Analytical closed-form solution of the Bagley-Torvik equation

TL;DR

This paper derives a new analytical closed-form solution for the Bagley-Torvik equation using Laplace transforms, enabling faster and more stable computations for arbitrary initial conditions and external forces.

Contribution

The paper presents a novel closed-form solution for the Bagley-Torvik equation, including a new inverse Laplace transform and explicit expressions for different external forces.

Findings

Solution expressed as sum of exponentials with error functions

Convolution integral with exponential-error kernel

Faster and more stable computation than previous methods

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.