Fractional Treatment of Vibration Equation Through Modern Analogy of Fractional Differentiations Using Integral Transforms

Abstract

Although the significance of the vibration equation has recently attracted the researchers because of the experimental, empirical or numerical analyses, there is a lack of modern fractional analytic approaches. The main aim of this investigation is to analyze the dual treatment of vibration equation for large membrane through the modern approaches of Caputo–Fabrizio and Atangana–Baleanu fractional operators. In order to analyze the fractional model of vibration equation for large membrane, an analytic study is carried out by using Laplace and Hankel transforms satisfying the imposed conditions. A comparative analysis of vibration equation is addressed by newly presented non-integer-order derivatives with and without singular kernel, namely Michele Caputo–Mauro Fabrizio and Atangana–Baleanu fractional derivatives. The analytical solutions are obtained via both fractional approaches and then separately expressed in terms of newly presented Wiman special function $$\varvec{E}_{\eta ,\xi } \left( z \right)$$ . The present fractional methods performed extremely well in terms of reliabilities and computational efficiencies. For the accuracy and validations of analytical treatment of fractional model of vibration equation for large membrane, a graphical comparison is made between Caputo–Fabrizio and Atangana–Baleanu fractional derivatives, which results in various similarities and differences on pertinent parameters involved in the vibration equation.