved the manuscript.
References
[1] Abdeljawad T., On conformable fractional calulus, preprint.Search in Google Scholar
[2] Anderson D.R., Avery R.I., Fractional-order boundary value problem with Sturm-Liouville boundary conditions, Electron. J. Differ. Equ., Volume 2015, 29, 1-10, 2015.10.1186/s13662-015-0413-ySearch in Google Scholar
[3] Anderson D.R., Ulness D.J., Properties of the Katugampola fractional derivative with potential application in quantum mechanics, J. Math. Phys 56, 063502, 2015.10.1063/1.4922018Search in Google Scholar
[4] Atangana A., Goufo E.F.D., Extension of Matched Asymptotic Method to Fractional Boundary Layers Problems, Mathematical Problems in Engineering, Volume 2014, 107535, 7 pages.10.1155/2014/107535Search in Google Scholar
[5] Atangana A., Noutchie S.C.O., Model of Break-Bone Fever via Beta-Derivatives, BioMed Research International, Volume 2014, 523159, 10 pages.10.1155/2014/523159Search in Google Scholar
[6] Babakhani A., Daftardar-Gejji V, On calculus of local fractional derivatives, J. Math. Anal. Appl. 270 (1), 66-79, 2002.10.1016/S0022-247X(02)00048-3Search in Google Scholar
[7] Batarfi H., Losada J., Nieto J.J., Shammakh W., Three-point boundary value problems for conformable fractional differential equations, Journal of function spaces, Volume 2015, 706383, 6 pages.10.1155/2015/706383Search in Google Scholar
[8] Çenesiz Y, Kurt A., The solution of time fractional heat equation with new fractional derivative definition, in Recent Advances in Applied Mathematics, Modelling and Simulation (eds N.E. Mastorakis, M. Demiralp, N. Mukhopadhyay and F. Mainardi) North Atlantic University Union, 195-198, 2014.Search in Google Scholar
[9] Chen Y, Yan Y, Zhang K., On the local fractional derivative, J. Math. Anal. Appl. 362 (1), 17-33, 2010.10.1016/j.jmaa.2009.08.014Search in Google Scholar
[10] Abu Hammad M., Khalil R., Legendre fractional differential equation and Legender fractional polynomials, Int. J. Appl. Math. Res. 3(3), 214-219, 2014.10.14419/ijamr.v3i3.2747Search in Google Scholar
[11] Hesameddini E., Asadollahifard E., Numerical solution of multi-order fractional differential equations via the sinc collocation method, Iran. J. Numer. Anal. Optim. 5 (1), 37-48, 2015.Search in Google Scholar
[12] Katumgapola U., A new fractional derivative with classical properties, preprint.Search in Google Scholar
[13] Khalil R., Al Horani M., Yousef A., Sababheh M., A new definition of fractional derivative, J. Comput. Appl. Math. 264. 65-70, 2014.10.1016/j.cam.2014.01.002Search in Google Scholar
[14] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.Search in Google Scholar
[15] Kolwankar K.M., Gangal A.D., Fractional differentiability of nowhere differentiable functions and dimension, Chaos 6, 505-513, 1996.10.1063/1.166197Search in Google Scholar PubMed
[16] Kolwankar K.M., Gangal A.D., Hölder exponents of irregular signals and local fractional derivatives, Pramana J. Phys. 48, 49-68, 1997.10.1007/BF02845622Search in Google Scholar
[17] Podlubny I., Fractional differential equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.Search in Google Scholar
[18] Ünal E., Gǒkdogan A., Çelik E., Solutions around a regular α singular point of a sequential conformable fractional differential equation, preprint.Search in Google Scholar
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