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Some New Results on the Stability of Fractional Integro-Differential Equations Under Uncertainty

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Recent Advances on Soft Computing and Data Mining (SCDM 2018)

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Abstract

In this research, we introduce the concept of \(E_\alpha \)-type stability for fractional integro-differential equations with uncertainty. We propose different types of fuzzy \(E_\alpha \) stabilities for some classes of fuzzy integro-differential equations of fractional order. Besides, we present some new findings on the existence and uniqueness of the solutions of fuzzy integro-differential equations of fractional order using the proposed new concept.

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References

  1. Ulam, S.M.: A Collection of Mathematical Problems. Interscience, New York (1960)

    MATH  Google Scholar 

  2. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27, 222–224 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  3. Rassias, Th.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    Google Scholar 

  4. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)

    Google Scholar 

  5. Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)

    MATH  Google Scholar 

  6. Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17, 1135–1140 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wang, G., Zhou, M., Sun, L.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 21, 1024–1028 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Rezaei, H., Jung, S.-M., Rassias, Th.M.: Laplace transform and Hyers–Ulam stability of linear differential equations. J. Math. Anal. Appl. 403, 244–251 (2013)

    Google Scholar 

  9. Abdollahpour, M.R., Aghayari, R., Rassias, Th.M.: HyersUlam stability of associated Laguerre differential equations in a subclass of analytic functions. J. Math. Anal. Appl. 437, 605–612 (2016)

    Google Scholar 

  10. Huang, J., Li, Y.: Hyers-Ulam stability of linear functional differential equations. J. Math. Anal. Appl. 426, 1192–1200 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kim, B., Jung, S.M.: Bessel’s differential equation and its Hyers–Ulam stability. J. Inequalities Appl. 2007(1), 1–8 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Kim, S.S., Cho, Y.J., Eshaghi Gordji, M.: On the generalized Hyers–Ulam–Rassias stability problem of radical functional equations. J. Inequalities Appl. 2012(1), 1–13 (2012)

    Article  MathSciNet  Google Scholar 

  13. Li, C.P., Zhang, F.R.: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193, 27–47 (2011)

    Article  Google Scholar 

  14. Li, Y., Chen, Y., Podlubny, I.: Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 19651969 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Li, Y., Chen, Y., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittageffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, J., Lv, L., Zhou, Y.: Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron. J. Qual. Theory Differ. Equ. 63, 1–10 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Wang, J., Lv, L., Zhou, Y.: New concepts and results in stability of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 2530–2538 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, J., Zhou, Y., Fečkan, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, 3389–3405 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, J., Li, X.: \(E_\alpha \)-Ulam type stability of fractional order ordinary differential equations. J. Appl. Math. Comput. 45, 449–459 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Agarwal, R.P., Lakshmikantham, V., Nieto, J.J.: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 72, 2859–2862 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Allahviranloo, T., Gouyandeh, Z., Armand, A.: Fuzzy fractional differential equations under generalized fuzzy Caputo derivative. J. Intell. Fuzzy Syst. 26, 1481–90 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Allahviranloo, T., Salahshour, S., Abbasbandy, S.: Explicit solutions of fractional differential equations with uncertainty. Soft Comput. 16, 297–302 (2012)

    Article  MATH  Google Scholar 

  23. Salahshour, S., Allahviranloo, T., Abbasbandy, S., Baleanu, D.: Existence and uniqueness results for fractional differential equations with uncertainty. Adv. Differ. Equ. 2012, 112 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mazandarani, M., Vahidian Kamyad, A.: Modified fractional Euler method for solving fuzzy fractional initial value problem. Commun. Nonlinear Sci. Numer. Simul. 18, 12–21 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mazandarani, M., Najariyan, M.: Type-2 fuzzy fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 19, 2354–2372 (2014)

    Article  MathSciNet  Google Scholar 

  26. Salahshour, S., Allahviranloo, T., Abbasbandy, S.: Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun. Nonlinear. Sci. Numer. Simul. 17, 1372–1381 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ahmadian, A., Chang, C.S., Salahshour, S.: Fuzzy approximate solutions to fractional differential equations under uncertainty: operational matrices approach. IEEE Trans. Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2016.2554156

  28. Ahmadian, A., Salahshour, S., Amirkhani, H., Baleanu, D., Yunus, R.: An efficient Tau method for numerical solution of a fuzzy fractional kinetic model and its application to oil palm frond as a promising source of xylose. J. Comput. Phys. 264, 562–564 (2015)

    Article  MATH  Google Scholar 

  29. Malinowski, M.T.: Random fuzzy fractional integral equations-theoretical foundations. Fuzzy Sets Syst. (In press). https://doi.org/10.1016/j.fss.2014.09.019

  30. Hoa, N.V.: Fuzzy fractional functional differential equations under Caputo gH-differentiability. Commun. Nonlinear Sci. Numer. Simul. 22, 1134–1157 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hoa, N.V.: Fuzzy fractional functional integral and differential equations. Fuzzy Sets Syst. (In press). https://doi.org/10.1016/j.fss.2015.01.009

  32. Mirmostafaee, A.K., Moslehian, M.: Fuzzy versions of Hyers–Ulam–Rassias theorem. Fuzzy Sets Syst. 159, 720–729 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mirmostafaee, A.K., Mirzavaziri, M., Moslehian, M.S.: Fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst. 159, 730–738 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shen, Y.: On the Ulam stability of first order linear fuzzy differential equations under generalized differentiability. Fuzzy Sets Syst. (In press). https://doi.org/10.1016/j.fss.2015.01.002

  35. Diamond, P., Kloeden, P.E.: Metric Spaces of Fuzzy Sets: Theory and Applications. World Scientific, Singapore (1994)

    Book  MATH  Google Scholar 

  36. Bede, B., Rudas, I.J., Bencsik, A.L.: First order linear fuzzy differential equations under generalized differentiability. Inform. Sci. 177, 1648–1662 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractionaldifferential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)

    Google Scholar 

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Acknowledgements

The authors acknowledge the financial support from Universiti Putra Malaysia under Putra-IPB grant: GP-IPB/2017/9542402.

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Authors and Affiliations

  1. Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia

    A. Ahmadian, N. Senu & F. Ismail

  2. Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran

    S. Salahshour

Authors
  1. A. Ahmadian
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  2. S. Salahshour
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  3. N. Senu
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  4. F. Ismail
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Corresponding author

Correspondence to S. Salahshour .

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Editors and Affiliations

  1. Faculty of Computer Science and Information Technology, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia

    Rozaida Ghazali

  2. Faculty of Computer Science and Information Technology, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia

    Mustafa Mat Deris

  3. Faculty of Computer Science and Information Technology, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia

    Nazri Mohd Nawi

  4. School of Information Technology, Deakin University, Geelong, Victoria, Australia

    Jemal H. Abawajy

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Ahmadian, A., Salahshour, S., Senu, N., Ismail, F. (2018). Some New Results on the Stability of Fractional Integro-Differential Equations Under Uncertainty. In: Ghazali, R., Deris, M., Nawi, N., Abawajy, J. (eds) Recent Advances on Soft Computing and Data Mining. SCDM 2018. Advances in Intelligent Systems and Computing, vol 700. Springer, Cham. https://doi.org/10.1007/978-3-319-72550-5_6

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  • DOI: https://doi.org/10.1007/978-3-319-72550-5_6

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