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Solving variable-order fractional differential algebraic equations via generalized fuzzy hyperbolic model with application in electric circuit modeling

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Abstract

In this paper, a new approach based on a generalized fuzzy hyperbolic model is used for the numerical solution of variable-order fractional differential algebraic equations. The fractional derivative is described in the Atangana–Baleanu sense that is a new derivative with fractional order based on the generalized Mittag–Leffler function. First, by using fuzzy solutions with adjustable parameters, the variable-order fractional differential algebraic equations are reduced to a problem consisting of solving a system of algebraic equations. For adjusting the parameters of fuzzy solutions, an unconstrained optimization problem is then considered. A learning algorithm is also presented for solving the unconstrained optimization problem. Finally, some numerical examples are given to verify the efficiency and accuracy of the proposed approach.

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References

  • Atangana A, Alqahtani RT (2016) Stability analysis of nonlinear thin viscous fluid sheet flow equation with local fractional variable-order derivative. J Comput Theor Nanosci 13:2710–17

    Google Scholar 

  • Bazaraa MS, Sherali HD, Shetty CM (2006) Nonlinear programming: theory and algorithms, 3rd edn. Wiley, Hoboken

    MATH  Google Scholar 

  • Bendtsen C, Thomsen PG (1999) Numerical solution of differential algebraic equations, technical report, Department of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark

  • Boulkaibet I, Belarbi K, Bououden S, Marwala T, Chadli M (2017) A new T–S fuzzy model predictive control for nonlinear processes. Expert Syst Appl 88:132–151

    Google Scholar 

  • Buckley JJ (1992) Universal fuzzy controllers. Automatica 28:1245–1248

    MathSciNet  MATH  Google Scholar 

  • Cao J, Qiu Y, Song G (2017) A compact finite difference scheme for variable order subdiffusion equation. Commun Nonlinear Sci Numer Simul 48:140–149

    MathSciNet  MATH  Google Scholar 

  • Chen YY, Chang YT, Chen BS (2009) Fuzzy solutions to partial differential equations: adaptive approach. IEEE Trans Fuzzy Syst 17(1):116–127

    Google Scholar 

  • Chen Y-M, Wei Y-Q, Liu D-Y, Yu H (2015) Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets. Appl Math Lett 46:83–88

    MathSciNet  MATH  Google Scholar 

  • Coronel-Escamilla A, Gómez-Aguilar JF, Torres L, Escobar-Jiménez RF (2018) A numerical solution for a variable-order reaction–diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys A 491:406–24

    MathSciNet  Google Scholar 

  • Cui Y, Zhang HG, Wang Y, Gao W (2016) Adaptive control for a class of uncertain strict-feedback nonlinear systems based on a generalized fuzzy hyperbolic model. Fuzzy Sets Syst 302:52–64

    MathSciNet  MATH  Google Scholar 

  • Dang QV et al (2017) Robust stabilizing controller design for Takagi–Sugeno fuzzy descriptor systems under state constraints and actuator satu-ration. Fuzzy Sets Syst 329:77–90

    MATH  Google Scholar 

  • Deng W, Zhao H, Zou L, Li G, Yang X, Wu D (2017a) A novel collaborative optimization algorithm in solving complex optimization problems. Soft Comput 21:4387–4398

    Google Scholar 

  • Deng W, Zhao H, Yang X, Xiong J, Sun M, Li B (2017b) Study on an improved adaptive PSO algorithm for solving multi-objective gate assignment. Appl Soft Comput 59:288–302

    Google Scholar 

  • Deng W, Xu J, Zhao H (2019) An improved ant colony optimization algorithm based on hybrid strategies for scheduling problem. IEEE Access 7:20281–20292

    Google Scholar 

  • Dong J, Fu Y (2017) A design method for T–S fuzzy systems with partly immeasurable premise variables subject to actuator saturation. Neurocomputing 225:164–173

    Google Scholar 

  • Fu Z, Chen W, Ling L (2015) Method of approximate particular solutions for constant- and variable-order fractional diffusion models. Eng Anal Bound Elem 57:37–46

    MathSciNet  MATH  Google Scholar 

  • Ghanbari B, Gómez-Aguilar JF (2018) Modeling the dynamics of nutrient–phytoplankton–zooplankton system with variable-order fractional derivatives. Chaos, Solitons Fractals 116:114–120

    MathSciNet  MATH  Google Scholar 

  • Ghanbari F, Ghanbari K, Mokhtary P (2018) Generalized Jacobi–Galerkin method for nonlinear fractional differential algebraic equations. Comput Appl Math 37(4):5456–5475

    MathSciNet  MATH  Google Scholar 

  • Ghomanjani F (2017) A new approach for solving fractional differential–algebraic equations. J Taibah Univ Sci 11:1158–1164

    Google Scholar 

  • Gómez-Aguilar JF (2018) Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations. Phys A 494:52–75

    MathSciNet  Google Scholar 

  • Jia Y, Xu M, Lin Y (2017) A numerical solution for variable order fractional functional differential equation. Appl Math Lett 64:125–130

    MathSciNet  MATH  Google Scholar 

  • Karabacak M, Celik E (2013) The numerical solution of fractional differential-algebraic equations (FDAEs). New Trends Math Sci 1(2):1–6

    Google Scholar 

  • Kosko B (1994) Fuzzy systems as universal approximators. IEEE Trans Comput 43(11):1329–1333

    MATH  Google Scholar 

  • Lee KY, El-Sharkawi MA (2008) Modern heuristic optimization techniques: theory and applications to power systems. Wiley, Hoboken

    Google Scholar 

  • Li X, Wu B (2017) A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations. J Comput Appl Math 311:387–393

    MathSciNet  MATH  Google Scholar 

  • Li X, Li H, Wu B (2017) A new numerical method for variable order fractional functional differential equations. Appl Math Lett 68:80–86

    MathSciNet  MATH  Google Scholar 

  • Liu H, Fu Y, Li B (2017) Discrete waveform relaxation method for linear fractional delay differential-algebraic equations. Discrete Dyn Nat Soc 6306570:9 pages

  • Liu S, Guo Z, Zhang HG (2008) Fuzzy hyperbolic neural network model and its application in H1 filter design, In: Sun F et al (eds) Proceedings of the 5th international symposium on neural networks: advances in neural networks, part I, LNCS 5263, Springer, Berlin, 2008, pp 222–230

  • Mei W, Bullo F (2017) Lasalle invariance principle for discrete-time dynamical systems: a concise and self-contained tutorial. arXiv preprint arXiv:1710.03710

  • Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York

    MATH  Google Scholar 

  • Mirzajani S, PourmahmoodAghababa M, Heydari A (2019) Adaptive T–S fuzzy control design for fractional-order systems withparametric uncertainty and input constraint. Fuzzy Sets Syst 365(15):22–39

    MATH  Google Scholar 

  • Muthukumar P, Balasubramaniam P, Ratnavelu K (2016) T–S fuzzy predictive control for fractional order dynamical systems and its applications. Nonlinear Dyn 86(2):751–763

    MATH  Google Scholar 

  • Nocedal J, Wright S (2006) Numerical Optimization, second edn. Springer- Verlag, Berlin, NewYork

    MATH  Google Scholar 

  • Pakdaman M, Effati S (2016) Approximating the solution of optimal control problems by fuzzy systems. Neural Process Lett 43(3):667–686

    Google Scholar 

  • Pham VT, Vaidyanathan S, Volos C, Kapitaniak T (2018) Nonlinear dynamical systems with self-excited and hidden attractors. Springer, Berlin

    MATH  Google Scholar 

  • Sh DM, Hassani H (2017) An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation. Nonlinear Dyn 88:1587–1598

    MathSciNet  MATH  Google Scholar 

  • Shen S, Liu F, Anh V, Turner I, Chen J (2013) A characteristic difference method for the variable-order fractional advection–diffusion equation. J Appl Math Comput 42:371–386

    MathSciNet  MATH  Google Scholar 

  • Shen H, Su L, Park JH (2017) Reliable mixed/passive control for T–S fuzzy delayed systems based on a semi-Markov jump model approach. Fuzzy Sets Syst 314:79–98

    MathSciNet  MATH  Google Scholar 

  • Shiri B, Baleanu D (2019) System of fractional differential algebraic equations with applications. Chaos Solitons Fractals 120:203–212

    MathSciNet  Google Scholar 

  • Solís-Pérez JE, Gómez-Aguilar JF, Atangana A (2018) Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag–Leffler laws. Chaos Solitons Fractals 114:175–185

    MathSciNet  MATH  Google Scholar 

  • Sun H, Chen W, Li CH, Chen Y (2012) Finite difference schemes for variable-order time fractional diffusion equation. Int J Bifurc Chaos 22:1250085

    MathSciNet  MATH  Google Scholar 

  • Sun Q, Wang Y, Yang J, Qiu Y, Zhang H (2014) Chaotic dynamics in smart grid and suppression scheme via generalized fuzzy hyperbolic model. Math Probl Eng 2014, Article ID 761271, 7 pages

  • Sun H, Zhang Y, Baleanu D, Chen W, Chen Y (2018) A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul 64:213–231

    Google Scholar 

  • Takagi T, Sugeno M (1985) Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern 15(1):116–132

    MATH  Google Scholar 

  • Wang LX (1992) Fuzzy systems are universal approximators. In: Proceedings 1EEE international conference on fuzzy systems (San Diego), pp 1163–1170

  • Wang L, Chen N (2014) The predictor-corrector solution for fractional order differential algebraic equation. In: CFDA’14 international conference on fractional differentiation and its applications

  • Wang SH, Jhu WL, Yung CF, Wang PF (2011) Numerical solutions of differential algebraic equations and its applications in solving TPPC problems. J Mar Sci Technol 19:76–88

    Google Scholar 

  • Wu ZG, Dong SH, Shi P, Su H, Huang T, Lu R (2017) Fuzzy-model-based nonfragile guaranteed cost control of nonlinear markov jump systems. IEEE Trans Syst Man Cybern Syst 47(8):1–10

    Google Scholar 

  • Xu T, Lu S, Chen W, Chen H (2018) Finite difference scheme for multi-term variable-order fractional diffusion equation. Adv Differ Equ 103:1–13

    MathSciNet  MATH  Google Scholar 

  • Yaghoobi S, Moghaddam BP, Ivaz K (2017) An efficient cubic spline approximation for variable-order fractional differential equations with time delay. Nonlinear Dyn 87:815–826

    MathSciNet  MATH  Google Scholar 

  • Yang XJ (2017) Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. Therm Sci 21:1161–1171

    Google Scholar 

  • Ying H (1994) Sufficient conditions on general fuzzy systems as function approximators. Automatica 30:521–525

    MATH  Google Scholar 

  • Zak SH (2003) Systems and control. Oxford University Press, Oxford

    Google Scholar 

  • Zeng XJ, Singh MG (1995) Approximation theory of fuzzy systems-MIMO case. IEEE Trans Fuzzy Syst 3(2):219–235

    Google Scholar 

  • Zhang X-S (2000) Neural networks in optimization. Kluwer Academic Publishers, Dordrecht

    MATH  Google Scholar 

  • Zhang HG, Yongbing Q (2001) Modeling, identification, and control of a class of nonlinear systems. IEEE Trans Fuzzy Syst 9(2):349–354

    MATH  Google Scholar 

  • Zhang M, Zhang H (2005) Modeling and control based on generalized fuzzy hyperbolic model. In: American control conference

  • Zhang M, Zhang H (2006) Robust adaptive fuzzy control scheme for nonlinear system with uncertainty. J Control Theory Appl 4(2):209–216

    MathSciNet  MATH  Google Scholar 

  • Zhang HG, Wang Z, Liu D (2003) Chaotifying fuzzy hyperbolic model using adaptive inverse optimal control approach. Int J Bifurc Chaos 12:32–43

    Google Scholar 

  • Zhang HG, Wang ZL, Li M, Quan B, Zhang MG (2004a) Generalized fuzzy hyperbolic model: a universal approximator. Acta Autom Sin 30(3):416–422

    MathSciNet  Google Scholar 

  • Zhang HG, Wang ZL, Li M, Quan B, Zhang MJ (2004b) Generalized fuzzy hyperbolic model: a universal approximator. Acta Autom Sin 30(3):416–422

    MathSciNet  Google Scholar 

  • Zhang M, Zhang H, Liu D (2004c) A generalized fuzzy hyperbolic modeling and control scheme. In: IEEE international conference on fuzzy systems

  • Zhang HG, Wang Z, Liu D (2005) Chaotifying fuzzy hyperbolic model using impulsive and nonlinear feedback control approaches. Int J Bifurc Chaos 15(8):2603–2610

    MathSciNet  MATH  Google Scholar 

  • Zhang JL, Zhang HG, Luo YH, Liang HJ (2013) Nearly optimal control scheme using adaptive dynamic programming based on generalized fuzzy hyperbolic model. Acta Autom Sin 39(2):142–148

    MathSciNet  MATH  Google Scholar 

  • Zhao H, Liu H, Xu J, Deng W (2019a) Performance prediction using high-order differential mathematical morphology gradient spectrum entropy and extreme learning machine. IEEE Trans Instrum Meas. https://doi.org/10.1109/TIM.2019.2948414

    Article  Google Scholar 

  • Zhao H, Zheng J, Xu J, Deng W (2019b) Fault diagnosis method based on principal component analysis and broad learning system. IEEE Access 7:99263–99272

    Google Scholar 

  • Zhao H, Zheng J, Deng W, Song Y (2020) Semi-supervised broad learning system based on manifold regularization and broad network. IEEE Trans Circuits Syst I Regul Pap 67(3):983–994

    MathSciNet  Google Scholar 

  • Zhu Q, Azar AT (2015) Complex system modelling and control through intelligent soft computations. Springer, Berlin

    MATH  Google Scholar 

  • Zurigat M, Momani SH, Alawneh A (2010) Analytical approximate solutions of systems of fractional algebraic differential equations by homotopy analysis method. Comput Math Appl 59:1227–1235

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

    Marzieh Mortezaee, Mehdi Ghovatmand & Alireza Nazemi

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  1. Marzieh Mortezaee
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  2. Mehdi Ghovatmand
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  3. Alireza Nazemi
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Correspondence to Mehdi Ghovatmand.

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Mortezaee, M., Ghovatmand, M. & Nazemi, A. Solving variable-order fractional differential algebraic equations via generalized fuzzy hyperbolic model with application in electric circuit modeling. Soft Comput 24, 16745–16758 (2020). https://doi.org/10.1007/s00500-020-04969-7

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