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On Hyers–Ulam Stability for Fractional Differential Equations Including the New Caputo–Fabrizio Fractional Derivative

  • Yasemin BaşcıEmail author
  • Süleyman Öğrekçi
  • Adil Mısır
Article
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Abstract

In this paper, we study the stability in the sense of Hyers–Ulam for the following fractional differential equations including the new Caputo–Fabrizio fractional derivative:
$$\begin{aligned} \left( ^{CF}D^{\alpha }y\right) \left( x\right) =f\left( x\right) \quad \qquad \quad \end{aligned}$$
and
$$\begin{aligned} \left( ^{CF}D^{\alpha } y \right) \left( x\right) -\lambda y\left( x\right) =f\left( x\right) . \end{aligned}$$
Finally, two examples are given to illustrate our results.

Keywords

Fractional differential equation the new Caputo–Fabrizio fractional derivative Hyers–Ulam stability laplace transform 

Mathematics Subject Classification

26A33 34D10 

Notes

Acknowledgements

The authors would like to thank the referees for their useful comments and remarks.

References

  1. 1.
    Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta. Appl. Math. 109, 973–1033 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alqifiary, Q.H., Jung, S.M.: Laplace transform and generalized Hyers–Ulam stability of linear differential equations. Electron. J. Differ. Equ. 2014, 1–11 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    András, S., Mészáros, A.R.: Ulam–Hyers stability of dynamic equations on time scales via Picard operators. Appl. Math. Comput. 219, 4853–4864 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular. Progr. Fract. Differ. Appl. 1(2), 73–85 (2015)Google Scholar
  5. 5.
    Goufo, E.F.D.: Application of Caputo–Fabrzio fractional derivative without singular kernel to Korteweg–de Vries–Burgers equation. Math. Model. Anal. 21(2), 188–198 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ibrahim, R.W.: Ulam stability of boundary value problem. Kragujevac J. Math. 37, 287–297 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Jung, S.M.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17(10), 1135–1140 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jung, S.M.: Hyers–Ulam stability of linear differential equations of first order III. J. Math. Anal. Appl. 311(1), 139–146 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jung, S.M.: Hyers–Ulam stability of linear differential equations of first order II. Appl. Math. Lett. 19(9), 854–858 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. North-Holland Mathematics Studies, Amsterdam (2006)CrossRefGoogle Scholar
  12. 12.
    Lungu, N., Popa, D.: Hyers–Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 385, 86–91 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Losada, J., Nieto, J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87–92 (2015)Google Scholar
  14. 14.
    Otrocol, D., Ilea, V.: Ulam stability for a delay differential equation. Cent. Eur. J. Math. 11, 1296–1303 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Peng, S., Wang, J.: Existence and Ulam-Hyers stability of ODEs involving two Caputo fractional derivative. Electron. J. Qual. Theory Differ. Equ. 52, 1–16 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Popa, D., Raşa, I.: On the Hyers–Ulam stability of the linear differential equation. J. Math. Anal. Appl. 381, 530–537 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rezaei, H., Jung, S.M., Rassias, T.M.: Laplace transform and Hyers–Ulam stability of linear differential equations. J. Math. Anal. Appl. 403, 244–251 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rus, I.A.: Ulam stability of ordinary differential equations. Stud. Univ. ’Babeş Bolyai’ Math. 54, 125–133 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Rus, I.A.: Ulam stabilities of ordinary differential equations in a Banach space. Carpathian J. Math. 26, 103–107 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Wang, J., Li, X.: \(\mathbb{E_{\alpha }}\)-Ulam stability of fractional order ordinary differential equations. J. Appl. Math. Comput. 45, 449–459 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wang, J., Li, X.: A uniform method to Ulam–Hyers stability for some linear fractional equations. Mediterr. J. Math. 13, 625–635 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wang, J., Lv, L., Zhou, Y.: Boundary value problems for for fractional differential equations involving Caputo derivative in Banach spaces. J. Appl. Math. Comput. 38, 209–224 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    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)MathSciNetzbMATHGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wang, C., Xu, T.Z.: Hyers–Ulam stability of fractional linear differential equations involving Caputo fractional derivatives. Appl. Math. 60(4), 383–393 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wang, J., Zhang, Y.: A class of nonlinear differential equations with fractional integrable impulses. Commun. Nonlinear Sci. Numer. Simul. 19, 3001–3010 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, J., Zhang, Y.: Ulam–Hyers–Mittag–Leffler stability of fractional-order delay differential equations. Optimization 63(8), 1181–1190 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. RWA 12, 262–272 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wang, J., Zhou, Y.: Mittag–Leffler–Ulam stabilities of fractional evolution equations. Appl. Math. Lett. 25, 723–728 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Yasemin Başcı
    • 1
    Email author
  • Süleyman Öğrekçi
    • 2
  • Adil Mısır
    • 3
  1. 1.Department of Mathematics, Faculty of Arts and SciencesBolu Abant Izzet Baysal UniversityBoluTurkey
  2. 2.Department of Mathematics, Faculty of Arts and SciencesAmasya UniversityAmasyaTurkey
  3. 3.Department of Mathematics, Faculty of SciencesGazi UniversityTeknikokullarTurkey

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