Forward and Inverse Problems of Time-Fractional Vibration Equation of Large Membranes in Uncertain Environment

The vibration of large membranes is a powerful part of engineering applications, such as in the components of drums, pumps, telephones, microphones, and other equipment’s. Various researchers have investigated the above topic, and the variables and parameters have been provided in a crisp/exact manner. Although this may be accurate in theory, it is possible that, in practice, it contains uncertainty due to inaccuracies in observations, maintenance-induced inaccuracies, and other sources of error. So, the primary objective of this paper is to solve this important problem numerically under interval and fuzzy uncertainty to have an uncertain solution and to study its behaviour.

In this study, we consider these uncertainties as fuzzy/intervals and employ a technique, namely the double parametric form of fuzzy numbers (DPFFNs), to solve the uncertain fractional vibration model of order \(\eta \hspace{1mm} (1 < \eta \le 2)\) $\eta (1<\eta \le 2)$ . In this titled problem has been solved for forward and inverse in various cases using the Adomian decomposition method (ADM).

The Adomian decomposition method has been successfully implemented along with the double parametric form to find forward and inverse problems of the time-fractional vibration equation of large membranes in an uncertain environment. The solution is expressed in compact or power series form, which is an advantage of this technique. In addition, this approach converges quickly to a precise solution. In this work, moving to the forward case, we have found fuzzy displacements, and in the inverse case, we found fuzzy velocities of the model problem successfully.

The forward and inverse problems of the time-fractional vibration equation of large membranes in an uncertain environment have been solved by the Adomian decomposition method. The present approach’s computational efficiency is good and reliable for providing an approximate numerical solution for various cases. The obtained results are illustrated graphically and compared to particular cases.