Table of Contents Author Guidelines Submit a Manuscript
Advances in Fuzzy Systems
Volume 2019, Article ID 6457548, 11 pages
https://doi.org/10.1155/2019/6457548
Research Article

Existence and Continuous Dependence on Initial Data of Solution for Initial Value Problem of Fuzzy Multiterm Fractional Differential Equation

1Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea
2Institute of Advanced Science, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea

Correspondence should be addressed to Kinam Sin; nc.ude.tih@52021fb51

Received 17 March 2019; Revised 13 May 2019; Accepted 15 May 2019; Published 19 June 2019

Academic Editor: Rustom M. Mamlook

Copyright © 2019 Huichol Choi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. S. Mansouri, M. Gachpazan, and O. S. Fard, “Existence, uniqueness and stability of fuzzy fractional differential equations with local Lipschitz and linear growth conditions,” Advances in Difference Equations, vol. 2017, p. 240, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  2. A. Rivaz, O. S. Fard, and T. A. Bidgoli, “On the existence and uniqueness of solutions for fuzzy fractional differential equations,” Tbilisi Mathematical Journal, vol. 10, no. 1, pp. 197–205, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  3. R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 6, pp. 2859–2862, 2010. View at Publisher · View at Google Scholar · View at Scopus
  4. S. Arshad and V. Lupulescu, “On the fractional differential equations with uncertainty,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 11, pp. 3685–3693, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. A. Khastan, J. J. Nieto, and R. Rodríguez-López, “Periodic boundary value problems for first-order linear differential equations with uncertainty under generalized differentiability,” Information Sciences, vol. 222, pp. 544–558, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  6. A. Khastan, J. J. Nieto, and R. Rodríguez-López, “Schauder fixed-point theorem in semilinear spaces and its application to fractional differential equations with uncertainty,” Fixed Point Theory and Applications, vol. 2014, article 21, 14 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  7. A. Khastan, J. J. Nieto, and R. Rodríguez-López, “Fuzzy delay differential equations under generalized differentiability,” Information Sciences, vol. 275, pp. 145–167, 2014. View at Publisher · View at Google Scholar · View at Scopus
  8. S. Salahshour, A. Ahmadian, F. Ismail, and D. Baleanu, “A fractional derivative with non-singular kernel for interval-valued functions under uncertainty,” Optik - International Journal for Light and Electron Optics, vol. 130, pp. 273–286, 2017. View at Publisher · View at Google Scholar · View at Scopus
  9. S. Salahshour, A. Ahmadian, S. Abbasbandy, and D. Baleanu, “-fractional derivative under interval uncertainty: theory, properties and applications,” Chaos, Solitons & Fractals, vol. 117, pp. 84–93, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  10. V. Lupulescu, “Fractional calculus for interval-valued functions,” Fuzzy Sets and Systems, vol. 265, pp. 63–85, 2015. View at Publisher · View at Google Scholar · View at Scopus
  11. M. T. Malinowski, “Random fuzzy fractional integral equations—theoretical foundations,” Fuzzy Sets and Systems, vol. 265, pp. 39–62, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. T. Allahviranloo, Z. Gouyandeh, A. Armand, and A. Hasanoglu, “On fuzzy solutions for heat equation based on generalized Hukuhara differentiability,” Fuzzy Sets and Systems, vol. 265, pp. 1–23, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  13. S. Salahshour, T. Allahviranloo, and S. Abbasbandy, “Solving fuzzy fractional differential equations by fuzzy Laplace transforms,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1372–1381, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. H. Wang, “Monotone iterative method for boundary value problems of fuzzy differential equations,” Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, vol. 30, no. 2, pp. 831–843, 2016. View at Publisher · View at Google Scholar · View at Scopus
  15. S. Salahshour, A. Ahmadian, and D. Baleanu, “Variation of constant formula for the solution of interval differential equations of non-integer order,” The European Physical Journal Special Topics, vol. 226, no. 16-18, pp. 3501–3512, 2017. View at Publisher · View at Google Scholar · View at Scopus
  16. A. Ahmadian, S. Salahshour, and C. S. Chan, “Fractional differential systems: a fuzzy solution based on operational matrix of shifted chebyshev polynomials and its applications,” IEEE Transactions on Fuzzy Systems, vol. 25, no. 1, pp. 218–236, 2017. View at Publisher · View at Google Scholar · View at Scopus
  17. K. Sin, M. Chen, H. Choi, and K. Ri, “Fractional Jacobi operational matrix for solving Fuzzy fractional differential equation,” Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, vol. 33, no. 2, pp. 1041–1052, 2017. View at Publisher · View at Google Scholar · View at Scopus
  18. K. Sin, C. Minghao, W. Chong et al., “Application of a spectral method to fractional Differential Equations under uncertainty,” Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, vol. 35, no. 4, pp. 4821–4835, 2018. View at Publisher · View at Google Scholar
  19. M. Mazandarani and A. V. Kamyad, “Modified fractional Euler method for solving fuzzy fractional initial value problem,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 1, pp. 12–21, 2013. View at Publisher · View at Google Scholar
  20. A. Ahmadian, F. Ismail, S. Salahshour, D. Baleanu, and F. Ghaemi, “Uncertain viscoelastic models with fractional order: a new spectral tau method to study the numerical simulations of the solution,” Communications in Nonlinear Science and Numerical Simulation, vol. 53, pp. 44–64, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  21. A. Ahmadian, H. S. Chan, S. Salahshour, and V. Vaitheeswaran, “FTFBE: A numerical approximation for fuzzy time-fractional Bloch equation,” in Proceedings of the 2014 IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2014, pp. 418–423, July 2014. View at Publisher · View at Google Scholar · View at Scopus
  22. S. Salahshour, T. Allahviranloo, S. Abbasbandy et al., “Existence and uniqueness results for fractional differential equations with uncertainty,” Advances in Difference Equations, vol. 2012, article 112, 12 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  23. Z.-P. Yang, T.-Z. Xu, and M. Qi, “The Cauchy problem for quaternion fuzzy fractional differential equations,” Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, vol. 29, no. 1, pp. 451–461, 2015. View at Publisher · View at Google Scholar · View at Scopus
  24. V. H. Ngo, “Fuzzy fractional functional integral and differential equations,” Fuzzy Sets and Systems, vol. 280, pp. 58–90, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  25. P. Prakash, J. J. Nieto, S. Senthilvelavan, and G. S. Priya, “Fuzzy fractional initial value problem,” Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, vol. 28, no. 6, pp. 2691–2704, 2015. View at Publisher · View at Google Scholar · View at Scopus
  26. A. Souahi, A. Guezane-Lakoud, and A. Hitta, “On the existence and uniqueness for high order fuzzy fractional differential equations with uncertainty,” Advances in Fuzzy Systems, vol. 2016, Article ID 5246430, 9 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  27. H. V. Ngo, V. Ho, and T. M. Duc, “Fuzzy fractional differential equations under CaputoKatugampola fractional derivative approach,” Fuzzy Sets and Systems, 2018. View at Publisher · View at Google Scholar
  28. H. V. Ngo, V. Lupulescu, and D. O'Regan, “A note on initial value problems for fractional fuzzy differential equations,” Fuzzy Sets and Systems, vol. 347, pp. 54–69, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  29. B. Bede and S. G. Gal, “Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 581–599, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus