An Efficient Perturbation Sumudu Transform Technique for the Time-Fractional Vibration Equation with a Memory Dependent Fractional Derivative in Liouville–Caputo Sense

The solution of a time-fractional vibration equation is obtained for the large membranes using powerful homotopy perturbation technique via Sumudu transform. The fractional derivative is taken in Liouville-Caputo sense. The numerical experiments by taking several initial conditions are conducted through some test examples. The results are discussed by taking distinct values of the wave velocity. The results show the competency and accuracy of this analytical scheme. The solution of fractional vibration equation by HPSTM for various orders of memory dependent derivative is compared with the published work and is discussed using figures and tables. The tables confirm that the absolute error between the succeeding approximations is negligible which confirm convergence of the obtained solution. The HPSTM scheme is competent also when the exact solution of a nonlinear differential equation is unknown and reduces time as well as size of the computation. It is useful for both small and large parameters. The outcomes disclose that the HPSTM is a reliable, accurate, attractive and an effective scheme. © 2021, The Author(s), under exclusive licence to Springer Nature India Private Limited.