Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation

Cheng Zhang

doi:doi:10.11648/j.acm.20221103.13

| Peer-Reviewed

Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation

Cheng Zhang

Published in Applied and Computational Mathematics (Volume 11, Issue 3)

Received: 26 April 2022 Accepted: 18 May 2022 Published: 31 May 2022

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The Zakharov-Kuznetsov equation is an important model to describes the nonlinear pulse propagation in plasma physics, which guides the characteristic of weakly nonlinear ion-acoustic waves in plasma composed of cold ions and hot isothermal electrons in a uniform magnetic field. In the current study, we investigate the generalized trigonometric solutions and new travelling wave solutions of the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation through the (G'/G)-expansion method and the Sech-Tanh expansion method. Before applying these, we imply the traveling wave transformation to convert the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation to a nonlinear differential equation (NLODE). By the aid of Mathematics software, the dynamical images such as three-dimensional (3D) graphs, two- dimensional (2D) graphs and contour surfaces of local solutions are plotted by choosing the appropriate parameters. The obtained solutions show the simplicity and efficiency of the two approaches that can be applied for nonlinear equations as well as linear ones. Furthermore, the accuracy of the solutions obtained by the two different methods is verified by the Adomain decomposition method (ADM) and showed in tables respectively. The study of ADM method in this paper indivates it is an effective mathematical tool to calculate the numerical solutions and to verify the accuracy of the solutions.

Published in	Applied and Computational Mathematics (Volume 11, Issue 3)
DOI	10.11648/j.acm.20221103.13
Page(s)	74-80
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

The (3+1)-dimensional Extended Quantum Zakharov–Kuznetsov Equation, The (G'/G)-Expansion Method, The Sech-Tanh Expansion Method, The ADM, The Analytical and Numerical Solutions

References

[1]	E. A. Kuznetsov, V. E. Zakharov (1974), On three dimensional solitons, Sov. Phys.—JETP, 39, 285–286.
[2]	W. M. Moslem, S. Ali, P. K. Shukla, X. Y. Tang, G Rowlands (2007), Solitary, explosive, and periodic solutions of the quantum Zakharov-Kuznetsov equation and its transverse instability, Phys. Plasmas, 14, 082308. https://doi.org/10.1063/1.2757612
[3]	R. Sabry, W. M. Moslem, F. Haas, S. Ali, P. K. Shukla (2008), Explosive, soliton, and shock in a quantum electron-positron-ion magnetoplasma, Phys. Plasmas, 15, 122308. https://doi.org/10.1063/1.3037265
[4]	K. K. Ali, R. Yilmazer, A. Yokus, et al (2020). Analytical solutions for the (3+1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov equation in plasma physics, Physica A, 548, 124327. https://doi.org/10.1016/j.physa.2020.124327
[5]	H. L. Zhen, B. Tian, Y. F. Wang, W. R. Sun, L. C. Liu (2014), Soliton solutions and chaotic motion of the extended Zakharov-Kuznetsov equations in a magnetized two-ion-temperature dustyplasma, Phys. Plasmas, 21, 073709. https://doi.org/10.1063/1.4885380
[6]	S. Kumar, D. Kumar (2019), Solitary wave solutions of (3+1)-dimensional extended Zakharov–Kuznetsov equation by Lie symmetry approach, Comput, Math. Appl, 77, 2096–2113. https://doi.org/10.1016/j.camwa.2018.12.009
[7]	A. R. Seadawy, Lu Dianchen (2016), Ion acoustic solitary wave solutions of three-dimensional nonlinear extended Zakharov–Kuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma, Results Phys, 6, 590–593.
[8]	E. M. E. Zayed, R. M. A. Shohib, A. Al-Nowehy (2019), On solving the (3+1)-dimensional NLEQZK equation and the (3+1)-dimensional NLmZK equation using the extended simplest equation method, Comput. Math. Appl. https://doi.org/10.1016/j.camwa.2019.05.007
[9]	Z. Yan (2009), Periodic solitary and rational wave solutions of the 3D extended quantum Zakharov–Kuznetsov equation in dense quantum plasmas, Phys. Lett. A, 373, 2432–2437. https://doi.org/10.1016/j.physleta.2009.04.018
[10]	Younis, M,. Zafar, A., Ul-Haq, K., Rahman, M. (2013), Travelling wave solutions of fractional order coupled Burgers’ equations by-expansion method, Am. J. Comput. Appl. Math, 3, 81-85.
[11]	Biswas, A., Bhrawy, A. H., Abdelkawy, M. A., Alshaery, A. A., Hilal, E. M. (2014), Symbolic computation of some nonlinear fractional differential equations, Rom. J. Phys, 59, 433-442.
[12]	Khan, K., Akbar, M. A., Koppelaar, H. (2015), Study of coupled nonlinear partial differential equations for finding exact analytical solutions, R. Soc. Open Sci, 2, 140406.
[13]	Khan, H., Shoaib, Baleanu, D., Kumam, P., Al-Zaidy, J. F. (2019), Families of travelling waves solutions for fractional-order extended shallow water wave equations, using an innovative analytical method. IEEE Access, 7, 107523-107532. https://doi.org/10.1098/rsos.140406
[14]	R. Attia, D. Lu, and M. Khater (2019), Chaos and relativistic energy momentum of the nonlinear time fractional Duffing equation, Mathematical and Computational Applications, vol. 24 (2019), 1-10. https://doi.org/10.3390/mca24010010
[15]	H. M. Baskonus, D. A. Koc¸ (2016), and H. Bulut, Dark and new travelling wave solutions to the nonlinear evolution equation, Optik, 127, 8043–8055. https://doi.org/10.1016/j.ijleo.2016.05.1321
[16]	F. Dusunceli (2019), New exponential and complex traveling wave solutions to the Konopelchenko-Dubrovsky model, Advances in Mathematical Physics, 7801247. https://doi.org/10.1155/2019/7801247
[17]	Z. Odiba (2020) t, An optimized decomposition method for nonlinear ordinary and partial differential equations, Physica A: Statistical Mechanics and Its Applications, 541, 123323. https:// doi.org/10.1016/j. physa. 2019.123323
[18]	H. O. Bakodah, A. A. Al Qarni, M. A. Banaja, Q. Zhou, S. P. Moshokoa, and A. Biswas (2017), Bright and dark irring optical solitons with improved Adomian decomposition method, Optik, 130, 1115––1123. https://doi.org/10.1016/j.ijleo.2016.11.123
[19]	G. ADOMIAN (1986), Nonlinear Stochastic Operator Equation, Orlando, FL: Academic Press.
[20]	G. ADOMIAN (1988), Applications of Nonlinear Stochastic Systems Theory to Physics Reidel, Dordrecht: Kluwer Academic Publisher.

Cite This Article

Plain Text BibTeX RIS

APA Style

Cheng Zhang. (2022). Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation. Applied and Computational Mathematics, 11(3), 74-80. https://doi.org/10.11648/j.acm.20221103.13

Copy | Download

ACS Style

Cheng Zhang. Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation. Appl. Comput. Math. 2022, 11(3), 74-80. doi: 10.11648/j.acm.20221103.13

Copy | Download

AMA Style

Cheng Zhang. Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation. Appl Comput Math. 2022;11(3):74-80. doi: 10.11648/j.acm.20221103.13

Copy | Download

@article{10.11648/j.acm.20221103.13,
  author = {Cheng Zhang},
  title = {Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {3},
  pages = {74-80},
  doi = {10.11648/j.acm.20221103.13},
  url = {https://doi.org/10.11648/j.acm.20221103.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221103.13},
  abstract = {The Zakharov-Kuznetsov equation is an important model to describes the nonlinear pulse propagation in plasma physics, which guides the characteristic of weakly nonlinear ion-acoustic waves in plasma composed of cold ions and hot isothermal electrons in a uniform magnetic field. In the current study, we investigate the generalized trigonometric solutions and new travelling wave solutions of the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation through the (G'/G)-expansion method and the Sech-Tanh expansion method. Before applying these, we imply the traveling wave transformation to convert the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation to a nonlinear differential equation (NLODE). By the aid of Mathematics software, the dynamical images such as three-dimensional (3D) graphs, two- dimensional (2D) graphs and contour surfaces of local solutions are plotted by choosing the appropriate parameters. The obtained solutions show the simplicity and efficiency of the two approaches that can be applied for nonlinear equations as well as linear ones. Furthermore, the accuracy of the solutions obtained by the two different methods is verified by the Adomain decomposition method (ADM) and showed in tables respectively. The study of ADM method in this paper indivates it is an effective mathematical tool to calculate the numerical solutions and to verify the accuracy of the solutions.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation
AU  - Cheng Zhang
Y1  - 2022/05/31
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221103.13
DO  - 10.11648/j.acm.20221103.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 74
EP  - 80
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221103.13
AB  - The Zakharov-Kuznetsov equation is an important model to describes the nonlinear pulse propagation in plasma physics, which guides the characteristic of weakly nonlinear ion-acoustic waves in plasma composed of cold ions and hot isothermal electrons in a uniform magnetic field. In the current study, we investigate the generalized trigonometric solutions and new travelling wave solutions of the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation through the (G'/G)-expansion method and the Sech-Tanh expansion method. Before applying these, we imply the traveling wave transformation to convert the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation to a nonlinear differential equation (NLODE). By the aid of Mathematics software, the dynamical images such as three-dimensional (3D) graphs, two- dimensional (2D) graphs and contour surfaces of local solutions are plotted by choosing the appropriate parameters. The obtained solutions show the simplicity and efficiency of the two approaches that can be applied for nonlinear equations as well as linear ones. Furthermore, the accuracy of the solutions obtained by the two different methods is verified by the Adomain decomposition method (ADM) and showed in tables respectively. The study of ADM method in this paper indivates it is an effective mathematical tool to calculate the numerical solutions and to verify the accuracy of the solutions.
VL  - 11
IS  - 3
ER  -

Copy | Download

Author Information

Cheng Zhang

School of Science, Beijing University of Posts and Telecommunications, Beijing, China

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Cheng Zhang. (2022). Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation. Applied and Computational Mathematics, 11(3), 74-80. https://doi.org/10.11648/j.acm.20221103.13

Copy | Download

ACS Style

Cheng Zhang. Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation. Appl. Comput. Math. 2022, 11(3), 74-80. doi: 10.11648/j.acm.20221103.13

Copy | Download

AMA Style

Cheng Zhang. Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation. Appl Comput Math. 2022;11(3):74-80. doi: 10.11648/j.acm.20221103.13

Copy | Download

@article{10.11648/j.acm.20221103.13,
  author = {Cheng Zhang},
  title = {Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {3},
  pages = {74-80},
  doi = {10.11648/j.acm.20221103.13},
  url = {https://doi.org/10.11648/j.acm.20221103.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221103.13},
  abstract = {The Zakharov-Kuznetsov equation is an important model to describes the nonlinear pulse propagation in plasma physics, which guides the characteristic of weakly nonlinear ion-acoustic waves in plasma composed of cold ions and hot isothermal electrons in a uniform magnetic field. In the current study, we investigate the generalized trigonometric solutions and new travelling wave solutions of the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation through the (G'/G)-expansion method and the Sech-Tanh expansion method. Before applying these, we imply the traveling wave transformation to convert the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation to a nonlinear differential equation (NLODE). By the aid of Mathematics software, the dynamical images such as three-dimensional (3D) graphs, two- dimensional (2D) graphs and contour surfaces of local solutions are plotted by choosing the appropriate parameters. The obtained solutions show the simplicity and efficiency of the two approaches that can be applied for nonlinear equations as well as linear ones. Furthermore, the accuracy of the solutions obtained by the two different methods is verified by the Adomain decomposition method (ADM) and showed in tables respectively. The study of ADM method in this paper indivates it is an effective mathematical tool to calculate the numerical solutions and to verify the accuracy of the solutions.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation
AU  - Cheng Zhang
Y1  - 2022/05/31
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221103.13
DO  - 10.11648/j.acm.20221103.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 74
EP  - 80
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221103.13
AB  - The Zakharov-Kuznetsov equation is an important model to describes the nonlinear pulse propagation in plasma physics, which guides the characteristic of weakly nonlinear ion-acoustic waves in plasma composed of cold ions and hot isothermal electrons in a uniform magnetic field. In the current study, we investigate the generalized trigonometric solutions and new travelling wave solutions of the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation through the (G'/G)-expansion method and the Sech-Tanh expansion method. Before applying these, we imply the traveling wave transformation to convert the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation to a nonlinear differential equation (NLODE). By the aid of Mathematics software, the dynamical images such as three-dimensional (3D) graphs, two- dimensional (2D) graphs and contour surfaces of local solutions are plotted by choosing the appropriate parameters. The obtained solutions show the simplicity and efficiency of the two approaches that can be applied for nonlinear equations as well as linear ones. Furthermore, the accuracy of the solutions obtained by the two different methods is verified by the Adomain decomposition method (ADM) and showed in tables respectively. The study of ADM method in this paper indivates it is an effective mathematical tool to calculate the numerical solutions and to verify the accuracy of the solutions.
VL  - 11
IS  - 3
ER  -

Copy | Download