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 & Continuous Dynamical Systems - S, 2022, 15 (3) : 519-534. doi: 10.3934/dcdss.2021149
References:
[1]

L. AbrunheiroL. Machado and N. Martins, The Herglotz variational problem on spheres and its optimal control approach, J. Math. Anal., 7 (2016), 12-22.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6] R. AlmeidaS. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations, Imperial College Press, London, 2015.  doi: 10.1142/p991.  Google Scholar
[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.  Google Scholar

[8]

T. M. AtanackovićS. 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.  Google Scholar

[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.  Google Scholar

[10]

D. BaleanuS. 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.  Google Scholar

[11]

R. S. BarbosaJ. A. T. Machado and I. M. Ferreira, PID controller tuning using fractional calculus concepts, Fract. Calc. Appl. Anal., 7 (2004), 119-134.   Google Scholar

[12]

L. Q. ChenW. 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.  Google Scholar

[13]

D. Craiem and R. L. Armentano, A fractional derivative model to describe arterial viscoelasticity, Biorheology, 44 (2007), 251-263.   Google Scholar

[14]

D. CraiemF. J. RojoJ. M. AtienzaR. 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.  Google Scholar

[15]

J. W. DengL. 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.  Google Scholar

[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.  Google Scholar

[17]

A. C. GalucioJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

B. GeorgievaR. 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.  Google Scholar

[22]

G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen", Göttingen, 1930. Google Scholar

[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.  Google Scholar

[24]

C. LiW. 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.  Google Scholar

[25]

J. A. T. Machado, Analysis and design of fractional-order digital control systems, Syst. Anal. Model. Simul., 27 (1997), 107-122.   Google Scholar

[26]

J. A. T. Machado, Discrete-time fractional-order controllers, Fract. Calc. Appl. Anal., 4 (2001), 47-66.   Google Scholar

[27]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[32]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[33]

S. PoosehR. 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.  Google Scholar

[34]

F. SabzikarM. M. Meerschaert and J. H. Chen, Tempered fractional calculus, J. Comput. Phys., 293 (2015), 14-28.  doi: 10.1016/j.jcp.2014.04.024.  Google Scholar

[35]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993  Google Scholar

[36]

S. P. S. SantosN. 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.  Google Scholar

[37]

S. P. S. SantosN. 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.  Google Scholar

[38]

S. P. S. SantosN. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

L. AbrunheiroL. Machado and N. Martins, The Herglotz variational problem on spheres and its optimal control approach, J. Math. Anal., 7 (2016), 12-22.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6] R. AlmeidaS. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations, Imperial College Press, London, 2015.  doi: 10.1142/p991.  Google Scholar
[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.  Google Scholar

[8]

T. M. AtanackovićS. 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.  Google Scholar

[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.  Google Scholar

[10]

D. BaleanuS. 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.  Google Scholar

[11]

R. S. BarbosaJ. A. T. Machado and I. M. Ferreira, PID controller tuning using fractional calculus concepts, Fract. Calc. Appl. Anal., 7 (2004), 119-134.   Google Scholar

[12]

L. Q. ChenW. 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.  Google Scholar

[13]

D. Craiem and R. L. Armentano, A fractional derivative model to describe arterial viscoelasticity, Biorheology, 44 (2007), 251-263.   Google Scholar

[14]

D. CraiemF. J. RojoJ. M. AtienzaR. 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.  Google Scholar

[15]

J. W. DengL. 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.  Google Scholar

[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.  Google Scholar

[17]

A. C. GalucioJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

B. GeorgievaR. 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.  Google Scholar

[22]

G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen", Göttingen, 1930. Google Scholar

[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.  Google Scholar

[24]

C. LiW. 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.  Google Scholar

[25]

J. A. T. Machado, Analysis and design of fractional-order digital control systems, Syst. Anal. Model. Simul., 27 (1997), 107-122.   Google Scholar

[26]

J. A. T. Machado, Discrete-time fractional-order controllers, Fract. Calc. Appl. Anal., 4 (2001), 47-66.   Google Scholar

[27]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[32]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[33]

S. PoosehR. 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.  Google Scholar

[34]

F. SabzikarM. M. Meerschaert and J. H. Chen, Tempered fractional calculus, J. Comput. Phys., 293 (2015), 14-28.  doi: 10.1016/j.jcp.2014.04.024.  Google Scholar

[35]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993  Google Scholar

[36]

S. P. S. SantosN. 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.  Google Scholar

[37]

S. P. S. SantosN. 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.  Google Scholar

[38]

S. P. S. SantosN. 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.  Google Scholar

[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.  Google Scholar

Table 1.  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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