# Numerical methods for solving first and second order fuzzy differential equations

## Citation

Afsharinafar, Reza (2014) Numerical methods for solving first and second order fuzzy differential equations. Doctoral thesis, Universiti Putra Malaysia.

## Abstract

Fuzzy differential equations (FDEs) with fuzzy initial conditions are studied as a suitable setting for the modeling of problems in science and engineering in which uncertainties or vagueness prevails. In this thesis, numerical methods are extended for solving first-order and second-order Fuzzy Initial Value Problems (FIVPs), which are interpreted by using the strongly generalized differentiability concepts. There are several interpretations of FDEs depending on the types of differentiability involved. Hukuhara difference is the starting point of fuzzy derivative and has been studied by several researchers. However, it has its drawbacks which resulted in the development of new ideas using different approaches for the solutions of FDEs with initial conditions. Consequently, it has inspired some researchers to present the analytical and numerical methods for the solutions of first-order and also, analytical approach for the solutions of first-order FIVPs. All the works are based on the Zadeh's extension principle and Hukuhara differentiability. In the first part, utilizing the characterization theorems, we transformed the FIVPs into an equivalent system of ordinary differential equations (ODEs). Then we generalized some explicit and implicit one-step methods such as Midpoint, Trapezoidal and Runge-Kutta (TIK) methods for solving first-order FIVPs under strongly generalized differentiability. The results under strongly generalized differentiability are more accurate compared to the Hukuhara differentiability. Next, we extend the multistep methods particularly the third-order Adams Moulton and fourth-order Adams Bashforth method as well as the Predictor-Corrector method of order three and four. They arc used for solving FIVPs under strongly generalized differentiability, which clearly shown that the results under strongly generalized differentiability is more accurate compared to other existing approaches. In the third part of the thesis, we extend the Diagonally Implicit RK method (DIRK) for FDEs under Hukuhara differentiability and the strongly generalized differentiability. Using some mathematical Lemmas, we showed that the approximate fuzzy solutions arc convergent to the exact fuzzy solutions. The numerical results arc compared with other existing methods and a complete error analysis, which guarantees pointwise convergence is also given. Finally, we extend some definitions of strongly generalized differentiability for the solutions of special second-order fuzzy differential equations. The second-order FDEs are transformed to an equivalent system of ODEs using characterization theorem. Then, the multiple solutions of the second-order FDEs are obtained using Runge- Kutta-Nystrom (RKN) method based on the stacking theorem. All the numerical methods are validated using several examples to depict their applicability and effectiveness.