Approximate Analytical and Numeric Solutions to a Forced Damped Gardner Equation

Salas, Alvaro H.; El-Tantawy, S. A.; Martinez H., Lorenzo J.

doi:https://doi.org/10.1155/2022/3240918

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Abstract Introduction Analysis and Discussion Conclusion Appendix Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 3240918 | https://doi.org/10.1155/2022/3240918

Approximate Analytical and Numeric Solutions to a Forced Damped Gardner Equation

Alvaro H. Salas,¹S. A. El-Tantawy,^2,3and Lorenzo J. Martinez H.⁴

Academic Editor: Syed Abbas

Received12 Nov 2021

Accepted25 Apr 2022

Published11 May 2022

Abstract

In this paper, some exact traveling wave solutions to the integrable Gardner equation are reported. The ansatz method is devoted for deriving some exact solutions in terms of Jacobi and Weierstrass elliptic functions. The obtained analytic solutions recover the solitary waves, shock waves, and cnoidal waves. Also, the relation between the Jacobi and Weierstrass elliptic functions is obtained. In the second part of this work, we derive some approximate analytic and numeric solutions to the nonintegrable forced damped Gardner equation. For the approximate analytic solutions, the ansatz method is considered. With respect to the numerical solutions, the evolution equation is solved using both the finite different method (FDM) and cubic B-splines method. A comparison between different approximations is reported.

1. Introduction

Partial differential equations (PDEs) and ordinary differential equations (ODEs) have received the attention of many researchers of applied mathematics and theoretical physics due to their great role in modeling many natural and engineering phenomena [1–16]. The investigation of the traveling wave solutions (TWSs) to the differential equations plays an important role in the study of nonlinear physical phenomena in different branches of science specially in fluid mechanics, optical fiber, plasma physics, ocean, and sea [1, 2, 7–16]. The following Gardner equation (or combined Korteweg–De Vries (KdV)-modified KdV (mKdV) equation or Extended KdV equation (EKdV)) is one of the most famous equations that was widely used in modeling many physical problems in general and in the physics of plasmas in particular [2, 17–19].where and represent the coefficients of the nonlinear and dispersion terms which are function of physical parameters related to the system under study. The Gardner equation (GE) has two nonlinear terms in the quadratic and cubic forms and the dissipative term is of third-order derivative. The GE is an integrable system and Miura transformation connects it to the KdV equation. The GE is a useful model to understand the propagation of acoustic waves in different plasma models [17–19]. This equation can be used for describing the small but finite amplitude of the nonlinear structures that can propagate with phase velocity. However, this equation and some related equations such as KdV equation and modified KdV equation can be used indirectly to describe waves that propagate with the group velocity such as rogue waves and dark and bright solitons. For example, there are many researchers interested in studying waves in plasma physics who used this equation to describe the rouge waves in different plasma models [20–23].

Exact solutions to the GE and some related equations are set up by using various methods [24–45]. Some solutions containing tanh and coth functions are proposed by the extended form of the tanh method [1]. Solitary wave and periodic solutions are constructed by aid of the projective Riccati equations. These solutions have various terms including trigonometric or hyperbolic functions in rational forms. In most published papers, the authors focused on the integrable GE. However, in many physical models there are many effects cannot be ignored such as the collisions between the charged and neutral particles in plasma physics as well as the collisions between the charged particles themselves. Also, some of the external periodic forces can be impacted on the physical model under consideration. If these effects are considered in this case, we can get a nonintegrable evolution equation (forced damped GE). In this paper, we shall proceed to obtain some approximations to the following forced damped GE:where gives the coefficient of the damping term which arises due to different collisions that take place in plasma physics and indicates the external forces. To our knowledge there is no one attempt for studying this equation. Thus, the main goal of this work is to find some approximations for this equation using different analytical and numerical methods.

2. Novel Exact Solutions to the Integrable Gardner Equation

In order to obtain some traveling wave solutions (TWSs) to (1), we make the traveling wave transformation , which leads to

Integrating (3) once over and denoting the constant of integration by , we finally get the following Helmholtz–Duffing equation:

Introducing the following ansatz,into (4) and after direct calculations, we obtainwhere the coefficients are given by

Solving the system will give us the values of as

Thus, the cnoidal wave solution to GE (1) reads

For letting , the soliton solution is recovered.

It is easy verified that the following solution is a rational solution:

To obtain a traveling wave solution in terms of sn and , we make the following substitution:

Inserting the ansatz (12) into GE (1) will give uswhere the coefficients are defined by

The solution of the algebraic system gives the values of coefficients :

Inserting the values given in (15) into ansatz (12), the solution in the form of Jacobi elliptic function sn is obtained:

Moreover, for , the shock wave solution is obtained:

Remark 1. Equation (4) can be written in the following form:The general solution to (18) can be expressed in the terms of Weierstrass elliptic function as following:By substituting ansatz solution (19) into the Helmholtz–Duffing equation (18) and after tedious calculations, we get,The constants and are determined from the initial conditions and . In particular, this gives the general solution to both Duffing and Helmholtz equations, respectivelyWe see that the GE (1) admits solutions in terms of the Weierstrass elliptic function. In general, any partial differential equation (pde) can be converted to the ode (18) via the traveling wave transformation, admits TWSs in terms of Weierstrass elliptic function . Also, Weierstrass function may be expressed in terms of the Jacobian cn function, which leads to cnoidal wave solutions to GE (1).
The relation between the Jacobian and Weierstrass elliptic functions readswithOn the other hand,withwhere is the least in magnitude root of the following cubic equation:Note that for , we have only one real root,Then,Let , the discriminant of the cubic (26) readsIn this case, our analysis depends on the sign of discriminant (29). Below, we will discuss three cases which depends on the sign of .

2.1. First Case:

For , the cubic (26) has three real roots in the following compact form:where .

2.2. Second Case:

For , the cubic (26) has only one real root:

2.3. Third Case:

Here, suppose that , we have two real roots for cubic (26):

It follows from (24) that the period of Weierstrass elliptic function readswhere is the greatest real root to the following cubic equation:

Using the identity (24) and the approximation (37), the following approximation is obtained:where is given by (25) and :where .

The Jacobian elliptic functions cn and sn may be approximated by means of the following expressions:

In the following Table 1, we can check the accuracy of the above obtained approximations.

We now give approximate expressions for inverse elliptic cosine and inverse elliptic cosine as follows:where .

In Table 2, the numerical values of approximations (40) and (41) are displayed.

3. Approximate Analytical Solution to the Forced Damped Gardner Equation

Let us suppose that is a solution to the GE (1):

We seek for approximate solution to the forced damped GE (2) in the ansatz form:

The functions , , and are to be determined later. Plugging ansatz (43) into (2) and taking the following value of into account:we finally obtain

The last expression suggests the choices:

We will choose the functions , , , and so that

Solving system (46) and (47), we obtain the following solutions:

Thus, an approximate analytical solution to the damped and forced GE will bewhere can found from (50) and is any analytic solution to the integrable GE (42).

4. FDM for Analyzing the Forced Damped Gardner Equation

To apply FDM for analyzing (2), we first write this equation in the following initial value problem (i.v.p.):where and .

The space-time domain are divided into subintervals with uniform size aswhere m and n are integer numbers.

According to the FDM, the following derivatives formulas are introduced:In the case, when or or or , we define , where is the analytical approximation defined in Section 3.

Now, we solve the following system of the nonlinear algebraic equations:with and and .

If we already solved system (55), then, we may construct an interpolation function with the data for and . This interpolation function will represent the approximate numerical solution to the evolution equation. Another numerical solution may be obtained using the NDSolve Mathematica command.

5. Cubic Splines (Odd-order B-splines) for Analyzing the Forced Damped Gardner Equation

The general odd B-splines of order are defined aswith , and . Here, for and 0 otherwise and .

Note that for , we obtain the so-called cubic B-splines as follows:with

Assuming that and with the help of Table 3, we get,

These formulas may be employed for solving PDEs like KdV, KdV-Burgers, MKdV, Gardner, and many third-order PDEs arising in different branches of science specially plasma physics.

Now, in order to solve the i.v.p. (52), we must solve the following system of nonlinear odes:where , for or . We must choose the value of in order to get the least residual error as possible.

6. Analysis and Discussion

We have obtained some analytical and numerical approximations the integrable GE (1) and nonintegrable forced damped GE (2). For analyzing the obtained solutions, we start by an exact solution to integrable GE (1). Some exact solutions to GE (1) such as the cnoidal wave solution given in (8) and the soliton solution given in (9) are introduced during the analysis the approximate solutions to the nonintegrable forced damped GE (2). Also, the following exact soliton solution is introduced to analyze (2).

Now, based on soliton solution (61) and according to the values , the profile of the approximate analytic soliton solution and the cubic B-splines soliton solution to the forced damped GE (2) is, respectively, presented in Figure 1. Also, the global residual errors in the whole domain for both approximate analytic solution and the numerical approximation using cubic B-splines for are, respectively, estimated as and . Note that the accuracy of the approximations depends on the values of physical parameters and the chosen exact solution.

(a)

(b)

(c)

(d)

7. Conclusion

In this work, some novel exact solutions to the integrable Gardner equation (GE) and approximations to the nonintegrable forced damped Gardner equations have been obtained. The most important obtained results can be briefly summarized in the following points:(1)In the first part, the integrable GE was reduced to the Helmholtz–Duffing equation using traveling wave transformation. After that, some exact solutions have been derived using the ansatz method. The obtained solutions have been derived in the form of Jacobi and Weierstrass elliptic functions. Moreover, the relation between Jacobi and Weierstrass elliptic functions has been presented. The obtained solutions can be recovered cnoidal waves, solitary waves, and shock waves to the GE.(2)In the second part, general formula for the approximate analytical solution to the forced damped GE has been derived in detail. This solution can be recovered from many nonlinear solutions that can be created and propagated in plasma physics. Based on this formula, the characteristics of many nonlinear structures in plasma physics such as solitary waves, shock waves, and cnoidal waves can be investigated.(3)In the third part, the evolution equation (the forced damped GE) has been analyzed using FDM in order to obtain an approximate numerical solution.(4)In the fourth-part, the cubic splines (Odd-order B-splines) were employed for analyzing the forced damped GE numerically.

Finally, the obtained solutions can help all researchers who are interested by studying the nonlinear structures in fluid mechanics, optical fiber, physics of plasmas, ocean and seas, and water tank waves.

Appendix

A. Appendix I. The coefficients

B. Appendix II: The coefficients

Data Availability

No data were used to support this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

A.-M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Beijing, China, 2009.
A.-M. Wazwaz, Partial Differential Equations: Methods and Applications, Lisse: Balkema, cop, Lisse, Netherlands, 2002.
P. G. Drazin, Philip. Solitons: An Introduction, Cambridge University Press, Cambridge, UK, 1989.
A. Yokus, “Construction of different types of traveling wavesolutions of the relativistic wave equation associatedwith the Schrödinger equation,” Mathematical Modelling and Numerical Simulationwith Applications, vol. 1, no. 1, p. 24, 2021.
View at: Google Scholar
A. Yokus, H. Durur, H. Ahmad, P. Thounthong, and Y.-F. Zhang, “Construction of exact traveling wave solutions of the Bogoyavlenskii equation by -expansion and -expansion techniques,” Results in Physics, vol. 19, p. 103409, 2020.
View at: Publisher Site | Google Scholar
A. Yokus and M. . Tuz, “An application of a new version of -expansion method,” AIP Conference Proceedings, vol. 1798, Article ID 020165, 2017.
View at: Google Scholar
W. Albalawi, R. Jahangir, W. Masood, S. A. Alkhateeb, S. A. El-Tantawy, and S. A. El-Tantawy, “Electron-acoustic (Un) modulated structures in a plasma having (r, q)-distributed electrons: solitons, super rogue waves, and breathers,” Symmetry, vol. 13, no. 11, p. 2029, 2021.
View at: Publisher Site | Google Scholar
W. Albalawi, S. A. El-Tantawy, and A. H. Salas, “On the rogue wave solution in the framework of a Korteweg-de Vries equation,” Results in Physics, vol. 30, Article ID 104847, 2021.
View at: Publisher Site | Google Scholar
B. S. Kashkari, S. A. El-Tantawy, A. H. Salas, and L. S. El-Sherif, “Homotopy perturbation method for studying dissipative nonplanar solitons in an electronegative complex plasma,” Chaos, Solitons & Fractals, vol. 130, Article ID 109457, 2020.
View at: Publisher Site | Google Scholar
J.-W. Xia, Y.-W. Zhao, and X. Lü, “Predictability, fast calculation and simulation for the interaction solutions to the cylindrical Kadomtsev-Petviashvili equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 90, Article ID 105260, 2020.
View at: Publisher Site | Google Scholar
S. Zhang and X. Zheng, “N-soliton solutions and nonlinear dynamics for two generalized Broer-Kaup systems,” Nonlinear Dynamics, vol. 107, no. 1, pp. 1179–1193, 2022.
View at: Publisher Site | Google Scholar
W. Albalawi, A. H. Salas, S. A. El-Tantawy, and A. A. A.-R. Youssef, “Approximate analytical and numerical solutions to the damped pendulum oscillator: Newton-Raphson and moving boundary methods,” Journal of Taibah University for Science, vol. 15, no. 1, pp. 479–485, 2021.
View at: Publisher Site | Google Scholar
S. A. El-Tantawy, “Nonlinear dynamics of soliton collisions in electronegative plasmas: the phase shifts of the planar KdV- and mkdV-soliton collisions,” Chaos, Solitons & Fractals, vol. 93, pp. 162–168, 2016.
View at: Publisher Site | Google Scholar
B. S. Kashkari and S. A. El-Tantawy, “Homotopy perturbation method for modeling electrostatic structures in collisional plasmas,” The European Physical Journal-Plus, vol. 136, p. 121, 2021.
View at: Google Scholar
Y.-L. Ma, A.-M. Wazwaz, and B.-Q. Li, “Novel bifurcation solitons for an extended Kadomtsev-Petviashvili equation in fluids,” Physics Letters A, vol. 413, Article ID 127585, 2021.
View at: Publisher Site | Google Scholar
L. ü Xing and Si-J. Chen, “Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types,” Nonlinear Dynamics, vol. 103, p. 947, 2021.
View at: Google Scholar
S. K. El-Labany, W. M. Moslem, K. A. Shnishin, and S. A. El-Tantawy, “Plasma with two-negative ions and immobile dust particles: planar and non-planar ion-acoustic wave propagation,” The European Physical Journal D, vol. 61, no. 2, pp. 409–420, 2011.
View at: Publisher Site | Google Scholar
S. A. El-Tantawy, N. A. El-Bedwehy, and W. M. Moslem, “Nonlinear ion-acoustic structures in dusty plasma with superthermal electrons and positrons,” Physics of Plasmas, vol. 18, no. 5, Article ID 052113, 2011.
View at: Publisher Site | Google Scholar
S. A. El-Tantawy and W. M. Moslem, “Nonlinear electrostatic excitations in electron-depleted electronegative dusty plasma with two-negative ion species,” Astrophysics and Space Science, vol. 337, no. 1, pp. 209–215, 2012.
View at: Publisher Site | Google Scholar
M. S. Ruderman, T. Talipova, and E. Pelinovsky, “Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions,” Journal of Plasma Physics, vol. 74, no. 5, pp. 639–656, 2008.
View at: Publisher Site | Google Scholar
M. S. Ruderman, “Freak waves in laboratory and space plasmas,” The European Physical Journal - Special Topics, vol. 185, no. 1, pp. 57–66, 2010.
View at: Publisher Site | Google Scholar
S. A. El-Tantawy, “Rogue waves in electronegative space plasmas: the link between the family of the KdV equations and the nonlinear Schrödinger equation,” Astrophysics and Space Science, vol. 361, no. 5, p. 164, 2016.
View at: Publisher Site | Google Scholar
N. H. Aljahdaly and S. A. El-Tantawy, “Simulation study on nonlinear structures in nonlinear dispersive media,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 30, no. 5, Article ID 053117, 2020.
View at: Publisher Site | Google Scholar
B. A. Kupershmidt, “A super Korteweg-de Vries equation: an integrable system,” Physics Letters A, vol. 102, no. 5-6, 1984.
View at: Google Scholar
E. Demler and A. Maltsev, “Semiclassical solitons in strongly correlated systems of ultracold bosonic atoms in optical lattices,” Annals of Physics, vol. 326, no. 7, pp. 1775–1805, 2011.
View at: Publisher Site | Google Scholar
Tom LycheCarla ManniHendrik Speleers, Splines and PDEs: From Approximation Theory to Numerical Linear Algebra, Springer, Berlin, Germany, 2017.
A. M. Kamchatnov, Y. H. Kuo, T. C. Lin et al., “Undular bore theory for the Gardner equation,” Physical review. E, Statistical, nonlinear, and soft matter physics, vol. 86, no. 3, Article ID 036605, 2012.
View at: Publisher Site | Google Scholar
R. Grimshaw, E. Pelinovsky, T. Taipova, and A. Sergeeva, “Rogue internal waves in the ocean: long wave model,” The European Physical Journal-Special Topics, vol. 185, no. 1, pp. 195–208, 2010.
View at: Publisher Site | Google Scholar
H. Hu, M. Tan, and X. Hu, “New interaction solutions to the combined KdV-mKdV equation from CTE method,” Journal of the Association of Arab Universities for Basic and Applied Sciences, vol. 21, no. 1, pp. 64–67, 2016.
View at: Publisher Site | Google Scholar
Y. Wei-Feng, L. Sen-Yue, Y. Jun, and H. Han-Wei, “Interactions between solitons and cnoidal periodic waves of the gardner equation,” Chinese Physics Letters, vol. 31, no. 7, Article ID 070203, 2014.
View at: Google Scholar
A. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Springer-Verlag, Berlin, Germany, 2009.
Y. Jia-Ren, P. Liu-Xian, and Z. Guang-Hui, “Soliton perturbations for a combined Kdv-MKdv equation,” Chinese Physics Letters, vol. 17, no. 9, pp. 625–627, 2000.
View at: Publisher Site | Google Scholar
A. Bekir, “On traveling wave solutions to combined KdV-mKdV equation and modified Burgers-KdV equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1038–1042, 2009.
View at: Publisher Site | Google Scholar
Z. Fu, S. Liu, and S. Liu, “New kinds of solutions to Gardner equation,” Chaos, Solitons & Fractals, vol. 20, no. 2, pp. 301–309, 2004.
View at: Publisher Site | Google Scholar
H. L. Liu, X. Q. Liu, and L. Niu, “A generalized -expansion method and its applications to nonlinear evolution equations,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 3811–3816, 2010.
View at: Google Scholar
M. A. Akbar, N. Hj, and M. Ali, “New solitary and periodic solutions of nonlinear evolution equation by Exp-function method,” World Applied Sciences Journal, vol. 17, no. 12, pp. 1603–1610, 2012.
View at: Google Scholar
H. Naher and F. A. Abdullah, “Some new solutions of the combined KdV-MKdV equation by using the improved G/G-expansion method,” World Applied Sciences Journal, vol. 16, no. 11, pp. 1559–1570, 2012.
View at: Google Scholar
A. J. A. M. Jawad, “New exact solutions of nonlinear partial differential equations using tan-cot function method,” Studies in Mathematical Sciences, vol. 5, no. 2, pp. 13–25, 2012.
View at: Google Scholar
N. Taghizade and A. Neirameh, “The solutions of TRLW and Gardner equations by-expansion method,” International Journal of Nonlinear Science, vol. 9, no. 3, pp. 305–310, 2010.
View at: Google Scholar
A.-M. Wazwaz, “New solitons and kink solutions for the Gardner equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 8, pp. 1395–1404, 2007.
View at: Publisher Site | Google Scholar
H. Nishiyama and T. Noi, “Conservative difference schemes for the numerical solution of the Gardner equation,” Computational and Applied Mathematics, vol. 35, no. 1, pp. 75–95, 2016.
View at: Publisher Site | Google Scholar
T. M. Rageh, G. Salem, and F. A. El-Salam, “Restrictive taylor approximation for gardner and KdV equations,” International Journal of Advances in Applied Mathematics and Mechanics, vol. 1, no. 3, pp. 1–10, 2014.
View at: Google Scholar
S. G. Rubin and R. A. Graves, “Cubic spline approximation for problems in fluid mechanics,” U.S. National Aeronautics and Space Administration, Washington, DC, USA, 1975, NASA TR R-436.
View at: Google Scholar
S. Hamdi, B. Morse, B. Halphen, and W. Schiesser, “Conservation laws and invariants of motion for nonlinear internal waves: part II,” Natural Hazards, vol. 57, no. 3, pp. 609–616, 2011.
View at: Publisher Site | Google Scholar
A. H. Salas, “Computing solutions to a forced KdV equation,” Nonlinear Analysis: Real World Applications, vol. 12, no. 2, 2011.
View at: Google Scholar

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Copyright © 2022 Alvaro H. Salas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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