Advertisement

Acta Mathematica Vietnamica

, Volume 42, Issue 4, pp 675–700 | Cite as

Ulam Stability for Fractional Partial Integro-Differential Equation with Uncertainty

  • Hoang Viet Long
  • Nguyen Thi Kim Son
  • Ha Thi Thanh Tam
  • Jen-Chih Yao
Article

Abstract

In this paper, the solvability of Darboux problems for nonlinear fractional partial integro-differential equations with uncertainty under Caputo gH-fractional differentiability is studied in the infinity domain J = [0,) × [0,). New concepts of Hyers-Ulam stability and Hyers-Ulam-Rassias stability for these problems are also investigated through the equivalent integral forms. A computational example is presented to demonstrate our main results.

Keywords

Hyers-Ulam stability Fuzzy solutions Global existence Fractional partial integro-differential equation 

Mathematics Subject Classification (2010)

47H10 47H04 03E72 46S40 

Notes

Acknowledgments

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2015.08.

References

  1. 1.
    Abbas, S., Benchohra, M., N’Guérékata, G. M.: Topics in Fractional Differential Equations. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  2. 2.
    Abbas, S., Benchohra, M., Petrusel, A.: Ulam stability for partial fractional differential inclusions via Picard operators theory. Electron. J. Qual. Theory Differ. Equ. 51, 1–13 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Allahviranloo, T., Gouyandeh, Z., Armand, A.: Fuzzy fractional differential equations under generalized fuzzy Caputo derivative. J. Intell. Fuzzy Syst. 26, 1481–1490 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Allahviranloo, T., Salahshour, S., Abbasbandy, S.: Explicit solutions of fractional differential equations with uncertainty. Soft Comput. 16, 297–302 (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Agarwal, R. P., Lakshmikantham, V., Neito, J. J.: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 72, 2859–2862 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Arshad, S., Lupulescu, V.: On the fractional differential equations with uncertainty. Nonlinear Anal. 74, 3685–3693 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bede, B., Stefanini, L.: Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. 230, 119–141 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Huang, J., Li, Y.: Hyers-Ulam stability of linear functional differential equations. J. Math. Anal. Appl. 426, 1192–1200 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hoa, N. V.: Fuzzy fractional functional differential equations under Caputo gH-differentiability. Commun. Nonlinear Sci. Numer. Simul. 22, 1134–1157 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hyers, D. H.: On the stability of linear functional equations. Proc. Natl. Acad. Sci. USA. 27, 222–224 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hukuhara, M.: Integration des Applications Measurables dont la Valuer est un Compact Convexe. Funkcialaj, Ekavacioy 10, 205–223 (1967)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V, Amsterdam (2006)zbMATHGoogle Scholar
  13. 13.
    Long, H.V., Nieto, J.J., Son, N.T.S.: New approach for studying nonlocal problems related to differential systems and partial differential equations in generalized fuzzy metric spaces. Fuzzy Sets Syst. doi: 10.1016/j.fss.2016.11.008
  14. 14.
    Long, H. V., Son, N. T. K., Tam, H. T. T.: The solvability of fuzzy fractional partial differential equations under Caputo gH-differentiability. Fuzzy Sets Syst. 309, 35–63 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Long, H. V., Son, N. T. K., Tam, H. T. T.: Global existence of solutions to fuzzy partial hyperbolic functional differential equations with generalized Hukuhara derivatives. J. Intell. Fuzzy Syst. 29, 939–954 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Long, H. V., Son, N. T. K., Tam, H. T. T., Cuong, B. C.: On the existence of fuzzy solutions for partial hyperbolic functional differential equations. Int. J. Comput. Intell. Syst. 7, 1159–1173 (2014)CrossRefGoogle Scholar
  17. 17.
    Long, H. V., Son, N. T. K., Ha, N. T. M., Son, L. H.: The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations. Fuzzy Optim. Decis. Making 13, 435–462 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mazandarani, M., Kamyad, A. V.: Modified fractional Euler method for solving fuzzy fractional initial value problem. Commun. Nonlinear Sci. Numer. Simul. 18, 12–21 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mazandarani, M., Najariyan, M.: Type-2 fuzzy fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 19, 2354–2372 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York (1993)zbMATHGoogle Scholar
  21. 21.
    Rassias, T. M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Obloza, M.: Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. 13, 259–270 (1993)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Petru, T. P., Petrusel, A., Yao, J. C.: Ulam-Hyers stability for operatorial equations and inclusions via nonself operators. Taiwan. J. Math. 5, 2195–2212 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Shen, Y., Wang, F.: A fixed point approach to the Ulam stability of fuzzy differential equations under generalized differentiability. J. Intell. Fuzzy Syst. 30, 3253–3260 (2016)CrossRefzbMATHGoogle Scholar
  26. 26.
    Shen, Y.: On the Ulam stability of first order linear fuzzy differential equations under generalized differentiability. Fuzzy Sets Syst. 280, 27–57 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Stefanini, L., Bede, B.: Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal. 71, 1311–1328 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Torrejón, R., Yong, J.: On a quasilinear wave equation with memory. Nonlinear Anal. 16, 61–78 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ulam, S. M.: Problems in Modern Mathematics. Wiley, New York (1960)zbMATHGoogle Scholar
  30. 30.
    Wang, C., Xu, T. Z.: Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives. Appl. Math. 60, 383–393 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017

Authors and Affiliations

  • Hoang Viet Long
    • 1
  • Nguyen Thi Kim Son
    • 2
  • Ha Thi Thanh Tam
    • 3
  • Jen-Chih Yao
    • 4
  1. 1.Faculty of Information TechnologyPeople’s Police University of Technology and LogisticsBac NinhVietnam
  2. 2.Department of MathematicsHanoi University of EducationHanoiVietnam
  3. 3.Department of Basic SciencesUniversity of Transport TechnologyHanoiVietnam
  4. 4.Center for General EducationChina Medical UniversityTaichungTaiwan

Personalised recommendations