Advances in Mathematical Physics

Advances in Mathematical Physics / 2020 / Article
Special Issue

Nonlinear Waves and Differential Equations in Applied Mathematics and Physics

View this Special Issue

Research Article | Open Access

Volume 2020 |Article ID 5809289 | https://doi.org/10.1155/2020/5809289

Mostafa M. A. Khater, Yu-Ming Chu, Raghda A. M. Attia, Mustafa Inc, Dianchen Lu, "On the Analytical and Numerical Solutions in the Quantum Magnetoplasmas: The Atangana Conformable Derivative ()-ZK Equation with Power-Law Nonlinearity", Advances in Mathematical Physics, vol. 2020, Article ID 5809289, 10 pages, 2020. https://doi.org/10.1155/2020/5809289

On the Analytical and Numerical Solutions in the Quantum Magnetoplasmas: The Atangana Conformable Derivative ()-ZK Equation with Power-Law Nonlinearity

Guest Editor: Xiao-Ling Gai
Received10 Jun 2020
Accepted31 Jul 2020
Published23 Sep 2020

Abstract

In this research paper, our work is connected with one of the most popular models in quantum magnetoplasma applications. The computational wave and numerical solutions of the Atangana conformable derivative ()-Zakharov-Kuznetsov (ZK) equation with power-law nonlinearity are investigated via the modified Khater method and septic-B-spline scheme. This model is formulated and derived by employing the well-known reductive perturbation method. Applying the modified Khater (mK) method, septic B-spline scheme to the ()-ZK equation with power-law nonlinearity after harnessing suitable wave transformation gives plentiful unprecedented ion-solitary wave solutions. Stability property is checked for our results to show their applicability for applying in the model’s applications. The result solutions are constructed along with their 2D, 3D, and contour graphical configurations for clarity and exactitude.

1. Introduction

In the existence of a magnetized e-p-i plasma [1], the ZK equation is one of the widely common methods to characterize the ion-acoustic solitary waves. The magnetized load-varying dusty plasma is the best location to look for alternate placed dust ion acoustic waves of nonthermal electrons with a vortex-like spread of velocity [2]. In a comprehensive computational analysis, the ZK method was used to spread the dust-acoustic waves in a magnetized dusty plasma [3] and to excite the electrostatic ion-acoustic lone wave in two dimensions of negative ion magnetoplasmas of superthermal electrons [4]. This plasma comprises of nonthermal ions and negatively charged mobile dust crystals, and q-distributed temperature electrons of distinct nonextensivity power [5]. The ZK equation’s mathematical formula found by the well-known reductive disruption process [6] is given bywhere , , , , , , and . Additionally, is Planck’s constant; is the lower-hybrid resonance frequency; , are the ion (electron) gyrofrequency; is the ion mass; and is the speed of light in vacuum.

Solving this kind of models has attracted many researchers in various areas, chemical physics [7], geochemistry [8], plasma physics [8], fluid mechanics [9], optical fiber [10], solid-state physics [11], and so on [1215]. Consequently, constructing the exact solutions of these mathematical models is an indispensable tool for detecting novel properties of them that can be used in their various applications. However, finding the exact solutions of them are not easy to process but is also considered a hard and complex process where there is no unified computational or numerical technique that is able to be applied to all nonlinear evolution (NLE) equations. Almost all computational and numerical techniques depend on an auxiliary equation that is considered a pivot tool in these techniques where all obtained solutions via these schemes are special cases of its general solutions [1624].

For the fractional models, many analytical and numerical methods with various fractional operators have been derived such as the exponential expansion method, Khater method, Kudryashov method, simplest equation method, -expansion method, Riccati expansion method, first integral method, tanh method, and the functional variable method [2534].

This paper studies the analytical and numerical solutions of the Atangana conformable derivative ()-ZK equation with power-law nonlinearity that is given by [3538].where , respectively, represent the nonlinearity and dispersion real valued constants. Also, is the evolution term while represents the power law nonlinearity parameter. Using the following wave transformation [39, 40] on Equation (1) where is an arbitrary constant yields

Integrating Equation (3) once with zero constant of the integration leads to

Through the balancing principle, the terms and force that . Thus, we employ another transformation on Equation (1) gives

Balancing between the terms of Equation (5) leads to

The outline of this research paper is given as follows. Section 2 employs the mK method and septic B-spline scheme to get the abundant explicit wave and numerical solutions of the Atangana conformable derivative ()-ZK equation with power-law nonlinearity. Section 3 investigates the stability of the results solutions. Section 4 shows and discusses the obtained results in our research paper. Section 5 gives the graphical demonstration of some of our solutions. Section 6 explains the conclusion of our study.

2. Implementation

In this section, we employ three recent analytical schemes to find the explicit wave solutions of the Atangana conformable derivative ()-ZK equation with power-law nonlinearity.

2.1. Ion-Acoustic Solitary Waves Solutions

This section gives a transitory elucidation of the mK method. We now explore a nontrivial solution for Equation (5) in the formwhere are arbitrary constants while is a function that satisfies the next ODEwhere are arbitrary constants. Exchanging the values of with Equation (6) along (7) and aggregation of all terms with the same power of then equating the gathering terms with zero lead to a system of equations. Solving this system yields

Family I

Family II

Family III

Family IV

Family VI

Thus, using the above families leads to the new exact solitary wave solutions to the Atangana conformable derivative ()-ZK equation with power-law nonlinearity in the next formulas.

For , we get

For , we get

For , we get

For , we get

For , we get

For , we get

For , we get

For , we get

For , we get

For , we get

where .

2.2. Numerical Solutions

Here, we use three different analytical solutions Equations (16), (19) and (20) to evaluate the numerical solutions of the Atangana conformable derivative ()-ZK equation with power-law nonlinearity. Employing the septic spline technique to Equation (5) with the following conditions gives its numerical solutions in the next formwhere follow the next conditions, respectively:

For , we get

Substituting Equation (32) into Equation (5) gives of equations. Resolving this system leads to the following values of exact, numerical, and absolute values or error.

3. Stability Characteristics

In this section, the stability property has been tested of the obtained results based on the Hamiltonian system characteristics. This system imposes a single condition to ensure the stability of the solution. This condition is given bywhere where is an arbitrary constants, is the frequency, and is an arbitrary constant.

Applying the stability check of Equation (20) with the following values of the parameters , leads to

Consequently, this solution is not stable and applying the same steps to other obtained solutions investigates their stability property.

4. Result and Discussion

Here, we discuss our obtained solutions of the Atangana conformable derivative ()-ZK equation with power-law nonlinearity that have been obtained through one of the most recent computational schemes in nonlinear evolution equation field (the mK method) via two main axes which are a comparison between our obtained computational solutions and other previous obtained solutions, while the second axis of this discussion is studying our exact and numerical solutions.(i)Computational solutions(1)Applying the modified Khater method to the Atangana conformable derivative ()-ZK equation with power-law nonlinearity has obtained sixty distinct traveling wave solutions(2)The difference between our obtained solutions and that have been obtained in [41] by Aminikhah et al. who had used the functional variable method; however, they have just found three solutions and accurate in their and our solutions, we can figure out the complete difference between these solutions that thing makes our solutions are novel(ii)Numerical solutions(1)Applying the septic B-spline scheme to the Atangana conformable derivative ()-ZK equation with power-law nonlinearity by using three of our obtained solutions in evaluating the initial and boundary conditions that give the ability of employing the septic B-spline scheme to the fractional model

5. Figure and Table Interpretation

This section illustrates our explained Figures 13 and Tables 13 with the abovementioned values of the parameters.(i)Figure 1 and Table 1 show the value of the exact and numerical solutions and absolute error of Equation (5) with Equation (16) in three distinct types of sketches to explain the convergence between the two types of solutions(ii)Figure 2 and Table 2 show the value of exact and numerical solutions and absolute error of Equation (5) with Equation (19) in three distinct types of sketches to illustrate the closer between the two types of solutions(iii)Figure 3 and Table 3 explain the value of exact, numerical solutions and absolute error of Equation (5) with Equation (20) in three distinct types of sketches to show the matching between the two types of solutions


Value of ExactNumericalAbsolute error

00. -0.408248 I0. -0.0000110311 I0.408237
0.00010. -0.408289 I0. -8.37587 I0.408281
0.00020. -0.40833 I0. -5.50221I0.408324
0.00030. -0.408371 I0. -2.53611 I0.408368
0.00040. -0.408412 I0. -1.13958 I0.408411
0.00050. -0.408453 I0. +1.58999 I0.408453
0.00060. -0.408493 I0. -1.01297 I0.408492
0.00070. -0.408534 I0. -2.41857 I0.408532
0.00080. -0.408575 I0. -5.26488 I0.40857
0.00090. -0.408616 I0. -7.96518 I0.408608
0.0010. -0.408657 I0. -0.0000103138 I0.408647


Value of ExactNumericalAbsolute error

00. -0.2 I0. -0.0000794823 I0.199921
0.00010. -0.19995 I0. -0.0000595992 I0.19989
0.00020. -0.1999 I0. -0.0000399605 I0.19986
0.00030. -0.19985 I0. -0.0000214 I0.199829
0.00040. -0.1998 I0. -8.76169 I0.199792
0.00050. -0.19975 I0. +1.47924 I0.199752
0.00060. -0.199701 I0. -7.5007 I0.199693
0.00070. -0.199651 I0. -0.0000199929 I0.199631
0.00080. -0.199601 I0. -0.0000376969 I0.199564
0.00090. -0.199552 I0. -0.000056177 I0.199495
0.0010. -0.199502 I0. -0.0000736582 I0.199428


Value of ExactNumericalAbsolute error

003.469453.46945
0.00010.00270.0007792360.00192076
0.00020.0054-8.67362-180.0054
0.00030.00810.004895440.00320455
0.00040.010800.0108
0.00050.01356.50521-190.0135
0.00060.016200.0162
0.00070.0189-4.33681-190.0189
0.00080.0216-8.67362-190.0216
0.00090.02429990.01415650.0101435
0.0010.02699990.02699993.46945

6. Conclusion

This paper has succeeded in the implementation of the mK method and septic B-spline scheme to the Atangana conformable derivative ()-ZK equation with power-law nonlinearity. Sixty distinct novel computational solutions have been obtained. Three of these solutions have been used to evaluate the initial and boundary conditions that have allowed the application of the numerical scheme. Calculating the absolute value of error between the exact and numerical is the aim of our study. Moreover, the stability of our obtained solutions has been illustrated based on the Hamiltonian system characteristics. The effectiveness and power of our two used schemes have been verified, and all obtained solutions have been also verified by putting them back in the original equation via Mathematica 12 software.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgments

The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, and 11601485).

References

  1. A. Lotekar, A. Kakad, and B. Kakad, “Generation of ion acoustic solitary waves through wave breaking in superthermal plasmas,” Physics of Plasmas, vol. 24, no. 10, 2017. View at: Publisher Site | Google Scholar
  2. K. Devi, A. Nag, J. Paul, and P. K. Karmakar, “Dynamics of sheath evolution in magnetized chargeuctuating dusty plasmas,” Chinese Journal of Physics, vol. 65, pp. 405–411, 2020. View at: Publisher Site | Google Scholar
  3. P. Ankiewicz, “Perceptions and attitudes of pupils towards technology: In search of a rigorous theoretical framework,” International Journal of Technology and Design Education, vol. 29, no. 1, pp. 37–56, 2019. View at: Publisher Site | Google Scholar
  4. M.-J. Lee, N. Ashikawa, and Y.-D. Jung, “Characteristics of ion-cyclotron surface waves in semi-bounded (r, q) distribution dusty plasmas,” Physics of Plasmas, vol. 25, article 062110, no. 6, 2018. View at: Publisher Site | Google Scholar
  5. M. M. Hossen, M. S. Alam, S. Sultana, and A. Mamun, “Low frequency nonlinear waves in electron depleted magnetized nonthermal plasmas,” The European Physical Journal D, vol. 70, no. 12, 2016. View at: Publisher Site | Google Scholar
  6. F. S. Khodadad, F. Nazari, M. Eslami, and H. Rezazadeh, “Soliton solutions of the conformable fractional Zakharov-Kuznetsov equation with dual-power law nonlinearity,” Optical and Quantum Electronics, vol. 49, no. 11, 2017. View at: Publisher Site | Google Scholar
  7. M. M. Khater, R. A. Attia, A.-H. Abdel-Aty, W. Alharbi, and D. Lu, “Abundant analytical and numerical solutions of the fractional microbiological densities model in bacteria cell as a result of diffusion mechanisms,” Solitons & Fractals, vol. 136, 2020. View at: Publisher Site | Google Scholar
  8. Z. Korpinar, M. Inc, and M. Bayram, “Theory and application for the system of fractional Burger equations with Mittag leffler kernel,” Applied Mathematics and Computation, vol. 367, 2020. View at: Publisher Site | Google Scholar
  9. C. Park, M. M. Khater, A.-H. Abdel-Aty et al., “Dynamical analysis of the nonlinear complex fractional emerging telecommunication model with higher–order dispersive cubic–quintic,” Alexandria Engineering Journal, vol. 59, no. 3, pp. 1425–1433, 2020. View at: Publisher Site | Google Scholar
  10. M. A. Abdelrahman, S. I. Ammar, K. M. Abualnaja, and M. Inc, “New solutions for the unstable nonlinear Schrödinger equation arising in natural science,” Aims Mathematics, vol. 5, no. 3, pp. 1893–1912, 2020. View at: Google Scholar
  11. M. M. Khater, B. Ghanbari, K. S. Nisar, and D. Kumar, “Novel exact solutions of the fractional Bogoyavlensky–Konopelchenko equation involving the Atangana-Baleanu-Riemann derivative,” Alexandria Engineering Journal, 2020, In press. View at: Publisher Site | Google Scholar
  12. Z.-y. Zhang, Z.-h. Liu, X.-j. Miao, and Y.-z. Chen, “New exact solutions to the perturbed nonlinear Schrödinger’s equation with kerr law nonlinearity,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 3064–3072, 2010. View at: Publisher Site | Google Scholar
  13. Z.-y. Zhang, Y.-x. Li, Z.-h. Liu, and X.-j. Miao, “New exact solutions to the perturbed nonlinear Schrdinger's equation with kerr law nonlinearity via modi_ed trigonometric function series method,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 8, pp. 3097–3106, 2011. View at: Publisher Site | Google Scholar
  14. Z.-y. Zhang, Z.-h. Liu, X.-j. Miao, and Y.-z. Chen, “Qualitative analysis and traveling wave solutions for the perturbed nonlinear Schrdinger's equation with kerr law nonlinearity,” Physics Letters A, vol. 375, no. 10, pp. 1275–1280, 2011. View at: Publisher Site | Google Scholar
  15. X.-j. Miao and Z.-y. Zhang, “The modified -expansion method and traveling wave solutions of nonlinear the perturbed nonlinear Schrdinger's equation with kerr law nonlinearity,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4259–4267, 2011. View at: Publisher Site | Google Scholar
  16. A.-H. Abdel-Aty, M. Khater, R. A. Attia, and H. Eleuch, “Exact traveling and nano-solitons wave solitons of the ionic waves propagating along microtubules in living cells,” Mathematics, vol. 8, no. 5, 2020. View at: Publisher Site | Google Scholar
  17. J. Li, R. A. Attia, M. M. Khater, and D. Lu, “The new structure of analytical and semi-analytical solutions of the longitudinal plasma wave equation in a magneto-electro-elastic circular rod,” Modern Physics Letters B, vol. 34, article 2050123, no. 12, 2020. View at: Publisher Site | Google Scholar
  18. M. M. Khater, R. A. Attia, S. S. Alodhaibi, and D. Lu, “Novel soliton waves of two uid nonlinear evolutions models in the view of computational scheme,” International Journal of Modern Physics B, vol. 34, article 2050096, no. 10, 2020. View at: Publisher Site | Google Scholar
  19. H. Rezazadeh, D. Kumar, A. Neirameh, M. Eslami, and M. Mirzazadeh, “Applications of three methods for obtaining optical soliton solutions for the Lakshmanan–Porsezian–Daniel model with Kerr law nonlinearity,” Pramana, vol. 94, no. 1, 2020. View at: Publisher Site | Google Scholar
  20. S. M. Mirhosseini-Alizamini, H. Rezazadeh, M. Eslami, M. Mirzazadeh, and A. Korkmaz, “New extended direct algebraic method for the Tzitzica type evolution equations arising in nonlinear optics,” Computational Methods for Differential Equations, vol. 8, no. 1, pp. 28–53, 2020. View at: Publisher Site | Google Scholar
  21. Z.-Y. Zhang, X.-Y. Gan, and D.-M. Yu, “Bifurcation Behaviour of the Travelling Wave Solutions of the Perturbed Nonlinear Schrödinger Equation with Kerr Law Nonlinearity,” Zeitschrift für Naturforschung A, vol. 66, no. 12, pp. 721–727, 2011. View at: Publisher Site | Google Scholar
  22. Z. Zhang, J. Huang, J. Zhong et al., “The extended (G/G)-expansion method and travelling wave solutions for the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity,” Pramana, vol. 82, no. 6, pp. 1011–1029, 2014. View at: Publisher Site | Google Scholar
  23. Z. Zai-Yun, G. Xiang-Yang, Y. De-Min, Z. Ying-Hui, and L. Xin-Ping, “A Note on Exact Traveling Wave Solutions of the Perturbed Nonlinear Schrödinger’s Equation with Kerr Law Nonlinearity,” Communications in Theoretical Physics, vol. 57, no. 5, p. 764, 2012. View at: Publisher Site | Google Scholar
  24. A. Korkmaz and K. Hosseini, “Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods,” Optical and Quantum Electronics, vol. 49, no. 8, 2017. View at: Publisher Site | Google Scholar
  25. 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, no. 1, 2020. View at: Publisher Site | Google Scholar
  26. C. Yue, A. Elmoasry, M. Khater et al., “On complex wave structures related to the nonlinear long-short wave interaction system: analytical and numerical techniques,” AIP Advances, vol. 10, no. 4, 2020. View at: Publisher Site | Google Scholar
  27. M. M. Khater, R. A. Attia, and D. Lu, “Computational and numerical simulations for the nonlinear fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation,” Physica Scripta, vol. 95, no. 5, 2020. View at: Publisher Site | Google Scholar
  28. N. A. Kudryashov, “Periodic and solitary waves of the Biswas–Arshed equation,” Optik, vol. 200, 2020. View at: Publisher Site | Google Scholar
  29. M. Torvattanabun, P. Juntakud, A. Saiyun, and N. Khansai, “The new exact solutions of the new coupled Konno-Oono equation by using extended simplest equation method,” Applied Mathematical Sciences, vol. 12, no. 6, pp. 293–301, 2018. View at: Publisher Site | Google Scholar
  30. M. Kaplan, A. Bekir, and A. Akbulut, “A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics,” Nonlinear Dynamics, vol. 85, no. 4, pp. 2843–2850, 2016. View at: Publisher Site | Google Scholar
  31. M. M. Khater, D. Lu, and E. H. Zahran, “Solitary Wave Solutions of the Benjamin-Bona- Mahoney-Burgers Equation with Dual Power-Law Nonlinearity,” Applied Mathematics & Information Sciences, vol. 11, no. 5, pp. 1–5, 2017. View at: Publisher Site | Google Scholar
  32. K. Hosseini, E. Y. Bejarbaneh, A. Bekir, and M. Kaplan, “New exact solutions of some nonlinear evolution equations of pseudoparabolic type,” Optical and Quantum Electronics, vol. 49, no. 7, 2017. View at: Publisher Site | Google Scholar
  33. H. Gündoğdu and Ö. F. Gözükızıl, “Applications of the decomposition methods to some nonlinear partial differential equations,” New Trends in Mathematical Sciences, vol. 6, no. 3, pp. 57–66, 2018. View at: Publisher Site | Google Scholar
  34. 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
  35. L. A ́valos-Ruiz, J. Go ́mez-Aguilar, A. Atangana, and K. M. Owolabi, “On the dynamics of fractional maps with power-law, exponential decay and Mittag–Leffler memory,” Chaos, Solitons & Fractals, vol. 127, pp. 364–388, 2019. View at: Publisher Site | Google Scholar
  36. M. N. Alam and C. Tunc, “Constructions of the optical solitons and other solitons to the conformable fractional Zakharov–Kuznetsov equation with power law nonlinearity,” Journal of Taibah University for Science, vol. 14, no. 1, pp. 94–100, 2020. View at: Publisher Site | Google Scholar
  37. M. Osman, H. Rezazadeh, and M. Eslami, “Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity,” Nonlinear Engineering, vol. 8, no. 1, pp. 559–567, 2019. View at: Publisher Site | Google Scholar
  38. Q. Jin, T. Xia, and J. Wang, “The Exact Solution of the Space-Time Fractional Modified Kdv-Zakharov-Kuznetsov Equation,” Journal of Applied Mathematics and Physics, vol. 5, no. 4, pp. 844–852, 2017. View at: Publisher Site | Google Scholar
  39. H. Yépez-Martínez and J. F. Gómez-Aguilar, “Fractional sub-equation method for Hirota–Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative,” Waves in Random and Complex Media, vol. 29, no. 4, pp. 678–693, 2019. View at: Publisher Site | Google Scholar
  40. H. Yepez-Martinez and J. Gomez-Aguilar, “Optical solitons solution of resonance nonlinear Schrödinger type equation with Atangana's-conformable derivative using sub-equation method,” Waves in Random and Complex Media, pp. 1–24, 2019, In press. View at: Publisher Site | Google Scholar
  41. H. Aminikhah, B. P. Ziabary, and H. Rezazadeh, “Exact traveling wave solutions of partial differential equations with power law nonlinearity,” Nonlinear Engineering, vol. 4, no. 3, 2015. View at: Publisher Site | Google Scholar

Copyright © 2020 Mostafa M. A. Khater 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.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views301
Downloads285
Citations

Related articles

Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.