Research Article
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Year 2023, , 65 - 75, 01.07.2023
https://doi.org/10.32323/ujma.1287524

Abstract

References

  • [1] B. Karaagac, A. Esen, Y. Ucar, N. M. Yagmurlu, A new outlook for analysis of Noyes-Field model for the nonlinear Belousov-Zhabotinsky reaction using operator splitting method, Computers Math. Appl. 136 (2023), 127-135.
  • [2] S. Kutluay, S. Ozer, N. M. Yagmurlu, A new highly accurate numerical scheme for Benjamin-Bona-Mahony-Burgers equation describing small amplitude long wave propagation, Mediterr. J. Math. 20(3) (2023), 1-24.
  • [3] S. Kutluay, N. M. Yagmurlu, A. S. Karakas, Operator time-splitting techniques combined with quintic B-spline collocation method for the generalized Rosenau–KdV equation, Numerical Meth. Partial Differential Equ., 35 (2019), 2221-2235.
  • [4] S. Kutluay, M. Karta, N. M. Yagmurlu, An Effective Numerical approach based on cubic Hermite B-spline collocation method for solving the 1D Heat conduction equation, New Trends Math. Sci., 10(4) (2022), 20-31.
  • [5] D. J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and a new type of long stationary wave, Philosophical Magazine 39, (1895) 422-443.
  • [6] N. J. Zabusky, A synergetic approach to problem of nonlinear dispersive wave propagation and interaction, in: W. Ames (Ed.). Proc. Symp. Nonlinear Partial Diff. Equ., Academic Press (1967) 223-258.
  • [7] S. B. G. Karakoc, A quartic subdomain finite element method for the modified KdV equation, Statistics, Optimization Information Comp. 6 (2018), 609-618.
  • [8] A. Bashan, A. Esen, Single soliton and double soliton solutions of the quadratic-nonlinear Korteweg-de Vries equation for small and long-times, Numerical Meth. Partial Differential Equ., 37(2) (2021), 1561-1582.
  • [9] S. B. G. Karakoc, Numerical solutions of the modified KdV equation with collocation method Malaya J. Mat., 6(4) (2018), 835-842.
  • [10] S. B. G. Karakoc, A. Saha, D. Sucu, A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability generalized Korteweg-de Vries equation, Chinese J. Phys., 68 (2020), 605 617.
  • [11] A. I. Zemlyanukhin, I. V. Andrianov, A. V. Bochkarev, L. I. Mogilevich, The generalized Schamel equation in nonlinear wave dynamics of cylindrical shells, Nonlinear Dyn. 98 (2019), 185-194.
  • [12] M. G .Kuzyk, Polymer Fiber Optics: Materials, Materials, Physics, and Applications, CRC Press, 2018.
  • [13] K. Shimoda, Introduction to Laser Physics. Springer-Verlag, Heidelberg, 1986.
  • [14] M. W. Coffey, On the integrability of Schamel’s modified Korteweg-de Vries equation, J. Phys. A: Mathematical and General 24(23) (1991).
  • [15] Y. Wu, Z. Liu, New types of nonlinear waves and bifurcation phenomena in Schamel-Korteweg-de Vries equation, Abstr. Appl. Anal., 2013 (2013), 1-18.
  • [16] J. Yang, S. Q. Tand, Exact traveling wave solutions of the Schamel-Korteweg-de Vries equation, J. Math. Sci. Adv. Appl., 31(25) (2015).
  • [17] E. Kengne, A. Lakhssassi, W. M. Liu, Nonlinear Schamel–Korteweg deVries equation for a modified Noguchi nonlinear electric transmission network: Analytical circuit modeling, Chaos, Solitons, Fractals 140 (2020), 110229.
  • [18] H. Schamel, Stationary solitary, snoidal and sinusoidal ion acoustic waves, Plasma Phys. 14(10) (1972), 905-924.
  • [19] K. U. Tariq, H. Rezazadeh, M. Zubair, M. S. Osman, L. Akinyemi, New exact and solitary wave solutions of nonlinear Schamel–KdV equation, Int. J. Appl. Comput. Math., 8(114) (2022), 1-16.
  • [20] Z. Pinar, A. Yildirim, S. T. Mohyud-Din, K. F. O˘guz, S. Djabrailov, A. Biswas, New exact solutions for Schamel-Korteweg-de-Vries equation, Studies in Nonlinear Sci. 3(3) (2012), 102-106.
  • [21] F. Kangalgil, Travelling wave solutions of the Schamel–Korteweg–de Vries and the Schamel equations, J. Egyptian Math. Soc., 24 (2016), 526-531.
  • [22] H. Schamel, A modified Korteweg-de Vries equation for ion-acoustic waves due to resonant electrons, J. Plasma Phys. 9(3) (1973), 377-387.
  • [23] Q. Cai, K. Tan, J. Li, Bifurcations and exact traveling wave solutions for the regularized Schamel equation, Open Math., 19 (2021), 1699-1712.
  • [24] O. Dönmez, D. Da˘ghan, Analytic solutions of the Schamel-KdV equation by using different methods: Application to a dusty space plasma, S¨uleyman Demirel Univ. J. Natural and Appl. Sci. 21(1) (2017), 208-215.
  • [25] I. B. Giresunlu, Y. S. Özkan, E. Yas¸ar, On the exact solutions, lie symmetry analysis, and conservation laws of Schamel–Korteweg–de Vries equation, Math. Methods Appl. Sci. 40(11) (2017), 3927-3936.
  • [26] K. U. Tariq, M. Inc, H. Y. Martinez, M. M. A. Khater, Explicit, periodic and dispersive soliton solutions to the Schamel-KdV equation with constant coefficients, Journal of Ocean Engineering and Science, In press.
  • [27] S. K. Mohanty, A. N. Dev, Recent trends in applied mathematics, Lect. Notes Mech. Engrg., 174 (2021), 109-136.
  • [28] J. Lee, R. Sakthivel, Exact travelling wave solutions of the Schamel-Korteweg- de Vries equation, Rep. Mathematical Phys., 68(2) (2011), 153-161.
  • [29] H. I. Abdel-Gawad, M. Tantawy, Exact solutions of the Shamel-Korteweg-de Vries equation with time dependent coefficients, Inf. Sci. Lett. 3 (2014), 103-109.
  • [30] N. Taghizadeh, M. Akbari, P. Esmaeelnejhad, Parirokh Esmaeelnejhad, application of Bernoulli Sub-ODE method for finding travelling wave solutions of Schrödinger Equation Power Law Nonlinearity, Appl. Appl. Math., 12(1) (2017), 596-603.
  • [31] Q. Feng, Traveling wave solution of (3+1) dimensional potential-YTSF equation by Bernoulli Sub-ODE method, Adv. Mater. Res., 403 (2012), 212-216.
  • [32] B. Zheng, Application of a Generalized Bernoulli Sub-ODE Method For Finding Traveling Solutions Of Some Nonlinear Equations, Wseas Trans. Math., 7(11) (2012), 618-626.
  • [33] B. Zheng, New analytical solutions for two equations by a proposed Sub-ODE method, International Conference on Computer Technology and Science (ICCTS 2012) 47 (2012), 360-364.
  • [34] M. A. Akbar, N. H. Mohd Ali, T. Tanjim, Outset of multiple soliton solutions to the nonlinear Schr¨odinger equation and the coupled Burgers equation, J. Phys. Commun., 3 (2019), 1-17.
  • [35] S. M. R. Islam, S. Khan, S. M. Y. Arafat, M. A. Akbar, Diverse analytical wave solutions of plasma physics and water wave equations, Results in Physics, 40 (2022), 1-8.
  • [36] M. A. Khani, M. A. Akbar, N. HJ. M. Ali, M. Abbas, The new auxiliary method in the solution of the generalized Burgers-Huxley equation, J. Prime Res. Math., 16(2) (2020), 16-26.
  • [37] M. M. A. Khater, A. R. Seadawy, D. Lu, Dispersive optical soliton solutions for higher order nonlinear Sasa-Satsuma equation in mono mode fibers via new auxiliary equation method, Superlattices and Microstructures 113 (2018), 346-358.

Exact Traveling Wave Solutions of the Schamel-KdV Equation with Two Different Methods

Year 2023, , 65 - 75, 01.07.2023
https://doi.org/10.32323/ujma.1287524

Abstract

The Schamel-Korteweg-de Vries (S-KdV) equation including a square root nonlinearity is very important pattern for the research of ion-acoustic waves in plasma and dusty plasma. As known, it is significant to discover the traveling wave solutions of such equations. Therefore, in this paper, some new traveling wave solutions of the S-KdV equation, which arises in plasma physics in the study of ion acoustic solitons when electron trapping is present and also it governs the electrostatic potential for a certain electron distribution in velocity space, are constructed. For this purpose, the Bernoulli Sub-ODE and modified auxiliary equation methods are used. It has been shown that the suggested methods are effective and give different types of function solutions as: hyperbolic, trigonometric, power,
exponential, and rational functions. The applied computational strategies are direct, efficient, concise and can be implemented in more complex phenomena with the assistant of symbolic computations. The results found in the paper are of great interest and may also be used to discover the wave sorts and
specialities in several plasma systems.

References

  • [1] B. Karaagac, A. Esen, Y. Ucar, N. M. Yagmurlu, A new outlook for analysis of Noyes-Field model for the nonlinear Belousov-Zhabotinsky reaction using operator splitting method, Computers Math. Appl. 136 (2023), 127-135.
  • [2] S. Kutluay, S. Ozer, N. M. Yagmurlu, A new highly accurate numerical scheme for Benjamin-Bona-Mahony-Burgers equation describing small amplitude long wave propagation, Mediterr. J. Math. 20(3) (2023), 1-24.
  • [3] S. Kutluay, N. M. Yagmurlu, A. S. Karakas, Operator time-splitting techniques combined with quintic B-spline collocation method for the generalized Rosenau–KdV equation, Numerical Meth. Partial Differential Equ., 35 (2019), 2221-2235.
  • [4] S. Kutluay, M. Karta, N. M. Yagmurlu, An Effective Numerical approach based on cubic Hermite B-spline collocation method for solving the 1D Heat conduction equation, New Trends Math. Sci., 10(4) (2022), 20-31.
  • [5] D. J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and a new type of long stationary wave, Philosophical Magazine 39, (1895) 422-443.
  • [6] N. J. Zabusky, A synergetic approach to problem of nonlinear dispersive wave propagation and interaction, in: W. Ames (Ed.). Proc. Symp. Nonlinear Partial Diff. Equ., Academic Press (1967) 223-258.
  • [7] S. B. G. Karakoc, A quartic subdomain finite element method for the modified KdV equation, Statistics, Optimization Information Comp. 6 (2018), 609-618.
  • [8] A. Bashan, A. Esen, Single soliton and double soliton solutions of the quadratic-nonlinear Korteweg-de Vries equation for small and long-times, Numerical Meth. Partial Differential Equ., 37(2) (2021), 1561-1582.
  • [9] S. B. G. Karakoc, Numerical solutions of the modified KdV equation with collocation method Malaya J. Mat., 6(4) (2018), 835-842.
  • [10] S. B. G. Karakoc, A. Saha, D. Sucu, A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability generalized Korteweg-de Vries equation, Chinese J. Phys., 68 (2020), 605 617.
  • [11] A. I. Zemlyanukhin, I. V. Andrianov, A. V. Bochkarev, L. I. Mogilevich, The generalized Schamel equation in nonlinear wave dynamics of cylindrical shells, Nonlinear Dyn. 98 (2019), 185-194.
  • [12] M. G .Kuzyk, Polymer Fiber Optics: Materials, Materials, Physics, and Applications, CRC Press, 2018.
  • [13] K. Shimoda, Introduction to Laser Physics. Springer-Verlag, Heidelberg, 1986.
  • [14] M. W. Coffey, On the integrability of Schamel’s modified Korteweg-de Vries equation, J. Phys. A: Mathematical and General 24(23) (1991).
  • [15] Y. Wu, Z. Liu, New types of nonlinear waves and bifurcation phenomena in Schamel-Korteweg-de Vries equation, Abstr. Appl. Anal., 2013 (2013), 1-18.
  • [16] J. Yang, S. Q. Tand, Exact traveling wave solutions of the Schamel-Korteweg-de Vries equation, J. Math. Sci. Adv. Appl., 31(25) (2015).
  • [17] E. Kengne, A. Lakhssassi, W. M. Liu, Nonlinear Schamel–Korteweg deVries equation for a modified Noguchi nonlinear electric transmission network: Analytical circuit modeling, Chaos, Solitons, Fractals 140 (2020), 110229.
  • [18] H. Schamel, Stationary solitary, snoidal and sinusoidal ion acoustic waves, Plasma Phys. 14(10) (1972), 905-924.
  • [19] K. U. Tariq, H. Rezazadeh, M. Zubair, M. S. Osman, L. Akinyemi, New exact and solitary wave solutions of nonlinear Schamel–KdV equation, Int. J. Appl. Comput. Math., 8(114) (2022), 1-16.
  • [20] Z. Pinar, A. Yildirim, S. T. Mohyud-Din, K. F. O˘guz, S. Djabrailov, A. Biswas, New exact solutions for Schamel-Korteweg-de-Vries equation, Studies in Nonlinear Sci. 3(3) (2012), 102-106.
  • [21] F. Kangalgil, Travelling wave solutions of the Schamel–Korteweg–de Vries and the Schamel equations, J. Egyptian Math. Soc., 24 (2016), 526-531.
  • [22] H. Schamel, A modified Korteweg-de Vries equation for ion-acoustic waves due to resonant electrons, J. Plasma Phys. 9(3) (1973), 377-387.
  • [23] Q. Cai, K. Tan, J. Li, Bifurcations and exact traveling wave solutions for the regularized Schamel equation, Open Math., 19 (2021), 1699-1712.
  • [24] O. Dönmez, D. Da˘ghan, Analytic solutions of the Schamel-KdV equation by using different methods: Application to a dusty space plasma, S¨uleyman Demirel Univ. J. Natural and Appl. Sci. 21(1) (2017), 208-215.
  • [25] I. B. Giresunlu, Y. S. Özkan, E. Yas¸ar, On the exact solutions, lie symmetry analysis, and conservation laws of Schamel–Korteweg–de Vries equation, Math. Methods Appl. Sci. 40(11) (2017), 3927-3936.
  • [26] K. U. Tariq, M. Inc, H. Y. Martinez, M. M. A. Khater, Explicit, periodic and dispersive soliton solutions to the Schamel-KdV equation with constant coefficients, Journal of Ocean Engineering and Science, In press.
  • [27] S. K. Mohanty, A. N. Dev, Recent trends in applied mathematics, Lect. Notes Mech. Engrg., 174 (2021), 109-136.
  • [28] J. Lee, R. Sakthivel, Exact travelling wave solutions of the Schamel-Korteweg- de Vries equation, Rep. Mathematical Phys., 68(2) (2011), 153-161.
  • [29] H. I. Abdel-Gawad, M. Tantawy, Exact solutions of the Shamel-Korteweg-de Vries equation with time dependent coefficients, Inf. Sci. Lett. 3 (2014), 103-109.
  • [30] N. Taghizadeh, M. Akbari, P. Esmaeelnejhad, Parirokh Esmaeelnejhad, application of Bernoulli Sub-ODE method for finding travelling wave solutions of Schrödinger Equation Power Law Nonlinearity, Appl. Appl. Math., 12(1) (2017), 596-603.
  • [31] Q. Feng, Traveling wave solution of (3+1) dimensional potential-YTSF equation by Bernoulli Sub-ODE method, Adv. Mater. Res., 403 (2012), 212-216.
  • [32] B. Zheng, Application of a Generalized Bernoulli Sub-ODE Method For Finding Traveling Solutions Of Some Nonlinear Equations, Wseas Trans. Math., 7(11) (2012), 618-626.
  • [33] B. Zheng, New analytical solutions for two equations by a proposed Sub-ODE method, International Conference on Computer Technology and Science (ICCTS 2012) 47 (2012), 360-364.
  • [34] M. A. Akbar, N. H. Mohd Ali, T. Tanjim, Outset of multiple soliton solutions to the nonlinear Schr¨odinger equation and the coupled Burgers equation, J. Phys. Commun., 3 (2019), 1-17.
  • [35] S. M. R. Islam, S. Khan, S. M. Y. Arafat, M. A. Akbar, Diverse analytical wave solutions of plasma physics and water wave equations, Results in Physics, 40 (2022), 1-8.
  • [36] M. A. Khani, M. A. Akbar, N. HJ. M. Ali, M. Abbas, The new auxiliary method in the solution of the generalized Burgers-Huxley equation, J. Prime Res. Math., 16(2) (2020), 16-26.
  • [37] M. M. A. Khater, A. R. Seadawy, D. Lu, Dispersive optical soliton solutions for higher order nonlinear Sasa-Satsuma equation in mono mode fibers via new auxiliary equation method, Superlattices and Microstructures 113 (2018), 346-358.
There are 37 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Articles
Authors

Seydi Battal Gazi Karakoç 0000-0002-2348-4170

Khalid K. Ali 0000-0002-7801-2760

Mona Mehanna 0000-0003-4349-1113

Publication Date July 1, 2023
Submission Date April 25, 2023
Acceptance Date June 9, 2023
Published in Issue Year 2023

Cite

APA Karakoç, S. B. G., Ali, K. K., & Mehanna, M. (2023). Exact Traveling Wave Solutions of the Schamel-KdV Equation with Two Different Methods. Universal Journal of Mathematics and Applications, 6(2), 65-75. https://doi.org/10.32323/ujma.1287524
AMA Karakoç SBG, Ali KK, Mehanna M. Exact Traveling Wave Solutions of the Schamel-KdV Equation with Two Different Methods. Univ. J. Math. Appl. July 2023;6(2):65-75. doi:10.32323/ujma.1287524
Chicago Karakoç, Seydi Battal Gazi, Khalid K. Ali, and Mona Mehanna. “Exact Traveling Wave Solutions of the Schamel-KdV Equation With Two Different Methods”. Universal Journal of Mathematics and Applications 6, no. 2 (July 2023): 65-75. https://doi.org/10.32323/ujma.1287524.
EndNote Karakoç SBG, Ali KK, Mehanna M (July 1, 2023) Exact Traveling Wave Solutions of the Schamel-KdV Equation with Two Different Methods. Universal Journal of Mathematics and Applications 6 2 65–75.
IEEE S. B. G. Karakoç, K. K. Ali, and M. Mehanna, “Exact Traveling Wave Solutions of the Schamel-KdV Equation with Two Different Methods”, Univ. J. Math. Appl., vol. 6, no. 2, pp. 65–75, 2023, doi: 10.32323/ujma.1287524.
ISNAD Karakoç, Seydi Battal Gazi et al. “Exact Traveling Wave Solutions of the Schamel-KdV Equation With Two Different Methods”. Universal Journal of Mathematics and Applications 6/2 (July 2023), 65-75. https://doi.org/10.32323/ujma.1287524.
JAMA Karakoç SBG, Ali KK, Mehanna M. Exact Traveling Wave Solutions of the Schamel-KdV Equation with Two Different Methods. Univ. J. Math. Appl. 2023;6:65–75.
MLA Karakoç, Seydi Battal Gazi et al. “Exact Traveling Wave Solutions of the Schamel-KdV Equation With Two Different Methods”. Universal Journal of Mathematics and Applications, vol. 6, no. 2, 2023, pp. 65-75, doi:10.32323/ujma.1287524.
Vancouver Karakoç SBG, Ali KK, Mehanna M. Exact Traveling Wave Solutions of the Schamel-KdV Equation with Two Different Methods. Univ. J. Math. Appl. 2023;6(2):65-7.

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