ABSTRACT (original article): 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 and . The computation of this new analytical solution is much faster and stable than other analytical solutions found in the literature. CITATION (original article): Juan Luis González-Santander, Alexander Apelblat (2025), Analytical closed-form solution of the Bagley-Torvik equation, arXiv:2501.05972. https://doi.org/10.48550/arXiv.2501.05972
The Bagley-Torvik equation considers the vertical motion y(t) of a rigid plate of mass m and area A immersed in a Newtonian fluid of infinite extent with density ρ and viscosity μ. The plate is connected by a massless spring of stiffness K to a fixed point, where an external force f(t) is applied to it for t > 0, see Fig_ 1.pdf. The displacement y(t) is referred to the equilibrium point at which the weight of the plate is compensated by the force exerted by the sring and the buoyancy force experimented by the plate. The motion of the plate is governerd by the following fractional-order differential equation: where and y(t) denotes the Caputo fractional derivative of order 3/2.
ℓ
0
ay''(t)+by(t)+cy(t)=f(t)t>0,
*
0
3/2
D
t
a=mb=2A
μρ
c=K,*
0
3/2
D
t
Useful Inverse Laplace Transform
Useful Inverse Laplace Transform
In order to calculate the solution of the Bagley-Torvik, it is very useful the following new inverse Laplace transform:For p/q positive rational, m non-negative integer, λ real number, the following inverse Laplace transform holds true: ;t=(),
-1
ℒ
λ
s
a+b+c
m
s
p/q
s
1/q-λ-1
t
N
∑
k=1
E
1/q,1/q-λ
r
k
1/q
t
P'()
r
k
where(k=1,...,N)aretheN=max(mq,p)differentrootsofthepolynomial:
r
k
P(r)=a+b+c,
mq
r
q
r
and(z)isthetwo-parameterMittag-Lefflerfunction.
E
α,β
Function definition of the Laplace transform
In[]:=
GL[α_,λ_,m_,s_]:=;
λ
s
a+b+c
m
s
α
s
Inverse Laplace transform
Inverse Laplace transform
Numerical inverse Laplace transform
In[]:=
gnum[α_,λ_,m_,t_]:=Module[{f,s,R},f[s_]:=GL[α,λ,m,s];R=InverseLaplaceTransform[f[s],s,t,Method->"Talbot"];R];
Analytical inverse Laplace transform for rational α = p/q
In[]:=
ganal[α_,λ_,m_,t_]:=Module{p,q,n,P,r,g},p=Numerator[Rationalize[α]];q=Denominator[Rationalize[α]];n=Max[mq,p];P[z_]:=a+b+c;r=z/.Solve[P[z]==0,z];g=;g//Chop;
mq
z
p
z
1/q-1-λ
t
n
∑
k=1
MittagLefflerE[1q,1q-λ,r[[k]]]
1/q
t
P'[r[[k]]]
Tests of the inverse Laplace transform
Tests of the inverse Laplace transform
Parameters
In[]:=
a=2.1;b=1.3;c=1.4;α=10/3;λ=1.3;m=2;t=3.4;
In[]:=
gnum[α,λ,m,t]
Out[]=
-0.508039
In[]:=
ganal[α,λ,m,t]
Out[]=
-0.508039
Solutions of Bagley-Torvik Equation
Solutions of Bagley-Torvik Equation
ThesolutionoftheBagley-Torvikequation:ay''[t]+by[t]+cy[t]=f[t],witha,b,c∈,andinitialconditionsy(0),y'(0),isgivenbyy(t)=(t)+(t),where(t)=f(t-τ)τ,takesintoaccounttheeffectoftheexternalforcef(t),and(t)=expterfc(-(k=1,2,3,4)are4differentrootsofthepolynomialP(r)=a+b+c.
0
3/2
D
t
y
f
y
c
y
f
t
∫
0
4
∑
k=1
expτerfc(-
2
r
k
r
k
τ
)(4+3b)
ar
k
r
k
y
c
4
∑
k=1
(+b)y(0)+y'(0)
ar
k
2
r
k
(4+3b)
ar
k
2
r
k
2
r
k
r
k
t
),takesintoaccounttheeffectoftheinitialconditions.Alsor
k
4
r
3
r
Non-homogeneous f[t]
Non-homogeneous function (constant)
In[]:=
C0=1;(*parameter*)f[t_]:=C0;Cl={C0};α={0};(*f(t)=1*)Attributes[f]={Listable};
Non-homogeneous function (potential)
In[]:=
Cl={1,1};α={0,1/2};(*Potentialsolutionparameters:f(t)=1+Cl[[l]]Ifα[[l]]==0,1,;Attributes[f]={Listable};
t
*)f[t_]:=Length[Cl]
∑
l=1
α[[l]]
t
Non-homogeneous function (sinusoidal)
In[]:=
ω=2.5;Ω=1;(*Sinusoidalparameters*)f[t_]:=ΩSin[ωt];Attributes[f]={Listable};
Non-homogeneous function (Bessel function first kind, order 0)
In[]:=
f[t_]:=BesselJ[0,t];Attributes[f]={Listable};
Paper Solutions
Paper Solutions
Bagley-Torvik parameters
In[]:=
a=1.3;b=2.6;c=3.4;
Rootsofthequarticpolynomial:P(r)=a+b+c
4
r
3
r
In[]:=
β=b/a;γ=c/a;δ=9+++;T[s_]:=-4γ--s;(*fork=1,2,3,4*)r={R+T[1],-R+T[-1],R-T[1],-R-T[-1]}-;
2
β
81-768γ
γ;R=4
β
1
2
2
β
2
1/3
δ
18
8γ
1/3
(12δ)
1
2
2
β
2
1/3
2
3δ
1/3
δ
18
3
β
8R
r
k
β
4
Homogeneous initial conditions: y(0) = y'(0) = 0
y0=0;(*Boundaryconditiony(0)=y0*)yp0=0;(*Boundaryconditiony'(0)=yp0*)
Non-homogeneous initial conditions: y(0) = y'(0) = 1
y0=1;(*Boundaryconditiony(0)=y0*)yp0=1;(*Boundaryconditiony'(0)=yp0*)
Solution (t) for non-homogeneous initial conditions
y
c
yc[t_]:=(4ar[[k]]+3b)Exp[t]Erfc[-r[[k]]
4
∑
k=1
(ar[[k]]+b)(y0+yp0)
2
r[[k]]
2
r[[k]]
2
r[[k]]
t
]//Chop;Solution y(t) (f[t] constant)
In[]:=
y[t_]:=C0(4ar[[k]]+3b)+yc[t]//Chop;Attributes[y]={Listable};
4
∑
k=1
Exp[t]Erfc[-r[[k]]
2
r[[k]]
t
]-13
r[[k]]
Solution y(t) (f[t] potential)
In[]:=
(*Solutiony(t)=(t)+(t)*)y[t_]:=yc[t]+Cl[[l]]Gamma[α[[l]]+1](4ar[[k]]+3b)//Chop;Attributes[y]={Listable};
y
c
y
f
Length[Cl]
∑
l=1
α[[l]]+1/2
t
4
∑
k=1
MittagLefflerE12,32+α[[l]],r[[k]]
t
2
r[[k]]
Solution y(t) (f[t] sinusoidal)
Solution y(t) (f[t] general)
Asymptotic solutions (this paper)
Asymptotic solutions (this paper)
Asymptotic solution for sinusoidal force (t -> +∞)
Asymptotic solution for constant force (t -> +∞)
Asymptotic solution for potential force (t -> +∞)
Solutions Given in the Literature
Solutions Given in the Literature
Numerical solutions
Numerical solutions
Podlubny's numerical solution
Analytical solutions
Analytical solutions
Auxiliary functions
2A Cohen-Rodriguez-Zagier convergence acceleration algorithm for alternating series
Arora's series solution (potential-type force)
Tests
Tests
Parameters for the computations
Checks
Plots
Plots
Constant Solution
Constant Solution
Constant solution with y(0) = y'(0) = 0
Constant solution with y(0) = y'(0) = 1
Potential solution with y(0) = y'(0) = 0
Potential solution with y(0) = y'(0) = 1
Sinusoidal Solution
Sinusoidal Solution
Sinusoidal solution with y(0) = y'(0) = 0
Sinusoidal solution with y(0) = y'(0) = 1
General Solution with Bessel Function
General Solution with Bessel Function
General solution with y(0) = y'(0) = 0
General solution with y(0) = y'(0) = 1
Asymptotic Approximation
Asymptotic Approximation
(* Asymptotic behaviour for t -> +∞ taking a sinusoidal solution with y(0) = y'(0) = 1 *)
Asymptotic behaviour for t -> +∞ taking a constant solution with y(0) = y'(0) = 1
Computational Time Performance
Computational Time Performance
ANALYTICAL COMPARISON: Pang vs. paper
ANALYTICAL COMPARISON: Pang vs. paper
General Solution with Bessel function and y(0) = y'(0) = 1
Sinusoidal Solution with y(0) = y'(0) = 1
Constant Solution with y(0) = y'(0) = 1
NUMERICAL COMPARISON: Numerical Podlubny vs. paper
NUMERICAL COMPARISON: Numerical Podlubny vs. paper
General Solution with Bessel function and y(0) = y'(0) = 1
Sinusoidal Solution with y(0) = y'(0) = 1
Constant Solution with y(0) = y'(0) = 1
Interactive plots of sinusoidal and constant forces
Interactive plots of sinusoidal and constant forces
Solution y(t) (f[t] sinusoidal)
Solution y(t) (f[t] constant)
Solution y(t) (f[t] potential)
Interactive plot for a sinusoidal force
Interactive plot for a constant force
Interactive plot for a constant force
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Analytical closed-form solution of the Bagley-Torvik equation
by Juan Luis González-Santander & Alexander Apelblat
Wolfram Community, STAFF PICKS, January 23, 2026
https://community.wolfram.com/groups/-/m/t/3623622
by Juan Luis González-Santander & Alexander Apelblat
Wolfram Community, STAFF PICKS, January 23, 2026
https://community.wolfram.com/groups/-/m/t/3623622