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Modeling in systems biology is often faced with challenges in terms of measurement
uncertainty. This is possibly either due to limitations of available data, environmental or
demographic changes. One of typical behavior that commonly appears in the systems biology is
a periodic behavior. Since uncertainties would get involved into the systems, the change of
solution behavior of the periodic system should be taken into account. To get insight into this
issue, in this work a simple mathematical model describing periodic behavior, i.e. a harmonic
oscillator model, is considered by assuming its initial value has uncertainty in terms of fuzzy
number. The system is known as Fuzzy Initial Value Problems. Some methods to determine the
solutions are discussed. First, solutions are examined using two types of fuzzy differentials,
namely Hukuhara Differential (HD) and Generalized Hukuhara Differential (GHD). Application
of fuzzy arithmetic leads that each type of HD and GHD are formed into α-cut deterministic
systems, and then are solved by the Runge-Kutta method. The HD type produces a solution with
increasing uncertainty starting from the initial condition. While, GHD type produces a periodic
solution but only until a certain time and above it the uncertainty becomes monotonic increasing.
Solutions of both types certainly do not provide the accuracy for harmonic oscillator model
which should show periodic behavior during its evolution. Therefore, we propose the third
method, so called fuzzy differential inclusions, to attack the problem. Using this method, we
obtain periodic solutions during its evolution.
Keywords: fuzzy initial value problems, fuzzy arithmetics, α-cut deterministic systems, fuzzy
differential inclusions. |
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