This paper reports the investigation of the existence and uniqueness of solution to Cauchy problems for a class of nonlinear fuzzy fractional differential equations with the Riemann-Liouville H-derivative. Further, Eq-Ulam type stability results are presented by using direct analysis methods. Finally, several numerical simulations are provided to verify the correctness and effectiveness of the main results.
AgarwalR., LakshmikanthamV., NietoJ., On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis72 (2010), 2859–2862.
2.
AllahviranlooT., SalahshourS., AbbasbandyS., Explicit solutions of fractional differential equations with uncertainty, Soft Computing16 (2012), 297–302.
3.
SalahshourS., AllahviranlooT., AbbasbandyS., Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Communications in Nonlinear Science and Numerical Simulation17 (2012), 1372–1381.
4.
BedeB., GalS., Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems147 (2004), 385–403.
5.
BedeB., GalS., Generalizations of the differentiability of fuzzynumber-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems151 (2005), 581–599.
6.
ChehlabiM., AllahviranlooT., Concreted solutions to fuzzy linear fractional differential equations, Applied Soft Computing44 (2016), 108–116.
7.
AllahviranlooT., GouyandehZ., ArmandA., Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal of Intelligent and Fuzzy Systems26 (2014), 1481–1490.
8.
ArmandA., AllahviranlooT., AbbasbandyS., GouyandehZ., Fractional relaxation-oscillation differential equations via fuzzy variational iteration method, Journal of Intelligent and Fuzzy Systems32 (2017), 363–371.
9.
WangY., SunS., HanZ., Existence of solutions to periodic boundary value problems for fuzzy fractional differential equations, International Journal of Dynamical Systems and Differential Equations7 (2017), 195–216.
10.
AgarwalR., BaleanuD., NietoJ., TorresD., ZhouY., A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, Journal of Computational and Applied Mathematics339 (2018), 3–29.
11.
HuangL., BaleanuD., MoZ., WuG., Fractional discrete-time diffusion equation with uncertainty: Applications of fuzzy discrete fractional calculus, Physica A508 (2018), 166–175.
12.
WangY., SunS., HanZ., On fuzzy fractional Schrödinger equations under Caputo’s H-differentiability, Journal of Intelligent and Fuzzy Systems34 (2018), 3929–3940.
13.
WangJ., LiX., Eα-Ulam type stability of fractional order ordinary differential equations, Journal of Applied Mathematics and Computing45 (2014), 449–459.
14.
ShenY., On the Ulam stability of first order linear fuzzy differential equations under generalized differentiability, Fuzzy Sets and Systems280 (2015), 27–57.
15.
ShenY., WangF., A fixed point approach to the Ulam stability of fuzzy differential equations under generalized differentiability, Journal of Intelligent and Fuzzy Systems30 (2016), 3253–3260.
16.
ShenY., Hyers-Ulam-Rassias stability of first order linear partial fuzzy differential equations under generalized differentiability, Advances in Difference Equations2015 (2015), 351.
17.
WangC., XuT., Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives, Applications of Mathematics60 (2015), 383–393.
18.
BrzḑkJ. and EghbaliNasrin, On approximate solutions of some delayed fractional differential equations, Applied Mathematics Letters54 (2016), 31–35.
19.
LakshmikanthamV., MohapatraR., Theory of Fuzzy Differential Equations and InclusionsTaylor and Francis, London, 2003.
20.
ChakravertyS., TapaswiniS., BeheraD., Fuzzy Arbitrary Order SystemJohn Wiley and Sons, Hoboken, 2016.
21.
BedeB., StefaniniL., Generalized differentiability of fuzzyvalued functions, Fuzzy Sets and Systems230 (2013), 119–141.
22.
LakshmikanthamV., GnanaB., VasundharaD., Theory of Set Differential Equations in Metric SpacesCambridge Scientific Publishers, Cambridge, 2006.
23.
GorenfloR., KilbasA., MainardiF. and RogosinS., Mittag-Leffler Functions, Related Topics and ApplicationsSpringer-Verlag, Berlin, 2014.
24.
WangJ., FečkanM. and ZhouY., Presentation of solutions of impulsive fractional Langevin equations and existence results: Impulsive fractional Langevin equations, European Physical Journal Special Topics222 (2013), 1857–1874.
25.
PengS. and WangJ., Cauchy problem for nonlinear fractional differential equations with positive constant coefficient, Journal of Applied Mathematics and Computing51 (2016), 341–351.
26.
YeH., GaoJ., and DingY., A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications328 (2007), 1075–1081.