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Most of the real systems in the world may contain uncertainties, which are possibly due to the
limitations of available data, complexity of the network of systems, and environmental or
demographic changes at the time of observation. One of the system behaviors that often appears
in mathematical modeling is periodic behavior, which often shows complex dynamic behavior,
depending on initial values and parameters. By accommodating the uncertainties in the model,
in-depth studies are needed to describe mathematical structure, methodology for determining
solution, and procedure for estimating parameters. Among the mathematical models that describe
periodic behavior is harmonic oscillator equation. In this paper, the model is assumed to have
uncertainty in the initial values in the form of fuzzy numbers, which is then called by fuzzy
harmonic oscillator equation. The model is examined by comparing three fuzzy differential
approaches, namely Hukuhara differential, generalized Hukuhara differential and fuzzy
differential inclusions. Applications of fuzzy arithmetic concepts to the models lead to a
deterministic alpha-cut systems, which are solved using extended Runge-Kutta method. In
contrast to the standard Runge-Kutta method, the extended Runge-Kutta method using the first
derivative approximation of the evaluation function to increase the accuracy of the solution.
Among the three fuzzy approaches, the fuzzy differential inclusion type is the most appropriate
approach to capture the periodic behavior of the equation. Next, it is shown how to estimate the
parameters of solution of the fuzzy differential inclusion type and simulation of fuzzy data using
lsqnonlin method.
Keywords: Parameter estimation; Hukuhara differential; Fuzzy differential inclusion; Extended
Runge-Kutta method; Fuzzy harmonic oscillator equation |
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