From the Boussinesq-type equation to the Static Beam equation
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From the Boussinesq-type Equation to the Static Beam Equation
Article · December 2019
DOI: 10.4316/JACSM.201902008
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Mathematics Section
50
From the Boussinesq-type Equation to the Static Beam Equation
R. Gilles BOKOLO
Department of Mathematics and Computer Sciences, University of Kinshasa, B.P. 190 Kinshasa XI, Kinshasa,
Democratic Republic of the Congo
rgillesbokolo@gmail.com
Abstract–In this paper, we investigate the Hopf-Cole
transformation to solve the Boussinesq-type equation. As a by-
product, a linearization of the later will generate the static beam
equation. Hence, we prove that soliton-type solutions profiles are
preserved.
Keywords: Boussinesq-type equation, static beam equation,
Hopf-Cole transformation, Solitons.
I. INTRODUCTION
In Fluid Mechanics, Boussinesq equations are mainly non-
linear partial differential equations that describe wave
equation systems obtained by Euler’s equations approximation
for incompressible fluid and irrotational flow at the free
surface. Also characterized both by a nonlinearity and by a
dispersive effect, these equations are given by
+=() (1)
which is a completely integrable nonlinear Partial
Differential Equation exhibiting a soliton-type solution [3].
(1) which allows waves travelling simultaneously in
opposing directions, can be reduced to the beam equation to
consider waves propagation in one horizontal direction.
Otherwise, Euler-Bernoulli beam theory is a simplification
of the linear theory of isotropic elasticity that provides a means
of calculating the load-carrying and deflection characteristics
of beams. Dynamic phenomena can be modeled using the
static beam equation by choosing appropriate forms of the load
distribution.
Beam equation, when the beam is homogeneous, is
described as
u+ E I u= 0 (2)
with the linear mass density of the beam, not necessarily a
constant. and ,that respectively represent the elastic
modulus and the area moment of inertia, are independent of .
Finally, () is the external load. The free distribution of a
beam can be accounted for by using the external distributed
load function: (,)=
In the next paragraphs, we will describe the Hopf-Cole
transformation to show the transition between (1) and (2).
After that linearization process embedded by some specific
conjectures, we will close our arguments by finding a
hyperbolic secant envelope soliton for the fourth-order linear
wave equation represented by a static beam-type equation.
II. LINEARIZATION
A)Hopf-Cole transformation
In [9], we describe the method underlying the Hopf-Cole
transformation. We will keep our comments brief and to the
point that we could easily apply steps for the aforementioned
method.
B) Application
Let be a smooth solution of(1). We define
=()
where : is a smooth function, not yet specified.
We will choose such as solve a linear equation. We
should have the following format
(,) = {[(,)]}
Where and are integers (2) determined as follows:
=()
=()+() (3)
=()
=()
+()
=()
+ 2()+()
+()
=()
+ 3()
+ 3()
+ 3()
+ 3()
+()+()
and therefore, (3) implies
=()+()[]
= ()+()
[()
3()
3()
3()
3()()]
= ()+()
[()
6()
3()
4()]
=+()+ 3()
+()+
()
+ 6()
+ 4() (4)
A quick observation of (4), shows that
+=()+ 3()
+()+
()
+ 6()
+ 4() (5)
DOI: 10.4316/JACSM.201902008
Journal of Applied Computer Science & Mathematics, Issue 2/2019, vol.13, No. 28, Suceava
51
Clearly, the right side of (5) correspond to (2) if ==
= 1
and the left side of (5) equal to zero. The later displays the
below conservation law
()+ 3()
+()+()
+ 6()
+ 4()= 0
In [8], the author shows that there exists many different
types of Nonlinearities. In the Geometric Nonlinearity, we
consider the total strain energy of the deformable elastic body,
which is in the form of extensional or stretching energy and
the bending energy. The extensional strain energy involves
higher order nonlinear terms than quadratic in the normal
displacement component, whereas the bending strain energy
remains quadratic in the displacement components. Thus,
when these energy expressions are used to derive the equations
of motion, they will turn out to be a set of nonlinear governing
differential equations.
Now, let make the conjecture that we prefer
nonlinearities ,
rather than
and which
have stronger nonlinearities.
That implies that we choose such as
()+ 3()
= 0 (6)
But in the present situation, let say that the nonlinearity is
also due to boundary conditions during the transformation.
We solve (6) by establishing = ±
with respect to .
So we see that if solve (1), then
= ±
“Hopf-Cole transformation”
Solve this Initial boundary Value Problem for the linearized
form of the Boussinesq-type equation, so called: the static
beam-type equation
+ = 0 ×]0, [
= ±
×{= 0} (7)
C) Theorem
Let be a solution of (7).
If we define
+ = (8)
Then = 0 (9)
Thus, we prove that (8) and (9) are equivalent to (7).
Proof
=(+ )(+ )
= +
= + +
As solve (7), then
+= 0 since =.
(8) and (9) are heat-type equations with a complex heat
coefficient of distribution. Which are close to the linear part of
the Ginsburg-Landau mathematical physical theory used to
describe superconductivity. This theory can also be given a
general geometric setting, placing it in the context of
Riemannian geometry, where in many cases exact solutions
can be given. In string theory, it is conventional to study the
Ginzburg–Landau functional for the manifold being a
Riemann surface, and taking n=1, i.e. a line bundle.
III. SOLITON-TYPE SOLUTION
Considering the following transformation
=
(7) becomes + = 0
Whose trail solution has the form
(,)=1+exp (++)
Where , and are arbitrary constants, with and
satisfying the following relation [9]:
0, 0 and += 0
Therefore,
(,)=
1 + exp(±
A
+
C
)
1 + exp±++
Simultaneously,(6) and (7) are automatically satisfied with
the substitution of the above (,), which is thus a special
solution for both equations.
As we had ()+ 3()
= 0
Finally, the set of exact solutions of the general form of (1)
are clearly (,)=+ 3
(,)=sech1
2(± +)
sech1
2±++
Which are the solitary waves having the sech profiles
[7]. and are parameters specifying the amplitude and the
speed of the wave.
CONCLUSIONS
The main aim of this paper keened on the linearization of
the Boussinesq-type equation to obtain the static beam
equation by the mean of the Hopf-Cole transformation. We
were able to prove that the soliton-type solutions is the
profile that better describe the Boussinesq-type solutions.
The originality of this works rely on the ability to juggle
between fluid mechanics and the theory of elasticity. Further
research will consist in use of the Ginsburg-Landau theory
with main aim to develop other characteristics of (8).
Mathematics Section
52
ACKNOWLEDGMENT
We thank Professor Walo Omana Rebecca who provided
insight and expertise that greatly illuminating development of
this research.
REFERENCES
[1]. An Lianjun and A. Piere, “A weakly nonlinear analysis of
elasto-plastic-microstructure models”, SIAM J.Appl.Math.55,
pp.136-155,1995
[2]. E.A.Witmer, “Elementary Bernoulli-Euler Beam theory”, MIT
Unified Engineering Course Notes, pp.5-114
[3]. Guowang CHEN and Zhijian YANG, “On the initial boundary
value problems of the fourth order nonlinear wave equation and
the Boussinesq type equation”, proceeding of the conference on
Nonlinear Partial Equations and Applications, pp. 20-26, 2002
[4]. J.Boussinesq, “Essaie sur la théorie des eaux courantes”,
Comptes rendus de l’academie des sciences, vol.23, p.1-
680,1877
[5]. J.Boussinesq, “Théorie des ondes et des remous qui se
propagent le long d’un canal rectangulaire horizontal, en
communiquant au liquide contenu dans ce canal des vitesses
sensiblement pareilles de la surface au fond”, Journal de
mathematiques pures et appliquées, vol.17, p.55-108,1872
[6]. J. L. Bona; W. G. Pritchard, L. R. Scott, “Solitary-wave
interaction”, Physics of Fluids, 23(3), pp.438-441, 1980
[7]. M. Remoissenet, “waves called solitons: Concepts and
experiements”, Springer, pp.11, 1999
[8]. Muthukrishnan Sathyamoorthy, “Nonlinear Analysis
Structures”, CRC Press, 1998, pp. 1–73.
[9]. R.Gilles Bokolo, “Hopf-Cole transformation with effect of the
dissipation coefficient on nonlinearity in the Benjamin-Bona-
Mahony (BBM) generalized pseudo-parabolic equation”,
International journal of Science and research (IJSR), volume 8
issue 6, pp.1707-1710, 2019
R. Gilles Bokolo received the B.S. and M.S. degrees in Applied Mathematics from the University of Kinshasa. With such experiences as
Relationship Manager Support at Citigroup DRC, MIS support in the Credit Department of Standardbank, and International Visitor Leadership
Program Alumni (IVLP) of the US State Department, to name but a few; Gilles now keep working as Teaching Assistant in the Department of
Mathematics and Computer Sciences of the University of Kinshasa.
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From the Boussinesq-type equation to the Static Beam equation
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