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
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
[email protected]
Abstract–In this paper, we investigate the Hopf-Cole
transformation to solve the Boussinesq-type equation. As a byproduct, 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 nonlinear 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)
𝑢𝑢𝑡𝑡𝑡𝑡 + 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 = 𝑎𝑎(𝑢𝑢2 )𝑥𝑥𝑥𝑥
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
(2)
μ utt + E I uxxxx = 0
with μ the linear mass density of the beam, not necessarily a
constant. 𝐸𝐸 and 𝐼𝐼 ,that respectively represent the elastic
modulus and the area moment of inertia, are independent of 𝑥𝑥.
Finally, 𝑞𝑞(𝑥𝑥) is the external load. The free distribution of a
beam can be accounted for by using the external distributed
load function:
𝑞𝑞(𝑥𝑥, 𝑡𝑡) = 𝜇𝜇 𝑢𝑢𝑡𝑡𝑡𝑡
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
DOI: 10.4316/JACSM.201902008
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 𝑢𝑢 be a smooth solution of(1). We define
𝜔𝜔 = 𝜙𝜙(𝑢𝑢)
where 𝜙𝜙: ℝ → ℝ is a smooth function, not yet specified.
We will choose 𝜙𝜙 such as 𝜔𝜔 solve a linear equation. We
should have the following format
𝜔𝜔(𝑥𝑥, 𝑡𝑡) = 𝜕𝜕𝑥𝑥𝑚𝑚 𝜕𝜕𝑡𝑡𝑛𝑛 {𝜙𝜙[𝑢𝑢(𝑥𝑥, 𝑡𝑡)]}
Where 𝑚𝑚 and 𝑛𝑛 are integers (≤ 2) determined as follows:
𝜔𝜔𝑡𝑡 = 𝜙𝜙 ′ (𝑢𝑢)𝑢𝑢𝑡𝑡
(3)
𝜔𝜔𝑡𝑡𝑡𝑡 = 𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑡𝑡2 + 𝜙𝜙 ′ (𝑢𝑢)𝑢𝑢𝑡𝑡𝑡𝑡
𝜔𝜔𝑥𝑥 = 𝜙𝜙 ′ (𝑢𝑢)𝑢𝑢𝑥𝑥
𝜔𝜔𝑥𝑥𝑥𝑥 = 𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥2 + 𝜙𝜙 ′ (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥
′′′ (𝑢𝑢)𝑢𝑢 3
′′
′′
𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥 = 𝜙𝜙
𝑥𝑥 + 2𝜙𝜙 (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥 𝑢𝑢𝑥𝑥 + 𝜙𝜙 (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥 𝑢𝑢𝑥𝑥
′
+ 𝜙𝜙 (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥
𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 = 𝜙𝜙 ′′′′ (𝑢𝑢)𝑢𝑢𝑥𝑥4 + 3𝜙𝜙 ′′′ (𝑢𝑢)𝑢𝑢𝑥𝑥2 𝑢𝑢𝑥𝑥𝑥𝑥 + 3𝜙𝜙 ′′′(𝑢𝑢)𝑢𝑢𝑥𝑥2 𝑢𝑢𝑥𝑥𝑥𝑥
2
+ 3𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥
+ 3𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥
′′ (𝑢𝑢)𝑢𝑢
′
+ 𝜙𝜙
𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥 + 𝜙𝜙 (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥
and therefore, (3) implies
𝜔𝜔𝑡𝑡𝑡𝑡 = 𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑡𝑡2 + 𝜙𝜙 ′ (𝑢𝑢)[𝑎𝑎∆𝑢𝑢2 − 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 ]
′′ (𝑢𝑢)𝑢𝑢 2
′
2
= 𝜙𝜙
𝑡𝑡 + 𝜙𝜙 (𝑢𝑢)𝑎𝑎∆𝑢𝑢
[𝜔𝜔
− 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 − 𝜙𝜙 ′′′′ (𝑢𝑢)𝑢𝑢𝑥𝑥4 − 3𝜙𝜙 ′′′ (𝑢𝑢)𝑢𝑢𝑥𝑥2 𝑢𝑢𝑥𝑥𝑥𝑥
2
− 3𝜙𝜙 ′′′ (𝑢𝑢)𝑢𝑢𝑥𝑥2 𝑢𝑢𝑥𝑥𝑥𝑥 − 3𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥
− 3𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥 − 𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥 ]
= 𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑡𝑡2 + 𝜙𝜙 ′ (𝑢𝑢)𝑎𝑎∆𝑢𝑢2
− [𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 − 𝜙𝜙 ′′′′ (𝑢𝑢)𝑢𝑢𝑥𝑥4 − 6𝜙𝜙 ′′′ (𝑢𝑢)𝑢𝑢𝑥𝑥2 𝑢𝑢𝑥𝑥𝑥𝑥
2
− 3𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥
− 4𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥 ]
2
′ (𝑢𝑢)𝑎𝑎∆𝑢𝑢 2
+ 3𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥
+ 𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑡𝑡2 +
𝜔𝜔𝑡𝑡𝑡𝑡 = −𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 + 𝜙𝜙
′′′′ (𝑢𝑢)𝑢𝑢 4
′′′ (𝑢𝑢)𝑢𝑢 2
′′ (𝑢𝑢)𝑢𝑢
(4)
𝜙𝜙
𝑥𝑥 + 6𝜙𝜙
𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥 + 4𝜙𝜙
𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥
A quick observation of (4), shows that
2
+ 𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑡𝑡2 +
𝜔𝜔𝑡𝑡𝑡𝑡 + 𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 = 𝜙𝜙 ′ (𝑢𝑢)𝑎𝑎∆𝑢𝑢2 + 3𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥
′′′′ (𝑢𝑢)𝑢𝑢 4
′′′ (𝑢𝑢)𝑢𝑢 2
′′ (𝑢𝑢)𝑢𝑢
(5)
𝜙𝜙
𝑥𝑥 + 6𝜙𝜙
𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥 + 4𝜙𝜙
𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥
50
Journal of Applied Computer Science & Mathematics, Issue 2/2019, vol.13, No. 28, Suceava
Clearly, the right side of (5) correspond to (2) if μ = 𝐸𝐸 =
𝐼𝐼 = 1
and the left side of (5) equal to zero. The later displays the
below conservation law
2
+ 𝜙𝜙 ′′(𝑢𝑢)𝑢𝑢𝑡𝑡2 + 𝜙𝜙 ′′′′ (𝑢𝑢)𝑢𝑢𝑥𝑥4
𝜙𝜙 ′ (𝑢𝑢)𝑎𝑎∆𝑢𝑢2 + 3𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥
′′′ (𝑢𝑢)𝑢𝑢 2
′′
+ 6𝜙𝜙
𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥 + 4𝜙𝜙 (𝑢𝑢)𝑢𝑢𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥 = 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
2
rather than 𝑢𝑢𝑥𝑥2 𝑢𝑢𝑥𝑥𝑥𝑥 and 𝑢𝑢𝑥𝑥 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥 which
nonlinearities ∆𝑢𝑢2 , 𝑢𝑢𝑥𝑥𝑥𝑥
have stronger nonlinearities.
That implies that we choose 𝜙𝜙 such as
2
=0
(6)
𝜙𝜙 ′ (𝑢𝑢)𝑎𝑎∆𝑢𝑢2 + 3𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥
But in the present situation, let say that the nonlinearity is
also due to boundary conditions during the transformation.
𝑎𝑎
We solve (6) by establishing 𝜙𝜙 = ±� 𝑖𝑖 with respect to 𝑢𝑢.
So we see that if 𝑢𝑢 solve (1), then
3
𝑎𝑎
𝜔𝜔 = ±� 𝑖𝑖 “Hopf-Cole transformation”
3
Solve this Initial boundary Value Problem for the linearized
form of the Boussinesq-type equation, so called: the static
beam-type equation
𝜔𝜔𝑡𝑡𝑡𝑡 + 𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 = 0 𝑖𝑖𝑛𝑛 ℂ𝑛𝑛 × ]0, ∞[
(7)
�
𝑎𝑎
𝜔𝜔 = ±� 𝑖𝑖 𝑜𝑜𝑜𝑜 ℂ𝑛𝑛 × {𝑡𝑡 = 0}
3
C) Theorem
Let 𝑤𝑤 be a solution of (7).
If we define
𝜔𝜔𝑡𝑡 + 𝑖𝑖𝜔𝜔𝑥𝑥𝑥𝑥 = 𝑣𝑣
Then
𝑣𝑣𝑡𝑡 − 𝑖𝑖𝑣𝑣𝑥𝑥𝑥𝑥 = 0
Thus, we prove that (8) and (9) are equivalent to (7).
Proof
𝑣𝑣𝑡𝑡 − 𝑖𝑖𝑣𝑣𝑥𝑥𝑥𝑥 = (𝜔𝜔𝑡𝑡 + 𝑖𝑖𝜔𝜔𝑥𝑥𝑥𝑥 )𝑡𝑡 − 𝑖𝑖(𝜔𝜔𝑡𝑡 + 𝑖𝑖𝜔𝜔𝑥𝑥𝑥𝑥 )𝑥𝑥𝑥𝑥
= 𝜔𝜔𝑡𝑡𝑡𝑡 + 𝑖𝑖𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥 − 𝑖𝑖𝜔𝜔𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑖𝑖 2 𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥
= 𝜔𝜔𝑡𝑡𝑡𝑡 + 𝑖𝑖𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥 − 𝑖𝑖𝜔𝜔𝑡𝑡𝑡𝑡𝑡𝑡 + 𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥
As 𝑤𝑤 solve (7), then
𝜔𝜔𝑡𝑡𝑡𝑡 + 𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 = 0 since 𝑖𝑖𝜔𝜔𝑥𝑥𝑥𝑥𝑥𝑥 = 𝑖𝑖𝜔𝜔𝑡𝑡𝑡𝑡𝑡𝑡 .
(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 𝑀𝑀 being a
Riemann surface, and taking n=1, i.e. a line bundle.
III. SOLITON-TYPE SOLUTION
Considering the following transformation
𝜔𝜔 = 𝜂𝜂𝑥𝑥
(7) becomes 𝜂𝜂𝑥𝑥𝑥𝑥𝑥𝑥 + 𝜂𝜂𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 = 0
Whose trail solution has the form
𝜂𝜂(𝑥𝑥, 𝑡𝑡) = 1 + exp(𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 + 𝐶𝐶)
Where 𝐴𝐴, 𝐵𝐵 and 𝐶𝐶 are arbitrary constants, with 𝐴𝐴 and 𝐵𝐵
satisfying the following relation [9]:
𝐴𝐴 ≠ 0, 𝐵𝐵 ≠ 0 and 𝐴𝐴4 + 𝐵𝐵2 = 0
Therefore,
2
1 + exp(± A 𝑖𝑖 𝑡𝑡 + C1)
𝜂𝜂(𝑥𝑥, 𝑡𝑡) = �
1 + exp�±√𝐵𝐵𝑖𝑖𝑖𝑖 + 𝐵𝐵𝐵𝐵 + 𝐶𝐶2 � 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
Simultaneously,(6) and (7) are automatically satisfied with
the substitution of the above 𝜂𝜂(𝑥𝑥, 𝑡𝑡), which is thus a special
solution for both equations.
2
=0
As we had 𝜙𝜙 ′ (𝑢𝑢)𝑎𝑎∆𝑢𝑢2 + 3𝜙𝜙 ′′ (𝑢𝑢)𝑢𝑢𝑥𝑥𝑥𝑥
Finally, the set of exact solutions of the general form of (1)
are clearly
2
𝑢𝑢(𝑥𝑥, 𝑡𝑡) = 𝜙𝜙 ′ 𝑎𝑎∆𝜔𝜔2 + 3𝜙𝜙 ′′ 𝜔𝜔𝑥𝑥𝑥𝑥
1
(𝐴𝐴𝐴𝐴 ± 𝐴𝐴2 𝑖𝑖 𝑡𝑡 + 𝐶𝐶1 )�
sech2 �
2
𝑢𝑢(𝑥𝑥, 𝑡𝑡) = �
1
sech2 � �±√𝐵𝐵𝑖𝑖𝑖𝑖 + 𝐵𝐵𝐵𝐵 + 𝐶𝐶2 �� 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
2
Which are the solitary waves having the sech2 profiles
[7]. 𝐴𝐴 and 𝐵𝐵are parameters specifying the amplitude and the
speed of the wave.
CONCLUSIONS
(8)
(9)
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).
51
Mathematics Section
ACKNOWLEDGMENT
We thank Professor Walo Omana Rebecca who provided
insight and expertise that greatly illuminating development of
this research.
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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-BonaMahony (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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