# Around Chaotic Disturbance and Irregularity for Higher Order Traveling Waves.

1. IntroductionIn the last decade, a great number of researchers have paid a particular attention to the study of solitary wave equations that undergo the influence of external perturbations. Most of physical dynamics related to the movement of liquids and waves are governed by Korteweg-de Vries (KdV) equation and its variants. Hence, for this particular equation, Cao et al. [1] as well as Grimshaw and Tian [2] have recently shown that a force combined together with dissipation can provoke a chaotic behavior usually detectable by other analysis like phase plane analysis or nonnegative Liapunov exponents. KdV equation and its variants are of infinite dimension and their use to address traveling waves or chaotic dynamics of low dimension is facilitated by numerical approximations, which have proven that correlation dimension established via Grassberger-Procaccia technique and information dimension obtained from formula of Kaplan-Yorke are both between two and three for steady traveling waves [3].

However, many authors (like, e.g., [4-7]) preferred to use numerical approach to analyze the KdV equation or its variants, especially the one with many levels of perturbations. Hence, it was shown in [4] that there is no periodic waves for the autonomous Korteweg-de Vries-Burgers equation of dimension two. We follow, in this paper, the same trend of numerical approach by making use of the recently developed fractional derivative with nonsingular kernel [8-13], to express a seventh order Korteweg-de Vries (KdV) equation with one perturbation level. This is the first instance where such a model is extended to the scope of fractional differentiation and fully investigated. We prove existence and uniqueness of a continuous solution. Before that, we shall give in the following section a brief review of the recent developments done in the theory of fractional differentiation.

2. Around the Nonsingular Kernel Differentiation with Fractional Order

The concept of fractional order derivative is seen by many authors as a great endeavor to ameliorate nonlinear mathematical models, widen their analysis, and expand their interpretation. Today's literature of the concept has been enriched with many innovative definitions more related to the complexity and diversity of natural phenomena surrounding us. There are fractional order derivatives of local type and also of nonlocal type [8, 14-17]. The Caputo's definition

[mathematical expression not reproducible], (1)

n - 1 < [gamma] [less than or equal to] n. remains the most commonly used in the applied science, followed by Riemann-Liouville's version given by

[mathematical expression not reproducible]. (2)

Recent observations by Caputo and Fabrizio [8] stated that the two definitions above better describe physical processes, related to fatigue, damage, and electromagnetic hysteresis, but do not genuinely depict some behavior taking place in multiscale systems and in materials with massive heterogeneities. Hence, the same authors introduced the following new version of fractional order derivative with no singular kernel:

Definition 1 (Caputo-Fabrizio fractional order derivative (CFFD)). Let u be a function in [H.sup.1](a;b); b > a; [gamma] [member of] [0; 1]; then, the Caputo-Fabrizio fractional order derivative (CFFD) is defined as

[mathematical expression not reproducible], (3)

where M([gamma]) is a normalization function such that M(0) = M(1) = 1.

Remark 2. Caputo and Fabrizio [8] substituted the kernel 1/[(t - [tau]).sup.[gamma]] appearing in (1) when n = 1 by the function exp(-[gamma](t - [tau])/(1 - [gamma])) and 1/[GAMMA](1 - [gamma]) by M([gamma])/(1 - [gamma]). This immediately removes the singularity at t = r that exists in the previous Caputo's expression.

For the function that does not belong to [H.sup.1](a;b), the CFFD is given by

[mathematical expression not reproducible]. (4)

Losada and Nieto [13] upgraded this definition of CFFD by proposing the following:

[mathematical expression not reproducible]. (5)

Unlike the classical version of Caputo fractional order derivative [14, 18], the CFFD with no singular kernel appears to be easier to handle. Furthermore, the CFFD verifies the following equalities:

[mathematical expression not reproducible], (6)

[mathematical expression not reproducible], (7)

with u any suitable function and a the starting point of the integrodifferentiation. The fractional integral related to the CFFD and proposed by Losada and Nieto reads as

[mathematical expression not reproducible], (8)

y [member of] [0, 1] t [greater than or equal to] 0. This antiderivative represents sort of average between the function u and its integral of order one. The Laplace transform of the CFFD reads

[mathematical expression not reproducible], (9)

where [??](v, s) is the Laplace transform L(u(v, t), s) of u(v, t).

Definition 3 (New Riemann-Liouville fractional order derivative (NRLFD)). As a response to the CFFD and being aware of the conflicting situations that exist between the classical Riemann-Liouville and Caputo derivatives, the classical Riemann-Liouville definition was modified [9, 10] in order to propose another definition known as the new Riemann-Liouville fractional derivative (NRLFD) without singular kernel and expressed for [gamma] [member of] [0, 1] as

[mathematical expression not reproducible]. (10)

Again, the NRLFD is without any singularity at t = [tau] compared to the classical Riemann-Liouville fractional order derivative and also verifies

[mathematical expression not reproducible]. (11)

Compared to (7), we note here the exact correspondence with u at [gamma] [right arrow] 0. The Laplace transform of the NRLFD reads as [9, 10]

[mathematical expression not reproducible]. (12)

Other versions and innovative definitions of fractional derivatives have since been introduced. This paper however uses the CFFD, so for more details about those recent definitions, please feel free to consult the articles and works mentioned above and also the references mentioned therein.

3. Existence and Uniqueness

In this section we prove the existence and uniqueness results for the seventh order Korteweg-de Vries equation (KdV) with one perturbation level, expressed with the CFFD and given by

[mathematical expression not reproducible], (13)

assumed to satisfy the initial condition

u (x, 0) = f (x), (14)

where [zeta] is the perturbation parameter and [sup.cf][D.sup.[gamma].sub.t] is the Caputo-Fabrizio fractional order derivative (CFFD) given in (5). Existence results for the model (13)-(14) here above are established by making use of the expression of the antiderivative (8). This yields

[mathematical expression not reproducible]. (15)

This can be rewritten as

[mathematical expression not reproducible]. (16)

Set now

[mathematical expression not reproducible]. (17)

The next step is to look for a real constant K [greater than or equal to] 0 such that

[mathematical expression not reproducible]. (18)

In fact

[mathematical expression not reproducible]. (19)

Well known properties for the norms give

[mathematical expression not reproducible]. (20)

Keeping in mind that u and v are bounded functions, then there exists real numbers [r.sub.1] >0 and [r.sub.2] >0 such that

[mathematical expression not reproducible]. (21)

Set r = max([r.sub.1], [r.sub.2]); hence,

[mathematical expression not reproducible]. (22)

Therefore, the Lipschitz condition holds for the partial derivatives [[partial derivative].sub.x]u and [[partial derivative].sub.x]v and there exists a real constant [K.sub.1] [greater than or equal to] 0 such that

[mathematical expression not reproducible]. (23)

where the bounded condition (14) has been exploited, whence

[parallel]Y (x, t, u, [zeta]) - Y (x, t, v, [zeta])[parallel] [less than or equal to] K[parallel]u - v[parallel] (24)

with

K = 12r[K.sub.1] + [K.sup.3.sub.1] + [K.sup.5.sub.1] + [zeta][K.sup.7.sub.1]. (25)

This proves that Y satisfies the Lipschitz condition and then, it allows us to state the following proposition.

Proposition 4. If the condition 1 > ((2(1-[gamma])/(2-[gamma])M([gamma]))K+ 2tK[gamma]/(2 - [gamma])M([gamma])) holds, then, there exists a unique and continuous solution to the seventh order Korteweg-de Vries equation with one perturbation level expressed with the CFFD given in (5):

[mathematical expression not reproducible]. (26)

Proof. Let us go back to the model (16) rewritten as

[mathematical expression not reproducible] (27)

which yields the recurrence formulation given as follows:

[mathematical expression not reproducible]. (28)

Let

[mathematical expression not reproducible], (29)

and it can be shown that [bar.u](x, t) = u(x, t) is a continuous solution. Indeed, if we take

[B.sub.n] (x, t) = [u.sub.n] (x, t) - [u.sub.n-1] (x, t) (30)

then, it is straightforward to see that

[u.sub.n] (x, t) = [n.summation over (p=0)] [B.sub.p] (x, t). (31)

More explicitly, we have

[mathematical expression not reproducible]. (32)

Passing this equation to the norm yields

[mathematical expression not reproducible]. (33)

Applying the Lipschitz condition to Y gives

[mathematical expression not reproducible] (34)

which can be rewritten as

[mathematical expression not reproducible]. (35)

After integrating, we make use of well known properties of the recursive technique from (35) to have

[mathematical expression not reproducible], (36)

with f(x) = u(x, 0), which explicitly shows the existence of the solution and that it is continuous.

The step forward is to prove that the solution of the model (26) is given by the function

[mathematical expression not reproducible]. (37)

For that, let

[Q.sub.n] (x, t) = [u.bar](x, t) - [u.sub.n] (x, t) with n [member of] N. (38)

Making use of (29), we should have [lim.sub.n[right arrow][infinity]][Q.sub.n] = 0. In other terms, the gap that exists between [u.bar](x, t) and [u.sub.n](x, t) should vanish as n [right arrow] [infinity]. Consider

[mathematical expression not reproducible], (39)

giving

[mathematical expression not reproducible]. (40)

Hence, [lim.sub.n[right arrow][infinity]][Q.sub.n] = 0 and from the right hand side, we have

[mathematical expression not reproducible]. (41)

Just take u(x, t) = [u.bar](x, t) as the solution of (26) that is continuous. Moreover, the Lipschitz condition for Y yields

[mathematical expression not reproducible]. (42)

This yields

[mathematical expression not reproducible] (43)

considering the initial condition and taking the limit as n [right arrow] 0 gives

[mathematical expression not reproducible]. (44)

Uniqueness. To prove that the solution is unique, we take two different functions u and v that satisfy the model (26); then,

[mathematical expression not reproducible], (45)

equivalently

[mathematical expression not reproducible]. (46)

This yields u = v if

[mathematical expression not reproducible], (47)

where we have used the Lipschitz condition for Y and this ends the proof.

4. Shape of Solitary Waves via Numerical Approximations

4.1. Shape of Solitary Waves for the Lower-Order Approximation. In this section, we are interested in waves traveling to a specific direction, and then, we consider solutions to the seventh order KdV equation

[mathematical expression not reproducible]. (48)

We start by considering the conventional case where [gamma] = 1 to have, using [sup.cf][D.sup.1.sub.t] f(t) = f'(t), the following model:

[mathematical expression not reproducible]. (49)

We investigate the traveling waves taking the form u(x, t) = u(x + ct) = u([eta]), where c is the speed of the wave. We assume that u does not depend on x independently from t but rather depends on the combined variable [eta] = x + ct. We also assume that the wave dies at infinity, meaning

[mathematical expression not reproducible]. (50)

Now, it is possible to transform the seventh order KdV equation (49) into an ordinary differential equation (ODE) by making use of the basic properties of differentiation. Then, [u.sub.x] = [u.sub.[eta]] x [[eta].sub.x] = [u.sub.[eta]] and [u.sub.t] = [u.sub.[eta]] x [[eta].sub.t] = [cu.sub.[eta]] which yield the following ODE:

[mathematical expression not reproducible]. (51)

The [eta]-integration of this equation once gives

[mathematical expression not reproducible], (52)

where we have ignored the constant of [eta]-integration that is null due to boundary conditions (50). If the higher order perturbation term [zeta][u.sub.[eta][eta][eta][eta][eta][eta]] is ignored, then we have

-cu - 3[u.sup.2] - [u.sub.[eta][eta]] + [u.sub.[eta][eta][eta][eta]] = 0. (53)

Then we can solve numerically this equation by transforming it into a system of four ODEs of order one as follows:

[mathematical expression not reproducible]. (54)

Numerical simulations are done in the phase-space (u, [u.sub.[eta]], [u.sub.[eta][eta]]) as shown in Figure 1. Let us now come back to the full model (48) with the nonsingular kernel derivative CFFD (given in (5)) and with no higher order perturbation parameter [zeta], given as

[mathematical expression not reproducible]. (55)

This fractional equation is solved numerically by making use of the Adams-Bashforth-Moulton type method also known as predictor-corrector (PECE) technique and is fully detailed in the article by Diethelm et al. [19]. Figures 1 and 2 represent the numerical simulations of solutions to the fractional model (55) with different values of the derivative order y. They clearly point out relative irregularities when the derivative order of the CFFD is [gamma] < 1.

4.2. Shape of Solitary Waves for the Higher Order Approximation. Now, the perturbation term [zeta][u.sub.[eta][eta][eta][eta][eta][eta][eta]] is considered to have (48):

[mathematical expression not reproducible] (56)

subject to localized initial condition

u(x, 0) = [square root of [12d/k]] [sech.sup.2] (x/k). (57)

Although we proved existence and uniqueness for this nonlinear problem, it still faces the challenge of providing an explicit expression of exact solution or approximated solution. It is almost impossible to use some analytical methods, like for instance, integral transform methods, the Green function technique, or the technique separation of variables. Hence, a semianalytical method like the Laplace iterative method can be a valuable tool to provide a special solution to the model (56)-(57) as shown in the following lines.

We start by applying the Laplace transform (9) on both sides of (56) to obtain

[mathematical expression not reproducible]. (58)

Let [theta]([gamma], s) = s + [gamma](1 - s); then,

[mathematical expression not reproducible]. (59)

Application of inverse Laplace transform [L.sup.-1] on both sides yields

[mathematical expression not reproducible]. (60)

This leads to the following recursive system

[mathematical expression not reproducible], (61)

then making the solution to be u(x, t) = [lim.sub.n[right arrow][infinity]][u.sub.n](x, t). Numerical approximations are performed according to the following steps:

(i) [u.sub.0](x, t) = u(x, 0) is considered as initial input:

Choose j as the number of terms to compute.

Name [u.sub.app] to be the approximate solution.

Set [u.sub.app] = u(x,0) = [square root of 12d/k] [sech.sup.2] (x/k) and [u.sub.app] = [u.sub.app].

(ii) For the other terms use

[mathematical expression not reproducible].

Compute [X.sub.n](x, t) = [X.sub.n-1] (x, t) + [u.sub.app]. to finally get [u.sub.app](x, t) = [X.sub.n](x, t) + [u.sub.app]

where

[mathematical expression not reproducible]. (62)

Note we can use 9(y, s), to obtain following results and some numerical approximations with related absolute errors are summarized in Tables 1 and 2 (which present some numerical approximations with related absolute errors as given by the sequences (63) and (65), respectively, representing the pure fractional case ([gamma] = 0.8) and the standard case ([gamma] = 1)).

[mathematical expression not reproducible]. (63)

Note that, for [gamma] = 1, we recover for the conventional model (49)

[mathematical expression not reproducible], (64)

the following well known result

[mathematical expression not reproducible]. (65)

This result corresponds to the similar one obtained in [20, 21] or in [22] via Adomian decomposition method. The other terms were computed following the same iterative approach and then, the functions w(x, t) admits the closed form found to be

[mathematical expression not reproducible]. (66)

The shape of numerical approximations is depicted in Figures 3-6, plotted for different values of the derivative order [gamma] and showing a spot of irregular perturbed motion in the pure fractional case ([gamma] = 0.80) compared to the conventional case ([gamma] = 1). Before going to conclusions, we can summarize the physical aspects of this work (issued from the mathematical analysis and simulations) as follows: analysis performed on the KdV model with no higher order perturbation term has shown the soliton solution via its related homoclinic orbit lying on a curved surface (Figures 1 and 2), and this remains true in the conventional case as well as the pure fractional case. However, irregular movements are more important in the latter case. Analysis performed on the seventh order KdV equation with one perturbations level has shown the shape of solitary waves characterized by motions with more irregular behaviors tending to become chaotic. The chaos is more likely to happen in pure fractional case compared to the conventional case (Figures 3-6). Hence, the main results of this paper, reflected by the six figures, insinuate that the regularity of a soliton can be perturbed by the nonsingular kernel derivative, which, combined with the perturbation parameter of the KdV model, may lead to chaos.

5. Concluding Remarks

We have made use of the recent version of derivative with nonsingular kernel to prove existence and uniqueness results for a model of seventh order Korteweg-de Vries (KdV) equation with one perturbation level. The unique solution is continuous. We then use numerical approximations to evaluate the behavior of the solution under the influence of external factors. It happened that when applied to equations of wave motion like the seventh order KdV equation, the new derivative acts as one of those external factors, which, combined with the perturbation term [zeta] of the model, causes the solution to be more irregular and unpredictable. This is the first instance where such a model is fully investigated and such result is exposed. This work differs from the previous ones within introduction of the nonsingular kernel derivative into a powerful model like Korteweg-de Vries's, which reveals another interesting feature that exists in the domain of wave motion as well as chaos theory.

https://doi.org/10.1155/2018/2391697

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the National Research Foundation (NRF) of South Africa, Grant no. 105932.

References

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Emile F. Doungmo Goufo (iD) (1) and Ignace Tchangou Toudjeu (2)

(1) Department of Mathematical Sciences, University of South Africa, Florida 0003, South Africa

(2) Department of Electrical Engineering, Tshwane University of Technology, Pretoria 0183, South Africa

Correspondence should be addressed to Emile F. Doungmo Goufo; franckemile2006@yahoo.ca

Received 27 February 2018; Accepted 29 March 2018; Published 3 June 2018

Academic Editor: S. K. Q. Al-Omari

Caption: Figure 1: Numerical plot in the phase-space (u, [u.sub.[eta]], [u.sub.[eta][eta]]) for the model KdV model (55) or model (48) with no higher order perturbation term, for [gamma] = 1 (conventional case). As expected in (a), the soliton solution is shown via its related homoclinic orbit to lie on a curved surface. In (b), the projection on the plane (u, [u.sub.[eta]].

Caption: Figure 2: Numerical plot in the phase-space (u, [u.sub.[eta]], [u.sub.[eta][eta]]) for the model KdV model (55) with no higher order perturbation term, for [gamma] = 0.75. As expected in (a), even in the pure fractional case, the soliton solution is shown via its related homoclinic orbit to lie on a curved surface, with some irregular movements compared to Figure 1. In (b), the projection on the plane (u, [n.sub.[eta]]).

Caption: Figure 3: The shape of approximated solution [u.sub.app] = [u.sub.3] in the conventional case ([gamma] = 1). It is plotted with respect to different fixed values of time and space at k = 2.5, d = 4.0, [zeta] = 0.5. The motion is quite standard and regular.

Caption: Figure 4: The shape of approximated solution [u.sub.app] = [u.sub.3] in the pure fractional case ([gamma] = 0.80), plotted with respect to different fixed values of time and space at k = 2.5, d = 4.0, [zeta] = 0.5. The motion is shown to behave irregularly compared to the conventional case and the dynamic tending to a chaotic one.

Caption: Figure 5: The shape of approximated solution [u.sub.app] = [u.sub.3] in the conventional case ([gamma] = 1). It is plotted with respect to different fixed values of time and space at k = 2.5, d = 4.0, [zeta] = 0.01. The motion is quite standard and regular.

Caption: Figure 6: The shape of approximated solution [u.sub.app] = [u.sub.3] in the pure fractional case ([gamma] = 0.80), plotted with respect to different fixed values of time and space at k = 2.5, d = 4.0, [zeta] = 0.01. The motion is shown to behave irregularly compared to the conventional case and the dynamic tending to a chaotic one.

Table 1: Some values for numerical and exact solutions to the seventh order KdV equation expressed with the CFFD given in (5) at k = 2.5, d = 4.0, and [gamma] = 0.80. For [gamma] = 0.80 Numerical Time (t) Spatial (x) Exact value value 0.5 -15.0 0.000098126 0.000224130 -7.0 0.056620000 0.056514000 0.0 0.100372300 0.100470000 +7.0 0.070555000 0.069355000 +15.0 0.000118180 0.000120180 1.0 -15.0 0.000091592 0.000284080 -7.0 0.054779000 0.054779000 0.0 0.100390000 0.100380000 +7.0 0.075546000 0.071755000 +15.0 0.000126620 0.000073380 2.0 -15.0 0.000081235 0.000181240 -7.0 0.048620000 0.069620000 0.0 0.100410000 0.100510000 +7.0 0.085083000 0.084954000 +15.0 0.000142760 0.000242760 4.0 -15.0 0.000065920 0.000132590 -7.0 0.039495000 0.039495000 0.0 0.100430000 0.090430000 +7.0 0.104610000 0.102610000 +15.0 0.000175920 0.000175920 8.0 -15.0 0.000045819 0.000224130 -7.0 0.027489000 0.058768000 0.0 0.090400000 0.100400000 +7.0 0.149720000 0.071755000 +15.0 0.000253100 0.000120180 10.0 -15.0 0.000038771 0.000224130 -7.0 0.023272000 0.058768000 0.0 0.090402000 0.091402000 +7.0 0.176380000 0.176380000 +15.0 0.000299110 0.00012018 For [gamma] = 0.80 Time (t) Spatial (x) Error made 0.5 -15.0 1.26 x [10.sup.-4] -7.0 1.06 x [10.sup.-4] 0.0 1.00 x [10.sup.-4] +7.0 1.20 x [10.sup.-3] +15.0 2.00 x [10.sup.-6] 1.0 -15.0 1.20 x [10.sup.-4] -7.0 00 0.0 1.00 x [10.sup.-5] +7.0 1.20 x [10.sup.-4] +15.0 2.00 x [10.sup.-4] 2.0 -15.0 1.00 x [10.sup.-4] -7.0 2.10 x [10.sup.-2] 0.0 1.01 x [10.sup.-4] +7.0 1.29 x [10.sup.-4] +15.0 1.00 x [10.sup.-4] 4.0 -15.0 1.26 x [10.sup.-3] -7.0 00 0.0 1.00 x [10.sup.-2] +7.0 2.00 x [10.sup.-3] +15.0 00 8.0 -15.0 1.26 x [10.sup.-3] -7.0 1.6 x [10.sup.-3] 0.0 1 x [10.sup.-2] +7.0 1.2 x [10.sup.-3] +15.0 1.01 x [10.sup.-5] 10.0 -15.0 1.26 x [10.sup.-4] -7.0 1.06 x [10.sup.-3] 0.0 1 x [10.sup.-3] +7.0 00 +15.0 2.1 x [10.sup.-5] Table 2: Some values for numerical and exact solutions to the seventh order KdV equation expressed with the CFFD given in (5) at k = 2.5, d = 4.0, and [gamma] = 1. For [gamma] = 1 Time (t) Spatial (x) Exact value Numerical value 0.5 -15.0 0.000099315 0.000109420 -70 0.593670000 0.593670000 0.0 0.151107000 0.151107000 +7.0 0.069718000 0.069718000 +15.0 0.000167 700 0.00009 6770 1.0 -15.0 0.00009159 2 0.00008 857 2 -7.0 0.054779000 0.054779000 0.0 0.151907000 0.151907000 +7.0 0.075546000 0.075546000 +15.0 0.0001266 20 0.000026620 2.0 -15.0 0.000077901 0.000237900 -7.0 0.046635000 0.046635000 0.0 0.152007000 0.152140000 +7.0 0.08 868 8000 0.08 868 8000 +15.0 0.0001488 70 0.000159170 4.0 -15.0 0.00005 635 2 0.001316 400 -7.0 0.03 378 5000 0.03 378 5000 0.0 0.152060000 0.162060000 +7.0 0.122130000 0.120120000 +15.0 0.0002057 90 0.000205790 8.0 -15.0 0.00002 948 8 0.001289500 -7.0 0.017711000 0.019311000 0.0 0.152060000 0.162060000 +7.0 0.230 430000 0.231630000 +15.0 0.0003 932 70 0.0004 033 70 10.0 -15.0 0.000021331 0.000020071 -7.0 0.012819000 0.012830000 0.0 0.161760000 0.162760000 +7.0 0.315 320000 0.315 320000 +15.0 0.0005 436 40 0.0005 436 40 For [gamma] = 1 Time (t) Spatial (x) Error made 0.5 -15.0 1.01 x [10.sup.-5] -70 00 0.0 00 +7.0 00 +15.0 2.00 x [10.sup.-5] 1.0 -15.0 3.02 x [10.sup.-6] -7.0 00 0.0 00 +7.0 00 +15.0 1.00 x [10.sup.-4] 2.0 -15.0 1.60 x [10.sup.-4] -7.0 00 0.0 1.31 x [10.sup.-4] +7.0 00 +15.0 1.03 x [10.sup.-5] 4.0 -15.0 1.26 x [10.sup.-3] -7.0 00 0.0 1.00 x [10.sup.-2] +7.0 2.01 x [10.sup.-3] +15.0 00 8.0 -15.0 1.26 x [10.sup.-3] -7.0 1.6 x [10.sup.-3] 0.0 1 x [10.sup.-2] +7.0 1.2 x [10.sup.-3] +15.0 1.01 x [10.sup.-5] 10.0 -15.0 1.26 x [10.sup.-6] -7.0 1.06 x [10.sup.-5] 0.0 1.00 x [10.sup.-3] +7.0 00 +15.0 00

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Title Annotation: | Research Article |
---|---|

Author: | Goufo, Emile F. Doungmo; Toudjeu, Ignace Tchangou |

Publication: | Journal of Mathematics |

Date: | Jan 1, 2018 |

Words: | 5011 |

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