Restricted accessResearch articleFirst published online 2019-06-11
On the initial value problem for random fuzzy differential equations with Riemann-Liouville fractional derivative: Existence theory and analytical solution
In this paper, the existence and uniqueness results of an initial value problem for random fuzzy differential equations with Riemann-Liouville fractional derivative (RFFDEs) are introduced. A method to find analytical solution of RFFDEs is also proposed. Finally, some initial value problems of linear random fractional fuzzy differential equation and the fish proportional harvesting model in fuzzy environment are given to show the efficiency of the method.
AgarwalR.P., LakshmikanthamV., NietoJ.J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis: (TMA)72 (2010), 2859–2862.
2.
AgarwalR.P., ArshadS., OReganD., LupulescuV., Fuzzy fractional integral equations under compactness type condition,, Frac Cal and App Anal15 (2012), 572–590.
3.
AgrazP.T., On Borel measurability and large deviations for fuzzy random variables,, Fuzzy Sets and Systems157 (2006), 2558–2568.
4.
AlikhaniR., BahramiF., Global solutions for nonlinear fuzzy fractional integral and integro differential equations, Commun Nonlinear Sci Numer Simulat18 (2013), 2007–2017.
5.
AllahviranlooT., GouyandehZ., ArmandA., Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal of Intelligent & Fuzzy Systems26 (2014), 1481–1490.
6.
AllahviranlooT., SalahshourS., AbbasbandyS., Explicit solutions of fractional differential equations with uncertainty, Soft Comput Fus Found Meth Appl16 (2012), 297–302.
7.
AnT.V., VuH., HoaN.V., Applications of contractive-like mapping principles to interval-valued fractional integro-differential equations, Journal of Fixed Point Theory and Applications19 (2017), 2577–2599.
8.
AnA.T.V., VuH., HoaN.V., A new technique to solve the initial value problems for fractional fuzzy delay differential equations, Advances in Difference Equations2017 (2017), 181.
9.
ArshadS., LupulescuV., On the fractional differential equations with uncertainty, Nonlinear Analysis: (TMA)74 (2011), 85–93.
10.
BedeB., StefaniniL., Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems230 (2013), 119–141.
11.
BhaskarT.G., LakshmikanthamV., LeelaS., Fractional differential equations with a Krasnoselskii– Krein type condition, Nonlinear Analysis: Hybrid Systems3 (2009), 734–737.
12.
ColubiA., Domínguez-MencheroJ.S., López-DíazM., RalescuD.A., A DE[0, 1] representation of random upper semicontinuous functions, Proc Amer Math Soc130 (2002), 3237–3242.
13.
HoaN.V., Fuzzy fractional functional differential equations under Caputo gH differentiability, Commun Nonlinear Sci Numer Simulat22 (2015), 1134–1157.
14.
HoaN.V., VuH., DucT.M., Fuzzy fractional differential equations under Caputo- Katugampola fractional derivative approach, Fuzzy Sets and Systems (2018).
15.
HoaN.V., Fuzzy fractional functional integral and differential equations, Fuzzy Sets and Systems280 (2015), 58–90.
16.
HoaN.V., LupulescuV., O’ReganD., Solving interval-valued fractional initial value problems under Caputo gH-fractional differentiability, Fuzzy Sets and Systems309 (2017), 1–34.
17.
HoaN.V., LupulescuV., O’ReganD., A note on initial value problems for fractional fuzzy differential equations, Fuzzy Sets and Systems347 (2018), 54–69.
18.
HoaN.V., Existence results for extremal solutions of interval fractional functional integro-differential equations, Fuzzy Sets and Systems347 (2018), 29–53.
19.
KalevaO., A note on fuzzy differential equations, Nonlinear Anal64 (2006), 895–900.
20.
KilbasA.A., SrivastavaH.M., TrujilloJ.J., Theory and applications of fractional differential equations, Amesterdam: Elsevier Science B.V, 2006.
21.
KlementE.P., PuriM.L., RalescuD.A., Limit theorems for fuzzy random variables, Proc Roy Soc Lond A407 (1986), 171–182.
22.
KruseR., MeyerK.D., Statistics with Vague Data, Kluwer, Dordrecht, 1987.
23.
KwakernaakH., Fuzzy random variables. Part I: Definitions and theorems, Inform Sci15 (1978), 1–29.
24.
LakshmikanthamV., Theory of fractional functional differential equations, Nonlinear Analysis: Theory, Methods & Applications69 (2008), 3337–3343.
25.
LakshmikanthamV., LeelaS., A Krasnoselskii-Krein-type uniqueness result for fractional differential equations, Nonlinear Analysis: TMA71 (2009), 3421–3424.
26.
LakshmikanthamV., MohapatraR.N., Theory of fuzzy differential equations and applications, Taylor & Francis, London, 2003.
27.
LakshmikanthamV., VatsalaA.S., Theory of fractional differential inequalities and applications, Communications in Applied Analysis11 (2007), 395–402.
28.
LongH.V., SonN.T.K., HoaN.V., Fuzzy fractional partial differential equations in partially ordered metric spaces, Iranian Journal of Fuzzy Systems14 (2017), 107–126.
29.
LongH.V., SonN.K., TamH.T., YaoJ.C., Ulam Stability for fractional partial integrodifferential equation with uncertainty, Acta Math Vietnam42 (2017), 675–700.
30.
LongH.V., SonN.K., TamH.T., The solvability of fuzzy fractional partial differential equations under Caputo gH-differentiability, Fuzzy Sets Systems309 (2017), 35–63.
31.
LongH.V., On random fuzzy fractional partial integro-differential equations under Caputo generalized Hukuhara differentiability, Computational and Applied Mathematics37 (2018), 2738–2765.
32.
LongH.V., DongN.P., An extension of Krasnoselskiis fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty, Journal of Fixed Point Theory and Applications20 (2018), 37.
33.
LupulescuV., HoaN.V., Existence and uniqueness of solutions for random fuzzy fractional integral and differential equations, Journal of Intelligent & Fuzzy Systems29 (2015), 27–42.
34.
LupulescuV., Fractional calculus for interval-valued functions, Fuzzy Set and Systems265 (2015), 63–85.
35.
MazandaraniM., KamyadA.V., Modied fractional Euler method for solving fuzzy fractional initial value problem, Communications in Nonlinear Science and Numerical Simulation18 (2013), 12–21.
36.
MazandaraniM., NajariyanM., Type-2 fuzzy fractional derivatives, Communications in Nonlinear Science and Numerical Simulation19 (2014), 2354–2372.
37.
MalinowskiM.T., On random fuzzy differential equations, Fuzzy Sets and Systems160 (2009), 3152–3165.
38.
MalinowskiM.T., Random fuzzy differential equations under generalized Lipschitz condition, Nonlinear Analysis: RealWorld Applications13 (2012), 860–881.
39.
MalinowskiM.T., Existence theorems for solutions to random fuzzy differential equations, Nonlinear Analysis: TMA73 (2010), 1515–1532.
40.
MalinowskiM.T., Random fuzzy fractional integral equations - theoretical foundations, Fuzzy sets and Systems265 (2015), 39–62.
41.
MalinowskiM.T., Fuzzy and set-valued stochastic differential equations with local Lipschitz condition, IEEE Transactions on Fuzzy Systems23 (2015), 1891–1898.
42.
MalinowskiM.T., AgarwalR.P., On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute352 (2015), 3014–3043.
43.
MalinowskiM.T., Some properties of strong solutions to stochastic fuzzy differential equations, Information Sciences252 (2013), 62–80.
44.
MalinowskiM.T., Itô type stochastic fuzzy differential equations with delay, Systems & Control Letters61 (2012), 692–701.
45.
MalinowskiM.T., Strong solutions to stochastic fuzzy differential equations of Itô type, Mathematical and Computer Modelling55 (2012), 918–928.
46.
MalinowskiM.T., Random fuzzy fractional integral equations-theoretical foundations, Fuzzy Sets and Systems265 (2015), 39–62.
47.
NguyenH.T., A note on the extension principle for fuzzy sets, J Math Anal Appl64 (1978), 369–380.
48.
ParkJ.Y., JeongJ.U., On random fuzzy functional differential equations, Fuzzy Sets and Systems223 (2013), 89–99.
49.
PodlubnyI., Fractional differential equation, San Diego:Academic Press, 1999.
50.
PuriM.L., RalescuD., Differential for fuzzy function, Journal of Mathematical Analysis and Applications91 (1983), 552–558.
51.
PuriM.L., RalescuD.A., The concept of normality for fuzzy random variables, Ann Probab13 (1985), 1373–1379.
52.
PuriM.L., RalescuD.A., Fuzzy random variables, J Math Anal Appl114 (1986), 409–422.
53.
SalahshourS., AllahviranlooT., AbbasbandyS., BaleanuD., Existence and uniqueness results for fractional differential equations with uncertainty, Advances in Difference Equations2012 (2012), 112.
54.
SalahshourS., AllahviranlooT., AbbasbandyS., Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Communications in Nonlinear Science and Numerical Simulation17 (2012), 1372–1381.
55.
SongS., WuC., Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations, Fuzzy Sets and Systems110 (2000), 55–67.
56.
VuH., DongL.S. and HoaN.V., Random fuzzy functional integro-differential equations under generalized Hukuhara differentiability, Journal of Intelligent & Fuzzy Systems27 (2014), 1491–1506.
57.
VuH., Ho, LupulescuV. and HoaN.V., Existence of extremal solutions to interval-valued delay fractional differential equations via monotone iterative technique, Journal of Intelligent & Fuzzy Systems34 (2018), 2177–2195.
58.
VuH., HoaN.V., SonN.T.K., O’ReganeD., Results on initial value problems for random fuzzy fractional functional differential equations, Filomat32 (2018), 2601–2624.
