%0 Journal Article
%T Numerical Solution of Uncertain Beam Equations Using Double Parametric Form of Fuzzy Numbers
%A Smita Tapaswini
%A S. Chakraverty
%J Applied Computational Intelligence and Soft Computing
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/764871
%X Present paper proposes a new technique to solve uncertain beam equation using double parametric form of fuzzy numbers. Uncertainties appearing in the initial conditions are taken in terms of triangular fuzzy number. Using the single parametric form, the fuzzy beam equation is converted first to an interval-based fuzzy differential equation. Next, this differential equation is transformed to crisp form by applying double parametric form of fuzzy number. Finally, the same is solved by homotopy perturbation method (HPM) to obtain the uncertain responses subject to unit step and impulse loads. Obtained results are depicted in terms of plots to show the efficiency and powerfulness of the methodology. 1. Introduction Recently, theory of differential equations plays a vital role to model physical and engineering problems such as in solid and fluid mechanics, viscoelasticity, biology, physics, and other areas of science. But in actual case, the parameters, variables, and initial conditions involved in the differential equations may be uncertain, or a vague estimation of those are found in general by some observation, experiment, experience, or maintenance induced error. So, to overcome the uncertainty and vagueness, one may use fuzzy environment in parameters, variables, and initial condition in place of crisp (fixed) ones. So, with these uncertainties the general differential equations turn into fuzzy differential equations (FDEs). In real-life application, it is too complicated to obtain the exact solution to fuzzy differential equations, so one may need a reliable and efficient numerical technique for the solution for fuzzy differential equations. There exist a large number of papers dealing with fuzzy differential equations and its applications in the open literature. Some are reviewed and cited here for better understanding of the present analysis. Chang and Zadeh [1] first introduced the concept of a fuzzy derivative, followed by Dubois and Prade [2] who defined and used the extension principle in their approach. The fuzzy differential equations and fuzzy initial value problems are studied by Kaleva [3, 4] and Seikkala [5]. Various numerical methods for solving fuzzy differential equations are introduced in [6¨C16]. Ma et al. [6] developed a scheme based on the classical Euler method to solve fuzzy ordinary differential equations. A two-dimensional differential transform method to solve fuzzy partial differential equations (FPDEs) has been studied in [7]. Abbasbandy and Allahviranloo [8] applied Taylor method for the solutions of fuzzy differential
%U http://www.hindawi.com/journals/acisc/2013/764871/