doi: 10.3934/dcdss.2020027

Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application

Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

* Permanent address: Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt. Email: Bahaa_gm@yahoo.com

Received  March 2018 Revised  May 2018 Published  March 2019

Fund Project: The author is supported by Taibah University, Dean of Scientific Research

In this paper, the generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's Derivatives are discussed. We consider the Hilfers generalized fractional derivative that in sense Atangana-Baleanu derivatives. We develop integration by parts formulas for the generalized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variational formulations can be obtained by setting different parameters to different values. The fractional Euler-Lagrange equations of fractional Lagrangians for constrained systems contains a fractional Hilfer-Atangana-Baleanu's derivatives with multi parameters are investigated. We also define fractional generalized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed. We present a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP.

Citation: G. M. Bahaa. Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020027
References:
[1]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Diff. Eq., 2016 (2016), Paper No. 232, 18 pp. doi: 10.1186/s13662-016-0949-5.

[2]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098-1107. doi: 10.22436/jnsa.010.03.20.

[3]

O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272 (2002), 368-379. doi: 10.1016/S0022-247X(02)00180-4.

[4]

O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynamics, 38 (2004), 323-337. doi: 10.1007/s11071-004-3764-6.

[5]

O. P. Agrawal and D. Baleanu, Hamiltonian formulation and direct numerical scheme for fractional optimal control problems, Journal of Vibration and Control, 13 (2007), 1269-1281. doi: 10.1177/1077546307077467.

[6]

R. P. AgarwalS. Baghli and M. Benchohra, Controllability for semilinear functional and neutral functional evolution equations with infinite delay in Frchet spaces, Appl. Math. Optim., 60 (2009), 253-274. doi: 10.1007/s00245-009-9073-1.

[7]

B. S. T. Alkahtani, Chua's circuit model with Atangana–Baleanu derivative with fractional order, Chaos, Solitons, Fractals, 89 (2016), 547-551.

[8]

N. Al-SaltiE. Karimov and K. Sadarangani, On a differential equation with caputo-fabrizio fractional derivative of order $1 < \beta\leq2$ and application to Mass-Spring-Damper system, Progr. Fract. Differ. Appl., 2 (2016), 257-263.

[9]

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.

[10]

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.

[11]

A. Atangana and J. F. Gomez Aguila, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3.

[12]

G. M. Bahaa, Fractional optimal control problem for variational inequalities with control constraints, IMA J. Math. Control and Inform., 35 (2018), 107-122. doi: 10.1093/imamci/dnw040.

[13]

G. M. Bahaa, Fractional optimal control problem for differential system with control constraints, Filomat, 30 (2016), 2177-2189. doi: 10.2298/FIL1608177B.

[14]

G. M. Bahaa, Fractional optimal control problem for infinite order system with control constraints, Adv. Diff. Eq., 2016 (2016), Paper No. 250, 16 pp. doi: 10.1186/s13662-016-0976-2.

[15]

G. M. Bahaa, Fractional optimal control problem for differential system with delay argument, Adv. Diff. Eq., 2017 (2017), Paper No. 69, 19 pp. doi: 10.1186/s13662-017-1121-6.

[16]

G. M. Bahaa, Fractional optimal control problem for variable-order differential systems, Fract. Calc. Appl. Anal., 20 (2017), 1447-1470. doi: 10.1515/fca-2017-0076.

[17]

D. Baleanu and S. I. Muslih, Lagrangian formulation on classical fields within Riemann-Liouville fractional derivatives, Phys. Scr., 72 (2005), 119-121. doi: 10.1238/Physica.Regular.072a00119.

[18]

D. Baleanu and T. Avkar, Lagrangian with linear velocities within Riemann-Liouville fractional derivatives, Nuovo Cimnto B, 119 (2004), 73-79.

[19]

D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 444–462, arXiv: 1712.01762v1, (2017). doi: 10.1016/j.cnsns.2017.12.003.

[20]

D. Baleanu and OM. P. Agrawal, Fractional Hamilton formalism within Caputo's derivative, Czechoslovak Journal of Physics, 56 (2000), 1087-1092. doi: 10.1007/s10582-006-0406-x.

[21]

D. BaleanuA. Jajarmi and M. Hajipour, A new formulation of the fractional optimal control problems involving Mittag Leffler nonsingular kernel, J Optim. Theory. Appl., 175 (2017), 718-737. doi: 10.1007/s10957-017-1186-0.

[22]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr Fract. Differ. Appl., 1 (2015), 73-75.

[23]

J. D. DjidaA. Atangana and I. Area, Numerical Computation of a Fractional Derivative with Non-Local and Non-Singular Kernel, Math. Model. Nat. Phenom., 12 (2017), 4-13. doi: 10.1051/mmnp/201712302.

[24]

J. D. Djida, G. M. Mophou and I. Area, Optimal control of diffusion equation with fractional time derivative with nonlocal and nonsingular mittag-leffler kernel, Journal of Optimization Theory and Applications, (2018), 1–18, arXiv: 1711.09070. doi: 10.1007/s10957-018-1305-6.

[25]

A. M. A. El-Sayed, On the stochastic fractional calculus operators, Journal of Fractional Calculus and Applications, 6 (2015), 101-109.

[26]

K. Ervin LenziA. Tateishi Angel and V. Haroldo Ribeiro, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 1-9.

[27]

S. F. Frederico Gastao and F. M. Torres Delfim, Fractional optimal control in the sense of Caputo and the fractional Noethers theorem., Int. Math. Forum, 3 (2008), 479-493.

[28]

J. F. Gomez-Aguilar, Irving-Mullineux oscillator via fractional derivatives with Mittag Leffler kernel, Chaos Soliton. Fract., 95 (2017), 179-186. doi: 10.1016/j.chaos.2016.12.025.

[29]

J. F. Gomez-Aguilar, Space time fractional diffusion equation using a derivative with nonsingular and regular kernel, Phys. A, 465 (2017), 562-572. doi: 10.1016/j.physa.2016.08.072.

[30]

J. F. Gomez-AguilarA. Atangana and J. F. Morales-Delgado, Electrical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives, Int. J. Circ. Theor. Appl., 45 (2017), 1514-1533. doi: 10.1002/cta.2348.

[31]

F. M. HafezA. M. A. El-Sayed and M. A. El-Tawil, On a stochastic fractional calculus, Fractional Calculus and Applied Analysis, 4 (2001), 81-90.

[32]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., Singapore, 2000. doi: 10.1142/9789812817747.

[33]

R. Hilfer, Threefold introduction to fractional derivatives, Anomalous Transport: Foundations and Applications, (2008), 17–73. doi: 10.1002/9783527622979.ch2.

[34]

J. Hristov, Transient heat diffusion with a non-singular fading memory, Therm Sci., 20 (2016), 757-762.

[35]

F. JaradT. Maraba and D. Baleanu, Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62 (2010), 609-614. doi: 10.1007/s11071-010-9748-9.

[36]

F. JaradT. Maraba and D. Baleanu, Higher order fractional variational optimal control problems with delayed arguments, Applied Mathematics and Computation, 218 (2012), 9234-9240. doi: 10.1016/j.amc.2012.02.080.

[37]

A. A. KilbasM. Saigo and K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Int. Tran. Spec. Funct., 15 (2004), 31-49. doi: 10.1080/10652460310001600717.

[38]

A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists. New York: Springer, 2008. doi: 10.1007/978-0-387-75894-7.

[39]

G. M. Mophou, Optimal control of fractional diffusion equation, Computers and Mathematics with Applications, 61 (2011), 68-78. doi: 10.1016/j.camwa.2010.10.030.

[40]

G. M. Mophou, Optimal control of fractional diffusion equation with state constraints, Computers and Mathematics with Applications, 62 (2011), 1413-1426. doi: 10.1016/j.camwa.2011.04.044.

[41]

N. OzdemirD. Karadeniz and B. B. Iskender, Fractional optimal control problem of a distributed system in cylindrical coordinates, Phys. Lett. A, 373 (2009), 221-226. doi: 10.1016/j.physleta.2008.11.019.

[42]

F. Riewe, Nonconservative Lagrangian and Hamiltonian Mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: 10.1103/PhysRevE.53.1890.

[43]

F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E, 55 (1997), 3581-3592. doi: 10.1103/PhysRevE.55.3581.

[44]

N. A. Sheikh, F. Ali, M. Saqib, et al. Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results Phys., 7 (2017), 789–800. doi: 10.1016/j.rinp.2017.01.025.

show all references

References:
[1]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Diff. Eq., 2016 (2016), Paper No. 232, 18 pp. doi: 10.1186/s13662-016-0949-5.

[2]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098-1107. doi: 10.22436/jnsa.010.03.20.

[3]

O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272 (2002), 368-379. doi: 10.1016/S0022-247X(02)00180-4.

[4]

O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynamics, 38 (2004), 323-337. doi: 10.1007/s11071-004-3764-6.

[5]

O. P. Agrawal and D. Baleanu, Hamiltonian formulation and direct numerical scheme for fractional optimal control problems, Journal of Vibration and Control, 13 (2007), 1269-1281. doi: 10.1177/1077546307077467.

[6]

R. P. AgarwalS. Baghli and M. Benchohra, Controllability for semilinear functional and neutral functional evolution equations with infinite delay in Frchet spaces, Appl. Math. Optim., 60 (2009), 253-274. doi: 10.1007/s00245-009-9073-1.

[7]

B. S. T. Alkahtani, Chua's circuit model with Atangana–Baleanu derivative with fractional order, Chaos, Solitons, Fractals, 89 (2016), 547-551.

[8]

N. Al-SaltiE. Karimov and K. Sadarangani, On a differential equation with caputo-fabrizio fractional derivative of order $1 < \beta\leq2$ and application to Mass-Spring-Damper system, Progr. Fract. Differ. Appl., 2 (2016), 257-263.

[9]

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.

[10]

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.

[11]

A. Atangana and J. F. Gomez Aguila, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3.

[12]

G. M. Bahaa, Fractional optimal control problem for variational inequalities with control constraints, IMA J. Math. Control and Inform., 35 (2018), 107-122. doi: 10.1093/imamci/dnw040.

[13]

G. M. Bahaa, Fractional optimal control problem for differential system with control constraints, Filomat, 30 (2016), 2177-2189. doi: 10.2298/FIL1608177B.

[14]

G. M. Bahaa, Fractional optimal control problem for infinite order system with control constraints, Adv. Diff. Eq., 2016 (2016), Paper No. 250, 16 pp. doi: 10.1186/s13662-016-0976-2.

[15]

G. M. Bahaa, Fractional optimal control problem for differential system with delay argument, Adv. Diff. Eq., 2017 (2017), Paper No. 69, 19 pp. doi: 10.1186/s13662-017-1121-6.

[16]

G. M. Bahaa, Fractional optimal control problem for variable-order differential systems, Fract. Calc. Appl. Anal., 20 (2017), 1447-1470. doi: 10.1515/fca-2017-0076.

[17]

D. Baleanu and S. I. Muslih, Lagrangian formulation on classical fields within Riemann-Liouville fractional derivatives, Phys. Scr., 72 (2005), 119-121. doi: 10.1238/Physica.Regular.072a00119.

[18]

D. Baleanu and T. Avkar, Lagrangian with linear velocities within Riemann-Liouville fractional derivatives, Nuovo Cimnto B, 119 (2004), 73-79.

[19]

D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 444–462, arXiv: 1712.01762v1, (2017). doi: 10.1016/j.cnsns.2017.12.003.

[20]

D. Baleanu and OM. P. Agrawal, Fractional Hamilton formalism within Caputo's derivative, Czechoslovak Journal of Physics, 56 (2000), 1087-1092. doi: 10.1007/s10582-006-0406-x.

[21]

D. BaleanuA. Jajarmi and M. Hajipour, A new formulation of the fractional optimal control problems involving Mittag Leffler nonsingular kernel, J Optim. Theory. Appl., 175 (2017), 718-737. doi: 10.1007/s10957-017-1186-0.

[22]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr Fract. Differ. Appl., 1 (2015), 73-75.

[23]

J. D. DjidaA. Atangana and I. Area, Numerical Computation of a Fractional Derivative with Non-Local and Non-Singular Kernel, Math. Model. Nat. Phenom., 12 (2017), 4-13. doi: 10.1051/mmnp/201712302.

[24]

J. D. Djida, G. M. Mophou and I. Area, Optimal control of diffusion equation with fractional time derivative with nonlocal and nonsingular mittag-leffler kernel, Journal of Optimization Theory and Applications, (2018), 1–18, arXiv: 1711.09070. doi: 10.1007/s10957-018-1305-6.

[25]

A. M. A. El-Sayed, On the stochastic fractional calculus operators, Journal of Fractional Calculus and Applications, 6 (2015), 101-109.

[26]

K. Ervin LenziA. Tateishi Angel and V. Haroldo Ribeiro, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 1-9.

[27]

S. F. Frederico Gastao and F. M. Torres Delfim, Fractional optimal control in the sense of Caputo and the fractional Noethers theorem., Int. Math. Forum, 3 (2008), 479-493.

[28]

J. F. Gomez-Aguilar, Irving-Mullineux oscillator via fractional derivatives with Mittag Leffler kernel, Chaos Soliton. Fract., 95 (2017), 179-186. doi: 10.1016/j.chaos.2016.12.025.

[29]

J. F. Gomez-Aguilar, Space time fractional diffusion equation using a derivative with nonsingular and regular kernel, Phys. A, 465 (2017), 562-572. doi: 10.1016/j.physa.2016.08.072.

[30]

J. F. Gomez-AguilarA. Atangana and J. F. Morales-Delgado, Electrical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives, Int. J. Circ. Theor. Appl., 45 (2017), 1514-1533. doi: 10.1002/cta.2348.

[31]

F. M. HafezA. M. A. El-Sayed and M. A. El-Tawil, On a stochastic fractional calculus, Fractional Calculus and Applied Analysis, 4 (2001), 81-90.

[32]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., Singapore, 2000. doi: 10.1142/9789812817747.

[33]

R. Hilfer, Threefold introduction to fractional derivatives, Anomalous Transport: Foundations and Applications, (2008), 17–73. doi: 10.1002/9783527622979.ch2.

[34]

J. Hristov, Transient heat diffusion with a non-singular fading memory, Therm Sci., 20 (2016), 757-762.

[35]

F. JaradT. Maraba and D. Baleanu, Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62 (2010), 609-614. doi: 10.1007/s11071-010-9748-9.

[36]

F. JaradT. Maraba and D. Baleanu, Higher order fractional variational optimal control problems with delayed arguments, Applied Mathematics and Computation, 218 (2012), 9234-9240. doi: 10.1016/j.amc.2012.02.080.

[37]

A. A. KilbasM. Saigo and K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Int. Tran. Spec. Funct., 15 (2004), 31-49. doi: 10.1080/10652460310001600717.

[38]

A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists. New York: Springer, 2008. doi: 10.1007/978-0-387-75894-7.

[39]

G. M. Mophou, Optimal control of fractional diffusion equation, Computers and Mathematics with Applications, 61 (2011), 68-78. doi: 10.1016/j.camwa.2010.10.030.

[40]

G. M. Mophou, Optimal control of fractional diffusion equation with state constraints, Computers and Mathematics with Applications, 62 (2011), 1413-1426. doi: 10.1016/j.camwa.2011.04.044.

[41]

N. OzdemirD. Karadeniz and B. B. Iskender, Fractional optimal control problem of a distributed system in cylindrical coordinates, Phys. Lett. A, 373 (2009), 221-226. doi: 10.1016/j.physleta.2008.11.019.

[42]

F. Riewe, Nonconservative Lagrangian and Hamiltonian Mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: 10.1103/PhysRevE.53.1890.

[43]

F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E, 55 (1997), 3581-3592. doi: 10.1103/PhysRevE.55.3581.

[44]

N. A. Sheikh, F. Ali, M. Saqib, et al. Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results Phys., 7 (2017), 789–800. doi: 10.1016/j.rinp.2017.01.025.

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