Non-probabilistic Solution of Uncertain Vibration Equation of Large Membranes Using Adomian Decomposition Method

Abstract

This paper proposes a new technique based on double parametric form of fuzzy numbers to handle the uncertain vibration equation for very large membrane for different particular cases. Uncertainties present in the initial condition and the wave velocity of free vibration are modelled through Gaussian convex normalised fuzzy set. Using the single parametric form of fuzzy number, the original fuzzy vibration equation is converted first to an interval fuzzy vibration equation. Next this equation is transformed to crisp form by applying double parametric form of fuzzy numbers. Finally the same governing equation is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds. The present methods are very simple and effective. Obtained results are depicted in terms of plots to show the efficiency and powerfulness of the present analysis. Results obtained by the methods with new techniques are compared with existing results in special cases.

Explicit Analytic Solution of Vibration Equation for large domain by mean of the Elzaki projected Differential Transform Method

“…FIGURE (4). (10) depicts that with the increase in t and c displacements increases for a fixed value of x = 25. Rate of increase of displacement is fast in Case V than in Case II.…”

“…Linear combination of the modes of structure can be used to explained vibrations. Alternatively propagation of wave travelling in a membrane structure, vibration can also cause the destruction of membrane structure in engineering, so characteristics of vibration of membrane and its dynamic response under the effect of external force become a great important scientific issue and number of researchers studied propagation, transmission and reflection of vibrations, like Tapaswini and Chakraverty studied non probabilistic solution of vibration equation using ADM [10], Yidrim studied the solution of vibration equation of a large membrane using HPM [7], Mohyud-din and Yildrim studied and analyzed the fractional vibrational equation for large membrane [9], further can studied in literature. In this paper we apply Elzaki projected differential transform method (EPDTM) [1,2,3] to solve the vibration equation and different cases has been discussed, numerical and graphical results are found with the help of Maple.…”

Uncertain dynamic responses of imprecisely defined arbitrary order fractionally damped beam subject to various loads

“…Fuzzy set theoretical concept was first developed by Zadeh (1965), which has been further used for the uncertainty analysis of various problems (Hanss and Turrin, 2010; Rao et al , 2010; Farkas et al , 2012; Behera and Chakraverty, 2013a, 2013b; Tapaswini and Chakraverty, 2014a, 2014b; Tapaswini et al , 2015) in a wide range. As both fractional and fuzzy play a vital role in the modelling and design process, various attempts have been made to combine the both.…”

Numerical solution of the imprecisely defined inverse heat conduction problem

“…Also, the fuzzy hyperbolic reaction-diffusion equation was solved by Tapaswini and Chakraverty [44] to analyze the uncertain forest fire. A new double parametric form of fuzzy number has been developed by Tapaswini and Chakraverty, [45] and then they applied the Adomian decomposition method to obtain the solution of uncertain vibration equation for very large membranes.…”

“…Our literature review reveals that fuzzy differential equations are always converted to two crisp differential equations in general to obtain the solution bounds. However, in the proposed methodology the fuzzy differential equation has been converted to a single crisp differential equation using double parametric form of fuzzy numbers [45] and then the corresponding crisp differential equation is solved to obtain the final fuzzy solution by substituting the parametric values. This paper is organized as follows.…”