Vibration Equation of Large Membranes in Uncertain Environment

The investigation of the time-fractional vibration equation (TFVE) of large membranes is significant because of its well-recognised practical applications. Several studies have been conducted on the above topic, and the variables and parameters have been provided in a crisp/exact manner. In practical application, these parameters may not be precisely understood but rather exist in an uncertain or approximate form. In this study, we analyse the uncertainties as an interval/fuzzy environment.

In this study, we analyse the uncertainties as interval/fuzzy and suggest a novel approach called the double parametric form of fuzzy numbers to resolve the uncertain fractional vibration model of order \((1 < \Theta \le 2)\) $(1<\mathrm{\Theta }\le 2)$ related to large membranes. The modified Adomian decomposition method (MADM) has finally been used to solve this problem.

The modified Adomian decomposition method has been successfully implemented along with the double parametric form to find vibration equation of large membranes in uncertain environment. Regarding computing efficiency, the present technique works well. It is shown that the approach is reliable for providing an approximate numerical solution for various cases. Plots are used to present the results, which are compared in particular cases.