PROPERTIES OF A NEW GENERALIZED CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE

Marc Jornet; Juan J. Nieto

doi:10.11948/20240079

2024 Volume 14 Issue 6

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2024 > 14(6): 3520-3538

Next Article Previous Article

Marc Jornet, Juan J. Nieto. PROPERTIES OF A NEW GENERALIZED CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3520-3538. doi: 10.11948/20240079

Citation:

Marc Jornet, Juan J. Nieto. PROPERTIES OF A NEW GENERALIZED CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3520-3538. doi: 10.11948/20240079

PROPERTIES OF A NEW GENERALIZED CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE

Marc Jornet^1, ,,
Juan J. Nieto^2,

1.
Departament de Matemàtiques, Universitat de València, 46100 Burjassot, Spain
2.
CITMAga, Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

Author Bio: Email: juanjose.nieto.roig@usc.es(J. J. Nieto)
Corresponding author: Email: marc.jornet@uv.es(M. Jornet)
Fund Project: The research of Juan J. Nieto was partially supported by the Agencia Estatal de Investigación (AEI) of Spain Grant PID2020-113275GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe", by the "European Union" and Xunta de Galicia, grant ED431C 2023/12 for Competitive Reference Research Groups (2023–2026)

Abstract

Abstract

We investigate properties of a new fractional derivative recently introduced in the literature, which aims at generalizing the well-known Caputo-Fabrizio operator. We study the null space of the generalized derivative, the associated fractional integral operator, the null space of this integral, the validity of a fundamental theorem of calculus, the equivalence of integral problems with ordinary differential equations, the existence and uniqueness of solution for integral problems, and the form the nonsingular kernel should have to ensure consistency with the fractional order. A complete example with power input function is analyzed, which gives rise to a novel non-elementary solution and new dynamics in terms of the famous Lambert function.
- Fractional calculus /
- fractional derivative /
- fractional integral operator /
- nonsingular kernel
MSC: 34A08, 26A33, 34A05

FullText(HTML)

References

[1]	S. Abbas, M. Benchohra, J. E. Lazreg, J. J. Nieto and Y. Zhou, Fractional Differential Equations and Inclusions, Classical and Advanced Topics, World Scientific, Singapore, 2023. Google Scholar
[2]	T. Alinei-Poiana, E. H. Dulf and L. Kovacs, Fractional calculus in mathematical oncology, Sci. Rep., 2023, 13, Paper No. 10083, 14 pp. Google Scholar
[3]	A. S. Alshomrani, M. Z. Ullah and D. Baleanu, Caputo SIR model for COVID-19 under optimized fractional order, Adv. Differ. Equ., 2021, 2021(1), Paper No. 185, 17 pp. Google Scholar
[4]	I. Area, H. Batarfi, J. Losada, J. J. Nieto, W. Shammakh and Á. Torres, On a fractional order Ebola epidemic model, Adv. Differ. Equ., 2015, 2015(1), 1–12. doi: 10.1186/s13662-014-0331-4 CrossRef Google Scholar
[5]	I. Area and J. J. Nieto, Fractional-order logistic differential equation with Mittag-Leffler-type kernel, Fractal Fract., 2021, 5(4), Paper No. 273, 13 pp. Google Scholar
[6]	I. Area and J. J. Nieto, On a quadratic nonlinear fractional equation, Fractal Fract., 2023, 7(6), Paper No. 469, 8 pp. Google Scholar
[7]	G. Ascione, Y. Mishura and E. Pirozzi, Fractional Deterministic and Stochastic Calculus, volume 4, De Gruyter Series in Probability and Stochastics, Walter de Gruyter, Berlin, 2024. Google Scholar
[8]	I. Baba and F. A. Rihan, A fractional-order model with different strains of COVID-19, Phys. A, 2022, 603, Paper No. 127813, 12 pp. Google Scholar
[9]	M. Bronstein, R. M. Corless, J. H. Davenport and D. J. Jeffrey, Algebraic properties of the Lambert W function from a result of Rosenlicht and of Liouville, Integral Transform. Spec. Funct., 2008, 19, 709–712. doi: 10.1080/10652460802332342 CrossRef Google Scholar
[10]	M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 2015, 1, 73–85. Google Scholar
[11]	J. Čermák, T. Kisela and L. Nechvátal, The Lambert function method in qualitative analysis of fractional delay differential equations, Fract. Calc. Appl. Anal., 2023, 26, 1545–1565. doi: 10.1007/s13540-023-00176-x CrossRef Google Scholar
[12]	I. Chatzigeorgiou, Bounds on the Lambert function and their application to the outage analysis of user cooperation, IEEE Communications Letters, 2013, 17(8), 1505–1508. doi: 10.1109/LCOMM.2013.070113.130972 CrossRef Google Scholar
[13]	R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Computational Math., 1996, 5(1), 329–359. doi: 10.1007/BF02124750 CrossRef Google Scholar
[14]	C. A. Cruz-López and G. Espinosa-Paredes, Fractional radioactive decay law and Bateman equations, Nucl. Eng. Technol., 2022, 54(1), 275–282. doi: 10.1016/j.net.2021.07.026 CrossRef Google Scholar
[15]	Z. Cui, Solutions of some typical nonlinear differential equations with Caputo-Fabrizio fractional derivative, AIMS Math., 2022, 7(8), 14139–14153. doi: 10.3934/math.2022779 CrossRef Google Scholar
[16]	A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. Google Scholar
[17]	M. Jornet, Power-series solutions of fractional-order compartmental models, Comput. Appl. Math., 2024, 43, Paper No. 67, 18 pp. Google Scholar
[18]	H. Joshi, B. K. Jha and M. Yavuz, Modelling and analysis of fractional-order vaccination model for control of COVID-19 outbreak using real data, Math. Bios. Eng., 2023, 20(1), 213–240. Google Scholar
[19]	S. Khajanchi, M. Sardar and J. J. Nieto, Application of non-singular kernel in a tumor model with strong Allee effect, Differ. Equ. Dyn. Syst., 2023, 31(3), 687–692. doi: 10.1007/s12591-022-00622-x CrossRef Google Scholar
[20]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of the Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006. Google Scholar
[21]	J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 2015, 1, 87–92. Google Scholar
[22]	J. Losada and J. J. Nieto, Fractional integral associated to fractional derivatives with nonsingular kernels, Prog. Fract. Differ. Appl., 2021, 7, 137–143. Google Scholar
[23]	F. Mainardi, E. Masina and J. L. González-Santander, A note on the Lambert W function: Bernstein and stieltjes properties for a creep model in linear viscoelasticity, Symmetry, 2023, 15(9), Paper No. 1654, 13 pp. Google Scholar
[24]	B. Melkamu and B. Mebrate, A fractional model for the dynamics of smoking tobacco using Caputo-Fabrizio derivative, J. Appl. Math., 2022, Paper No. 2009910, 10 pp. Google Scholar
[25]	F. Ndaïrou, I. Area, J. J. Nieto, C. J. Silva and D. F. Torres, Fractional model of COVID-19 applied to Galicia, Spain and Portugal, Chaos Soliton. Fract., 2021, 144, Paper No. 110652, 7 pp. Google Scholar
[26]	J. J. Nieto, Fractional Euler numbers and generalized proportional fractional logistic differential equation, Fract. Calc. Appl. Anal., 2022, 25(3), 876–886. doi: 10.1007/s13540-022-00044-0 CrossRef Google Scholar
[27]	J. J. Nieto, Solution of a fractional logistic ordinary differential equation, Appl. Math. Lett., 2022, 123, Paper No. 107568, 5 pp. Google Scholar
[28]	A. Omame, M. Abbas and A. Abdel-Aty, Assessing the impact of SARS-CoV2 infection on the dynamics of dengue and HIV via fractional derivatives, Chaos Soliton. Fract., 2022, 162, Paper No. 112427, 22 pp. Google Scholar
[29]	M. Onitsuka and I. I. EL-Fassi, Generalized Caputo-Fabrizio fractional differential equation, J. Appl. Anal. Comput., 2024, 14(2), 964–975. Google Scholar
[30]	I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 1st edn, Volume 198, Academic Press, New York, 1998. Google Scholar
[31]	D. Prodanov, Computational aspects of the approximate analytic solutions of the SIR model: Applications to modelling of COVID-19 outbreaks, Nonlinear Dyn., 2023, 111, 15613–15631. doi: 10.1007/s11071-023-08656-8 CrossRef Google Scholar
[32]	J. Sabatier, Non-singular kernels for modelling power law type long memory behaviours and beyond, Cybernet. Syst., 2020, 51(4), 383–401. doi: 10.1080/01969722.2020.1758470 CrossRef Google Scholar
[33]	J. Sabatier, Fractional state space description: A particular case of the Volterra equations, Fractal Fract., 2020, 4(2), Paper No. 23, 14 pp. Google Scholar
[34]	V. Tartaglione, C. Farges and J. Sabatier, Fractional behaviours modelling with Volterra equations: Application to a lithium-ion cell and comparison with a fractional model, Fractal Fract., 2022, 6(3), Paper No. 137, 14 pp. Google Scholar
[35]	Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL, 2020. Google Scholar
[36]	B. Yang, Z. Luo, X. Zhang, Q. Tang and J. Liu, Trajectories and singular points of two-dimensional fractional-order autonomous systems, Adv. Math. Phys., 2022, Paper No. 3722011, 9 pp. Google Scholar
[37]	Z. Yong, Basic Theory of Fractional Differential Equations, 3rd edn, World Scientific, Singapore, 2023. Google Scholar