Extended Runge-Kutta fourth order method with polynomial interpolation technique for fuzzy population models

Abstract

Uncertainty quantification plays an increasingly important role in the mathematical modeling of physical phenomena. One alternative of the mathematical modelings is provided by fuzzy sets. The main research of this thesis is the study of numerical method in solving fuzzy differential equations (FDEs). In this thesis, the problem of FDEs in one-dimensional problem and two-dimensional problem were considered, namely fuzzy logistic differential equation and fuzzy predatorprey systems. The problems were solved using extended Runge-Kutta fourth order (ERK4) method. Nevertheless, due to the lacking of numerical methods available for solving polynomial type of FDEs, the ERK4 method is incorporated with polynomial interpolation technique in order to reduce the high degree of polynomials during multiplication operation. Parameter estimation provides tools for the efficient use of data in the estimation of the parameters that appears in the mathematical models. Thus, this study presents the parameter estimation using two techniques of minimization which are center difference differentiation and robust gradient minimization. Stability analysis and convergence proof of the approximation methods had been carried out. Hence, this research is carried out in order to solve these problems. The obtained numerical results prove that the ERK4 method with incorporated polynomial interpolation technique produce higher accuracy results and may become an alternative method for other uncertainty problems.