The research paper published by IJSER journal is about Solving the oscillating electric dipole equation by Adomian decomposition technique 1

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# Solving the oscillating electric dipole equation by Adomian decomposition technique

HANGNILO R., ADANHOUNME V.

Abstract

For modeling the electric organ discharge of Malapterurus electricus catfish we have set up the electromechanical scheme of the catfish as an electric dipole the charges of which are linked by a spring of stiffness k. W e call this particular system of charges an oscillating electric dipole. Taking into account the main forces applied to this dipole and applying the Newton’s second law of motion we obtain a relation that we call oscillating electric dipole equation. Our paper reports the resolution of such a nonlinear differential equation. The oscillating electric dipole approach is applied to electric behavior of Malapterurus electricus catfish in order to design the electric equivalent scheme of the fish.

Keywords: Electric charges, oscillating electric dipole equation, electric fish, electric organ discharge, Adomian decomposition technique.

## Introduction

figure 1. The potential at any point due to a group of two point charges is found by calculating the potential Vn, n = 1, 2 due to each charge, as if the other charge was not present and added the quantities so obtained:

V(M) = = , (1)

Where q1, q2 are the magnitude of the charges, (q1
= q2 = q) and r1 and r2 stand for the distance from each charge to the point M. We limit consideration to points such that r >> 2a. Then follow these approximate relations from figure 1:
Electrostatics remain the branch of physics which studies the phenomena related to fixed electric charges and is a part of Electromagnetism, [1]. As one characterizes the universal gravitation force by associating with the body the mass m which determines its weight, one can characterize the electric state of the body by its electric charge q. There are two kinds of charge. Matter as we ordinarily experience it can be regarded as composed of three kinds of elementary particles, the proton, the neutron and the electron. The electric properties of matter can be explained by its electronic structure. A positive and negative
charge of equal magnitude q placed a distance 2a

and , (2)

We need only to find V(r, ѳ) which reduces to:

(3)

M(r, ϴ )
apart constitute an electric dipole, see figure 1.
We derive an analytical expression for the electric
potential V (M) at any point M of the space due

-

A(- a)

Ѳ +

O B(+ a) X
to an electric dipole provided only that the point is not too close to the dipole, [2]. The point M is
specified by giving the quantities r and ѳ in

Fig. 1. An electric dipole

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The research paper published by IJSER journal is about Solving the oscillating electric dipole equation by Adomian decomposition technique 2

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When the two charges of an electric dipole are
linked by a spring of stiffness k as it is shown in figure 2, we call it the oscillating electric dipole, OED. This study is devoted to solving the equation of the oscillating electric dipole. In section 2, we establish the equation. In section 3 we provide the analytical solution to the oscillating dipole equation, [3], [4]. In section 4 the electric potential we have obtained by means
of that analytical solution is applied for modeling
the electric behavior of Malapterurus electricus catfish. In section 5 follows the conclusion.

## 2. MATHEMATICAL MODEL

second law, the equation of motion of the
oscillating electric dipole can be written as:

, (7)

For the oscillating electric dipole, , where is the distance between the charges at the equilibrium and . It is well known that the gravitational force is about 1031 lower than Coulomb one and can be neglected. The equation (7) is equivalent to:

, (8)

We consider two equal electric charges q of
opposite sign possessing equal masse m. The forces acting on these charges are in order: the gravitational force FG, the force of viscous friction FV, the repelling force of the spring FR and the generalized Lorentz force FL. The analytical
expression of each force is as following:

Where,

, (9)

M(r, ϴ )

, , , (4) (4)

Ѳ

- +

, (5)

A(- a)

(5) O

B(a) X

, , (6)

Where d is the distance between the center of gross masses m1 and m2 mainly resulting masses of protons and neutrons contained in the nucleus of every atoms of the considered charges q1 and q2, G = 6.67 10-11 N.m2/kg2, [5]; the coefficient of viscous friction, the linear velocity; the stiffness of the spring and the elongation; stands for the Coulomb force and ε0 8.85.10-12

Farad/m, [5]. Each moving charge will create a magnetic field B as it is shown by Batailler [6] throughout Rowland’s experience. When an electric charge q moves with the speed in a magnetic field it experiences an additional force on top of Coulomb one. In what follows, we neglect the component as we adopt the assumption that the magnetic field is negligible
as is small [2]. According to Newton’s

Fig. 2. An oscillating electric

#### 2.1 Definition The equation (8) is called the oscillating electric dipole equation

We impose the initial conditions to be:

= u, , (10)

## 3. Analytical solution

In this section we provide the analytical solution to (8) where is the unknown function depending on the time, taking into account (10). Then the equation (8) is equivalent to the non linear integral equation, [7]:

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The research paper published by IJSER journal is about Solving the oscillating electric dipole equation by Adomian decomposition technique 3

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, (11) (11)

Where:

W (g(s), h(s)) = ;

, (18)

Therefore the electric potential of the OED is as

; , (12)

And g, h are the particular solutions of homogeneous equation related to the equation (8)
following:

, (19)

for , (13)

Thus we can write the non linear functional equation (11) in the form:

, (14)

Where,

## 4. Applying the solution of the Equation in(1M3)alapterurus electricus EOD modeling

We have investigated the electric behavior of Malapterurus electricus to test the oscillating electric dipole that we consider as the equivalent electromechanical model of the catfish. The model leads to th(1e4n)on linear differential
equation of second order (8). As we have
characterized the electric shocks of Malapterurus electricus catfish in our Laboratory [9] some data of the Electric Organ Discharge, EOD of the

catfish are available: these EOD are similar to

, (15)

mono or two fold al

(15) ing electric waves

G is a nonlinear operator from a Hilbert space H into H; p is a given function in H. We assume that (14) has a unique solution. The Adomian technique [8] defines the solution of (14) by the decomposition series using the following scheme:

, (16)

frequency and magnitude of which vary depending on the fish. Taking into account (19) we realize a computer simulation in Matlab. The results of the simulation and the EOD of the catfish are very close each to other as we can see in figure 3 and seem like the output voltage of a Graetz bridge rectifier. Therefore the design of Malapterurus electricus catfish’s EOD simulator is conceivable.

,(17)

The 2-term approximation of can be written in the form, for

(16)

a b

Fig. 3. a- Computer simulation of the model

b- Electric organ discharge of the catfish

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The research paper published by IJSER journal is about Solving the oscillating electric dipole equation by Adomian decomposition technique 4

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5. Conclusion

The EOD of Catfish in general and Malapterurus electricus ‘one in particular are powerful and can supply some applications in energy. Malapterurus electricus EOD overcome
sometime 350 Volts [10] and the catfish is able to give so many EOD per second, [9]: it is a true source of energy worthy of interest. In order for Malapterurus electricus catfish energy supplying to be we must set up the equivalent electric scheme of the fish. For that purpose we have solved the non linear second order differential equation given by the electromechanical model that we got for the fish. Each parameter α, β and γ of (8) plays a role: when α increases the magnitude of the electric shocks of the model decreases and when β increases the frequency of these shocks increases. The magnitude of the shocks is proportional to γ. The electric equivalent scheme we got for the fish allows us
to design a mosquitoes ‘larvae killer in fresh
water to control malaria disease burden. Elsewhere we succeed with the help of the scheme in the design of a 12 Volts accumulator
battery charger with the energy of the catfish. We hope to use the electric shocks of the model to
cure some sickness as Egyptian did formerly with the help of Malapterurus electricus catfish.

### References

[1] M. Hulin, JP. Maury, ‘’Les bases de l’électromagnétisme’’, Dunod, Paris, 345p, 1991.

[2] D. Halliday, R. Resnick, ‘’Fundamentals of physics’’,

John Wiley and Sons Inc, USA, 815p, 1974.

[3] N. Piskounov, ‘’Calcul différentiel et intégral tome 2’’,

MIR Moscou, 614p, 1980

[4] A. Philippov, ‘’Recueil de problèmes d’équations

différentielles’’, MIR Moscou, 136p, 1976

[5] J. Orear, ‘’Physics-2’’, MIR Moscow, 622p, 1981

*6+ G. Batailler, ‘’ Electricité fondamentale 2-

Electromagnétisme’’, Armand Colin, Paris, 168p, 1970

[7] AN. Kolmogorov, SV. Fomin, ‘’Elements of function theory and functional analysis’’, Moscow, 496p, 1988

[8] GA. Adomian, ‘’Review of the decomposition method in applied mathematics’’, J. Math. Anal. Appl., 135, pp.

501-544, 1988

[9] SA., Adédjouma, R. Hangnilo, JM. Zonou, GA. Mensah,

‘’Determination of the electric energy characteristics produced by the Malapterurus electricus, an electric catfish of fresh water in Benin’’, Revue Cames-Série A, 12 (1), pp. 52-57, 2011

[10] KB. Augustinsson, AG. Johnels, ‘’The Acetylcholine system of electric organ of Malapterurus electricus’’, J. Physiol. 140, pp. 498-500, 1958

Hangnilo Robert received the M.Sc. degree in Energy

Power & Motion Control in Energy Institut of Moscow and the Ph.D. degree in the University of Havre in France.

He is currently Assistant Professor of Electromagnetism and Energy Systems Implementations at the ‚Ecole Polytechnique d’Abomey-Calavi‛ - University of Abomey- Calavi in Benin Republic.

His principal research interests are Modeling of Energy systems and vector Control in Malaria.

e-mail : hangnilo555@yahoo.fr

Villevo Adanhounme received the M.Sc. and Ph.D. degrees in mathematics from the Russiqn People University of Moscow. Russian Federation.He is currently Assistant Professor of variational calculus and advanced probability at the International Chair of Mathematical Physics and Applications – University of Abomey Calavi. Benin.

His principal research interests are applied mechanics , Partial differential equations and optimal at the

International Chair of Mathematical Physics and

Applications.

e-mail : adanhuonm@yahoo.fr

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