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A robust stability criterion in the one-dimensional subdiffusion equation with Caputo–Fabrizio fractional derivative

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Abstract

In this paper, we present a robust stability criterion for the one-dimensional subdiffusion equation with Caputo–Fabrizio fractional derivative. The criterion is obtained by extending a concept of stability under constant-acting perturbations that is regularly applied to systems of differential equations of integer order. We assume the existence of uncertainty in the subdiffusion equation due to the effect of external sources that are represented by Fourier series whose generalized Fourier coefficients are absolutely continuous and bounded functions. The results obtained suggest that the robust stability criterion allows us to guarantee that the solution of the subdiffusion equation, as well as its Caputo–Fabrizio fractional derivative and its first partial derivative with respect to the longitudinal axis, are bounded by a constant whose value is initially established. The results obtained are illustrated numerically.

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References

  1. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering, vol. 111. Academic Press Inc, New York (1974)

    Google Scholar 

  2. Baleanu, D., Fernandez, A.: On fractional operators and their classifications. Mathematics (2019). https://doi.org/10.3390/math7090830

    Article  Google Scholar 

  3. Sales Teodoro, G., Tenreiro Machado, J.A., Capelas de Oliveira, E.: 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  Google Scholar 

  4. Rossikhin, Y.A., Shitikova, M.V.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50(1), 15–67 (1997). https://doi.org/10.1115/1.3101682

    Article  Google Scholar 

  5. Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018). https://doi.org/10.1016/j.cnsns.2018.04.019

    Article  Google Scholar 

  6. Patnaik, S., Hollkamp, J.P., Semperlotti, F.: Applications of variable-order fractional operators: a review. Proc. R. Soc. A. 476, 20190498 (2020). https://doi.org/10.1098/rspa.2019.0498

    Article  MathSciNet  Google Scholar 

  7. Podlubny, I.: Fractinal Differential Equations, vol. Mathematics in Science and Engineering. Academic Press (1999)

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

    Google Scholar 

  9. Oprzȩdkiewicz, K., Mitkowski, M., Gawin, E., Dziedzic, K.: The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process. Bull. Pol. Acad. Tech. 66(4), 501–507 (2018). https://doi.org/10.24425/124267

    Article  Google Scholar 

  10. Zhang, S., Hu, L., Sun, S.: The uniqueness of solution for initial value problems for fractional differential equation involving the Caputo-Fabrizio derivative. J. Nonlinear Sci. Appl. 11(3), 428–436 (2018). https://doi.org/10.22436/jnsa.011.03.11

    Article  MathSciNet  Google Scholar 

  11. Toprakseven, S.: The existence and uniqueness of initial-boundary value problems of the fractional Caputo-Fabrizio differential equations. Univers. J. Math. Appl. 2(2), 100–106 (2019). https://doi.org/10.32323/ujma.549942

    Article  Google Scholar 

  12. Li, H., Cheng, J., Li, H.-B., Zhong, S.-M.: Stability analysis of a fractional-order linear system described by the Caputo–Fabrizio derivative. Mathematics (2019). https://doi.org/10.3390/math7020200

    Article  Google Scholar 

  13. Abbas, S., Benchohra, M., Nieto, J.J.: Caputo–Fabrizio fractional differential equations with instantaneous impulses. AIMS Math. 6(3), 2932–2946 (2021). https://doi.org/10.3934/math.2021177

    Article  MathSciNet  Google Scholar 

  14. Atangana, A.: On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation. Appl. Math. Comput. 273, 948–956 (2016). https://doi.org/10.1016/j.amc.2015.10.021

    Article  MathSciNet  Google Scholar 

  15. Cuahutenango-Barro, B., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: Application of fractional derivative with exponential law to bi-fractional-order wave equation with frictional memory kernel. Eur. Phys. J. Plus 132, 515 (2017). https://doi.org/10.1140/epjp/i2017-11796-9

    Article  Google Scholar 

  16. Seminara, S., Troparevsky, M., Fabio, M., Mura, G.L.: Anomalous diffusion with Caputo–Fabrizio time derivative: an inverse problem. Trend Comput. Appl. Math. 23(3), 515–529 (2022). https://doi.org/10.5540/tcam.2022.023.03.00515

    Article  MathSciNet  Google Scholar 

  17. Korpinar, Z.: On numerical solutions for the Caputo–Fabrizio fractional heat-like equation. Therm. Sci. 22(1), 87–95 (2018). https://doi.org/10.2298/TSCI170614274K

    Article  MathSciNet  Google Scholar 

  18. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science Inc. (2006)

  19. Tateishi, A.A., Ribeiro, H.V., Lenzi, E.K.: The role of fractional time-derivative operators on anomalous diffusion. Front. Phys. (2017). https://doi.org/10.3389/fphy.2017.00052

    Article  Google Scholar 

  20. Elsgolts, L.: Differential Equations and the Calculus of Variations. Mir, Moscow (1977)

    Google Scholar 

  21. Temoltzi-Ávila, R.: On a robust stability criterion in the subdiffusion equation with Caputo–Dzherbashian fractional derivative. Bol. Soc. Mat. Mex. 29(3), 74 (2023). https://doi.org/10.1007/s40590-023-00548-6

    Article  MathSciNet  Google Scholar 

  22. Zhermolenko, V.N., Temoltzi-Ávila, R.: Bulgakov problem for a hyperbolic equation and robust stability. Mosc. Univ. Mech. Bull. 76(4), 95–104 (2021). https://doi.org/10.3103/S0027133021040051

    Article  Google Scholar 

  23. Temoltzi-Ávila, R.: A robust stability criterion on the time-conformable fractional heat equation in a axisymmetric cylinder. SeMA J. 80(4), 687–700 (2023). https://doi.org/10.1007/s40324-022-00317-x

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  25. Abdeljawad, T.: Fractional operators with exponential kernels and a Lyapunov type inequality. Adv. Differ. Equ. (2017). https://doi.org/10.1186/s13662-017-1285-0

    Article  MathSciNet  Google Scholar 

  26. Caputo, M., Fabrizio, M.: On the notion of fractional derivative and applications to the hysteresis phenomena. Meccanica 52, 3043–3052 (2017). https://doi.org/10.1007/s11012-017-0652-y

    Article  MathSciNet  Google Scholar 

  27. Atanacković, T.M., Pilipović, S., Zorica, D.: Properties of the Caputo–Fabrizio fractional derivative and its distributional settings. Fract. Calc. Appl. Anal. 21(1), 29–44 (2018). https://doi.org/10.1515/fca-2018-0003

    Article  MathSciNet  Google Scholar 

  28. Al-Refai, M., Pal, K.: New aspects of Caputo–Fabrizio fractional derivative. Progr. Fract. Differ. Appl. 5(2), 157–166 (2019). https://doi.org/10.18576/pfda/050206

    Article  Google Scholar 

  29. Nchama, G.A.M.: Properties of Caputo–rizio fractional operators. New Trend Math. Sci. 8(1), 1–25 (2020). https://doi.org/10.20852/ntmsci.2020.393https://doi.org/10.20852/ntmsci.2020.393

  30. Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29(1), 129–143 (2002). https://doi.org/10.1023/A:1016547232119

    Article  MathSciNet  Google Scholar 

  31. Sene, N.: Fractional diffusion equation with new fractional operator. Alex. Eng. J. 59(5), 2921–2926 (2020). https://doi.org/10.1016/j.aej.2020.03.027

    Article  Google Scholar 

  32. Sene, N.: Fractional diffusion equation described by the Atangana–Baleanu fractional derivative and its approximate solution. J. Frac. Calc. Nonlinear Syst. 2(1), 60–75 (2021). https://doi.org/10.48185/jfcns.v2i1.214

    Article  Google Scholar 

  33. Sene, N.: Fractional diffusion equation with reaction term described by the Caputo–Liouville generalized fractional derivated. J. Fractional Calc. Appl. 13(1), 42–57 (2022)

    MathSciNet  Google Scholar 

  34. Al-Saltí, N., Karímov, E., Kerbal, S.: Boundary-value problems for fractional heat equation involving Caputo–Fabrizio derivative. New Trend. Math. Sci. 4(4), 79–89 (2016). https://doi.org/10.20852/ntmsci.2016422308

    Article  Google Scholar 

  35. Wang, H., Zhang, X., Luo, Z., Liu, J.: Analysis of numerical method for diffusion equation with time-fractional Caputo–Fabrizio derivative. J. Math. 2023, 7906656 (2023). https://doi.org/10.1155/2023/7906656

    Article  MathSciNet  Google Scholar 

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

    R. Temoltzi-Ávila

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Temoltzi-Ávila, R. A robust stability criterion in the one-dimensional subdiffusion equation with Caputo–Fabrizio fractional derivative. Ricerche mat (2024). https://doi.org/10.1007/s11587-024-00861-w

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