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Fractional Derivatives with Mittag-Leffler Kernel pp 35–47Cite as

Variable Order Mittag–Leffler Fractional Operators on Isolated Time Scales and Application to the Calculus of Variations

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Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 194))

Abstract

We introduce new fractional operators of variable order in isolated time scales with Mittag–Leffler kernels. This allows a general formulation of a class of fractional variational problems involving variable-order difference operators. Main results give fractional integration by parts formulas and necessary optimality conditions of Euler–Lagrange type.

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References

  1. Abdeljawad, T.: On delta and nabla Caputo fractional differences and dual identities. Discret. Dyn. Nat. Soc. 1, 1–12 (2013)

    MathSciNet  Google Scholar 

  2. Abdeljawad, T., Baleanu, D.: Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels. Adv. Differ. Equ. 1, 1–18 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Abdeljawad, T., Baleanu, D.: On fractional derivatives with exponential kernel and their discrete versions. Rep. Math. Phys. 80(1), 11–27 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abdeljawad, T., Jarad, F., Baleanu, D.: A semigroup-like property for discrete Mittag-Leffler functions. Adv. Differ. Equ. 72, 1–7 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Almeida, R., Tavares, D., Torres, D.F.M.: The Variable-Order Fractional Calculus of Variations. Springer Briefs in Applied Sciences and Technology. Springer, Cham (2018)

    MATH  Google Scholar 

  6. Atanackovic, T.M., Pilipovic, S.: Hamilton’s principle with variable order fractional derivatives. Fract. Calc. Appl. Anal. 14(1), 94–109 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Atangana, A.: Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology. Academic, London (2018)

    MATH  Google Scholar 

  8. Atangana, A., Baleanu, D.: New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)

    Article  Google Scholar 

  9. Atangana, A., Gómez-Aguilar, J.F.: A new derivative with normal distribution kernel: theory, methods and applications. Phys. A Stat. Mech. Appl. 476, 1–14 (2017)

    Article  MathSciNet  Google Scholar 

  10. Atangana, A., Gómez-Aguilar, J.F.: Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus 133, 1–22 (2018)

    Article  Google Scholar 

  11. Atangana, A., Gómez-Aguilar, J.F.: Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 114, 516–535 (2019)

    Article  MathSciNet  Google Scholar 

  12. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus. Series on Complexity, Nonlinearity and Chaos, vol. 3. World Scientific Publishing Co. Pte. Ltd., Hackensack (2012)

    Google Scholar 

  13. Bastos, N.R.O., Ferreira, R.A.C., Torres, D.F.M.: Necessary optimality conditions for fractional difference problems of the calculus of variations. Discret. Contin. Dyn. Syst. 29(2), 417–437 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bastos, N.R.O., Ferreira, R.A.C., Torres, D.F.M.: Discrete-time fractional variational problems. Signal Process. 91(3), 513–524 (2011)

    Article  MATH  Google Scholar 

  15. Bastos, N.R.O., Mozyrska, D., Torres, D.F.M.: Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform. Int. J. Math. Comput. 11, 1–9 (2011)

    MathSciNet  Google Scholar 

  16. Bayour, B., Torres, D.F.M.: Complex-valued fractional derivatives on time scales. In: Differential and Difference Equations with Applications. Springer Proceedings in Mathematics and Statistics, vol. 164, pp. 79–87 (2015)

    Chapter  Google Scholar 

  17. Benkhettou, N., Brito da Cruz, A.M.C., Torres, D.F.M.: Nonsymmetric and symmetric fractional calculi on arbitrary nonempty closed sets. Math. Methods Appl. Sci. 39(2), 261–279 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Benkhettou, N., Hammoudi, A., Torres, D.F.M.: Existence and uniqueness of solution for a fractional Riemann-Liouville initial value problem on time scales. J. King Saud Univ. Sci. 28(1), 87–92 (2016)

    Article  Google Scholar 

  19. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015)

    Google Scholar 

  20. Caputo, M., Fabrizio, M.: Applications of new time and spatial fractional derivatives with exponential kernels. Prog. Fract. Differ. Appl. 2(1), 1–11 (2016)

    Article  Google Scholar 

  21. Coimbra, C.F.M.: Mechanics with variable-order differential operators. Ann. Phys. 12(11–12), 692–703 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Coronel-Escamilla, A., Gómez-Aguilar, J.F., Torres, L., Escobar-Jiménez, R.F., Valtierra-Rodríguez, M.: Synchronization of chaotic systems involving fractional operators of Liouville-Caputo type with variable-order. Phys. A Stat. Mech. Appl. 487, 1–21 (2017)

    Article  MathSciNet  Google Scholar 

  23. Coronel-Escamilla, A., Gómez-Aguilar, J.F., Torres, L., Valtierra-Rodríguez, M., Escobar-Jiménez, R.F.: Design of a state observer to approximate signals by using the concept of fractional variable-order derivative. Digit. Signal Process. 69, 127–139 (2017)

    Article  Google Scholar 

  24. Coronel-Escamilla, A., Gómez-Aguilar, J.F., Torres, L., Escobar-Jiménez, R.F.: A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys. A Stat. Mech. Appl. 491, 406–424 (2018)

    Article  MathSciNet  Google Scholar 

  25. Diaz, G., Coimbra, C.F.M.: Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation. Nonlinear Dyn. 56(1–2), 145–157 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gómez-Aguilar, J.F.: Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations. Phys. A Stat. Mech. Appl. 494, 52–75 (2018)

    Article  MathSciNet  Google Scholar 

  27. Gómez-Aguilar, J.F.: Novel analytical solutions of the fractional Drude model. Optik 168, 728–740 (2018)

    Article  Google Scholar 

  28. Gómez-Aguilar, J.F., Atangana, A.: New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications. Eur. Phys. J. Plus 132(1), 1–13 (2017)

    Article  Google Scholar 

  29. Gómez-Aguilar, J.F., Escobar-Jiménez, R.F., López-López, M.G., Alvarado-Martínez, V.M.: Analysis of projectile motion: a comparative study using fractional operators with power law, exponential decay and Mittag-Leffler kernel. Eur. Phys. J. Plus 133(3), 1–26 (2018)

    Article  Google Scholar 

  30. Goodrich, C., Peterson, A.C.: Discrete Fractional Calculus. Springer, Cham (2015)

    Book  MATH  Google Scholar 

  31. Hang, Y., Liu, Y., Xu, X., Chen, Y., Mo, S.: Sensitivity analysis based on Markovian integration by parts formula. Math. Comput. Appl. 22(4), 1–12 (2017)

    MathSciNet  Google Scholar 

  32. Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29(1–4), 57–98 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87–92 (2015)

    Google Scholar 

  34. Morales-Delgado, V.F., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: On the solutions of fractional order of evolution equations. Eur. Phys. J. Plus 132(1), 1–17 (2017)

    Article  Google Scholar 

  35. Nelsen, R.B.: Proof without words: integration by parts. Math. Mag. 64(2), 1–13 (1991)

    Article  MathSciNet  Google Scholar 

  36. Pei, B., Xu, Y., Yin, G., Zhang, X.: Averaging principles for functional stochastic partial differential equations driven by a fractional Brownian motion modulated by two-time-scale Markovian switching processes. Nonlinear Anal. Hybrid Syst. 27, 107–124 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ramirez, L.E.S., Coimbra, C.F.M.: On the selection and meaning of variable order operators for dynamic modeling. Int. J. Differ. Equations 1, 1–16 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ramirez, L.E.S., Coimbra, C.F.M.: On the variable order dynamics of the nonlinear wake caused by a sedimenting particle. Phys. D 240(13), 1111–1118 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ross, B., Samko, S.: Fractional integration operator of variable order in the Hölder spaces \(H^{\lambda (x)}\). Int. J. Math. Math. Sci. 18(4), 777–788 (1995)

    MATH  Google Scholar 

  40. Samko, S.G.: Fractional integration and differentiation of variable order. Anal. Math. 21(3), 213–236 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  41. Samko, S.G., Ross, B.: Integration and differentiation to a variable fractional order. Integr. Transform. Spec. Funct. 1(4), 277–300 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  42. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives (translated from the 1987 Russian original). Gordon and Breach Science Publishers, Yverdon (1993)

    MATH  Google Scholar 

  43. Sheng, K., Zhang, W., Bai, Z.: Positive solutions to fractional boundary-value problems with p-Laplacian on time scales. Bound. Value Probl. 1(70), 1–17 (2018)

    MathSciNet  Google Scholar 

  44. Solís-Pérez, J.E., Gómez-Aguilar, J.F., Atangana, A.: Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws. Chaos Solitons Fractals 114, 175–185 (2018)

    Article  MathSciNet  Google Scholar 

  45. Tang, J.: Computation of an infinite integral using integration by parts. Math. Methods Appl. Sci. 41(3), 929–935 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  46. Tavares, D., Almeida, R., Torres, D.F.M.: Caputo derivatives of fractional variable order: numerical approximations. Commun. Nonlinear Sci. Numer. Simul. 3, 69–87 (2016)

    Article  MathSciNet  Google Scholar 

  47. Tavares, D., Almeida, R., Torres, D.F.M.: Constrained fractional variational problems of variable order. IEEE/CAA J. Autom. Sin. 4(1), 80–88 (2017)

    Article  MathSciNet  Google Scholar 

  48. Tavares, D., Almeida, R., Torres, D.F.M.: Fractional Herglotz variational problems of variable order. Discret. Contin. Dyn. Syst. Ser. S 11(1), 143–154 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  49. Tavares, D., Almeida, R., Torres, D.F.M.: Combined fractional variational problems of variable order and some computational aspects. J. Comput. Appl. Math. 339, 374–388 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  50. Zúñiga-Aguilar, C.J., Romero-Ugalde, H.M., Gómez-Aguilar, J.F., Escobar-Jiménez, R.F., Valtierra-Rodríguez, M.: Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks. Chaos Solitons Fractals 103, 382–403 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Abdeljawad is grateful to Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), number RG-DES-2017-01-17; Torres to the support of FCT within the R&D unit CIDMA, UID/MAT/04106/2019.

Author information

Authors and Affiliations

  1. Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh, 11586, Saudi Arabia

    Thabet Abdeljawad

  2. Mechatronic Engineering Department, University of Turkish Aeronautical Association, 06790, Ankara, Turkey

    Raziye Mert

  3. Department of Mathematics, Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193, Aveiro, Portugal

    Delfim F. M. Torres

Authors
  1. Thabet Abdeljawad
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  2. Raziye Mert
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  3. Delfim F. M. Torres
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Correspondence to Thabet Abdeljawad .

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Editors and Affiliations

  1. CONACYT-Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    José Francisco Gómez

  2. CONACYT-Instituto de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Lizeth Torres

  3. Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    Ricardo Fabricio Escobar

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Abdeljawad, T., Mert, R., Torres, D.F.M. (2019). Variable Order Mittag–Leffler Fractional Operators on Isolated Time Scales and Application to the Calculus of Variations. In: Gómez, J., Torres, L., Escobar, R. (eds) Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control, vol 194. Springer, Cham. https://doi.org/10.1007/978-3-030-11662-0_3

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  • DOI: https://doi.org/10.1007/978-3-030-11662-0_3

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  • Online ISBN: 978-3-030-11662-0

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