Hybrid runge-kutta method for solving linear fuzzy delay differential equations with unknown state-delays

Abstract

In this research, a new method to solve the Fuzzy Delay Differential Equations (FDDEs) with unknown state-delays constrained optimization problem is introduced. This method is based on the coupling of second and third orders Runge- Kutta (RK) method called hybrid RK method. The main goal of this thesis is to identify the unknown state-delays using experimental data. RK methods are chosen because they are well-established and can be easily modified to overcome the discontinuities which occur in Delay Differential Equations (DDEs) especially outside uniform nodes with delay step-size. Numerical results of FDDEs from the hybrid RK methods are compared with exact solutions derived from stepwise approach using Maple software. The relative errors are calculated for the purpose of accuracy checking on these numerical schemes. In this study, a dynamic optimization problem in which the state-delays are decision variables is also imposed; with its formulated cost function. The gradient of the cost function is computed by solving auxiliary FDDEs. By exploiting the results, the state-delay identification problem can be solved efficiently and accurately using a gradient-based optimization method. In addition, a C program has been developed based on hybrid RK methods for solving these problems. Consequently, the results show that the new hybrid scheme is an efficient numerical technique in solving all the problems above with acceptable errors.