Analytical Solution of General Bagley-Torvik Equation

Abstract

Bagley-Torvik equation appears in viscoelasticity problems where fractional derivatives seem to play an important role concerning empirical data. There are several works treating this equation by using numerical methods and analytic formulations. However, the analytical solutions presented in the literature consider particular cases of boundary and initial conditions, with inhomogeneous term often expressed in polynomial form. Here, by using Laplace transform methodology, the general inhomogeneous case is solved without restrictions in boundary and initial conditions. The generalized Mittag-Leffler functions with three parameters are used and the solutions presented are expressed in terms of Wiman’s functions and their derivatives.

New existence and uniqueness result for fractional Bagley-Torvik differential equation

“…Equation (1.1) was initially introduced in [25] and is thoroughly discussed in [23]. Particularly, the investigators have studied the analytical and numerical solutions of equation (1.1); see for instance [12,14,21,22]. In [7], the equivalence between the Caputo and Riemann derivatives are discussed and pointed out that they are identical in describing the linear viscoelastic material just under two minimal restrictions.…”

Analytical solution of the generalized Bagley–Torvik equation

“…Podlubny has presented the analytical solution to the general B-T equation (where α = 3 2 ) with zero initial conditions by the Green function [1]. In [43], the analytical solution to the general B-T equation (where α = 3 2 ) was presented for general initial conditions. Motivated by the above articles, we devote this paper to discussing the well-posed problems and the analytical solution of the generalized B-T equation with the fractional order 0 < α < 2.…”