New forms of Bézier control points and b-spline methods for solving fuzzy ordinary differential equations

Fuzzy ordinary differential equations (FODEs) are widely used to model many problems in science and engineering and have been studied by many researchers. Specific problems require the solutions of FODEs, which satisfy fuzzy initial conditions for fuzzy initial value problems (FIVPs) and fuzzy boundary conditions for fuzzy boundary value problems (FBVPs). Such problems can be found in physics, engineering, population models, nuclear reactor dynamics, medical problems, neural networks, and control theory. However, most FIVPs and FBVPs cannot be solved exactly. Furthermore, exact solutions may also be too difficult to obtain for some problems. Therefore, numerical and approximate methods are necessary to be considered for the solutions. In the last two decades, the development of approximate methods to solve these equations has been an important area of research. Therefore, this thesis needs to modify and formulate new, efficient, and more accurate methods, which are the main focus. This thesis proposes new approximate methods based on fuzzy set theory properties with the Bézier control points method (BCPM) and B-Spline method (BSM) to solve first- and nth-order linear and nonlinear FODEs involving FIVPs and FBVPs. We also conduct the error and convergence analysis of the method. We modified and formulated these methods to solve linear and nonlinear nth-order FIVPs and FBVPs directly without reducing them into a first-order system, as done by most other researchers. Some test examples are given to illustrate the proposed methods' feasibility and accuracy. In general, the numerical results indicate that the proposed BCPM and BSM provide better accuracy than the existing approximate-analytical methods. Hence, the BCPM and BSM are viable alternative methods for solving FIVPs and FBVPs.