A new non-conformable derivative based on Tsallis’s q- exponential function
DOI:
https://doi.org/10.22481/intermaths.v2i2.10101Palavras-chave:
Cálculo Fracionário, Derivada não compatível, Função q-exponencialResumo
Neste artigo, uma nova derivada do tipo local é proposta e algumas propriedades básicas são estudadas. Esta nova derivada satisfaz algumas propriedades do cálculo de ordem inteira, por exemplo linearidade, regra do produto, regra do quociente e a regra da cadeia. Devido à função exponencial generalizada de Tsallis, podemos estender alguns dos resultados clássicos, a saber: teorema de Rolle, teorema do valor médio. Apresentamos a correspondente Q-integral a partir da qual surgem novos resultados. Especificamente, generalizamos a propriedade de inversão do teorema fundamental do cálculo e provamos um teorema associado à integração clássica por partes. Finalmente, apresentamos uma aplicação envolvendo equações diferenciais lineares por meio da Q-derivada.
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B. Ross, “A Brief History and Exposition of the Fundamental Theory of Fractional Calculus”, In Fractional Calculus and Its Applications; Lecture Notes in Mathematics No. 457; Ross, B., Ed.; Springer: Heidelberg, Germany, 1975. https://doi.org/10.1007/BFb0067096
E. C. de Oliveira, J. A. Tenreiro Machado, “A review of definitions for fractional derivatives and integral”, Mathematical Problems in Engineering, 2014. https://doi.org/10.1155/2014/238459
T. Abdeljawad, “The flaw in the conformable calculus: It is conformable because it is not fractional”, Fractional Calculus and Applied Analysis, 22(2), 242–254, 2019. https://doi.org/10.1515/fca-2019-0016
G. S. Teodoro, J. A. T. Machado, E. C. De Oliveira, “A review of definitions of fractional derivatives and other operators”, Journal of Computational Physics, 388, 195–208, 2019. https://doi.org/10.1016/j.jcp.2019.03.008
V. E. Tarasov, “No nonlocality. No fractional derivative”, Communications in Nonlinear Science and Numerical Simulation, 62, 157–163, 2018. https://doi.org/10.1016/j.cnsns.2018.02.019
R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, “A new definition of fractional derivative” Journal of Computational and Applied Mathematics, 264, 65–70, 2014. https://doi.org/10.1016/j.cam.2014.01.002
A. Atangana, D. Baleanu, A. Alsaedi, “Analysis of time-fractional Hunter-Saxton equation: a model of neumatic liquid crystal”, Open Physics, 14(1), 145–149, 2016. https://doi.org/10.1515/phys-2016-0010
J. V. D. C. Sousa, E. C. De Oliveira, On the local M-derivative, Progress in Fractional Differentiation and Applications, 4(4), 479–492, 2018. e. arXiv preprint arXiv:1704.08186
A. A. Abdelhakim, J. A. T. Machado, “A critical analysis of the conformable derivative”, Nonlinear Dynamics, 95(4), 3063–3073, 2019. https://doi.org/10.1007/s11071-018-04741-5
P. M. Guzman, G. Langton, L. M. Lugo, J. Medina, J. N. Valdes, “A new definition of a fractional derivative of local type”, Journal of Mathematical Analysis. Ilirias Research Istitute, 9(2), 88–98, 2018.
T. Abdeljawad, “On conformable fractional calculus”, Journal of computational and Applied Mathematics, 279, 57–66, 2015. https://doi.org/10.1016/j.cam.2014.10.016
F. S. Silva, D. M. Moreira, M. A. Moret, “Conformable Laplace transform of fractional differential equations”, Axioms, 7(3):55, 2018. https://doi.org/10.3390/axioms7030055
J. J. Rosales, F. A. Godinez, V. Banda, G. H. Valencia, “Analysis of the Drude model in view of the conformable derivative”, Optik, 178, 1010–1015, 2019. https://doi.org/10.1016/j.ijleo.2018.10.079
B. Yan, S. He, “Dynamics and complexity analysis of the conformable fractional-order two-machine interconnected power system”, Mathematical Methods in the Applied Sciences, 44(3), 2439–2454. 2019. https://doi.org/10.1002/mma.5937
R. Shah, H. Khan, M. Arif, P. Kumam, “Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations”, Entropy. 21(4):335, 2019. https://doi.org/10.3390/e21040335
S. Yang, L. Wang, S. Zhang, “Conformable derivative: application to non-Darcian flow in low-permeability porous media”, Applied Mathematics Letters, 79, 105–110, 2018. https://doi.org/10.1016/j.aml.2017.12.006
H. Suyari, H. Matsuzoe, A. M. Scarfone, “Advantages of q-logarithm representation over q-exponential representation from the sense of scale and shift on nonlinear systems”, The European Physical Journal Special Topics, 229(5), 773–785, 2020. https://doi.org/10.1140/epjst/e2020-900196-x
C. Tsallis, “Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World”, Springer: Berlin, 2009.
F. Nielsen, Geometric Theory of Information. Heidelberg: Springer, 2014. https://doi.org/10.1007/978-3-319-05317-2
C. Tsallis, “Possible generalization of Boltzmann–Gibbs statistics”, Journal of statistical physics, 52, 479–487, 1988. https://doi.org/10.1007/BF01016429
D. R. Anderson, D. J. Ulness, “Newly defined conformable derivatives”, Advances in Dynamical Systems and Applications, 10(2), 109–137, 2015.
E. A. Marchisotto, G. A. Zakeri, “An invitation to integration in finite terms”, The College Mathematics Journal, 1994, 25(4), 295–308, 1994. https://doi.org/10.1080/07468342.1994.11973625
A. D. Polyanin, A. V. Manzhirov, Handbook of integral equations. CRC press, 2008.
A. Fleitas, J. A. Mendez-Bermudez, J. N. Valdes, J. S. Almira, “On fractional Liénard–type systems”, Revista Mexicana de Física, 65(6), 618–625, 2019. https://doi.org/10.31349/revmexfis.65.618
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