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Uncertain fractional differential equations on a time scale under Granular differentiability concept

Abstract

In this work, the Caputo q-fractional initial value problem of order \(\alpha \in (0,1)\) in uncertain environment is investigated. The uncertainties are considered as possibility sets using fuzzy sets. The concepts of horizontal membership function and granular difference are used to give a new definition for fuzzy fractional integral and derivative, called granular Riemann–Liouville q-fractional integral and granular Caputo q-fractional derivative. The existence and uniqueness of solution for the Caputo q-fractional initial value problem under granular differentiability concept is established. Finally, some illustrative examples are also presented.

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References

  • Abdalla B, Abdeljawad T, Nieto JJ (2016) A monotonicity result for the \( q-\) fractional operator. J. Math. Anal. 7:83–92

    MathSciNet  MATH  Google Scholar 

  • Abdeljawad T, Baleanu D (2011) Caputo \(q-\)fractional initial value problems and a \(q-\)analogue Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 16:4682–4688

    MathSciNet  MATH  Google Scholar 

  • Abdeljawad T, Alzabut JO (2013) The \(q-\)fractional analogue for Gronwall-type inequality. J Funct. Spaces Appl. 7 (2013) (Article ID 543839)

  • Abdeljawad T, Alzabut J (2018) On Riemann-Liouville fractional q-difference equations and their application to retarded logistic type model. Math. Methods Appl. Sci 41(18):8953–8962

    MathSciNet  MATH  Google Scholar 

  • Ahmad B, Ntouyas S, Alsaedi A (2014) Nonlinear \(q\)-fractional differential equations with nonlocal and sub-strip type boundary conditions. Electron. J. Qual. Theory Differ. Equ. 26:1–12

    MathSciNet  MATH  Google Scholar 

  • Agarwal RP, Lakshmikantham V, Nieto JJ (2010) On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. (TMA) 72:2859–2862

    MathSciNet  MATH  Google Scholar 

  • Agarwal RP, Arshad S, O’Regan D, Lupulescu V (2012) Fuzzy fractional integral equations under compactness type condition. Fract. Calc. Appl. Anal. 15:572–590

    MathSciNet  MATH  Google Scholar 

  • Alikhani R, Bahrami F (2013) Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations. Commun. Nonlinear Sci. Numer. Simul. 18:2007–2017

    MathSciNet  MATH  Google Scholar 

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

    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 

  • Annaby MH, Mansour ZS (2012) \(q-\)fractional Calculus and Equations. Springer, Berlin 2056: 318 pages

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Atici FM, Eloe PW (2007) Fractional \(q-\)calculus on a time scale. J. Nonlinear Math. Phys. 14:333–44

    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 

  • Diethelm K (2010) The Analysis of Fractional Differential Equations An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin 2004: 247 pages

    MATH  Google Scholar 

  • Ernst T (2000) The History of \(q-\)calculus and a New Method. Department of Mathematics, Uppsala University, Sweden

    Google Scholar 

  • Evans RM, Katugampola UN, Edwards DA (2017) Applications of fractional calculus in solving Abel-type integral equations: surface-volume reaction problem. Comput. Math. Appl. 73:1346–1362

    MathSciNet  MATH  Google Scholar 

  • Grecksch W, Roth C, Anh VV (2009) \(Q-\)fractional Brownian motion in infinite dimensions with application to fractional Black–Scholes market. Stoch. Anal. Appl. 27:149–175

    MathSciNet  MATH  Google Scholar 

  • Gomes LT, Barros LC (2015) A note on the generalized difference and the generalized differentiability. Fuzzy Sets Syst. 280:142–145

    MathSciNet  MATH  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, 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 (2018a) Existence results for extremal solutions of interval fractional functional integro-differential equations. Fuzzy Sets Syst. 347:29–53

    MathSciNet  MATH  Google Scholar 

  • Hoa NV, Lupulescu V, O’Regan D (2018b) 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 (2018c) Fuzzy fractional differential equations under Caputo–Katugampola fractional derivative approach. Fuzzy Sets Syst. https://doi.org/10.1016/j.fss.2018.08.001

    Article  MATH  Google Scholar 

  • Jarad F, Abdeljawad T, Baleanu D (2013) Stability of \(q-\)fractional non-autonomous systems. Nonlinear Anal. Real World Appl. 14:780–784

    MathSciNet  MATH  Google Scholar 

  • Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amesterdam

    MATH  Google Scholar 

  • Koca I (2015) A method for solving differential equations of \(q-\)fractional order. Appl. Math. Comput. 266:1–5

    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 NTK, Tam HTT, Yao JC (2017b) Ulam stability for fractional partial integro-differential equation with uncertainty. Acta Math. Vietnam 42:675–700

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Long HV, Dong NP (2018) An extension of Krasnoselskii’s fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty. J. Fixed Point Theory Appl. 20:37

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Malinowski MT (2015) Random fuzzy fractional integral equations-theoretical foundations. Fuzzy Sets Syst. 265:39–62

    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  MATH  Google Scholar 

  • Mazandarani M, Pariz N, Kamyad AV (2018a) Granular differentiability of fuzzy-number-valued functions. IEEE Trans. Fuzzy Syst. 26:310–323

    Google Scholar 

  • Mazandarani M, Naser P (2018b) Sub-optimal control of fuzzy linear dynamical systems under granular differentiability concept. ISA Trans. 76:1–17

    Google Scholar 

  • Mazandarani M, Yi Z (2018c) Fuzzy Bang-Bang control problem under granular differentiability. J. Franklin Inst. 355(12):4931–4951

    MathSciNet  MATH  Google Scholar 

  • Ord GN (1983) Fractal space-time: a geometric analogue of relativistic quantum mechanics. J. Phys. A Math. General 16:1869

    MathSciNet  Google Scholar 

  • Piegat A, Landowski M (2015) Horizontal membership function and examples of its applications. Int. J. Fuzzy Syst. 17:22–30

    MathSciNet  Google Scholar 

  • Podlubny I (1999) Fract. Differ. Equ. Academic Press, San Diego

    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 

  • Rajković PM, Marinković SD, Stanković MS (2007) Fractional integrals and derivatives in \(q-\)calculus. Appl. Anal. Discret. Math. 1(1):311–323

    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 

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Acknowledgements

The authors would like to express deep gratitude to the Associate Editor Professor Rosana Jafelice and anonymous referees for their valuable comments and suggestions which have greatly improved the paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2018.311.

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Correspondence to Ngo Van Hoa.

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Communicated by Rosana Sueli da Motta Jafelice.

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Vu, H., Hoa, N.V. Uncertain fractional differential equations on a time scale under Granular differentiability concept. Comp. Appl. Math. 38, 110 (2019). https://doi.org/10.1007/s40314-019-0873-x

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  • DOI: https://doi.org/10.1007/s40314-019-0873-x

Keywords

  • Horizontal membership function
  • Granular difference
  • Granular derivative
  • Riemann–Liouville q-fractional integral
  • Caputo q-fractional derivative

Mathematics Subject Classification

  • 03E72
  • 05A30
  • 33D05
  • 39A13
  • 74H20
  • 74H25