Some New Families of Exact Solitary Wave Solutions for Pseudo-Parabolic Type Nonlinear Models

Hussain, Akhtar; Ali, Hassan; Usman, M.; Zaman, F. D.; Park, Choonkil

doi:https://doi.org/10.1155/2024/5762147

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Research Article | Open Access

Volume 2024 | Article ID 5762147 | https://doi.org/10.1155/2024/5762147

Some New Families of Exact Solitary Wave Solutions for Pseudo-Parabolic Type Nonlinear Models

Akhtar Hussain,¹Hassan Ali,¹M. Usman,²F. D. Zaman,¹and Choonkil Park³

Academic Editor: M. M. Bhatti

Received13 Feb 2023

Revised18 Jul 2023

Accepted23 Mar 2024

Published31 Mar 2024

Abstract

The objective of the current study is to provide a variety of families of soliton solutions to pseudo-parabolic equations that arise in nonsteady flows, hydrostatics, and seepage of fluid through fissured material. We investigate a class of such equations, including the one-dimensional Oskolkov (1D OSK), the Benjamin-Bona-Mahony (BBM), and the Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation. The Exp (-)-expansion method is used for new hyperbolic, trigonometric, rational, exponential, and polynomial function-based solutions. These solutions of the pseudo-parabolic class of partial differential equations (PDEs) studied here are new and novel and have not been reported in the literature. These solutions depict the hydrodynamics of various soliton shapes that can be utilized to study the nature of traveling wave solutions of other nonlinear PDE’s.

1. Introduction

Nonlinear problems have always been of interest to researchers [1–3] due to various applications to practical problems. While several approximate and numerical methods are available, analytical solutions always provide a benchmark for such methods. Several ansatz-based methods like mapping method [4, 5], the Jacobi elliptic function method [6], the new extended direct algebraic method [7], the tanh-coth method [8], the simple equation method [9], the symmetry method [10, 11], and many others are used for handling exact solutions to nonlinear PDEs. Pseudo-parabolic equations, which appear in many branches of mathematics and physics, have a one-time derivative in the highest-order term. They arise, for example, in the study of the flow of fluid in fractured rocks, the consolidation of clay, the shear of second-order fluids, thermodynamics, and the propagation of long, low-amplitude waves.

To solve a nonlinear pseudo-parabolic equation, a numerical approach has been established [12]. The stability of numerical approximations to backward-time parabolic and pseudo-parabolic problems and a relationship between parabolic and pseudo-parabolic difference schemes were also explored, along with a relationship between parabolic and pseudo-parabolic difference schemes. The approach suggested by Sobolev [13] may be used to identify the source of new issues in mathematical physics, and [14] explained why these problems are referred to as pseudo-parabolic problems. Pseudo-parabolic equations apply to the study of a variety of significant physical phenomena, including the seepage of homogeneous fluids through a fissured rock [15], the accumulation of populations, and the conduction of heat between bodies kept at two different temperatures. They are characterized by the occurrence of a time derivative appearing in the highest-order term [16]. Such equations, which comprise the nonlinear pseudo-parabolic differential, are used in many branches of physics and mathematics to explain many physical implications [17]. In this direction, Camassa [18] has obtained some soliton solutions for the pseudo-parabolic equations. Wazwaz [19, 20] has considered analytical solutions to the same class of equations. Johnson [21] discussed some models arising from water waves. The authors [22, 23] discussed the trigonometric and hyperbolic-type soliton solutions of the same class. To provide readers with additional information, reference [24] is cited.

The generalized pseudo-parabolic equations are one specific instance of this class. We consider the following generalized form of Benjamin Bona Mahony equation:where is a -smooth nonlinear function, is a positive constant, is a real constant, and represents the velocity of fluid in the horizontal direction. Peregrine [25] and Benjamin et al. [26] suggested the regular long-wave equation for the widely used KdV equation as the specific case of with in equation (1), i.e.,

The equation for BBMPB is obtained by substituting in the following equation (1) to get

The following equation is obtained for in the above equation, which is the following BBM equation:in which and is a parameter. We also consider the one-dimensional OSK equation which models incompressible viscoelastic Kelvin-Voigt fluid [27]

This article aims to find new families of exact soliton solutions for the nonlinear pseudo-parabolic type models arising in mathematical physics using the Exp (-)-expansion method [28–30]. This method is an extremely powerful tool for dealing with soliton solutions of nonlinear PDEs. It provides the hyperbolic, trigonometric, rational, exponential, and polynomial functions-based soliton solutions to the nonlinear PDEs. This is the actual limitation of the used methodology here. For the more general soliton solutions, we need to enhance our methodology also. Akcagil et al. [31] reported the solutions to the class of trigonometric, hyperbolic, and rational soliton solutions. We want to build on earlier research to advance our quest for a wealth of fresh traveling wave solutions. Our findings include the dark, bright, rational, exponential, polynomial, and solitary wave solutions. The solutions provided in this research are exclusively novel and valid and have not been previously presented for this class of equations.

The structure of this paper is as follows: In Section 2, we present a layout for the Exp (-)-expansion method. In Sections 3–5, the BBMPB, the 1D OSK and BBM equations respectively have been studied to obtain various solutions using this method. The graphical representations of these solutions are presented in Section 6. In Section 7 the conclusion and some possible directions of future study are mentioned.

2. Floor Plan for the Exp -Expansion Method

This section deals with the brief floor plan for the Exp -expansion method [28–30] to find the explicit soliton solutions. We give here the main steps of the method. We consider an explicit form of the nonlinear PDE as follows:where is the dependent variable.

Step 1. To reduce the number of independent variables of the equation (6), we introduce the following traveling wave transformation:where is a nonzero real parameter indicating wave speed. Then, by adopting the traveling wave transformation, the nonlinear PDE (6) becomesNote that for cases, the invariance of the transformation (7) serves as the existence criterion for the traveling wave solution. Also, the term traveling wave is due to the time behavior of the dependent variable.

Step 2. The general solution to (8) is appropriated as a polynomial in exp where are the coefficients to be determined later, and is the solution of the following equation:where , , and are real constant parameters and can be determined by the homogeneous balance principle. There are five cases for the ; Case I: When and , Case II: For and , Case III: For and , Case IV: For , and , Case V: For , (or ),

Step 3. Substituting the equation (9) into (8), the left hand side of the ODE (8) becomes a polynomial in . By comparing the coefficients of both sides, we get a system of algebraic equations that is solvable by some symbolic software like Mathematica 11.3 or Maple.

Step 4. Putting the values of from equations (11)–(15) one by one in (9), we get solutions for ODE (8). Replacing by , we get solutions for our PDE (6).
Now we are going to apply this method for some important nonlinear PDEs to have explicit soliton solutions.

3. The Benjamin-Bona-Mahony-Peregrine-Burgers Equation

Here, we first look at the equation given by

By using traveling wave transformation , we obtain the following nonlinear ODE:

By comparing the highest order of linear term with the highest degree of nonlinear term in (17), we decide the order of as . Based on this order, we deduce that the solution of (17) is of the following type:

Such that satisfies (10). By substituting (18) into (17), we transform the right-hand side of the equation into a polynomial in , and then by comparing coefficients the following system arises:

We solve this system with Mathematica 11.3 to get the following set of parameters.

First set

Inserting these parameters into (18), we obtain

Thus, the solutions for (10) becomes Case I: When and , Case II: For and , Case III: For and , Case IV: For , , and , Case V: For , , (or ),

Now by putting (22)–(26) one by one into (21) the solutions for our ODE are given by

Family 5. When and ,

Family 6. For and ,

Family 7. For and ,

Family 8. For , , and ,

Family 9. For , , (or ),where in all above cases.
Second set is as follows:Inserting these parameters into (18), we obtain.Thus, the solutions for (10) becomes Case I: When and , Case II: For and , Case III: For and , Case IV: For , and , Case V: For , (or ),Now by putting (34)–(38) one by one into (33) the solutions for our ODE are given as follows.

Family 10. When and ,

Family 11. For and ,

Family 12. For and ,

Family 13. For , and ,

Family 14. For , (or ),where in all above cases.
Third set is as follows:Inserting these parameters into (18), we obtainThus, the solutions for (10) becomes Case I: When and , Case II: For and , Case III: For and , Case IV: For , and , Case V: For , , (or )Now by putting (34)–(38) one by one into (33) the solutions for our ODE are given by

Family 15. When and ,

Family 16. For and ,

Family 17. For and ,

Family 18. For , and ,

Family 19. For , (or ),where in all above cases.

4. The One-Dimensional Oskolkov (OSK) Equation

The second model, , is given by

By using the wave transformation , we obtain the following nonlinear ODE:

By comparing the highest order of linear term with the highest degree of nonlinear term in (57), we decide the order of as . Based on this order, we deduce that the solution of (57) is of the type.

Such that satisfies (10). By substituting (58) into (57), we transform the right-hand side of the equation into a polynomial in , and then by comparing coefficients the following system arises:

We solve this system for desired constants with Mathematica 11.3 that leads to the following set of parameters:

Inserting these parameters into (58), we obtain:

Thus, the solutions for (10) becomes Case I: When and , Case II: For and , Case III: For and , Case IV: For , and , Case V: For , (or ),

Now by putting (62)–(66) one by one into (61) the solutions for our ODE are given by

Family 20. When and ,

Family 21. For. . and ,

Family 22. For and ,

Family 23. For , and ,

Family 24. For , (or ),where in all above cases and for all families.

5. The Benjamin-Bona-Mahony Equation

The third equation, , is given by

By using the traveling wave transformation , we obtain the following nonlinear ODE:

By comparing the highest order of linear term with the highest degree of nonlinear term in (73), we decide the order of as . Based on this order, we deduce that the solution of (73) is of the following type:

Such that satisfies (10). By substituting (74) into (73), we transform the right-hand side of the equation into a polynomial in , and then by comparing coefficients the following system arises:

We solve this system with Mathematica 11.3 for desired constants that lead to the following two sets of parameters.

First set is as follows:

Inserting these parameters into (74), we obtain

Thus, the solution for (10) becomes Case I: When and , Case II: For and , Case III: For and , Case IV: For , and , Case V: For , (or ),

Now by putting (78)–(82) one by one into (77) the solutions for our ODE are given by

Family 25. When and ,

Family 26. For and ,

Family 27. For and ,

Family 28. For , and ,

Family 29. For , (or ),where and in all above cases.
Second set is as follows:Inserting these parameters into (88), we obtainThus, the solution for (10) becomes as follows: Case I: When and , Case II: For and , Case III: For and , Case IV: For , , and , Case V: For , , (or ),Now by putting (90)–(94) one by one into (89) the solutions for our ODE are given by

Family 30. When and ,

Family 31. For and ,

Family 32. For and ,

Family 33. For , and ,

Family 34. For , (or ),where and in all above cases.

6. Graphical Structures of Some Solitons

This section offers a brief graphical summary of the solutions to the pseudo-parabolic equations discussed here. The wave profile of the BBMPB equation, the BBM equation, and the OSK equation are the main topics of our discussion. Our newly developed families of soliton solutions for nonlinear pseudo-parabolic models represent hyperbolic, trigonometric, rational, exponential, and polynomial functions. The waveform characteristics conform to the properties of some known solitons including the solitary waves, dark, bright, rational, exponential, and polynomial solutions provided in this research. For a particular set of parameters that are listed alongside, Wolfram Mathematica 11.3 simulations were used to create all of these visualizations. The plots include 3D, 2D, contour plots, and density plots. The plots of the solutions are shown in Figures 1–5.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

7. Discussion of the Obtained Results

Using the Exp -expansion method, new solitonic families for nonlinear pseudo-parabolic type models have been effectively found. Our focus was on the BBMPB equation, the BBM equation, and the OSK equation. In this study, we have successfully derived the new families of soliton solutions for the nonlinear pseudo-parabolic models. We have obtained the solutions hyperbolic, trigonometric, rational, exponential, and polynomial functions. The dynamics of the solutions show that the obtained solitons are solitary waves, dark, bright, rational, exponential, and polynomial functions-based. The results obtained from the authors [22, 23] contain only dark and bright solitons. So, our results are more general and new and have many applications in the field of fluid dynamics, nonsteady flows, seepage of fluid through fissured material, ocean engineering, and physical sciences.

8. Conclusions

To integrate the pseudo-parabolic type equations in this study, new applications of the Exp (-)-expansion method were used. It proved to be effective to apply the Exp (-)-expansion method to find new analytical solutions to the pseudo-parabolic equations. This technique establishes the solutions of the pseudo-parabolic equations in terms of hyperbolic, trigonometric, rational, exponential, and polynomial functions. Previously, only hyperbolic function-based solutions were given by the authors [22, 23], while the solutions found in [31] were trigonometric, hyperbolic, and rational functions. These solutions exhibited the characteristics of dark and bright soliton solutions. However, our results showed that the solitary waves, dark, bright, rational, exponential, and polynomial solutions provided in this research are exclusively novel and valid and have not been previously presented for this class of equations. These precise solutions capture the dynamics of various soliton wave shapes, and they may be used to evaluate, compare, and numerical studies in the area. Additionally, the approach utilized in this work can be applied to other mathematical and physics-related problems. It appears that more study is necessary for the advancement of fresh, effective analytical techniques for solving partial differential equations. By exploiting the improved capacities of such techniques, more problems in science and engineering that occur in the real world can be solved effectively. This can serve as one of the finest incentives for scientists to concentrate more on this remarkable area of study.

Data Availability

All data generated or analyzed during this study are included in this published article [and its supplementary information files].

Conflicts of Interest

The authors declare that there are no conflicts of interest.

References

A. Hussain, Y. Chahlaoui, M. Usman, F. D. Zaman, and C. Park, “Optimal system and dynamics of optical soliton solutions for the Schamel KdV equation,” Scientific Reports, vol. 13, no. 1, Article ID 15383, 2023.
View at: Publisher Site | Google Scholar
A. Hussain, Y. Chahlaoui, F. D. Zaman, T. Parveen, and A. M. Hassan, “The Jacobi elliptic function method and its application for the stochastic NNV system,” Alexandria Engineering Journal, vol. 81, pp. 347–359, 2023.
View at: Publisher Site | Google Scholar
A. Hussain, T. Parveen, B. A. Younis, H. U. Ahamd, T. F. Ibrahim, and M. Sallah, “Dynamical behavior of solitons of the (2+1)-dimensional Konopelchenko Dubrovsky system,” Scientific Reports, vol. 14, no. 1, p. 147, 2024.
View at: Publisher Site | Google Scholar
E. V. Krishnan and A. Biswas, “Solutions to the Zakharov-Kuznetsov equation with higher order nonlinearity by mapping and ansatz methods,” Physics of Wave Phenomena, vol. 18, no. 4, pp. 256–261, 2010.
View at: Publisher Site | Google Scholar
N. A. Kudryashov and M. B. Soukharev, “Popular Ansatz methods and Solitary wave solutions of the Kuramoto-Sivashinsky equation,” Regular and Chaotic Dynamics, vol. 14, no. 3, pp. 407–419, 2009.
View at: Publisher Site | Google Scholar
E. Fan and J. Zhang, “Applications of the Jacobi elliptic function method to special-type nonlinear equations,” Physics Letters A, vol. 305, no. 6, pp. 383–392, 2002.
View at: Publisher Site | Google Scholar
S. M. Mirhosseini-Alizamini, H. Rezazadeh, K. Srinivasa, and A. Bekir, “New closed form solutions of the new coupled Konno-Oono equation using the new extended direct algebraic method,” Pramana, vol. 94, pp. 52–2, 2020.
View at: Publisher Site | Google Scholar
A. M. Wazwaz, “The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1467–1475, 2007.
View at: Publisher Site | Google Scholar
T. A. Nofal, “Simple equation method for nonlinear partial differential equations and its applications,” Journal of the Egyptian Mathematical Society, vol. 24, no. 2, pp. 204–209, 2016.
View at: Publisher Site | Google Scholar
S. M. Al-Omari, A. Hussain, M. Usman, and F. D. Zaman, “Invariance analysis and closed-form solutions for the beam equation in timoshenko model,” Malaysian Journal of Mathematical Sciences, vol. 17, no. 4, pp. 587–610, 2023.
View at: Publisher Site | Google Scholar
M. Usman, A. Hussain, F. Zaman, and N. Abbas, “Symmetry analysis and invariant solutions of generalized coupled Zakharov-Kuznetsov equations using optimal system of Lie subalgebra,” International Journal of Mathematics and Computer in Engineering, vol. 0, no. 0, pp. 53–70, 2024.
View at: Publisher Site | Google Scholar
W. H. Ford and T. W. Ting, “Stability and convergence of difference approximations to pseudo-parabolic partial differential equations,” Mathematics of Computation, vol. 27, no. 124, pp. 737–743, 1973.
View at: Publisher Site | Google Scholar
S. L. Sobolev, “On a new problem of mathematical physics. Selected Works of SL Sobolev: volume I,” Mathematical Physics, Computational Mathematics, and Cubature Formulas, pp. 279–332, 2006.
View at: Google Scholar
T. Wu Ting, “Parabolic and pseudo-parabolic partial differential equations,” Journal of the Mathematical Society of Japan, vol. 21, no. 3, pp. 440–453, 1969.
View at: Publisher Site | Google Scholar
G. I. Barenblatt, I. P. Zheltov, and I. N. Kochina, “Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata],” Journal of Applied Mathematics and Mechanics, vol. 24, no. 5, pp. 1286–1303, 1960.
View at: Publisher Site | Google Scholar
R. E. Showalter and T. W. Ting, “Pseudo-parabolic partial differential equations,” SIAM Journal on Mathematical Analysis, vol. 1, no. 1, pp. 1–26, 1970.
View at: Publisher Site | Google Scholar
O. A. Ilhan, A. Esen, H. Bulut, and H. M. Baskonus, “Singular solitons in the pseudo-parabolic model arising in nonlinear surface waves,” Results in Physics, vol. 12, pp. 1712–1715, 2019.
View at: Publisher Site | Google Scholar
R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993.
View at: Publisher Site | Google Scholar
A. M. Wazwaz, “New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa-Holm equations,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 130–141, 2007.
View at: Publisher Site | Google Scholar
A. M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Springer Science \&Business Media, Berlin, Germany, 2010.
R. S. Johnson, “Camassa–Holm, Korteweg–de Vries and related models for water waves,” Journal of Fluid Mechanics, vol. 455, pp. 63–82, 2002.
View at: Publisher Site | Google Scholar
Ö. F. Gözükızıl and Ş Akçağıl, “The tanh-coth method for some nonlinear pseudo-parabolic equations with exact solutions,” Advances in Difference Equations, vol. 2013, no. 1, pp. 143–148, 2013.
View at: Publisher Site | Google Scholar
A. Hussain, M. Usman, and F. Zaman, “Lie group analysis, solitons, self-adjointness and conservation laws of the nonlinear elastic structural element equation,” Journal of Taibah University for Science, vol. 18, no. 1, Article ID 2294554, 2024.
View at: Publisher Site | Google Scholar
A. P. Oskolkov, “Nonlocal problems for one class of nonlinear operator equations that arise in the theory of Sobolev type equations,” Journal of Soviet mathematics, vol. 64, no. 1, pp. 724–735, 1993.
View at: Publisher Site | Google Scholar
D. H. Peregrine, “Calculations of the development of an undular bore,” Journal of Fluid Mechanics, vol. 25, no. 2, pp. 321–330, 1966.
View at: Publisher Site | Google Scholar
T. B. Benjamin, J. L. Bona, and J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems. Philosophical Transactions of the Royal Society of London. Series A,” Mathematical and Physical Sciences, vol. 272, no. 1220, pp. 47–78, 1972.
View at: Google Scholar
A. P. Oskolkov, “On stability theory for solutions of semilinear dissipative equations of the Sobolev type,” Journal of Mathematical Sciences, vol. 77, no. 3, pp. 3225–3231, 1995.
View at: Publisher Site | Google Scholar
K. Khan and M. A. Akbar, “Application of the exp-expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation,” World Applied Sciences Journal, vol. 24, no. 10, pp. 1373–1377, 2013.
View at: Google Scholar
S. R. Islam, K. Khan, and M. A. Akbar, “Study of exp(-Φ(ξ))-expansion method for solving nonlinear partial differential equations,” British Journal of Mathematics and Computer Science, vol. 5, no. 3, pp. 397–407, 2015.
View at: Publisher Site | Google Scholar
M. M. Khater, “Exact traveling wave solutions for the generalized Hirota-Satsuma couple KdV system using the Exp(-ϕξ)-expansion method,” Cogent Mathematics, vol. 3, no. 1, Article ID 1172397, 2016.
View at: Publisher Site | Google Scholar
S. Akcagil, T. Aydemir, and O. F. Gozukizil, “Exact travelling wave solutions of nonlinear pseudo-parabolic equations by using the G'/G-Expansion Method,” New Trends in Mathematical Science, vol. 4, no. 4, pp. 51–66, 2016.
View at: Publisher Site | Google Scholar

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Copyright © 2024 Akhtar Hussain 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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