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Differential Equations With Fuzzy Parameters

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  • M. Oberguggenberger
  • S. Pittschmann

Abstract

This paper is concerned with systems of ordinary differential equations with fuzzy parameters. Applying the Zadeh extension principle to the equations, we introduce the notions of fuzzy solutions and of componentwise fuzzy solutions. The fuzzy extension of the solution operator is shown to provide the unique fuzzy solution as well as the maximal componentwise fuzzy solution. A numerical algorithm based on monotonicity properties of membership functions is presented, together with a proof of convergence. In an interplay of interval analysis and possibility theory, these methods allow to process subjective information on the possible fluctuations of parameters in models involving ordinary differential equations. This is demonstrated in two engineering applications: a queueing model for earthwork and a model of oscillations of bell-towers.

Suggested Citation

  • M. Oberguggenberger & S. Pittschmann, 1999. "Differential Equations With Fuzzy Parameters," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 5(3), pages 181-202, September.
    • Handle: RePEc:taf:nmcmxx:v:5:y:1999:i:3:p:181-202
    DOI: 10.1076/mcmd.5.3.181.3683
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    Cited by:

    1. Mawia Osman & Yonghui Xia & Omer Abdalrhman Omer & Ahmed Hamoud, 2022. "On the Fuzzy Solution of Linear-Nonlinear Partial Differential Equations," Mathematics, MDPI, vol. 10(13), pages 1-49, June.
    2. Yilmaz, Yilmaz, 2009. "Fréchet differentiation of nonlinear operators between fuzzy normed spaces," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 473-484.
    3. Animesh Mahata & Sankar Prasad Mondal & Ali Ahmadian & Fudiah Ismail & Shariful Alam & Soheil Salahshour, 2018. "Different Solution Strategies for Solving Epidemic Model in Imprecise Environment," Complexity, Hindawi, vol. 2018, pages 1-18, May.
    4. Chalco-Cano, Y. & Román-Flores, H., 2008. "On new solutions of fuzzy differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 112-119.
    5. Nadjafikhah, M. & Bakhshandeh-Chamazkoti, R., 2009. "Fuzzy differential invariant (FDI)," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1677-1683.

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