Numerical And Approximate- Analytical Solution Of Fuzzy Initial Value Problems
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Date
2015-04
Authors
Al-Jassar, Ali Fareed Jameel
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Abstract
Fuzzy differential equations (FDEs) are used for the modeling of some problems in science and engineering and have been studied by many researchers. Certain problems require the solution of FDEs which satisfy fuzzy initial conditions giving rise to fuzzy initial problems (FIVPs). Examples of such problems can be found in physics, engineering, population models, nuclear reactor dynamics, medical problems, neural networks and control theory. However, most fuzzy initial value problems cannot be solved exactly. Furthermore, exact analytical solutions obtained may also be so difficult to evaluate and therefore numerical and approximate- analytical methods may be necessary to evaluate the solution. In the last two decades, the development of numerical and approximate -analytical techniques to solve these equations has been an important area of research. There is a need to formulate new, efficient, more accurate techniques and this is the area of focus of this thesis. In this thesis, we propose a new numerical method based on fifth order Runge-Kutta method with six stages to solve first and high order linear and nonlinear FIVPs involving ordinary differential equations. We also conduct the error and convergence analysis of the method. In addition, we have also studied several approximate-analytical methods- Variation Iterative Method, Homotopy Perturbation Method have been formulated and applied to solve linear and nonlinear first order FIVPs involving ordinary differential equations. Also, we formulated and employed these methods to solve linear and nonlinear high order FIVPs directly without reducing into a first order system as was done by most other researchers. We have also proposed another two approximate–analytical methods - Homotopy Analysis Method and Optimal Homotopy Asymptotic Method - to obtain an approximate solution of linear and nonlinear first and high order FIVPs together with an empirical study of the convergence. Some test examples are given to illustrate the proposed methods to show their feasibility and accuracy.
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Computer science