A new conformable fractional derivative and applications
DOI:
https://doi.org/10.17268/sel.mat.2022.02.12Keywords:
Fractional derivatives, Fractional calculusAbstract
The motivation for this paper comes from other papers treating the fractional derivatives. We introduce
a new definition of fractional derivative which obeys classical properties including linearity, product rule,
quotient rule, power rule, chain rule, Rolle’s theorem, mean value theorem and Taylor series. Usage of the
defined derivative is given in the example section which shows how our derivative can be used in solving
differential equations. Comparison of our derivative with the derivative defined by Abdejjawad and overall
conclusions are given in the conclusion section.
References
Abdeljawad T. On Conformable Fractional Calculus. J. of Computational and Applied Mathematics. 2015; 279:57-66.
Afzal W, Abbas M, Macías-Díaz JE, Treantã S. Some H-Godunova–Levin Function Inequalities Using Center Radius (Cr) Order Relation. Fractal Fract. 2022; 6:518.
Afzal W, Alb Lupac A, Shabbir K. Hermite–Hadamard and Jensen-Type Inequalities for Harmonical (h1, h2)-Godunova–Levin Interval-Valued Functions. Mathematics 2022; 10(16):2970.
Afzal W, Shabbir K, Treantâ S, Nonlaopon K. Jensen and Hermite-Hadamard type inclusions for harmonical h-Godunova-Levin functions. AIMS Math. 2023; 8(2):3303–3321.
Caputo M. Linear models of dissipation whose q is almost frequency independent-ii. Geophysical J. of the Royal Astronomical Society. 1967; 13(5):529–539.
Davison M, Essex C. Fractional differential equations and initial value problems. The Mathematical Scientist. 1998; 23(2):108–116.
Grunwald AK. Uber begrenzte derivationen und deren anwendung. Zeitschrift fur Mathematik und Physik. 1867; 12:441–480.
Guzman PM, Langton G, Lugo-Motta L, Medina J, N'apoles-Valdes JE. A new definition of a fractional derivative of local type. J. Math. Anal. 2018; 9:88-96
Jumarie G. An approach to differential geometry of fractional order via modified Riemann-Liouville derivative. Acta Mathematica Sinica. 2012; 28(9):1741–1768.
Jumarie G. On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling. Central European J. of Physics. 2013; 11(6):617–633.
Jumarie G. On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion. App. Math. Lett. 2005; 18(7):817–826.
Khalil R, Al-Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. J. of Comp. and App. Math. 2014; 264:65-70.
Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006.
Kommum P, Ali A, Shah K, Ali Khan R. Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order diferential equations. J. Nonlinear Sci. Appl. 2017; 10:2986-2997
Letnikov AV. Theory of differentiation with an arbitrary index. Sbornik Mathematics(Russian), 1868; 3:1–66.
Andrei L. Some differential subordinations using Ruscheweyh derivative and Salagean operator. Advances in Difference Equations, 2013, no 252.
Machado JT, Kiryakova V, Mainardi F. Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation. 2011; 16(3):1140–1153
Meerschaert MM, Mortensen J, Wheatcraft SW. Fractional vector calculus for fractional advection-dispersion. Physica A: Statistical Mechanics and Its Applications. 2006. 367(8):181–190.
Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: John Wiley 'I&' Sons; 1993.
Monje CA, Chen Y, Vinagre BM, Xue D, Feliu V. Fractional-Order Systems and Controls: Fundamentals and applications. London: Springer; 2010.
Napoles-Valdes JE, Quevedo MN. The derivative notion revised: The fractional case. The Mathematics Enthusiast. 2019; 16(1): 18.
Ortiguera MD, Tenreiro JA. What is a fractional derivative?, J. of Computational Physics. 2015; 293(15):4-13.
Parraga P, Vivas-Cortez M, Larreal O. Conformable fractional derivatives and applications to Newtonian dynamic and cooling body law. Selecciones Matematicas. 2022; 9(1):34–43. http://dx.doi.org/10.17268/sel.mat.2022.01.03
Riesz M. L’integrale de Riemann-Liouville et le probleme de Cauchy. Acta Mathematica. 1949; 81(1):1–222.
Riesz M. L’integrale de Riemann-Liouville et le probleme de Cauchy pour l’equation des ondes. Bulletin de la Societe Mathematique de Francè. 1939; 67:153–170.
Stojiljković V, Ramaswamy R, Ashour Abdelnaby OA, Radenović S. Riemann-Liouville Fractional Inclusions for Convex Functions Using Interval Valued Setting. Mathematics 2022; 10:3491. https://doi.org/10.3390/math10193491
Stojiljković V, Ramaswamy R, Alshammari F, Ashour OA, Alghazwani MLH, Radenović S. Hermite–Hadamard Type Inequalities Involving (k-p) Fractional Operator for Various Types of Convex Functions. Fractal Fract. 2022; 6:376. https://doi.org/10.3390/fractalfract6070376
Stojiljković V, Ramaswamy R, Abdelnaby OAA, Radenović S. Some Novel Inequalities for LR-(k,h-m)-p Convex Interval Valued Functions by Means of Pseudo Order Relation. Fractal Fract. 2022; 6:726. https://doi.org/10.3390/fractalfract6120726
Weyl H. Bemerkungen zum begriff des differentialquotienten gebrochener ordung vierteljahresschr. Naturforschende Gesellschaft in Zurich. 1917; 62:296–302.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Selecciones Matemáticas
This work is licensed under a Creative Commons Attribution 4.0 International License.
The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.
