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A robust stability criterion on the time-conformable fractional heat equation in a axisymmetric cylinder

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Abstract

In this paper we consider a time-conformable fractional heat equation that admits heat sources that belong to a prefixed set of functions. Assuming that the time-conformable fractional heat equation is defined on an axisymmetric cylinder, we obtain a robust stability criterion for a class of solutions that can be expressed as a Fourier series. The robust stability criterion is obtained by considering an extension of the definition of stability under constant-acting perturbations that is regularly used in systems of ordinary differential equations. It is also shown that the robust stability criterion obtained is independent of the order of the conformable fractional derivative of the time-conformable fractional heat equation. The results obtained are illustrated numerically by means of an example.

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References

  1. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    MATH  Google Scholar 

  2. de Oliveira, E.C., Tenreiro Machado, J.A.: A review of definitions for fractional derivatives and integral. Math. Probl. Eng. 2014, 1–6 (2014). https://doi.org/10.1155/2014/238459

    Article  MathSciNet  MATH  Google Scholar 

  3. Sales Teodoro, G., Tenreiro Machado, J.A., Oliveira, Capelas, de Oliveira, E.C.: A review of definitions of fractional derivatives and other operators. J Comput Phys. 388, 195–208 (2019). https://doi.org/10.1016/j.jcp.2019.03.008

    Article  MathSciNet  MATH  Google Scholar 

  4. Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014). https://doi.org/10.1016/j.cam.2014.01.002

    Article  MathSciNet  MATH  Google Scholar 

  5. Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015). https://doi.org/10.1016/j.cam.2014.10.016

    Article  MathSciNet  MATH  Google Scholar 

  6. Atangana, A., Baleanu, D., Alsaedi, A.: New properties of conformable derivative. Open Math. 13(1), 889–898 (2015). https://doi.org/10.1515/math-2015-0081

    Article  MathSciNet  MATH  Google Scholar 

  7. Bayour, B., Torres, D.F.M.: Existence of solution to a local fractional nonlinear differential equation. J. Comput. Appl. Math. 312, 127–133 (2017). https://doi.org/10.1016/j.cam.2016.01.014

    Article  MathSciNet  MATH  Google Scholar 

  8. Souahi, A., Ben Makhlouf, A., Hammami, M.A.: Stability analysis of conformable fractional-order nonlinear systems. Indag. Math. 28(6), 1265–1274 (2017). https://doi.org/10.1016/j.indag.2017.09.009

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhong, W., Wang, L.: Basic theory of initial value problems of conformable fractional differential equations. Adv. Differ. Equ. (2018). https://doi.org/10.1186/s13662-018-1778-5

    Article  MathSciNet  MATH  Google Scholar 

  10. Atangana, A., Baleanu, D., Alsaedi, A.: Analysis of time-fractional Hunter-Saxton equation: a model of neumatic liquid crystal. Open Phys. 14(1), 145–149 (2016). https://doi.org/10.1515/phys-2016-0010

    Article  Google Scholar 

  11. Zhao, D., Luo, M.: General conformable fractional derivative and its physical interpretation. Calcolo 54(3), 903–917 (2017). https://doi.org/10.1007/s10092-017-0213-8

    Article  MathSciNet  MATH  Google Scholar 

  12. Abu-Shady, M., Kaabar, M.K.A.: A generalized definition of the fractional derivative with applications. Math. Probl. Eng. 2021, 9444803 (2021). https://doi.org/10.1155/2021/9444803

    Article  Google Scholar 

  13. Gözütok, N., Gözütok, U.: Multi-variable conformable fractional calculus. Filomat 32(1), 45–53 (2018). https://doi.org/10.2298/FIL1801045G

    Article  MathSciNet  MATH  Google Scholar 

  14. Yépez-Martínez, H., Gómez-Aguilar, J.F., Atangana, A.: First integral method for non-linear differential equations with conformable derivative. Math. Model. Nat. Phenom. (2018). https://doi.org/10.1051/mmnp/2018012

    Article  MathSciNet  MATH  Google Scholar 

  15. Tajadodi, H., Khan, Z.A., ur Rehman Irshad, A., Gómez-Aguilar, J.F.: Exact solutions of conformable fractional differential equations. Results Phys. 22, 103916 (2021). https://doi.org/10.1016/j.rinp.2021.103916

    Article  Google Scholar 

  16. Aderyani, S.R., Saadati, R., Vahidi, J., Gómez-Aguilar, J.F.: The exact solutions of conformable time-fractional modified nonlinear Schrödinger equation by first integral method and functional variable method. Opt. Quantum Electron. 54(4), 218 (2022). https://doi.org/10.1007/s11082-022-03605-y

    Article  Google Scholar 

  17. Yépez-Martínez, H., Pashrashid, A., Gómez-Aguilar, J.F., Akinyemi, L., Rezazadeh, H.: The novel soliton solutions for the conformable perturbed nonlinear Schrödinger equation. Mod. Phys. Lett. B 36(08), 2150597 (2022). https://doi.org/10.1142/S0217984921505977

    Article  Google Scholar 

  18. Çenesiz, Y., Kurt, A.: The solutions of time and space conformable fractional heat equations with conformable Fourier transform. Acta Univ. Sapientiae Matem. 7(2), 130–140 (2016). https://doi.org/10.1515/ausm-2015-0009

    Article  MathSciNet  MATH  Google Scholar 

  19. Avci, D., Eroglu, I., Ozdemir, N.: Conformable heat problem in a cylinder. In: International Conference on Fractional Differentiation and Its Applications, Novi Sad, Serbia, pp. 572–588 (2016)

  20. Avci, D., Iskender, E., Ozdemir, N.: Conformable heat equation on a radial symmetric plate. Therm. Sci. 21(2), 819–926 (2017). https://doi.org/10.2298/TSCI160427302A

    Article  MATH  Google Scholar 

  21. Muneshwar, R., Bondar, K.L., Shirole, Y.H.: Solution of linear and non-linear partial differential equations of fractional order. Proyecciones 40(5), 1179–1195 (2021). https://doi.org/10.22199/issn.0717-6279-4396

    Article  MathSciNet  MATH  Google Scholar 

  22. Elsgolts, L.: Differential equations and the calculus of variations. Mir, Moscow (1977)

    Google Scholar 

  23. Ladyzhenskaya, O.A.: The boundary value problems of mathematical physics. Springer, New York (1985)

    Book  MATH  Google Scholar 

  24. Yang, S., Xue, X., Xiong, X.: A modified quasi-boundary value method for a backward problem for the inhomogeneous time conformable fractional heat equation in a cylinder. Inverse Probl. Sci. Eng. 29(9), 1323–1342 (2020). https://doi.org/10.1080/17415977.2020.1849179

    Article  MathSciNet  MATH  Google Scholar 

  25. Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, London (1952)

    Google Scholar 

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  1. Área Académica de Matemáticas y Física, Universidad Autónoma del Estado de Hidalgo, Pachuca–Tulancigo Km 4.5, Mineral de la Reforma, 42184, Hidalgo, México

    R. Temoltzi-Ávila

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Temoltzi-Ávila, R. A robust stability criterion on the time-conformable fractional heat equation in a axisymmetric cylinder. SeMA 80, 687–700 (2023). https://doi.org/10.1007/s40324-022-00317-x

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