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Mittag-Leffler input stability of fractional differential equations and its applications
March  2020, 13(3): 881-888. doi: 10.3934/dcdss.2020051

## 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 and Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. 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. [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. [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. [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. [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. [6] T. Busso, Variable dose-response relationship between exercise training and performance, Med Sci Sports Exerc, 35 1188–1195. [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. [8] S. Eassom, Critical reflections on olympic ideology, Ontario: The Centre for Olympic Studies, (1994), 120–123. [9] A. Finn, Running with the Kenyans. p. chapter 2. Mangan, J A (2014), Sport in Latin American Society: Past and Present, (2012), 93. [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. [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. [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. [13] R. H. Morton, J. R. Fitz-Clarke and E. W. Banister, Modeling human performance in running, J Appl Physiol, 69 (1990), 1171-1177. [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. [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. [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.

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. [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. [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. [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. [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. [6] T. Busso, Variable dose-response relationship between exercise training and performance, Med Sci Sports Exerc, 35 1188–1195. [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. [8] S. Eassom, Critical reflections on olympic ideology, Ontario: The Centre for Olympic Studies, (1994), 120–123. [9] A. Finn, Running with the Kenyans. p. chapter 2. Mangan, J A (2014), Sport in Latin American Society: Past and Present, (2012), 93. [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. [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. [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. [13] R. H. Morton, J. R. Fitz-Clarke and E. W. Banister, Modeling human performance in running, J Appl Physiol, 69 (1990), 1171-1177. [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. [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. [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.
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