PARAMETER ESTIMATION OF FUZZY ORDINARY DIFFERENTIAL EQUATIONS - Ahsar K, Muhammad
Most of systems in the real world contain uncertainties. Those uncertainties can be caused by limited available data, network complexity of the system, environmental or demographic changes during doing experiments. Therefore, a mathematical modeling is needed as an effort to understand the system phenomenon, to simulate pre-experiments before actually being implemented, or to study dynamics of system that is often difficult or even not manageable by experiments. By accommodating uncertainty factors in initial values and parameters in the model, an in depth study is needed to describe the structure of mathematics, the methodology for determining solutions, and procedures for estimating parameters. To get insight into these, in this dissertation we take an exponential growth model, a logistic growth model, a Goodwin model and a forced Van Der Pol model, as our study objects. These models are all assumed to have uncertainties in the initial values in the form of fuzzy numbers, known as fuzzy models. They are examined using three fuzzy differential approaches, namely Hukuhara Differential and its generalizations, and Fuzzy Differential Inclusions. Applications of the concept of fuzzy arithmetic to these models lead to alpha-cut deterministic systems. The alpha-cut deterministic system is then solved using two numerical methods: the classical Runge-Kutta and the extended Runge-Kutta methods. Among these three fuzzy approaches, fuzzy differential inclusion is the most appropriate approach to capture the periodic behavior of equations, for both numerical methods. By choosing the concept of fuzzy differential inclusion, a procedure to estimate parameters is presented for a set of fuzzy data simulation using the nonlinear least squares method.