Fractional Calculus: Historical Notes Apuntes Históricos del Cálculo Fraccionario
Keywords:
Fractional derivatives e integral, generalized derivative, fractional calculus Integrales y derivadas fraccionarias, derivadas generalizadas, cálculo fraccional
Abstract
In this paper, we present some historical notes to Generalized Calculus, sometimes called Local Fractional Calculus, and highlight some properties and applications of these new mathematical tools
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References
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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
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[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.
Published
2021-12-31
How to Cite
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. Retrieved from https://investigaciones.uniatlantico.edu.co/revistas/index.php/MATUA/article/view/3358
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