Solving a Fuzzy Initial Value Problem of a Harmonic Oscillator Model

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 alpha-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 an oscillatory 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 during its evolution. Therefore, we propose the third method, so called Fuzzy Differential Inclusions (FDI), to attack the problem. Using this method, we obtain oscillatory solutions during its evolution.