New fractional derivative with sigmoid function as the kernel and its models. (English) Zbl 07848621
Summary: Based on the idea of the fractional derivative with respect to another function, a new fractional derivative operator with sigmoid function as the kernel in this article, is proposed for the first time. Then, we make use of this new fractional operator to model various nonlinear phenomena from different fields of applications in science, such as the population growth, the shallow water wave phenomena and reaction-diffusion processes, and so on. As a result, we hope that the new fractional operator can be used to discover more evolutionary mechanisms of these phenomena.
MSC:
| 26Axx | Functions of one variable |
| 34Axx | General theory for ordinary differential equations |
| 35Kxx | Parabolic equations and parabolic systems |
References:
| [1] | Switzerland · Zbl 0818.26003 |
| [2] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory & applications of fractional differential equations, North-HollandMathematical Studies, 2006, Elsevier: Elsevier Amsterdam · Zbl 1092.45003 |
| [3] | Hilfer, R.; Butzer, P. L.; Westphal, U., An introduction to fractional calculus, Appl. Fract. Cal. Phys. World Scientific, 1-85, 2010 · Zbl 0987.26005 |
| [4] | Liu, J. G.; Zhang, Y. F., Analytical study of exact solutions of the nonlinear Korteweg-de Vries equation with space-time fractional derivatives, Mod. Phys. Lett. B., 32, 1850012, 2018 |
| [5] | Saad, K. M.; Alshareef, E. H.; Alomari, A. K.; Baleanu, D.; Gómez-Aguilarf, J. F., On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burger’s equations using homotopy analysis transform method, Chin. J. Phys, 63, 149-162, 2020 · Zbl 07829561 |
| [6] | T.R. Malthus, An Essay on the Principle of Population, 1872. |
| [7] | Ferdi, Y., Some applications of fractional order calculus to design digital filters for biomedical signal processing, J. Mech. Medi. Biol., 12, 02, 1240008, 2012 |
| [8] | Mainardi, F., Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, World Scientific, 2010 · Zbl 1210.26004 |
| [9] | Feng, Y. Y.; Yang, X. J.; Liu, J. G., On overall behavior of maxwell mechanical model by the combined Caputo fractional derivative, Chin. J. Phys., 66, 269-276, 2020 · Zbl 07832374 |
| [10] | Liu, J. G.; Yang, X. J.; Feng, Y. Y.; Zhang, H. Y., Analysis of the time fractional nonlinear diffusion equation from diffusion process, J. Appl. Anal. Comput., 10, 3, 1060C1072, 2020 · Zbl 1480.35394 |
| [11] | Das, S., Analytical solution of a fractional diffusion equation by variational iteration method, Comput. Math. Appl., 57, 3, 483-487, 2009 · Zbl 1165.35398 |
| [12] | Das, S., A note on fractional diffusion equations, Chaos Solitons & Fract., 42, 4, 2074-2079, 2009 · Zbl 1198.65137 |
| [13] | Yang, X. J., A new integral transform operator for solving the heat-diffusion problem, Appl. Math. Lett., 64, 193-197, 2017 · Zbl 1353.35018 |
| [14] | Liu, J. G.; Yang, X. J.; Feng, Y. Y., On integrability of the time fractional nonlinear heat conduction equation, J. Geom. Phys., 144, 190-198, 2019 · Zbl 1439.35540 |
| [15] | Saleh, X.; Kassem, M.; Mabrouk, S. M., Exact solutions of nonlinear fractional order partial differential equations via singular manifold method, Chin. J. Phys., 61, 290-300, 2019 · Zbl 07824993 |
| [16] | Rezazadeh, H.; Tariq, H.; Eslami, M.; Mirzazadeh, M.; Zhou, Q., New exact solutions of nonlinear conformable time-fractional Phi-4 equation, Chin. J. Phys., 56, 6, 2805-2816, 2018 · Zbl 07822195 |
| [17] | Riemann, B., Versuch einer allgemeinen auffassung der integration und differentiation, 14 Janvier 1847, Bernhard Riemanns Gesammelte Mathematische Werke, 14, 353-362, 1892 |
| [18] | Caputo, M.; Int, J., Linear models of dissipation whose \(q\) is almost frequency independent-II, Geop., 13, 5, 529-539, 1967 |
| [19] | Yang, X. J.; Tenreiro Machado, J. A.; Baleanu, D., Anomalous diffusion models with general fractional derivatives within the kernels of the extended mittag-leffler type functions, Rom. Rep. Phys., 69, S1-S9, 2017 |
| [20] | Yang, X. J., General Fractional Derivatives: Theory, Methods and Applications, 2019, CRC Press: CRC Press New York, USA · Zbl 1417.26001 |
| [21] | Caputo, M.; Fabrizio, M., A new definition of fractional derivative without singular kernel, Prog. Fract. Diff. Appl., 1, 2, 1-13, 2015 |
| [22] | Arafa, A. A.; Hagag, A. M., A new analytic solution of fractional coupled Ramani equation, Chin. J. Phys, 60, 388-406, 2019 · Zbl 07823600 |
| [23] | Almeida, R., A caputo fractional derivative of a function with respect to another function, Commun. Nonl. Sci. Numer. Simul., 44, 460-481, 2017 · Zbl 1465.26005 |
| [24] | Oliveira, D. S.; de Oliveira, E. C., Hilfer-Katugampola fractional derivatives, Comput. Appl. Math., 37, 3, 3672-3690, 2018 · Zbl 1401.26014 |
| [25] | Sousa, J. V.C.; de Oliveira, E. C., On the \(\psi \)-Hilfer fractional derivative, Commun. Nonl. Sci. Numer. Simul., 60, 72-91, 2018 · Zbl 1470.26015 |
| [26] | Gu, H.; Trujillo, J. J., Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257, 344-354, 2015 · Zbl 1338.34014 |
| [27] | Miller, K. S., The Weyl Fractional Calculus: Fractional Calculus and its Applications, 1975, Springer: Springer Berlin, Heidelberg · Zbl 0305.44004 |
| [28] | Anderson, D. R.; Ulness, D. J., Properties of the Katugampola fractional derivative with potential application in quantum mechanics, J. Math. Phys., 56, 6, 063502, 2015 · Zbl 1323.26008 |
| [29] | Pagnini, G., Erdélyi-Kober fractional diffusion, Frac. Cal. Appl. Anal., 15, 1, 117-127, 2012 · Zbl 1276.26021 |
| [30] | Agrawal, O. P., Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A: Math. Theo., 40, 24, 6287, 2007 · Zbl 1125.26007 |
| [31] | Almeida, R., Fractional variational problems with the Riesz-Caputo derivative, Appl. Math. Lett., 25, 2, 142-148, 2012 · Zbl 1236.49044 |
| [32] | Teodoro, G. S.; Machado, J. A.T.; De Oliveira, E. C., A review of definitions of fractional derivatives and other operators, J. Comput. Phys., 388, 195-208, 2019 · Zbl 1452.26008 |
| [33] | Tommiska, M. T., Efficient digital implementation of the sigmoid function for reprogrammable logic, IEE P.-Comput. Dig. Tech., 150, 6, 403-411, 2003 |
| [34] | Marreiros, A. C.; Daunizeau, J.; Kiebel, S. J.; Friston, K. J., Population dynamics: variance and the sigmoid activation function, Neuroimage., 42, 1, 147-157, 2008 |
| [35] | Hassan, N.; Akamatsu, N., A new approach for contrast enhancement using sigmoid function, Int. Arab. J. Inf. Tech., 1, 2, 221-225, 2004 |
| [36] | Yong, P. C.; Nordholm, S.; Dam, H. H., Optimization and evaluation of sigmoid function with a priori SNR estimate for real-time speech enhancement, Speech. commun., 55, 2, 358-376, 2013 |
| [37] | Miura, R. M., Korteweg-de Vries equation and generalizations. i. a remarkable explicit nonlinear transformation, J. Math. Phys., 9, 8, 1202-1204, 1968 · Zbl 0283.35018 |
| [38] | Mittal, R. C.; Singhal, P., Numerical solution of Burger’s equation, Commun. numer. meth. eng., 9, 5, 397-406, 1993 · Zbl 0782.65147 |
| [39] | Thacker, W. C., Some exact solutions to the nonlinear shallow-water wave equations, J. Fluid. Mech., 107, 499-508, 1981 · Zbl 0462.76023 |
| [40] | de Pablo, A.; Vázquez, J. L., Travelling waves and finite propagation in a reaction-diffusion equation, J. Diff. Equ., 93, 1, 19-61, 1991 · Zbl 0784.35045 |
| [41] | Benguria, R. D.; Depassier, M. C., Speed of fronts of the reaction-diffusion equation, Phys. Rev. Lett., 77, 6, 1171, 1996 · Zbl 0856.35058 |
| [42] | Grace, S.; Agarwal, R.; Wong, P.; Zafer, A., On the oscillation of fractional differential equations, Fract. Cal. Appl. Anal., 15, 2, 222-231, 2012 · Zbl 1273.34007 |
| [43] | Vivek, D.; Elsayed, E. M.; Kanagarajan, K., On the oscillation of fractional differential equations via \(\psi \)-Hilfer fractional derivative, Eng. Appl. Sci. Lett., 2, 3, 1-6, 2019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
