A note on “Numerical solutions for linear system of first-order fuzzy differential equations with fuzzy constant coefficients” (original) (raw)

Partial differential equations are used for modeling various physical phenomena. Unfortunately, many problems are dynamical and too complicated, developing an accurate differential equation model for such problems require complex and time consuming algorithms hardly implementable in Leveque R. J (2005). For a long time, scientist goal was to develop constructive and effective methods that reliably compute the partial differential equation with more accuracy as possible. In classical mathematics, various kinds of transforms (Fourier, Laplace, integral, wavelet) are used as powerful methods for construction of approximation models and for solution of differential or integral-differential equations Perfilieva I. (2004). Fuzzy set theory is composed of an organized body of mathematical tools particularly well-suited for handling incomplete information, the un-sharpness of classes of objects or situations, or the gradualness of preference profiles, in a flexible way. It offers a unifying framework for modeling various types of information ranging from precise numerical, interval-valued data, to symbolic and linguistic knowledge, with a stress on semantics rather than syntax. Zimmermann H. J (2001) Achieving high levels of precision is a very important subject in all science fields, getting a satisfied precision depending basically on the way we deal with elements in the problem Universe. For many years we were depended on crisp set theory "classical set theory " to deal with elements and sets which belong to the problem Universe, but in real world there are many application problems which can't be described nor handled by the crisp set theory Leondes C. (1998).The study of fuzzy differential equations is rapidly expanding as a new branch of fuzzy mathematics. Both theory and applications have been actively discussed over the last few years. According to Vorobiev and Seikkala (1986), the term 'fuzzy differential equation' was first coined in 1978. Since then, it has been a subject of interest among scientists and engineers. In the literature, the study of fuzzy differential equations has several interpretations. The first one is based on the notion of Hukuhara derivative (R. Goetschel (1986 et al)). Under this interpretation, the existence and uniqueness of the solution of fuzzy differential equations have been extensively studied by (S. Song,(2000), S. Seikkala, (1987))The concept of Hukuhara derivative was further explored by Kaleva (1987) and Seikkala (1986). Subsequently, the theory of fuzzy differential equations has been developed and fuzzy initial value problems have been studied. However, this approach produces many solutions that have an increasing length of support as the independent variable increases (T.G. Bhaskar,et al (2012)). Moreover, different formulations of the same fuzzy differential equation might lead to different solutions.