The Bagley-Torvik equation is the linear fractional differential equation
 |
(1)
|
where 
is the second derivative with respect to 
, 
, 
, and 
are constants with 
, 
is the forcing term, and the fractional term uses the Caputo fractional derivative of order
with lower terminal 0. The equation models the displacement of a spring-mounted rigid
plate immersed in a Newtonian fluid. The fractional term accounts for the dependence
of the fluid force on the plate's past motion (Torvik and Bagley 1984, González-Santander
and Apelblat 2026).
Write 
and 
for the Laplace transforms of 
and 
,
respectively. With zero initial displacement and velocity, the Laplace
transform of the equation gives
 |
(2)
|
For initial conditions 
and 
,
the Caputo fractional derivative gives
 |
(3)
|
Use the principal square root for the powers of 
in the right half-plane where the transforms exist and the denominator is nonzero
(González-Santander and Apelblat 2026). For sufficiently smooth 
, the Caputo
fractional derivative of 
equals the Riemann-Liouville
fractional derivative of 
.
Thus the two operators agree on 
for zero initial conditions.
