# American Institute of Mathematical Sciences

March  2022, 15(3): 519-534. doi: 10.3934/dcdss.2021149

## Optimality conditions involving the Mittag–Leffler tempered fractional derivative

 1 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal 2 Center for Computational and Stochastic Mathematics, Instituto Superior Técnico and Department of Mathematics, University of Trás-os-Montes e Alto Douro, UTAD, 5000-801, Vila Real, Portugal

* Corresponding author: Ricardo Almeida

Received  November 2019 Revised  February 2021 Published  March 2022 Early access  November 2021

Fund Project: R. Almeida is supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UIDB/04106/2020. M. L. Morgado acknowledges the financial support of the Portuguese FCT - Fundação para a Ciência e a Tecnologia, through projects UIDB/04621/2020 and UIDP/04621/2020

In this work we study problems of the calculus of the variations, where the differential operator is a generalization of the tempered fractional derivative. Different types of necessary conditions required to determine the optimal curves are proved. Problems with additional constraints are also studied. A numerical method is presented, based on discretization of the variational problem. Through several examples, we show the efficiency of the method.

Citation: Ricardo Almeida, M. Luísa Morgado. Optimality conditions involving the Mittag–Leffler tempered fractional derivative. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 519-534. doi: 10.3934/dcdss.2021149
##### References:
 [1] L. Abrunheiro, L. Machado and N. Martins, The Herglotz variational problem on spheres and its optimal control approach, J. Math. Anal., 7 (2016), 12-22. [2] O. P. Agrawal, Formulation of Euler–Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4. [3] O. P. Agrawal, Fractional variational calculus and the transversality conditions, J. Phys. A: Math. Gen., 39 (2006), 10375-10384.  doi: 10.1088/0305-4470/39/33/008. [4] R. Almeida and A. B. Malinowska, Fractional variational principle of Herglotz, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2367-2381.  doi: 10.3934/dcdsb.2014.19.2367. [5] R. Almeida and M. L. Morgado, Analysis and numerical approximation of tempered fractional calculus of variations problems, J. Comput. Appl. Math., 361 (2019), 1-12.  doi: 10.1016/j.cam.2019.04.010. [6] R. Almeida, S. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations, Imperial College Press, London, 2015.  doi: 10.1142/p991. [7] R. Almeida, D. Tavares and D. F. M. Torres, The Variable Order Fractional Calculus of Variations, SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 2019. doi: 10.1007/978-3-319-94006-9. [8] T. M. Atanacković, S. Konjik and S. Pilipović, Variational problems with fractional derivatives: Euler–Lagrange equations, J. Phys. A: Math. Theor., 41 (2008), 095201.  doi: 10.1088/1751-8113/41/9/095201. [9] B. Baeumer and M. M. Meerschaert, Tempered stable Lévy motion and transient super-diffusion, J. Comput. Appl. Math., 233 (2010), 2438-2448.  doi: 10.1016/j.cam.2009.10.027. [10] D. Baleanu, S. I. Muslih and E. M. Rabei, On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative, Nonlinear Dynam., 53 (2008), 67-74.  doi: 10.1007/s11071-007-9296-0. [11] R. S. Barbosa, J. A. T. Machado and I. M. Ferreira, PID controller tuning using fractional calculus concepts, Fract. Calc. Appl. Anal., 7 (2004), 119-134. [12] L. Q. Chen, W. J. Zhao and J. W. Zu, Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law, J. Sound Vib., 278 (2004), 861-871.  doi: 10.1016/j.jsv.2003.10.012. [13] D. Craiem and R. L. Armentano, A fractional derivative model to describe arterial viscoelasticity, Biorheology, 44 (2007), 251-263. [14] D. Craiem, F. J. Rojo, J. M. Atienza, R. L. Armentano and G. V. Guinea, Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries, Phys. Med. Biol., 53 (2008), 4543-4554.  doi: 10.1088/0031-9155/53/17/006. [15] J. W. Deng, L. J. Zhao and Y. J. Wu, Fast predictor-corrector approach for the tempered fractional ordinary differential equations, Numer. Algorithms, 74 (2017), 717-754.  doi: 10.1007/s11075-016-0169-9. [16] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010. doi: 10.1007/978-3-642-14574-2. [17] A. C. Galucio, J. F. Deu and R. Ohayon, A fractional derivative viscoelastic model for hybrid active-passive damping treatments in time domain - application to sandwich beams, J. Intell. Mater. Syst. Struct., 16 (2005), 33-45.  doi: 10.1177/1045389X05046685. [18] R. Garrappa, Numerical evaluation of two and three parameter Mittag-Leffler functions, SIAM J. Numerical Anal., 53 (2015), 1350-1369.  doi: 10.1137/140971191. [19] B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273.  doi: 10.12775/TMNA.2002.036. [20] B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314.  doi: 10.12775/TMNA.2005.034. [21] B. Georgieva, R. Guenther and T. Bodurov, Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem, J. Math. Phys., 44 (2003), 3911-3927.  doi: 10.1063/1.1597419. [22] G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen", Göttingen, 1930. [23] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B. V., Amsterdam, 2006. [24] C. Li, W. Deng and L. Zhao, Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1989-2015.  doi: 10.3934/dcdsb.2019026. [25] J. A. T. Machado, Analysis and design of fractional-order digital control systems, Syst. Anal. Model. Simul., 27 (1997), 107-122. [26] J. A. T. Machado, Discrete-time fractional-order controllers, Fract. Calc. Appl. Anal., 4 (2001), 47-66. [27] R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, 2006. [28] A. B. Malinowska, T. Odzijewicz and D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations, SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 2015. [29] A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012.  doi: 10.1142/p871. [30] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, Wiley, New York, 1993. [31] T. F. Nonnenmacher and R. Metzler, Applications of fractional calculus techniques to problems in biophysics, Applications of Fractional Calculus in Physics, (2000), 377–428. doi: 10.1142/9789812817747_0008. [32] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. [33] S. Pooseh, R. Almeida and D. F. M. Torres, Discrete direct methods in the fractional calculus of variations, Comput. Math. Appl., 66 (2013), 668-676.  doi: 10.1016/j.camwa.2013.01.045. [34] F. Sabzikar, M. M. Meerschaert and J. H. Chen, Tempered fractional calculus, J. Comput. Phys., 293 (2015), 14-28.  doi: 10.1016/j.jcp.2014.04.024. [35] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993 [36] S. P. S. Santos, N. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz type, Vietnam J. Math., 42 (2014), 409-419.  doi: 10.1007/s10013-013-0048-9. [37] S. P. S. Santos, N. Martins and D. F. M. Torres, Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem, Discrete Contin. Dyn. Syst., 35 (2015), 4593-4610.  doi: 10.3934/dcds.2015.35.4593. [38] S. P. S. Santos, N. Martins and D. F. M. Torres, Noether currents for higher-order variational problems of Herglotz type with time delay, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 91-102.  doi: 10.3934/dcdss.2018006. [39] F. Zeng and C. Li, A new Crank–Nicolson finite element method for the time-fractional subdiffusion equation, Appl. Num. Mathem., 121 (2017) 82–95. doi: 10.1016/j.apnum.2017.06.011.

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##### References:
 [1] L. Abrunheiro, L. Machado and N. Martins, The Herglotz variational problem on spheres and its optimal control approach, J. Math. Anal., 7 (2016), 12-22. [2] O. P. Agrawal, Formulation of Euler–Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4. [3] O. P. Agrawal, Fractional variational calculus and the transversality conditions, J. Phys. A: Math. Gen., 39 (2006), 10375-10384.  doi: 10.1088/0305-4470/39/33/008. [4] R. Almeida and A. B. Malinowska, Fractional variational principle of Herglotz, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2367-2381.  doi: 10.3934/dcdsb.2014.19.2367. [5] R. Almeida and M. L. Morgado, Analysis and numerical approximation of tempered fractional calculus of variations problems, J. Comput. Appl. Math., 361 (2019), 1-12.  doi: 10.1016/j.cam.2019.04.010. [6] R. Almeida, S. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations, Imperial College Press, London, 2015.  doi: 10.1142/p991. [7] R. Almeida, D. Tavares and D. F. M. Torres, The Variable Order Fractional Calculus of Variations, SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 2019. doi: 10.1007/978-3-319-94006-9. [8] T. M. Atanacković, S. Konjik and S. Pilipović, Variational problems with fractional derivatives: Euler–Lagrange equations, J. Phys. A: Math. Theor., 41 (2008), 095201.  doi: 10.1088/1751-8113/41/9/095201. [9] B. Baeumer and M. M. Meerschaert, Tempered stable Lévy motion and transient super-diffusion, J. Comput. Appl. Math., 233 (2010), 2438-2448.  doi: 10.1016/j.cam.2009.10.027. [10] D. Baleanu, S. I. Muslih and E. M. Rabei, On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative, Nonlinear Dynam., 53 (2008), 67-74.  doi: 10.1007/s11071-007-9296-0. [11] R. S. Barbosa, J. A. T. Machado and I. M. Ferreira, PID controller tuning using fractional calculus concepts, Fract. Calc. Appl. Anal., 7 (2004), 119-134. [12] L. Q. Chen, W. J. Zhao and J. W. Zu, Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law, J. Sound Vib., 278 (2004), 861-871.  doi: 10.1016/j.jsv.2003.10.012. [13] D. Craiem and R. L. Armentano, A fractional derivative model to describe arterial viscoelasticity, Biorheology, 44 (2007), 251-263. [14] D. Craiem, F. J. Rojo, J. M. Atienza, R. L. Armentano and G. V. Guinea, Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries, Phys. Med. Biol., 53 (2008), 4543-4554.  doi: 10.1088/0031-9155/53/17/006. [15] J. W. Deng, L. J. Zhao and Y. J. Wu, Fast predictor-corrector approach for the tempered fractional ordinary differential equations, Numer. Algorithms, 74 (2017), 717-754.  doi: 10.1007/s11075-016-0169-9. [16] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010. doi: 10.1007/978-3-642-14574-2. [17] A. C. Galucio, J. F. Deu and R. Ohayon, A fractional derivative viscoelastic model for hybrid active-passive damping treatments in time domain - application to sandwich beams, J. Intell. Mater. Syst. Struct., 16 (2005), 33-45.  doi: 10.1177/1045389X05046685. [18] R. Garrappa, Numerical evaluation of two and three parameter Mittag-Leffler functions, SIAM J. Numerical Anal., 53 (2015), 1350-1369.  doi: 10.1137/140971191. [19] B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273.  doi: 10.12775/TMNA.2002.036. [20] B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314.  doi: 10.12775/TMNA.2005.034. [21] B. Georgieva, R. Guenther and T. Bodurov, Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem, J. Math. Phys., 44 (2003), 3911-3927.  doi: 10.1063/1.1597419. [22] G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen", Göttingen, 1930. [23] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B. V., Amsterdam, 2006. [24] C. Li, W. Deng and L. Zhao, Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1989-2015.  doi: 10.3934/dcdsb.2019026. [25] J. A. T. Machado, Analysis and design of fractional-order digital control systems, Syst. Anal. Model. Simul., 27 (1997), 107-122. [26] J. A. T. Machado, Discrete-time fractional-order controllers, Fract. Calc. Appl. Anal., 4 (2001), 47-66. [27] R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, 2006. [28] A. B. Malinowska, T. Odzijewicz and D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations, SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 2015. [29] A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012.  doi: 10.1142/p871. [30] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, Wiley, New York, 1993. [31] T. F. Nonnenmacher and R. Metzler, Applications of fractional calculus techniques to problems in biophysics, Applications of Fractional Calculus in Physics, (2000), 377–428. doi: 10.1142/9789812817747_0008. [32] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. [33] S. Pooseh, R. Almeida and D. F. M. Torres, Discrete direct methods in the fractional calculus of variations, Comput. Math. Appl., 66 (2013), 668-676.  doi: 10.1016/j.camwa.2013.01.045. [34] F. Sabzikar, M. M. Meerschaert and J. H. Chen, Tempered fractional calculus, J. Comput. Phys., 293 (2015), 14-28.  doi: 10.1016/j.jcp.2014.04.024. [35] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993 [36] S. P. S. Santos, N. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz type, Vietnam J. Math., 42 (2014), 409-419.  doi: 10.1007/s10013-013-0048-9. [37] S. P. S. Santos, N. Martins and D. F. M. Torres, Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem, Discrete Contin. Dyn. Syst., 35 (2015), 4593-4610.  doi: 10.3934/dcds.2015.35.4593. [38] S. P. S. Santos, N. Martins and D. F. M. Torres, Noether currents for higher-order variational problems of Herglotz type with time delay, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 91-102.  doi: 10.3934/dcdss.2018006. [39] F. Zeng and C. Li, A new Crank–Nicolson finite element method for the time-fractional subdiffusion equation, Appl. Num. Mathem., 121 (2017) 82–95. doi: 10.1016/j.apnum.2017.06.011.
Maximum of the committed absolute error in the approximation of the solution
 $N$ 5 10 20 40 $E$ 8.81 $\times 10^{-3}$ 3.61 $\times 10^{-3}$ 1.40 $\times 10^{-3}$ 5.24 $\times 10^{-4}$
 $N$ 5 10 20 40 $E$ 8.81 $\times 10^{-3}$ 3.61 $\times 10^{-3}$ 1.40 $\times 10^{-3}$ 5.24 $\times 10^{-4}$
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