Abstract
The main purpose of this paper is to introduce fuzzy Adams–Bashforth (A–B) and fuzzy Adams–Moulton (A–M) methods based on the generalized Hukuhara (gH)-differentiability and employ them as the predictor and corrector, respectively. The local truncation error, stability and convergence of these methods are discussed in the sequel. Finally, some fuzzy linear and nonlinear initial value problems (IVPs) are solved. The numerical results obtained here show that our methods provide a suitable approximation for the exact solution.
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References
Abbasbandy S, Allahviranloo T (2002) Numerical solution of fuzzy differential equation by Taylor method. Comput Methods Appl Math 2(2):113–124
Abbasbandy S, Allahviranloo T (2004) Numerical solution of fuzzy differential equation by Runge-Kutta method. Nonlinear Stud 11(1):117–129
Ahmady N (2019) A numerical method for solving fuzzy differential equations with fractional order. Int J Ind Math 11(2):71–77
Allahviranloo T, Ahmady N, Ahmady E (2007) Numerical solution of fuzzy differential equation by predictor-corrector method. Inform Sci 177(7):1633–1647
Allahviranloo T, Armand A, Gouyandeh Z (2014) Fuzzy fractional differential equations under generalized fuzzy Caputo derivative. J Intell Fuzzy Syst 26(3):1481–1490
Allahviranloo T, Gholami S (2012) Note on generelized Hukuhara differentiability of interval-valued functions and interval differential equations. J Fuzzy Set Valued Anal 1–4
Allahviranloo T, Gouyandeh Z, Armand A (2015) A method for solving fuzzy differential equations based on fuzzy Taylor expansion. Undefined 1:1–16
Allahviranloo T, Kiani NA, Motamedi N (2009) Solving fuzzy differential equations by differential transformation method. Inform Sci 179(7):956–966
Babakordi F, Allahviranloo T (2021) A new method for solving fuzzy Bernoulli differential equation. J Math Ext 15(4):1–20
Barai SV, Nair RS (2004) Neuro-fuzzy models for constructability analysis. J Inf Technol Constr (ITcon) 9(4):65–73
Branco PJC, Dente JA (2000) On using fuzzy logic to integrate learning mechanisms in an electro-hydraulic system. I. Actuator’s fuzzy modeling. IEEE Trans Syst Man Cybern 30(3):305–316
Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst. 151(3):581–599
Bede B, Gal SG (2006) Remark on the new solutions of fuzzy differential equations, Chaos Solit. Fractals, unpublished
Bede B, Stefanini L (2011) Solution of fuzzy differential equations with generelized differentiability using LU-parametric representation, Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-11) 785–790
Bede B, Stefanini L (2013) Generelized differentiability of fuzzy-valued functions. Fuzzy Sets Syst 230:119–141
Chalco-Cano Y, Román-Flores H (2008) On new solutions of fuzzy differential equations. Chaos Solit Fractals 38(1):112–119
Chalco-Cano Y, Román-Flores H, Jiménez-Gamero M-D (2011) Generalized derivative and \(\pi \)-derivative for set-valued functions. J Inf Sci 181(11):2177–2188
Chang SSL, Zadeh LA (1972) On fuzzy mapping and control. IEEE Trans Syst Cybern 2(1):30–34
Chen X, Gu H, Wang X (2020) Existence and uniqueness for fuzzy differential equation with Hilfer-Katugampola fractional derivative, Adv Difference Equ Paper No. 241, 16 pp
Gear CW (1971) Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall, Englewood Clifs
Guo M, Xiaoping X, Li R (2003) Impulsive functional differential inclusions and fuzzy population models. Fuzzy Sets Syst 138(3):601–615
Hukuhara M (1967) Integration des applications mesurables dont la valeur est un compact convex. Funkc Ekvacioj 10(3):205–233
Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317
Liu X-M, Jiang J, Hong L (2021) A numerical method to solve a fuzzy differential equation via differential inclusions. Fuzzy Sets Syst 404:38–61
Miri Karbasaki M, Balooch Shahriari M, Sedaghatfar O (2023) The fuzzy D’Alembert solutions of the fuzzy wave equation under generalized differentiability. J Mahani Math Res 12(1):91–126
Ma M, Friedman M, Kandel A (1999) Numerical solutions of fuzzy differential equations. Fuzzy Sets Syst 105(1):133–138
Mansouri SS, Gachpazan M, Ahmady N, Ahmady E (2022) On the existence and uniqueness of fuzzy differential equations with monotone condition. J Math Ext 16(5):1–17
Mon D-L, Cheng C-H, Lin J-C (1994) Evaluating weapon system using fuzzy analytic hierarchy process based on entropy weight. Fuzzy Sets Syst 62(2):127–134
Negoiţă CV, Virgil C, Ralescu DA (1975) Applications of Fuzzy Sets to Scstems Analysis. Springer, New York
Park JY, Han HK (1999) Existence and uniqueness theorem for a solution of fuzzy differential equations. Internat J Math & Math Sci 22(2):271–279
Puri ML, Ralescu DA (1983) Differentials of fuzzy functions. J Math Anal Appl 91(2):552–558
Rostami M, Kianpour M, Bashardoust E (2011) A numerical algorithm for solving nonlinear fuzzy differential equations. J Math Comput Sci 2(4):667–671
Shahryari N, Abbasbandy S (2021) Numerical solution of second-order hybrid fuzzy differential equations by generalized differentiability. Int J Ind Math 13(4):451–464
Stefanini L, Bede B (2009) Generelized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal 71(3–4):1311–1328
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Safikhani, L., Vahidi, A., Allahviranloo, T. et al. Multi-step gH-difference-based methods for fuzzy differential equations. Comp. Appl. Math. 42, 27 (2023). https://doi.org/10.1007/s40314-022-02167-9
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DOI: https://doi.org/10.1007/s40314-022-02167-9
Keywords
- Generalized Hukuhara difference
- Fuzzy Adams–Bashforth method
- Fuzzy Adams–Moulton method
- Local truncation error