Battery discharging model on fractal time sets
-
Karmina Kamal Ali
and
Resat Yilmazer
Abstract
This article is devoted to propose and investigate the fractal battery discharging model, which is one of the well-known models with a memory effect. It is presented as to how non-locality affects the behavior of solutions and how the current state of the system is affected by its past. Firstly, we present a local fractal solution. Then we solve the non-local fractal differential equation and examine the memory effect that includes the Mittag-Leffler function with one parameter. For that aim, the local fractal and non-local fractal Laplace transforms are used to achieve fractional solutions. In addition, the simulation analysis is performed by comparing the underlying fractal derivatives to the classical ones in order to understand the significance of the results. The effects of the fractal parameter and the fractional parameter are discussed in the conclusion section.
-
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
-
Research funding: None declared.
-
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] B. B. Mandelbrot, The Fractal Geometry of Nature, New York, W. H. Freeman, 1982.Search in Google Scholar
[2] A. Bunde and S. Havlin, Fractals in Science, Berlin Heidelberg, Springer, 2013.Search in Google Scholar
[3] K. J. Falconer, The Geometry of Fractal Sets, Cambridge, Cambridge University Press, 1986.10.1017/CBO9780511623738Search in Google Scholar
[4] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, New York, John Wiley & Sons, 2004.10.1002/0470013850Search in Google Scholar
[5] K. Falconer, Techniques in Fractal Geometry, New York, John Wiley & Sons, 1999.Search in Google Scholar
[6] G. A. Edgar, Integral, Probability, and Fractal Measures, New York, Springer, 1998.10.1007/978-1-4757-2958-0Search in Google Scholar
[7] S. Rice, “Fractal modelling: growth and form in biology,” Science, vol. 266, pp. 664–666, 1994. https://doi.org/10.1126/science.266.5185.664-a.Search in Google Scholar PubMed
[8] R. S. Strichartz, Differential Equations on Fractals, Princeton, Princeton University Press, 2018.10.2307/j.ctv346nvvSearch in Google Scholar
[9] M. T. Barlow and E. A. Perkins, “Brownian motion on the Sierpinski gasket,” Probab. Theor. Relat. Fields, vol. 79, pp. 543–623, 1988. https://doi.org/10.1007/bf00318785.Search in Google Scholar
[10] F. H. Stillinger, “Axiomatic basis for spaces with noninteger dimension,” J. Math. Phys., vol. 18, pp. 1224–1234, 1977. https://doi.org/10.1063/1.523395.Search in Google Scholar
[11] A. S. Balankin, “A continuum framework for mechanics of fractal materials I: from fractional space to continuum with fractal metric,” Eur. Phys. J. B, vol. 88, 2015. https://doi.org/10.1140/epjb/e2015-60189-y.Search in Google Scholar
[12] M. Zubair, M. J. Mughal, and Q. A. Naqvi, Electromagnetic Fields and Waves in Fractional Dimensional Space, Berlin Heidelberg, Springer, 2012.10.1007/978-3-642-25358-4Search in Google Scholar
[13] L. Nottale and J. Schneider, “Fractals and nonstandard analysis,” J. Math. Phys., vol. 25, pp. 1296–1300, 1984. https://doi.org/10.1063/1.526285.Search in Google Scholar
[14] T. Gregory Dewey, Fractals in Molecular Biophysics, Oxford, Oxford University Press, 1998.10.1093/oso/9780195084474.001.0001Search in Google Scholar
[15] M. Czachor, “Waves along fractal coastlines: from fractal arithmetic to wave equations,” Acta Phys. Pol. B, vol. 50, p. 813, 2019. https://doi.org/10.5506/aphyspolb.50.813.Search in Google Scholar
[16] J. Kigami, Analysis on Fractals, Cambridge, Cambridge University Press, 2001.10.1017/CBO9780511470943Search in Google Scholar
[17] V. E. Tarasov, Fractional Dynamics, Berlin, Heidelberg, Springer, 2010.10.1007/978-3-642-14003-7Search in Google Scholar
[18] A. A. Iliasov, M. I. Katsnelson, and S. Yuan, “Hall conductivity of a Sierpiński carpet,” Phys. Rev. B, vol. 101, p. 045413, 2020. https://doi.org/10.1103/physrevb.101.045413.Search in Google Scholar
[19] J. Wu and C. Wang, “Fractal Stokes’theorem based on integrals on fractal manifolds,” Fractals, vol. 28, p. 2050010, 2020. https://doi.org/10.1142/s0218348x20500103.Search in Google Scholar
[20] M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time Scales, New York, Springer Science & Business Media, 2002.10.1007/978-0-8176-8230-9Search in Google Scholar
[21] A. Parvate and A. D. Gangal, “Calculus on fractal subsets of real-line I: Formulation,” Fractals, vol. 17, pp. 53–81, 2009. https://doi.org/10.1142/s0218348x09004181.Search in Google Scholar
[22] A. Parvate and A. D. Gangal, “Calculus on fractal subsets of real line II: conjugacy with ordinary calculus,” Fractals, vol. 19, pp. 271–290, 2011. https://doi.org/10.1142/s0218348x11005440.Search in Google Scholar
[23] A. Parvate, S. Satin, and A. D. Gangal, “Calculus on fractal curves in Rn,” Fractals, vol. 19, pp. 15–27, 2011. https://doi.org/10.1142/s0218348x1100518x.Search in Google Scholar
[24] A. K. Golmankhaneh and D. Baleanu, “Non-local integrals and derivatives on fractal sets with applications,” Open Phys., vol. 14, pp. 542–548, 2016. https://doi.org/10.1515/phys-2016-0062.Search in Google Scholar
[25] A. K. Golmankhaneh and C. Tunç, “Sumudu transform in fractal calculus,” Appl. Math. Comput., vol. 350, pp. 386–401, 2019. https://doi.org/10.1016/j.amc.2019.01.025.Search in Google Scholar
[26] C. Tunç, A. K. Golmankhaneh, and U. Branch, “On stability of a class of second alpha-order fractal differential equations,” AIMS math., vol. 5, pp. 2126–2142, 2020. https://doi.org/10.3934/math.2020141.Search in Google Scholar
[27] A. K. Golmankhaneh and C. Tunç, “Stochastic differential equations on fractal sets,” Stochastics, vol. 92, pp. 1244–1260, 2019. https://doi.org/10.1080/17442508.2019.1697268.Search in Google Scholar
[28] A. K. Golmankhaneh and C. Cattani, “Fractal logistic equation,” Fractal Fract., vol. 3, p. 41, 2019. https://doi.org/10.3390/fractalfract3030041.Search in Google Scholar
[29] A. K. Golmankhaneh and A. Fernandez, “Random variables and stable distributions on fractal Cantor sets,” Fractal Fract., vol. 3, p. 31, 2019. https://doi.org/10.3390/fractalfract3020031.Search in Google Scholar
[30] A. K. Golmankhaneh and C. Tunc, “Analogues to Lie method and Noether’s theorem in fractal calculus,” Fractal Fract., vol. 3, p. 25, 2019. https://doi.org/10.3390/fractalfract3020025.Search in Google Scholar
[31] A. K. Golmankhaneh, “Statistical mechanics involving fractal temperature,” Fractal Fract., vol. 3, p. 20, 2019. https://doi.org/10.3390/fractalfract3020020.Search in Google Scholar
[32] A. Golmankhaneh and A. Fernandez, “Fractal calculus of functions on Cantor Tartan spaces,” Fractal Fract., vol. 2, p. 30, 2018. https://doi.org/10.3390/fractalfract2040030.Search in Google Scholar
[33] A. K. Golmankhaneh, A. Fernandez, A. K. Golmankhaneh, and D. Baleanu, “Diffusion on middle-ξ Cantor sets,” Entropy, vol. 20, p. 504, 2018. https://doi.org/10.3390/e20070504.Search in Google Scholar PubMed PubMed Central
[34] A. K. Golmankhaneh, “About Kepler’s third law on fractal-time spaces,” Ain Shams Eng. J., vol. 9, pp. 2499–2502, 2018. https://doi.org/10.1016/j.asej.2017.06.005.Search in Google Scholar
[35] Y. Sato, S. Takeuchi, and K. Kobayakawa, “Cause of the memory effect observed in alkaline secondary batteries using nickel electrode,” J. Power Sources, vol. 93, pp. 20–24, 2001. https://doi.org/10.1016/s0378-7753(00)00506-1.Search in Google Scholar
[36] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Netherlands, Elsevier, 1998.Search in Google Scholar
[37] T. P. J. Crompton, Battery Reference Book, Netherlands, Elsevier, 2000.Search in Google Scholar
[38] U. Hullmeine, A. Winsel, and E. Voss, “Effect of previous charge/discharge history on the capacity of the PbO2/PbSO4 electrode: the hysteresis or memory effect,” J. Power Sources, vol. 25, pp. 27–47, 1989. https://doi.org/10.1016/0378-7753(89)80120-x.Search in Google Scholar
[39] J-M. Tarascon, A. S. Gozdz, C. Schmutz, F. Shokoohi, and P. C. Warren, “Performance of Bellcore’s plastic rechargeable Li-ion batteries,” Solid State Ionics, vol. 86, pp. 49–54, 1996. https://doi.org/10.1016/0167-2738(96)00330-x.Search in Google Scholar
[40] S. Westerlund, “Dead matter has memory!,” Phys. Scr., vol. 43, p. 174, 1991. https://doi.org/10.1088/0031-8949/43/2/011.Search in Google Scholar
[41] R. Herrmann, Fractional Calculus: An Introduction for Physicists, Singapore, World Scientific, 2014.10.1142/8934Search in Google Scholar
[42] R. DiMartino and W. Urbina, “On Cantor-like sets and Cantor-Lebesgue singular functions,” arXiv preprint arXiv:1403.6554, 2014.Search in Google Scholar
[43] K. Welch, A Fractal Topology of Time: Deepening into Timelessness, Austin, TX, Fox Finding Press, 2020.Search in Google Scholar
[44] A. K. Golmankhaneh and K. Welch, “Equilibrium and non-equilibrium statistical mechanics with generalized fractal derivatives: a review,” Mod. Phys. Lett. A, vol. 36, no. 14, p. 2140002, 2021. https://doi.org/10.1142/s0217732321400022.Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Modeling and assessment of the flow and air pollutants dispersion during chemical reactions from power plant activities
- Stochastic dynamics of dielectric elastomer balloon with viscoelasticity under pressure disturbance
- Unsteady MHD natural convection flow of a nanofluid inside an inclined square cavity containing a heated circular obstacle
- Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations
- Battery discharging model on fractal time sets
- Adaptive neural network control of second-order underactuated systems with prescribed performance constraints
- Optimal control for dengue eradication program under the media awareness effect
- Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations
- Modeling and simulations of a Zika virus as a mosquito-borne transmitted disease with environmental fluctuations
- Mathematical analysis of the impact of vaccination and poor sanitation on the dynamics of poliomyelitis
- Anti-sway method for reducing vibrations on a tower crane structure
-
Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized
- Convergence analysis of online learning algorithm with two-stage step size
- An estimative (warning) model for recognition of pandemic nature of virus infections
- Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation
- Global exponential stability of periodic solution of delayed discontinuous Cohen–Grossberg neural networks and its applications
- An efficient class of fourth-order derivative-free method for multiple-roots
- Numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
- Construction of breather solutions and N-soliton for the higher order dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising from wave patterns
- Delay-dependent robust stability analysis of uncertain fractional-order neutral systems with distributed delays and nonlinear perturbations subject to input saturation
- Construction of complexiton-type solutions using bilinear form of Hirota-type
- Inverse estimation of time-varying heat transfer coefficients for a hollow cylinder by using self-learning particle swarm optimization
- Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting
- Lump solutions to a generalized nonlinear PDE with four fourth-order terms
- Quantum motion control for packaging machines
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Modeling and assessment of the flow and air pollutants dispersion during chemical reactions from power plant activities
- Stochastic dynamics of dielectric elastomer balloon with viscoelasticity under pressure disturbance
- Unsteady MHD natural convection flow of a nanofluid inside an inclined square cavity containing a heated circular obstacle
- Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations
- Battery discharging model on fractal time sets
- Adaptive neural network control of second-order underactuated systems with prescribed performance constraints
- Optimal control for dengue eradication program under the media awareness effect
- Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations
- Modeling and simulations of a Zika virus as a mosquito-borne transmitted disease with environmental fluctuations
- Mathematical analysis of the impact of vaccination and poor sanitation on the dynamics of poliomyelitis
- Anti-sway method for reducing vibrations on a tower crane structure
-
Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized
- Convergence analysis of online learning algorithm with two-stage step size
- An estimative (warning) model for recognition of pandemic nature of virus infections
- Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation
- Global exponential stability of periodic solution of delayed discontinuous Cohen–Grossberg neural networks and its applications
- An efficient class of fourth-order derivative-free method for multiple-roots
- Numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
- Construction of breather solutions and N-soliton for the higher order dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising from wave patterns
- Delay-dependent robust stability analysis of uncertain fractional-order neutral systems with distributed delays and nonlinear perturbations subject to input saturation
- Construction of complexiton-type solutions using bilinear form of Hirota-type
- Inverse estimation of time-varying heat transfer coefficients for a hollow cylinder by using self-learning particle swarm optimization
- Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting
- Lump solutions to a generalized nonlinear PDE with four fourth-order terms
- Quantum motion control for packaging machines
