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MittagLeffler input stability of fractional differential equations and its applications
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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 
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 MittagLeffler law. We derived the exact solutions using the Laplace transform for the nondelay version.
References:
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A. Atangana, Fractalfractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Fractals, 102 (2017), 396406. doi: 10.1016/j.chaos.2017.04.027. 
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A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and nonMarkovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688706. doi: 10.1016/j.physa.2018.03.056. 
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A. Atangana and B. Dumitru, New fractional derivatives with nonlocal and nonsingular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763769. 
[4] 
A. Atangana and J. F. GmezAguilar, 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), 447454. doi: 10.1016/j.chaos.2016.02.012. 
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T. Busso, Variable doseresponse 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 Transactions on Systems, Man and Cybernetics SMC6(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), 5574. 
[11] 
L. K. Ervin, A. A. Tateishi and R. V. Haroldo, The role of fractional timederivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 19. 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), 142153. doi: 10.1016/j.cnsns.2017.04.024. 
[13] 
R. H. Morton, J. R. FitzClarke and E. W. Banister, Modeling human performance in running, J Appl Physiol, 69 (1990), 11711177. 
[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), 171179. 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), 94102. 
[16] 
H. YépezMartínez and J. F. GómezAguilar, Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and MittagLeffler kernel, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 13, 17 pp. doi: 10.1051/mmnp/2018002. 
show all references
References:
[1] 
A. Atangana, Fractalfractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Fractals, 102 (2017), 396406. 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 nonMarkovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688706. doi: 10.1016/j.physa.2018.03.056. 
[3] 
A. Atangana and B. Dumitru, New fractional derivatives with nonlocal and nonsingular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763769. 
[4] 
A. Atangana and J. F. GmezAguilar, 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), 447454. doi: 10.1016/j.chaos.2016.02.012. 
[6] 
T. Busso, Variable doseresponse 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 Transactions on Systems, Man and Cybernetics SMC6(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), 5574. 
[11] 
L. K. Ervin, A. A. Tateishi and R. V. Haroldo, The role of fractional timederivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 19. 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), 142153. doi: 10.1016/j.cnsns.2017.04.024. 
[13] 
R. H. Morton, J. R. FitzClarke and E. W. Banister, Modeling human performance in running, J Appl Physiol, 69 (1990), 11711177. 
[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), 171179. 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), 94102. 
[16] 
H. YépezMartínez and J. F. GómezAguilar, Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and MittagLeffler kernel, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 13, 17 pp. doi: 10.1051/mmnp/2018002. 
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