Solving a Fuzzy Initial Value Problem of a Harmonic Oscillator Model
M_Ahsar_Pres_Symomath_2016Solving a Fuzzy Initial Value Problem of a Harmonic Oscillator Model
(Citation: AIP Conference Proceedings 1825, 020011 (2017); doi: 10.1063/1.4978980)
Muhammad Ahsar K.1,2,*, Agus Yodi Gunawan1, Mochamad Apri1, and Kuntjoro Adji Sidarto1
1Department of Mathematics, Institut Teknologi Bandung, Bandung, Indonesia
2Faculty of Mathematics and Natural Sciences, Universitas Lambung Mangkurat, Banjarbaru, Indonesia
*m_ahsar@yahoo.com
Abstract
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.
References
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Note: Telah diseminarkan pada The 4rd Symposium of BioMathematics (Symomath 2016)
