# American Institute of Mathematical Sciences

• Previous Article
European option valuation under the Bates PIDE in finance: A numerical implementation of the Gaussian scheme
• DCDS-S Home
• This Issue
• Next Article
Mittag-Leffler input stability of fractional differential equations and its applications

## Inclusion of fading memory to Banister model of changes in physical condition

 1 Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 28095, Riyadh 11437, Saudi Arabia 2 Department of mathematics, AMITY School of Engineering and Technology, AMITY University Rajasthan, Jaipur -302022, India 3 Nature Science Department, Community College of Riyadh, King Saud University, P.O. Box 28095, Riyadh 11437, Saudi Arabia 4 Department of Mathematics, Faculty of Science, Fayoum University, Fayoum, Egypt

* Corresponding author: Ravi Shanker Dubey

Received  May 2018 Revised  June 2018 Published  March 2019

Fund Project: The authors would like to extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No (RG-1438-086)

We introduced the fading memory effect to the model portraying the prediction in physical condition. The classical model is known as the Banister model. We presented the existence and uniqueness conditions of the exact solutions of this model using three different memory including the bad memory induces by the power law and the good memories induced by exponential decay law and the Mittag-Leffler law. We derived the exact solutions using the Laplace transform for the non-delay version.

Citation: Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020051
##### References:
 [1] A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Fractals, 102 (2017), 396-406. doi: 10.1016/j.chaos.2017.04.027. Google Scholar [2] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706. doi: 10.1016/j.physa.2018.03.056. Google Scholar [3] A. Atangana and B. Dumitru, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. Google Scholar [4] A. Atangana and J. F. Gmez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166.Google Scholar [5] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order, Fractals, 89 (2016), 447-454. doi: 10.1016/j.chaos.2016.02.012. Google Scholar [6] T. Busso, Variable dose-response relationship between exercise training and performance, Med Sci Sports Exerc, 35 1188–1195.Google Scholar [7] T. Calvert, E. Banister, M. Savage and T. Bach, A systems model of the effects of training on physical performance, IEEE Transac-tions on Systems, Man and Cybernetics SMC-6(2), (1976), 94–102.Google Scholar [8] S. Eassom, Critical reflections on olympic ideology, Ontario: The Centre for Olympic Studies, (1994), 120–123.Google Scholar [9] A. Finn, Running with the Kenyans. p. chapter 2. Mangan, J A (2014), Sport in Latin American Society: Past and Present, (2012), 93.Google Scholar [10] G. Fulton and A. Bairner, Sport, space and national identity in ireland: The GAA, croke park and rule 42, Policy, 11 (2007), 55-74. Google Scholar [11] L. K. Ervin, A. A. Tateishi and R. V. Haroldo, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 1-9. doi: 10.3389/fphy.2017.00052. Google Scholar [12] J. A. T. Machado and A. M. Lopes, On the mathematical modeling of soccer dynamics, Communications in Nonlinear Science and Numerical Simulation, 53 (2017), 142-153. doi: 10.1016/j.cnsns.2017.04.024. Google Scholar [13] R. H. Morton, J. R. Fitz-Clarke and E. W. Banister, Modeling human performance in running, J Appl Physiol, 69 (1990), 1171-1177. Google Scholar [14] K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo–Fabrizio derivative in Riemann–Liouville sense, Fractals, 99 (2017), 171-179. doi: 10.1016/j.chaos.2017.04.008. Google Scholar [15] T. W. Calvert, E. W. Banister, M. V. Savage and T. Bach, A systems model of the effects of training on physical performance, IEEE Transactions on Systems, Man, and Cybernetics, 6 (1976), 94-102. Google Scholar [16] H. Yépez-Martínez and J. F. Gómez-Aguilar, Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 13, 17 pp. doi: 10.1051/mmnp/2018002. Google Scholar

show all references

##### References:
 [1] A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Fractals, 102 (2017), 396-406. doi: 10.1016/j.chaos.2017.04.027. Google Scholar [2] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706. doi: 10.1016/j.physa.2018.03.056. Google Scholar [3] A. Atangana and B. Dumitru, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. Google Scholar [4] A. Atangana and J. F. Gmez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166.Google Scholar [5] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order, Fractals, 89 (2016), 447-454. doi: 10.1016/j.chaos.2016.02.012. Google Scholar [6] T. Busso, Variable dose-response relationship between exercise training and performance, Med Sci Sports Exerc, 35 1188–1195.Google Scholar [7] T. Calvert, E. Banister, M. Savage and T. Bach, A systems model of the effects of training on physical performance, IEEE Transac-tions on Systems, Man and Cybernetics SMC-6(2), (1976), 94–102.Google Scholar [8] S. Eassom, Critical reflections on olympic ideology, Ontario: The Centre for Olympic Studies, (1994), 120–123.Google Scholar [9] A. Finn, Running with the Kenyans. p. chapter 2. Mangan, J A (2014), Sport in Latin American Society: Past and Present, (2012), 93.Google Scholar [10] G. Fulton and A. Bairner, Sport, space and national identity in ireland: The GAA, croke park and rule 42, Policy, 11 (2007), 55-74. Google Scholar [11] L. K. Ervin, A. A. Tateishi and R. V. Haroldo, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 1-9. doi: 10.3389/fphy.2017.00052. Google Scholar [12] J. A. T. Machado and A. M. Lopes, On the mathematical modeling of soccer dynamics, Communications in Nonlinear Science and Numerical Simulation, 53 (2017), 142-153. doi: 10.1016/j.cnsns.2017.04.024. Google Scholar [13] R. H. Morton, J. R. Fitz-Clarke and E. W. Banister, Modeling human performance in running, J Appl Physiol, 69 (1990), 1171-1177. Google Scholar [14] K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo–Fabrizio derivative in Riemann–Liouville sense, Fractals, 99 (2017), 171-179. doi: 10.1016/j.chaos.2017.04.008. Google Scholar [15] T. W. Calvert, E. W. Banister, M. V. Savage and T. Bach, A systems model of the effects of training on physical performance, IEEE Transactions on Systems, Man, and Cybernetics, 6 (1976), 94-102. Google Scholar [16] H. Yépez-Martínez and J. F. Gómez-Aguilar, Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 13, 17 pp. doi: 10.1051/mmnp/2018002. Google Scholar
 [1] Marilena Filippucci, Andrea Tallarico, Michele Dragoni. Simulation of lava flows with power-law rheology. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 677-685. doi: 10.3934/dcdss.2013.6.677 [2] Mehmet Yavuz, Necati Özdemir. Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 995-1006. doi: 10.3934/dcdss.2020058 [3] Frank Jochmann. Power-law approximation of Bean's critical-state model with displacement current. Conference Publications, 2011, 2011 (Special) : 747-753. doi: 10.3934/proc.2011.2011.747 [4] Ebenezer Bonyah, Samuel Kwesi Asiedu. Analysis of a Lymphatic filariasis-schistosomiasis coinfection with public health dynamics: Model obtained through Mittag-Leffler function. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 519-537. doi: 10.3934/dcdss.2020029 [5] José A. Carrillo, Yanghong Huang. Explicit equilibrium solutions for the aggregation equation with power-law potentials. Kinetic & Related Models, 2017, 10 (1) : 171-192. doi: 10.3934/krm.2017007 [6] Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 867-880. doi: 10.3934/dcdss.2020050 [7] Afaf Bouharguane. On the instability of a nonlocal conservation law. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 419-426. doi: 10.3934/dcdss.2012.5.419 [8] Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581 [9] JÓzsef Balogh, Hoi Nguyen. A general law of large permanent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5285-5297. doi: 10.3934/dcds.2017229 [10] Ambroise Vest. On the structural properties of an efficient feedback law. Evolution Equations & Control Theory, 2013, 2 (3) : 543-556. doi: 10.3934/eect.2013.2.543 [11] Jean Daniel Djida, Juan J. Nieto, Iván Area. Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 609-627. doi: 10.3934/dcdss.2020033 [12] Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229 [13] Asim Aziz, Wasim Jamshed. Unsteady MHD slip flow of non Newtonian power-law nanofluid over a moving surface with temperature dependent thermal conductivity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 617-630. doi: 10.3934/dcdss.2018036 [14] Alberto Bressan, Graziano Guerra. Shift-differentiabilitiy of the flow generated by a conservation law. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 35-58. doi: 10.3934/dcds.1997.3.35 [15] Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255 [16] Robert I. McLachlan, G. R. W. Quispel. Discrete gradient methods have an energy conservation law. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1099-1104. doi: 10.3934/dcds.2014.34.1099 [17] Julien Jimenez. Scalar conservation law with discontinuous flux in a bounded domain. Conference Publications, 2007, 2007 (Special) : 520-530. doi: 10.3934/proc.2007.2007.520 [18] Zengjing Chen, Weihuan Huang, Panyu Wu. Extension of the strong law of large numbers for capacities. Mathematical Control & Related Fields, 2019, 9 (1) : 175-190. doi: 10.3934/mcrf.2019010 [19] Roman VodiČka, Vladislav MantiČ. An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1539-1561. doi: 10.3934/dcdss.2017079 [20] Steinar Evje, Kenneth Hvistendahl Karlsen. Global weak solutions for a viscous liquid-gas model with singular pressure law. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1867-1894. doi: 10.3934/cpaa.2009.8.1867

2018 Impact Factor: 0.545