A novel analytical iterative approach to time-fractional Vibration equation using Aboodh transform

Abstract The main aim of the work is to achieve the approximate analytical solution of a time-fractional vibration problem for large membranes via Aboodh transform iterative method (ATIM). The ATIM method is the combination of Daftardar-Gejji's New iterative method and Aboodh transform. The fractional derivative operator is considered in the sense of Liouville-Caputo. The approximate-analytical solutions of the proposed equation for multiple situations have been determined by utilizing the initial values, which has accelerated the quick convergence of the series solution. The proposed method performs extremely well in terms of simplicity and effectiveness. Using the fixed point theorem the existence and uniqueness of the proposed problem have been stated. The validity and applicability of the technique are shown for obtaining approximate analytical solutions for different particular cases which are presented numerically and graphically. Many science and engineering problems can be solved analytically and numerically using the proposed approach.