Analytical closed-form solution of the Bagley-Torvik equation

Abstract

We calculate the solution of the Bagley-Torvik equation for arbitrary initial conditions and arbitrary external force as the sum of two terms. The first one is a linear combination of exponentials with error functions, and the second one is a convolution integral whose kernel is a linear combination of exponentials with error functions. The derivation of the solution is carried out by using the Laplace transform method and the calculation of a new inverse Laplace transform. The aforementioned convolution integral can be calculated for the cases of a sinusoidal- or a potential-type external force. In addition, we calculate the asymptotic behaviour of the solution for $t\rightarrow 0^{+}$ and $t\rightarrow +\infty$. The computation of this new analytical solution is much faster and stable than other analytical solutions found in the literature.