Skip to main content
Log in

On the initial value problem for fuzzy differential equations of non-integer order \(\alpha \in (1,2)\)

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

The purpose of this paper is to present some basic theories of an initial value problem of fuzzy fractional differential equations involving the Caputo-fuzzy-type concept of fractional derivative in the case of the order \(\alpha \in (1,2).\) The existence and uniqueness results of the solution for the given problem are presented. Finally, some examples are given to illustrate our main results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  • Agarwal RP, Lakshmikantham V, Nieto JJ (2010) On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal Theory Methods Appl 72:2859–62

    MathSciNet  MATH  Google Scholar 

  • Ahmad MZ, Hasan MK, De Baets B (2013) Analytical and numerical solutions of fuzzy differential equations. Inf Sci 236:156–167

    MathSciNet  MATH  Google Scholar 

  • Ahmadian A, Salahshour S, Chan CS (2017a) Fractional differential systems: a fuzzy solution based on operational matrix of shifted Chebyshev polynomials and its applications. IEEE Trans Fuzzy Syst 25:218–236

    Google Scholar 

  • Ahmadian A, Ismail F, Salahshour S, Baleanu D, Ghaemi F (2017b) Uncertain viscoelastic models with fractional order: a new spectral tau method to study the numerical simulations of the solution. Commun Nonlinear Sci Numer Simul 53:44–64

    MathSciNet  Google Scholar 

  • Allahviranloo T, Ghanbari B (2020) On the fuzzy fractional differential equation with interval Atangana–Baleanu fractional derivative approach. Chaos Solitons Fractals 130:109397

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  • Allahviranloo T, Abbasbandy S, Sedaghatfar O, Darabi P (2012b) A new method for solving fuzzy integro-differential equation under generalized differentiability. Neural Comput Appl 21:191–196

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Allahviranloo T, Gouyandeh Z, Armand A (2014) Fuzzy fractional differential equations under generalized fuzzy Caputo derivative. J Intell Fuzzy Syst 26:1481–1490

    MathSciNet  MATH  Google Scholar 

  • An TV, Vu H, Hoa NV. The existence of solutions for an initial value problem of Caputo-Hadamard-type fuzzy fractional differential equations of order \(\alpha \in (1,2)\). J Intell Fuzzy Syst(Preprint) 1–4

  • An TV, Vu H, Hoa NV (2017a) Applications of contractive-like mapping principles to interval-valued fractional integro-differential equations. J Fixed Point Theory Appl 19:2577–2599

    MathSciNet  MATH  Google Scholar 

  • An TV, Vu H, Hoa NV (2017) A new technique to solve the initial value problems for fractional fuzzy delay differential equations. Adv Differ Equ 2017:181

    MathSciNet  MATH  Google Scholar 

  • An TV, Vu H, Hoa NV (2019) Hadamard-type fractional calculus for fuzzy functions and existence theory for fuzzy fractional functional integro-differential equations. J Intell Fuzzy Syst 36:3591–605

    Google Scholar 

  • Arshad S, Lupulescu V (2011) On the fractional differential equations with uncertainty. Nonlinear Anal 74:85–93

    MathSciNet  MATH  Google Scholar 

  • Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151:581–599

    MathSciNet  MATH  Google Scholar 

  • Bede B, Stefanini L (2013) Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst 230:119–141

    MathSciNet  MATH  Google Scholar 

  • Bede B, Rudas IJ, Bencsik AL (2007) First order linear fuzzy differential equations under generalized differentiability. Inf Sci 177:1648–1662

    MathSciNet  MATH  Google Scholar 

  • Bhaskar TG, Lakshmikantham V, Leela S (2009) Fractional differential equations with a Krasnoselskii–Krein type condition. Nonlinear Anal Hybrid Syst 3:734–737

    MathSciNet  MATH  Google Scholar 

  • Chalco-Cano Y, Rufián-Lizana A, Román-Flores H, Jiménez-Gamero MD (2013) Calculus for interval-valued functions using generalized Hukuhara derivative and applications. Fuzzy Sets Syst 219:49–67

    MathSciNet  MATH  Google Scholar 

  • Fard OS, Salehi M (2014) A survey on fuzzy fractional variational problems. J Comput Appl Math 271:71–82

    MathSciNet  MATH  Google Scholar 

  • Gasilov NA, Amrahov SE, Fatullayev AG (2014) Solution of linear differential equations with fuzzy boundary values. Fuzzy Sets Syst 257:169–183

    MathSciNet  MATH  Google Scholar 

  • Gomes LT, Barros LC (2015) A note on the generalized difference and the generalized differentiability. Fuzzy Sets Syst 280:142–5

    MathSciNet  MATH  Google Scholar 

  • Hasan S, Alawneh A, Al-Momani M, Momani S (2017) Second order fuzzy fractional differential equations under Caputo’s H-differentiability. Appl Math Inf Sci 11:1–12

    MathSciNet  Google Scholar 

  • Hoa NV (2015a) Fuzzy fractional functional integral and differential equations. Fuzzy Sets Syst 280:58–90

    MathSciNet  MATH  Google Scholar 

  • Hoa NV (2015b) Fuzzy fractional functional differential equations under Caputo gH-differentiability. Commun Nonlinear Sci Numer Simul 22:1134–1157

    MathSciNet  MATH  Google Scholar 

  • Hoa NV (2018) Existence results for extremal solutions of interval fractional functional integro-differential equations. Fuzzy Sets Syst 347:29–53

    MathSciNet  MATH  Google Scholar 

  • Hoa NV, Ho V (2019) A survey on the initial value problems of fuzzy implicit fractional differential equations. Fuzzy Sets Syst. https://doi.org/10.1016/j.fss.2019.10.012

    Article  Google Scholar 

  • Hoa NV, Lupulescu V, O’Regan D (2017) Solving interval-valued fractional initial value problems under Caputo gH-fractional differentiability. Fuzzy Sets Syst 309:1–34

    MathSciNet  MATH  Google Scholar 

  • Hoa NV, Lupulescu V, O’Regan D (2018) A note on initial value problems for fractional fuzzy differential equations. Fuzzy Sets Syst 347:54–69

    MathSciNet  MATH  Google Scholar 

  • Hoa NV, Vu H, Duc TM (2019) Fuzzy fractional differential equations under Caputo–Katugampola fractional derivative approach. Fuzzy Sets Syst 375:70–99

    MathSciNet  MATH  Google Scholar 

  • Khastan A, Nieto JJ, Rodríguez-López R (2014a) Fuzzy delay differential equations under generalized differentiability. Inf Sci 275:145–67

    MathSciNet  MATH  Google Scholar 

  • Khastan A, Nieto JJ, Rodríguez-López R (2014) Schauder fixed-point theorem in semilinear spaces and its application to fractional differential equations with uncertainty. Fixed Point Theory Appl 2014:21

    MathSciNet  MATH  Google Scholar 

  • Lakshmikantham V, Leela S (2009) A Krasnoselskii–Krein-type uniqueness result for fractional differential equations. Nonlinear Anal 71:3421–3424

    MathSciNet  MATH  Google Scholar 

  • Long HV (2018) On random fuzzy fractional partial integro-differential equations under Caputo generalized Hukuhara differentiability. Comput Appl Math 37:2738–2765

    MathSciNet  MATH  Google Scholar 

  • Long HV, Son NTK, Hoa NV (2017a) Fuzzy fractional partial differential equations in partially ordered metric spaces. Iran J Fuzzy Syst 14:107–126

    MathSciNet  MATH  Google Scholar 

  • Long HV, Son NK, Tam HT (2017b) The solvability of fuzzy fractional partial differential equations under Caputo gH-differentiability. Fuzzy Sets Syst 309:35–63

    MathSciNet  MATH  Google Scholar 

  • Lupulescu V (2015) Fractional calculus for interval-valued functions. Fuzzy Sets Syst 265:63–85

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Mazandarani M, Najariyan M (2014) Type-2 fuzzy fractional derivatives. Commun Nonlinear Sci Numer Simul 19:2354–72

    MathSciNet  Google Scholar 

  • Noeiaghdam Z, Allahviranloo T, Nieto JJ (2019) \(Q\)-fractional differential equations with uncertainty. Soft Comput 2019:1–18

    Google Scholar 

  • Prakash P, Nieto JJ, Senthilvelavan S, Sudha Priya G (2015) Fuzzy fractional initial value problem. J Intell Fuzzy Syst 28:2691–2704

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Son NTK (2018) A foundation on semigroups of operators defined on the set of triangular fuzzy numbers and its application to fuzzy fractional evolution equations. Fuzzy Sets Syst 347:1–28

    MathSciNet  MATH  Google Scholar 

  • Son NTK, Thao HTP (2019) On Goursat problem for fuzzy delay fractional hyperbolic partial differential equations. J Intell Fuzzy Syst 36:6295–6306

    Google Scholar 

  • Stefanini L (2010) A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst 161:1564–1584

    MathSciNet  MATH  Google Scholar 

  • Stefanini L, Bede B (2009) Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal 71:1311–1328

    MathSciNet  MATH  Google Scholar 

  • Yoruk F, Bhaskar TG, Agarwal RP (2013) New uniqueness results for fractional differential equations. Appl Anal 92:259–269

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is very grateful to the referees for their valuable suggestions, which helped to improve the paper significantly. This research was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 107.02-2017.319.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ngo Van Hoa.

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.

Additional information

Communicated by A. Di Nola.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hoa, N.V. On the initial value problem for fuzzy differential equations of non-integer order \(\alpha \in (1,2)\). Soft Comput 24, 935–954 (2020). https://doi.org/10.1007/s00500-019-04619-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-019-04619-7

Keywords

Navigation