This article employs the Laplace transform approach to solve the Bagley–Torvik equation including Caputo's fractional derivative. Laplace transform is a powerful method for enabling solving integer and non-integer order ODEs and PDEs in engineering and science. Using the Laplace transform for solving integer and non-integer order ODEs and PDEs, however, sometimes leads to solutions in the Laplace domain that are difficult to convert back into the time domain using analytical techniques. Therefore, the best option to convert the Laplace domain solution back into real domain is to use numerical inversion techniques. But it's important to remember that the inverse Laplace transform is not a well-posed problem. There isn't a single, general approach that can address every challenge in this matter. Numerous algorithms are devised for numerically inverting the Laplace transform. In this paper, we have evaluated three numerical inverse Laplace transform techniques. These methods include the Fourier series method, the Gauss–Hermite quadrature method, and Zakian's method. Different test problems are used to evaluate these inversion methods. The effectiveness and efficiency of these methods are demonstrated through graphical representations and tabular data. The comparison of the obtained results with the results of other methods available in literature led us to conclude that the proposed methods exhibit rapid convergence with optimal accuracy without any time instability.
Santina et al. (Fri,) studied this question.