Fractional Calculus: Historical Notes Apuntes Históricos del Cálculo Fraccionario
Palabras clave:
Fractional derivatives e integral, generalized derivative, fractional calculus Integrales y derivadas fraccionarias, derivadas generalizadas, cálculo fraccional
Resumen
En este trabajo, presentamos algunos apuntes históricos del cálculo fraccionario Local, y destacamos algunas propiedades y aplicaciones de estas nuevas herramientas matemicas
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Referencias
[1] T. Abdeljawad. On conformable fractional calculus, J. of Computational and Applied Mathematics
279, 2015, 57-66.
[2] A. Atangana, Derivative with a New Parameter Theory, Methods and Applications. Academic Press,
2016.
[3] A. Atangana, Extension of rate of change concept: From local to nonlocal operators with applications.,
Results in Physics, 19, 2020, Article 103515,1-523.
[4] D. Baleanu, COMMENTS ON: ”The failure of certain fractional calculus operators in two physical
models”by M. Ortigueira, V. Martynyuk, M. Fedula, J.A.T. Machado Fractional Calculus and Applied
Analysis, (2020) , 1-5
[5] D. Baleanu, A. Fernandez, On Fractional Operators and Their Classifications, Mathematics.
7(9)(2019) 830, 1-10
[6] Capelas de Oliveira, E., Tenreiro Machado, J.A. A Review of Definitions for Fractional Derivatives and
Integral. Mathematical Problems in Engineering 2014,(2014), Article 238459, 1-6.
[7] W. Chen. Time-space fabric underying anomalous diffusion, Chaos, Solitons and Fractals 28, 2006,
923-929.
[8] J. G. Delgado, J. E. Napoles ´ Valdez, , E. P. Reyes , M. Vivas-Cortez. The Minkowski Inequality for
Generalized Fractional Integrals. Appl. Math. Inf. Sci. , 15(1), 2021, 1–7
[9] A. Fleitas, J. E. Napoles, ´ J. M. Rodr´ıguez, J. M. Sigarreta. On the generalized fractional derivative,
submited.
[10] F. Gao, X. Yang, Z. Kang. Local fractional Newton method derived from modified local fractional
calculus, Proc. of CSO 2009, 2009, 228– 232.
[11] P. M. Guzman, ´ P. Korus, ´ J. E. Napoles ´ Valdes, ´ Generalized Integral Inequalities of Chebyshev Type,
Fractal Fract. 4, (2020), 100 , 1-7
[12] Guzman, ´ P. M., Langton, G., Lugo, L. M., Medina, J. and Napoles ´ Valdes, ´ J. E., A new definition of a
fractional derivative of local type. J. Math. Anal., 9(2), 2018, 88-98.
[13] P. M. Guzman, ´ L. M. Lugo, J. E. Napoles, ´ On the stability of solutions of fractional non conformable
differential equations, Stud. Univ. Babes-Bolyai Math. 65 2020, 495-502
[14] P. M. Guzman, ´ J. E. Napoles, ´ Y. Gasimov, Integral inequalities within the framework of generalized
fractional integrals, Fractional Differential Calculus, 11, (1) 2021 , 69-84
[15] J. H. He. A new fractal derivation, Thermal Science 15(1), 2011, 145-147.
[16] A. Karci, Chain Rule for Fractional Order Derivatives, Science Innovation, 4(6), 2015, 63-67.
[17] U. N. Katugampola, A new fractional derivative with classical properties, arXiv:1410.6535v2
[math.CA] 2014, 1-15.
[18] V. N. Katugampola. A new fractional derivative with classical properties, J. of the Amer. Math. Soc.
2014, in press, arXiv:1410.6535.
[19] Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative. J. Comput. Appl. Math., 264, 2014, 65-70.
[20] K. M. Kolwankar, A. D. Gangal. Fractional differentiability of nowhere differentiable functions and
dimensions, Chaos 6 (4), 1996, 505-513.
[21] K. M. Kolwankar, A. D. Gangal. Holder exponents of irregular signals and local fractional derivatives,
Pramana, 48(1), 1997, 49-68.
[22] K. M. Kolwankar, A. D. Gangal. Local fractional Fokker-Planck equation, Phys. Rev. Lett. 80, 1996,
214-2020.
[23] P. Korus, ´ L. M. Lugo, J. E. Napoles ´ Valdes, ´ Integral inequalities in a generalized context, Studia
Scientiarum Mathematicarum Hungarica, 57(3), 2020, 312-320
[24] L. M. Lugo, J. E. Napoles ´ Valdes, ´ M. Vivas-Cortez, On the oscillatory behaviour of some forced nonlinear generalized differential equation, Revista Investigacion´ Operacional, 42(2), 2021, 267-278
[25] L. M. Lugo, J. E. Napoles ´ Valdes, ´ M. Vivas-Cortez, A multi-index generalized derivative. Some introductory notes, submited
[26] Machado, J.T., Kiryakova, V., Mainardi, F. Recent history of fractional calculus. Commun. Nonlinear
Sci. Numer. Simul. 16, 2011, 1140–1153.
[27] Mart´ınez, F., Mohammed, P.O., Napoles ´ Valdes, ´ J.E. Non Conformable Fractional Laplace Transform.,
Kragujevac J. of Math. 46(3), 2022, 341-354.
[28] Mart´ınez, F., Napoles ´ Valdes, ´ J. E., Towards a Non-Conformable Fractional Calculus of N-variables,
Journal of Mathematics and Applications, JMA 43, 2020, 87-98
[29] A. B. Mingarelli, On generalized and fractional derivatives and their applications to classical mechanics, arXiv:submit/2328320 [math-ph], 2018.
[30] J. E. Napoles, ´ Oscillatory Criteria for Some non Conformable Differential Equation with Damping,
Discontinuity, Nonlinearity, and Complexity 10(3), 2021, 461-469
[31] J. E. Napoles ´ Valdes, ´ Y. S. Gasimov, A. R. Aliyeva, On the Oscillatory Behavior of Some Generalized
Differential Equation, Punjab University Journal of Mathematics, 53(1), 202,) 71-82
[32] Napoles ´ Valdes, ´ J. E., Guzman, ´ P. M. and Lugo, L. M., Some New Results on Nonconformable Fractional Calculus, Advances in Dynamical Systems and Applications, 13(2), 2018, 167–175.
[33] Napoles, ´ J.E., Guzman, ´ P. M., Lugo, L. M., Kashuri, A., The local generalized derivative and Mittag
Leffler function, Sigma J Eng & Nat Sci 38 (2), 2020, 1007–1017
[34] J. E. Napoles, ´ M. N. Quevedo, On the Oscillatory Nature of Some Generalized Emden-Fowler Equation, Punjab University Journal of Mathematics, 52(6), 2020, 97–106
[35] J. E. Napoles, ´ M. N. Quevedo, A. R. Gomez ´ Plata, ONn the asymptotic behaviour of a generalized
nonliear differential equation , Sigma J Eng & Nat Sci 38(4), 2020, 2109–2121
[36] J. E. Napoles, ´ J. M. Rodr´ıguez, J. M. Sigarreta, On Hermite-Hadamard type inequalities for nonconformable integral operators., Symmetry 11, 2019, 1108.
[37] J. E. Napoles, ´ C. Tunc, On the boundedness and oscillation of non conformable Lienard Equation,
Journal of Fractional Calculus and Applications, 11(2), 2020, 92-101.
[38] A. Parvate, A. D. Gangal. Calculus on fractal subsets of real line - I: formulation. Fractals 17 (1), 2009,
53-81.
[39] E. Reyes-Luis, G. Fernandez-Anaya, ´ J. Chav´ ez-Carlos, L. Diago-Cisneros, R. Munoz-V ˜ ega, A twoindex generalization of conformable operators with potential applications in engineering and physics,
Revista Mexicana de FAsica ˜ 67 (3), 2021, 429-442.
[40] A. Tallafha, S. Al Hihi, Total and directional fractional derivatives, Intern.l J. Pure and Applied Math.,
107(4), 2016, 1037-1051.
[41] S. Umarov, S. Steinberg, Variable order differential equations with piecewise constant order–function
and diffusion with changing modes. Z. Anal Anwend 28(4),2009, 431–450.
[42] J. Vanterler da C. Sousa, E. Capelas de Oliveira, A new truncated M-fractional derivative type unifying
some fractional derivative types with classical properties , arXiv:1704.08187 [math.CA].
[43] J. Vanterler da C. Sousa, E. Capelas de Oliveira, M-fractional derivative with classical properties,
Submitted, (2017).
[44] M. Vivas-Cortez, P. Korus, ´ J. E. Napoles ´ Valdes, ´ Some generalized Hermite-Hadamard-Fejer´ inequality for convex functions, Advances in Difference Equations 2021, 2021, Article 199, 1-11
[45] M. Vivas-Cortez, J. E. Napoles ´ Valdes, ´ L. M. Lugo, On a Generalized Laplace Transform, Appl. Math.
Inf. Sci. 15(5), 2021, 667-675
[46] M. Vivas-Cortez, J. E. Hernandez ´ Hernandez, ´ On a Hardy’s inequality for a fractional integral operator. Annals of the University of Craiova, Mathematics and Computer Science Series, 45(2), 2018,
232-242
[47] M. Vivas-Cortez, J. E. Hernandez ´ Hernandez, ´ Hermite-Hadamard Inequalities type for Raina’s fractional integral operator using η−convex functions. Revista de Matematica: ´ Teor´ıa y Aplicaciones, 26(1),
2019, 1-19
[48] M. Vivas-Cortez, A. Kashuri, R. Liko, J. E. Hernandez ´ Hernandez, ´ Trapezium-Type Inequalities for an
Extension of Riemann-Liouville Fractional Integrals Using Raina’s Special Function and Generalized
Coordinate Convex Functions, Axioms, 9, 2020, 1-17
[49] M. Vivas-Cortez , A. Kashuri, J. E. Hernandez ´ Hernandez, ´ Trapezium-Type Inequalities for Raina’s
Fractional Integrals Operator Using Generalized Convex Functions. Symmetry, 12, 2020, 1–17
[50] M. Vivas-Cortez, M. Aamir Ali, H. Budak, H. Kalsoom, P. Agarwal. Some New Hermite-Hadamard
and related inequalities for convex functions via (p, q)−integral. Entropy, 23, 2021, 1–21
[51] X. J. Yang, Local Fractional Integral Transforms. Progress in Nonlinear Science, 4, 2011, 1-225.
[52] X. J. Yang, Local Fractional Functional Analysis and Its Applications. Asian Academic publisher
Limited, Hong kong, China. 2011.
[53] X. J. Yang et al., Transport Equations in Fractal Porous Media within Fractional Complex Transform
Method, Proceedings, Romanian Academy, Series A, 14(4),2013, 287-292
[54] X. J. Yang et al., Local Fractional Integral Transforms and Their Applications, Academic Press, New
York, USA, 2015
[55] X. J. Yang et al., A New Local Fractional Derivative and Its Application to Diffusion in Fractal Media,
Computational Method for Complex Science, 1 (1), 2016), 1-5
[56] D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation. Calcolo
54 , 2017, 903–917.
[57] Zulfeqarr, F., (2017) Ujlayan, A. and Ahuja, A. A new fractional derivative and its fractional integral
with some applications, arXiv: 1705.00962v1 [math.CA] 26.
279, 2015, 57-66.
[2] A. Atangana, Derivative with a New Parameter Theory, Methods and Applications. Academic Press,
2016.
[3] A. Atangana, Extension of rate of change concept: From local to nonlocal operators with applications.,
Results in Physics, 19, 2020, Article 103515,1-523.
[4] D. Baleanu, COMMENTS ON: ”The failure of certain fractional calculus operators in two physical
models”by M. Ortigueira, V. Martynyuk, M. Fedula, J.A.T. Machado Fractional Calculus and Applied
Analysis, (2020) , 1-5
[5] D. Baleanu, A. Fernandez, On Fractional Operators and Their Classifications, Mathematics.
7(9)(2019) 830, 1-10
[6] Capelas de Oliveira, E., Tenreiro Machado, J.A. A Review of Definitions for Fractional Derivatives and
Integral. Mathematical Problems in Engineering 2014,(2014), Article 238459, 1-6.
[7] W. Chen. Time-space fabric underying anomalous diffusion, Chaos, Solitons and Fractals 28, 2006,
923-929.
[8] J. G. Delgado, J. E. Napoles ´ Valdez, , E. P. Reyes , M. Vivas-Cortez. The Minkowski Inequality for
Generalized Fractional Integrals. Appl. Math. Inf. Sci. , 15(1), 2021, 1–7
[9] A. Fleitas, J. E. Napoles, ´ J. M. Rodr´ıguez, J. M. Sigarreta. On the generalized fractional derivative,
submited.
[10] F. Gao, X. Yang, Z. Kang. Local fractional Newton method derived from modified local fractional
calculus, Proc. of CSO 2009, 2009, 228– 232.
[11] P. M. Guzman, ´ P. Korus, ´ J. E. Napoles ´ Valdes, ´ Generalized Integral Inequalities of Chebyshev Type,
Fractal Fract. 4, (2020), 100 , 1-7
[12] Guzman, ´ P. M., Langton, G., Lugo, L. M., Medina, J. and Napoles ´ Valdes, ´ J. E., A new definition of a
fractional derivative of local type. J. Math. Anal., 9(2), 2018, 88-98.
[13] P. M. Guzman, ´ L. M. Lugo, J. E. Napoles, ´ On the stability of solutions of fractional non conformable
differential equations, Stud. Univ. Babes-Bolyai Math. 65 2020, 495-502
[14] P. M. Guzman, ´ J. E. Napoles, ´ Y. Gasimov, Integral inequalities within the framework of generalized
fractional integrals, Fractional Differential Calculus, 11, (1) 2021 , 69-84
[15] J. H. He. A new fractal derivation, Thermal Science 15(1), 2011, 145-147.
[16] A. Karci, Chain Rule for Fractional Order Derivatives, Science Innovation, 4(6), 2015, 63-67.
[17] U. N. Katugampola, A new fractional derivative with classical properties, arXiv:1410.6535v2
[math.CA] 2014, 1-15.
[18] V. N. Katugampola. A new fractional derivative with classical properties, J. of the Amer. Math. Soc.
2014, in press, arXiv:1410.6535.
[19] Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative. J. Comput. Appl. Math., 264, 2014, 65-70.
[20] K. M. Kolwankar, A. D. Gangal. Fractional differentiability of nowhere differentiable functions and
dimensions, Chaos 6 (4), 1996, 505-513.
[21] K. M. Kolwankar, A. D. Gangal. Holder exponents of irregular signals and local fractional derivatives,
Pramana, 48(1), 1997, 49-68.
[22] K. M. Kolwankar, A. D. Gangal. Local fractional Fokker-Planck equation, Phys. Rev. Lett. 80, 1996,
214-2020.
[23] P. Korus, ´ L. M. Lugo, J. E. Napoles ´ Valdes, ´ Integral inequalities in a generalized context, Studia
Scientiarum Mathematicarum Hungarica, 57(3), 2020, 312-320
[24] L. M. Lugo, J. E. Napoles ´ Valdes, ´ M. Vivas-Cortez, On the oscillatory behaviour of some forced nonlinear generalized differential equation, Revista Investigacion´ Operacional, 42(2), 2021, 267-278
[25] L. M. Lugo, J. E. Napoles ´ Valdes, ´ M. Vivas-Cortez, A multi-index generalized derivative. Some introductory notes, submited
[26] Machado, J.T., Kiryakova, V., Mainardi, F. Recent history of fractional calculus. Commun. Nonlinear
Sci. Numer. Simul. 16, 2011, 1140–1153.
[27] Mart´ınez, F., Mohammed, P.O., Napoles ´ Valdes, ´ J.E. Non Conformable Fractional Laplace Transform.,
Kragujevac J. of Math. 46(3), 2022, 341-354.
[28] Mart´ınez, F., Napoles ´ Valdes, ´ J. E., Towards a Non-Conformable Fractional Calculus of N-variables,
Journal of Mathematics and Applications, JMA 43, 2020, 87-98
[29] A. B. Mingarelli, On generalized and fractional derivatives and their applications to classical mechanics, arXiv:submit/2328320 [math-ph], 2018.
[30] J. E. Napoles, ´ Oscillatory Criteria for Some non Conformable Differential Equation with Damping,
Discontinuity, Nonlinearity, and Complexity 10(3), 2021, 461-469
[31] J. E. Napoles ´ Valdes, ´ Y. S. Gasimov, A. R. Aliyeva, On the Oscillatory Behavior of Some Generalized
Differential Equation, Punjab University Journal of Mathematics, 53(1), 202,) 71-82
[32] Napoles ´ Valdes, ´ J. E., Guzman, ´ P. M. and Lugo, L. M., Some New Results on Nonconformable Fractional Calculus, Advances in Dynamical Systems and Applications, 13(2), 2018, 167–175.
[33] Napoles, ´ J.E., Guzman, ´ P. M., Lugo, L. M., Kashuri, A., The local generalized derivative and Mittag
Leffler function, Sigma J Eng & Nat Sci 38 (2), 2020, 1007–1017
[34] J. E. Napoles, ´ M. N. Quevedo, On the Oscillatory Nature of Some Generalized Emden-Fowler Equation, Punjab University Journal of Mathematics, 52(6), 2020, 97–106
[35] J. E. Napoles, ´ M. N. Quevedo, A. R. Gomez ´ Plata, ONn the asymptotic behaviour of a generalized
nonliear differential equation , Sigma J Eng & Nat Sci 38(4), 2020, 2109–2121
[36] J. E. Napoles, ´ J. M. Rodr´ıguez, J. M. Sigarreta, On Hermite-Hadamard type inequalities for nonconformable integral operators., Symmetry 11, 2019, 1108.
[37] J. E. Napoles, ´ C. Tunc, On the boundedness and oscillation of non conformable Lienard Equation,
Journal of Fractional Calculus and Applications, 11(2), 2020, 92-101.
[38] A. Parvate, A. D. Gangal. Calculus on fractal subsets of real line - I: formulation. Fractals 17 (1), 2009,
53-81.
[39] E. Reyes-Luis, G. Fernandez-Anaya, ´ J. Chav´ ez-Carlos, L. Diago-Cisneros, R. Munoz-V ˜ ega, A twoindex generalization of conformable operators with potential applications in engineering and physics,
Revista Mexicana de FAsica ˜ 67 (3), 2021, 429-442.
[40] A. Tallafha, S. Al Hihi, Total and directional fractional derivatives, Intern.l J. Pure and Applied Math.,
107(4), 2016, 1037-1051.
[41] S. Umarov, S. Steinberg, Variable order differential equations with piecewise constant order–function
and diffusion with changing modes. Z. Anal Anwend 28(4),2009, 431–450.
[42] J. Vanterler da C. Sousa, E. Capelas de Oliveira, A new truncated M-fractional derivative type unifying
some fractional derivative types with classical properties , arXiv:1704.08187 [math.CA].
[43] J. Vanterler da C. Sousa, E. Capelas de Oliveira, M-fractional derivative with classical properties,
Submitted, (2017).
[44] M. Vivas-Cortez, P. Korus, ´ J. E. Napoles ´ Valdes, ´ Some generalized Hermite-Hadamard-Fejer´ inequality for convex functions, Advances in Difference Equations 2021, 2021, Article 199, 1-11
[45] M. Vivas-Cortez, J. E. Napoles ´ Valdes, ´ L. M. Lugo, On a Generalized Laplace Transform, Appl. Math.
Inf. Sci. 15(5), 2021, 667-675
[46] M. Vivas-Cortez, J. E. Hernandez ´ Hernandez, ´ On a Hardy’s inequality for a fractional integral operator. Annals of the University of Craiova, Mathematics and Computer Science Series, 45(2), 2018,
232-242
[47] M. Vivas-Cortez, J. E. Hernandez ´ Hernandez, ´ Hermite-Hadamard Inequalities type for Raina’s fractional integral operator using η−convex functions. Revista de Matematica: ´ Teor´ıa y Aplicaciones, 26(1),
2019, 1-19
[48] M. Vivas-Cortez, A. Kashuri, R. Liko, J. E. Hernandez ´ Hernandez, ´ Trapezium-Type Inequalities for an
Extension of Riemann-Liouville Fractional Integrals Using Raina’s Special Function and Generalized
Coordinate Convex Functions, Axioms, 9, 2020, 1-17
[49] M. Vivas-Cortez , A. Kashuri, J. E. Hernandez ´ Hernandez, ´ Trapezium-Type Inequalities for Raina’s
Fractional Integrals Operator Using Generalized Convex Functions. Symmetry, 12, 2020, 1–17
[50] M. Vivas-Cortez, M. Aamir Ali, H. Budak, H. Kalsoom, P. Agarwal. Some New Hermite-Hadamard
and related inequalities for convex functions via (p, q)−integral. Entropy, 23, 2021, 1–21
[51] X. J. Yang, Local Fractional Integral Transforms. Progress in Nonlinear Science, 4, 2011, 1-225.
[52] X. J. Yang, Local Fractional Functional Analysis and Its Applications. Asian Academic publisher
Limited, Hong kong, China. 2011.
[53] X. J. Yang et al., Transport Equations in Fractal Porous Media within Fractional Complex Transform
Method, Proceedings, Romanian Academy, Series A, 14(4),2013, 287-292
[54] X. J. Yang et al., Local Fractional Integral Transforms and Their Applications, Academic Press, New
York, USA, 2015
[55] X. J. Yang et al., A New Local Fractional Derivative and Its Application to Diffusion in Fractal Media,
Computational Method for Complex Science, 1 (1), 2016), 1-5
[56] D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation. Calcolo
54 , 2017, 903–917.
[57] Zulfeqarr, F., (2017) Ujlayan, A. and Ahuja, A. A new fractional derivative and its fractional integral
with some applications, arXiv: 1705.00962v1 [math.CA] 26.
Publicado
2021-12-31
Cómo citar
Vivas Cortez, M., Guzman, P. M., Lugo, L. M., & Nápoles V, J. E. (2021). Fractional Calculus: Historical Notes Apuntes Históricos del Cálculo Fraccionario. Revista MATUA ISSN: 2389-7422, 8(2), 35-45. Recuperado a partir de https://investigaciones.uniatlantico.edu.co/revistas/index.php/MATUA/article/view/3358
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