Research Article
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Year 2022, Volume: 71 Issue: 1, 116 - 132, 30.03.2022
https://doi.org/10.31801/cfsuasmas.884025

Abstract

References

  • Ablowitz, M. J., Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (Vol. 149), Cambridge University Press, 1991. https://doi.org/10.1017/cbo9780511623998
  • Hirota, R., Exact n-soliton solutions of the wave equation of long waves in shallowwater and in nonlinear lattices, Journal of Mathematical Physics, 14(7) (1973), 810-814. https://doi.org/10.1063/1.1666400
  • Cariello, F., Tabor, M., Similarity reductions from extended Painlev´e expansions for nonintegrable evolution equations, Physica D: Nonlinear Phenomena, 53(1) (1991), 59-70. https://doi.org/10.1016/0167-2789(91)90164-5
  • Fan, E., Extended tanh-function method and its applications to nonlinear equations, Physics Letters A, 227(4) (2000), 212-218. https://doi.org/10.1016/S0375-9601(00)00725-8
  • Liu, S., Fu, Z., Liu, S., Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Physics Letters A, 289(1) (2001), 69-74. https://doi.org/10.1016/S0375-9601(01)00580-1
  • Wang, M., Solitary wave solutions for variant Boussinesq equations, Physics Letters A, 199(3-4) (1995), 169-172. https://doi.org/10.1016/0375-9601(95)00092-H
  • Wang, M., Exact solutions for a compound KdV-Burgers equation, Physics Letters A, 213(5-6) (1996), 279-287. https://doi.org/10.1016/0375-9601(96)00103-X
  • Wang, M., Zhou, Y., Li, Z., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A, 216(1-5) (1996), 67-75. https://doi.org/10.1016/0375-9601(96)00283-6
  • Kudryashov, N. A., Exact solutions of the generalized Kuramoto-Sivashinsky equation, Physics Letters A, 147(5-6) (1990), 287-291. https://doi.org/10.1016/0375-9601(90)90449X.
  • He, J. H., Wu, X. H., Exp-function method for nonlinear wave equations, Chaos, Solitons Fractals, 30(3) (2006), 700-708. https://doi.org/10.1016/j.chaos.2006.03.020
  • Rogers, C., Shadwick, W. F., B¨acklund Transformations and Their Applications. Academic Press, New York, USA, 1982.
  • Yang, L., Liu, J., Yang, K., Exact solutions of nonlinear PDE, nonlinear transformations and reduction of nonlinear PDE to a quadrature, Physics Letters A, 278(5) (2001), 267-270. https://doi.org/10.1016/S0375-9601(00)00778-7
  • Yan, Z., Zhang, H., New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water, Physics Letters A, 285(5) (2001), 355-362. https://doi.org/10.1016/S0375-9601(01)00376-0
  • Yan, Z., New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Physics Letters A, 292(1) (2001), 100-106. https://doi.org/10.1016/S0375-9601(01)00772-1
  • Wang, M., Li, X., Zhang, J., The (G’/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372(4) (2008), 417-423. https://doi.org/10.1016/j.physleta.2007.07.051
  • Zhang, S., Tong, J. L., Wang, W., A generalized (G’/G)-expansion method for the mKdV equation with variable coefficients, Physics Letters A, 372(13) (2008), 2254-2257. https://doi.org/10.1016/j.physleta.2007.11.026
  • Zhang, J. L., Wang, M. L., Li, X. Z., The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrodinger equation, Physics Letters A, 357(3) (2006), 188-195. https://doi.org/10.1016/j.physleta.2006.03.081
  • Wang, M., Li, X., Zhang, J., Various exact solutions of nonlinear Schrodinger equation with two nonlinear terms, Chaos, Solitons Fractals, 31(3) (2007), 594-601. https://doi.org/10.1016/j.chaos.2005.10.009
  • Li, X., Wang, M., A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms, Physics Letters A, 361(1) (2007), 115-118. https://doi.org/10.1016/j.physleta.2006.09.022
  • Wang, M., Li, X., Zhang, J., Sub-ODE method and solitary wave solutions for higher order nonlinear Schr¨odinger equation, Physics Letters A, 363(1) (2007), 96-101. https://doi.org/10.1016/j.physleta.2006.10.077
  • Islam, M., Akbar, M. A., Azad, A. K., A Rational (G’/G)-expansion method and its application to the modified KdV-Burgers equation and the (2+ l)-dimensional Boussinesq equation, Nonlinear Studies, 22(4) (2015), 635-645.
  • Konno, K., Ichikawa Y. H., A modified Korteweg de Vries equation for ion acoustic waves, Journal of the Physical Society of Japan, 37(6) (1974), 1631-1636. https://doi.org/10.1143/JPSJ.37.1631
  • Narayanamurti, V., Varma, C. M., Nonlinear propagation of heat pulses in solids, Physical Review Letters, 25(16) (1970), 1105. https://doi.org/10.1103/PhysRevLett.25.1105
  • Tappert, F. D., Varma, C. M., Asymptotic theory of self-trapping of heat pulses in solids, Physical Review Letters, 25(16) (1970), 1108. https://doi.org/10.1103/PhysRevLett.25.1108
  • Yomba, E., The extended Fan’s sub-equation method and its application to KdV–MKdV, BKK and variant Boussinesq equations, Physics Letters A, 336(6) (2005), 463-476. https://doi.org/10.1016/j.physleta.2005.01.027
  • Fan, E., Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos, Solitons Fractals, 16(5) (2003), 819-839. https://doi.org/10.1016/S0960-0779(02)00472-1
  • Wadati, M., Wave propagation in nonlinear lattice, II. Journal of the Physical Society of Japan, 38(3) (1975), 681-686. https://doi.org/10.1143/JPSJ.38.673
  • Mohamad, M. N. B., Exact solutions to the combined KdV and MKdV equation, Mathematical Methods in the Applied Sciences, 15(2) (1992), 73-78. https://doi.org/10.1002/mma.1670150202
  • Zayed, E. M. E., Gepreel, K. A., The (G’/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics, Journal of Mathematical Physics, 50(1) (2009), 013502. https://doi.org/10.1063/1.3033750
  • Mei, J. Q., Zhang, H. Q., Jiang, D. M., New exact solutions for a reaction-diffusion equation and a Quasi-Camassa Holm equation, Appl. Math. E-Notes, 4 (2004), 85-91.
  • Wu, Y., Geng, X., Hu, X., Zhu, S., A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations, BKK and variant Boussinesq equations, Physics Letters A, 255(4-6) (1999), 259-264. https://doi.org/10.1016/S0375-9601(99)00163-2
  • Inan, I. E., Duran, S., Ugurlu, Y., $Tan(F(\frac{\xi }{2}))$-expansion method for traveling wave solutions of AKNS and Burgers-like equations, Optik, 138 (2017), 15-20. https://doi.org/10.1016/j.ijleo.2017.02.087
  • Ekici, M., Ayaz, F., Solution of model equation of completely passive natural convection by improved differential transform method, Research on Engineering Structures and Materials, 3(1) (2017), 1-10. http://dx.doi.org/10.17515/resm2015.10me0818
  • Ekici, M., Ünal, M., Application of the Exponential Rational Function Method to Some Fractional Soliton Equations, In Emerging Applications of Differential Equations and Game Theory, (pp. 13-32), IGI Global, 2020.
  • Ünal, M., Ekici, M., The double (G’/G, 1/G)-expansion method and its applications for some nonlinear partial differential equations, Journal of the Institute of Science and Technology, 11(1) (2021), 599-608. https://doi.org/10.21597/jist.767930
  • Islam, M. T., Akter, M. A., Distinct solutions of nonlinear space–time fractional evolution equations appearing in mathematical physics via a new technique, Partial Differential Equations in Applied Mathematics, 3 (2021), 100031. https://doi.org/10.1016/j.padiff.2021.100031
  • Islam, M. T., Akter, M. A., Exact analytic wave solutions to some nonlinear fractional differential equations for the shallow water wave arise in physics and engineering, Journal of Research in Engineering and Applied Sciences, 6(1) (2021), 11-18.
  • Islam, T., Akter, A., Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics, Arab Journal of Mathematical Sciences, 26(1/2) (2020), Doi: 10.1108/AJMS-09.2020-0078
  • Akbar, M. A., Ali, N. H. M., Islam, M. T., Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics, AIMS Mathematics, 4(3) (2019), 397-411. doi: 10.3934/math.2019.3.397
  • Islam, M. T., Akbar, M. A., Azad, M. A. K., Closed-form travelling wave solutions to the nonlinear space-time fractional coupled Burgers’ equation, Arab Journal of Basic and Applied Sciences, 26(1) (2019), 1-11. https://doi.org/10.1080/25765299.2018.1523702

Application of the rational (G' /G)-expansion method for solving some coupled and combined wave equations

Year 2022, Volume: 71 Issue: 1, 116 - 132, 30.03.2022
https://doi.org/10.31801/cfsuasmas.884025

Abstract

In this paper, we explore the travelling wave solutions for some nonlinear partial differential equations by using the recently established rational (G' /G)-expansion method. We apply this method to the combined KdV-mKdV equation, the reaction-diffusion equation and the coupled Hirota-Satsuma KdV equations. The travelling wave solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are also derived from the travelling waves. We have also given some figures for the solutions.

References

  • Ablowitz, M. J., Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (Vol. 149), Cambridge University Press, 1991. https://doi.org/10.1017/cbo9780511623998
  • Hirota, R., Exact n-soliton solutions of the wave equation of long waves in shallowwater and in nonlinear lattices, Journal of Mathematical Physics, 14(7) (1973), 810-814. https://doi.org/10.1063/1.1666400
  • Cariello, F., Tabor, M., Similarity reductions from extended Painlev´e expansions for nonintegrable evolution equations, Physica D: Nonlinear Phenomena, 53(1) (1991), 59-70. https://doi.org/10.1016/0167-2789(91)90164-5
  • Fan, E., Extended tanh-function method and its applications to nonlinear equations, Physics Letters A, 227(4) (2000), 212-218. https://doi.org/10.1016/S0375-9601(00)00725-8
  • Liu, S., Fu, Z., Liu, S., Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Physics Letters A, 289(1) (2001), 69-74. https://doi.org/10.1016/S0375-9601(01)00580-1
  • Wang, M., Solitary wave solutions for variant Boussinesq equations, Physics Letters A, 199(3-4) (1995), 169-172. https://doi.org/10.1016/0375-9601(95)00092-H
  • Wang, M., Exact solutions for a compound KdV-Burgers equation, Physics Letters A, 213(5-6) (1996), 279-287. https://doi.org/10.1016/0375-9601(96)00103-X
  • Wang, M., Zhou, Y., Li, Z., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A, 216(1-5) (1996), 67-75. https://doi.org/10.1016/0375-9601(96)00283-6
  • Kudryashov, N. A., Exact solutions of the generalized Kuramoto-Sivashinsky equation, Physics Letters A, 147(5-6) (1990), 287-291. https://doi.org/10.1016/0375-9601(90)90449X.
  • He, J. H., Wu, X. H., Exp-function method for nonlinear wave equations, Chaos, Solitons Fractals, 30(3) (2006), 700-708. https://doi.org/10.1016/j.chaos.2006.03.020
  • Rogers, C., Shadwick, W. F., B¨acklund Transformations and Their Applications. Academic Press, New York, USA, 1982.
  • Yang, L., Liu, J., Yang, K., Exact solutions of nonlinear PDE, nonlinear transformations and reduction of nonlinear PDE to a quadrature, Physics Letters A, 278(5) (2001), 267-270. https://doi.org/10.1016/S0375-9601(00)00778-7
  • Yan, Z., Zhang, H., New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water, Physics Letters A, 285(5) (2001), 355-362. https://doi.org/10.1016/S0375-9601(01)00376-0
  • Yan, Z., New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Physics Letters A, 292(1) (2001), 100-106. https://doi.org/10.1016/S0375-9601(01)00772-1
  • Wang, M., Li, X., Zhang, J., The (G’/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372(4) (2008), 417-423. https://doi.org/10.1016/j.physleta.2007.07.051
  • Zhang, S., Tong, J. L., Wang, W., A generalized (G’/G)-expansion method for the mKdV equation with variable coefficients, Physics Letters A, 372(13) (2008), 2254-2257. https://doi.org/10.1016/j.physleta.2007.11.026
  • Zhang, J. L., Wang, M. L., Li, X. Z., The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrodinger equation, Physics Letters A, 357(3) (2006), 188-195. https://doi.org/10.1016/j.physleta.2006.03.081
  • Wang, M., Li, X., Zhang, J., Various exact solutions of nonlinear Schrodinger equation with two nonlinear terms, Chaos, Solitons Fractals, 31(3) (2007), 594-601. https://doi.org/10.1016/j.chaos.2005.10.009
  • Li, X., Wang, M., A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms, Physics Letters A, 361(1) (2007), 115-118. https://doi.org/10.1016/j.physleta.2006.09.022
  • Wang, M., Li, X., Zhang, J., Sub-ODE method and solitary wave solutions for higher order nonlinear Schr¨odinger equation, Physics Letters A, 363(1) (2007), 96-101. https://doi.org/10.1016/j.physleta.2006.10.077
  • Islam, M., Akbar, M. A., Azad, A. K., A Rational (G’/G)-expansion method and its application to the modified KdV-Burgers equation and the (2+ l)-dimensional Boussinesq equation, Nonlinear Studies, 22(4) (2015), 635-645.
  • Konno, K., Ichikawa Y. H., A modified Korteweg de Vries equation for ion acoustic waves, Journal of the Physical Society of Japan, 37(6) (1974), 1631-1636. https://doi.org/10.1143/JPSJ.37.1631
  • Narayanamurti, V., Varma, C. M., Nonlinear propagation of heat pulses in solids, Physical Review Letters, 25(16) (1970), 1105. https://doi.org/10.1103/PhysRevLett.25.1105
  • Tappert, F. D., Varma, C. M., Asymptotic theory of self-trapping of heat pulses in solids, Physical Review Letters, 25(16) (1970), 1108. https://doi.org/10.1103/PhysRevLett.25.1108
  • Yomba, E., The extended Fan’s sub-equation method and its application to KdV–MKdV, BKK and variant Boussinesq equations, Physics Letters A, 336(6) (2005), 463-476. https://doi.org/10.1016/j.physleta.2005.01.027
  • Fan, E., Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos, Solitons Fractals, 16(5) (2003), 819-839. https://doi.org/10.1016/S0960-0779(02)00472-1
  • Wadati, M., Wave propagation in nonlinear lattice, II. Journal of the Physical Society of Japan, 38(3) (1975), 681-686. https://doi.org/10.1143/JPSJ.38.673
  • Mohamad, M. N. B., Exact solutions to the combined KdV and MKdV equation, Mathematical Methods in the Applied Sciences, 15(2) (1992), 73-78. https://doi.org/10.1002/mma.1670150202
  • Zayed, E. M. E., Gepreel, K. A., The (G’/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics, Journal of Mathematical Physics, 50(1) (2009), 013502. https://doi.org/10.1063/1.3033750
  • Mei, J. Q., Zhang, H. Q., Jiang, D. M., New exact solutions for a reaction-diffusion equation and a Quasi-Camassa Holm equation, Appl. Math. E-Notes, 4 (2004), 85-91.
  • Wu, Y., Geng, X., Hu, X., Zhu, S., A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations, BKK and variant Boussinesq equations, Physics Letters A, 255(4-6) (1999), 259-264. https://doi.org/10.1016/S0375-9601(99)00163-2
  • Inan, I. E., Duran, S., Ugurlu, Y., $Tan(F(\frac{\xi }{2}))$-expansion method for traveling wave solutions of AKNS and Burgers-like equations, Optik, 138 (2017), 15-20. https://doi.org/10.1016/j.ijleo.2017.02.087
  • Ekici, M., Ayaz, F., Solution of model equation of completely passive natural convection by improved differential transform method, Research on Engineering Structures and Materials, 3(1) (2017), 1-10. http://dx.doi.org/10.17515/resm2015.10me0818
  • Ekici, M., Ünal, M., Application of the Exponential Rational Function Method to Some Fractional Soliton Equations, In Emerging Applications of Differential Equations and Game Theory, (pp. 13-32), IGI Global, 2020.
  • Ünal, M., Ekici, M., The double (G’/G, 1/G)-expansion method and its applications for some nonlinear partial differential equations, Journal of the Institute of Science and Technology, 11(1) (2021), 599-608. https://doi.org/10.21597/jist.767930
  • Islam, M. T., Akter, M. A., Distinct solutions of nonlinear space–time fractional evolution equations appearing in mathematical physics via a new technique, Partial Differential Equations in Applied Mathematics, 3 (2021), 100031. https://doi.org/10.1016/j.padiff.2021.100031
  • Islam, M. T., Akter, M. A., Exact analytic wave solutions to some nonlinear fractional differential equations for the shallow water wave arise in physics and engineering, Journal of Research in Engineering and Applied Sciences, 6(1) (2021), 11-18.
  • Islam, T., Akter, A., Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics, Arab Journal of Mathematical Sciences, 26(1/2) (2020), Doi: 10.1108/AJMS-09.2020-0078
  • Akbar, M. A., Ali, N. H. M., Islam, M. T., Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics, AIMS Mathematics, 4(3) (2019), 397-411. doi: 10.3934/math.2019.3.397
  • Islam, M. T., Akbar, M. A., Azad, M. A. K., Closed-form travelling wave solutions to the nonlinear space-time fractional coupled Burgers’ equation, Arab Journal of Basic and Applied Sciences, 26(1) (2019), 1-11. https://doi.org/10.1080/25765299.2018.1523702
There are 40 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Research Articles
Authors

Mustafa Ekici 0000-0003-2494-8229

Metin Ünal 0000-0002-4462-0872

Publication Date March 30, 2022
Submission Date February 20, 2021
Acceptance Date July 29, 2021
Published in Issue Year 2022 Volume: 71 Issue: 1

Cite

APA Ekici, M., & Ünal, M. (2022). Application of the rational (G’ /G)-expansion method for solving some coupled and combined wave equations. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(1), 116-132. https://doi.org/10.31801/cfsuasmas.884025
AMA Ekici M, Ünal M. Application of the rational (G’ /G)-expansion method for solving some coupled and combined wave equations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. March 2022;71(1):116-132. doi:10.31801/cfsuasmas.884025
Chicago Ekici, Mustafa, and Metin Ünal. “Application of the Rational (G’ /G)-Expansion Method for Solving Some Coupled and Combined Wave Equations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 1 (March 2022): 116-32. https://doi.org/10.31801/cfsuasmas.884025.
EndNote Ekici M, Ünal M (March 1, 2022) Application of the rational (G’ /G)-expansion method for solving some coupled and combined wave equations. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 1 116–132.
IEEE M. Ekici and M. Ünal, “Application of the rational (G’ /G)-expansion method for solving some coupled and combined wave equations”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 1, pp. 116–132, 2022, doi: 10.31801/cfsuasmas.884025.
ISNAD Ekici, Mustafa - Ünal, Metin. “Application of the Rational (G’ /G)-Expansion Method for Solving Some Coupled and Combined Wave Equations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/1 (March 2022), 116-132. https://doi.org/10.31801/cfsuasmas.884025.
JAMA Ekici M, Ünal M. Application of the rational (G’ /G)-expansion method for solving some coupled and combined wave equations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:116–132.
MLA Ekici, Mustafa and Metin Ünal. “Application of the Rational (G’ /G)-Expansion Method for Solving Some Coupled and Combined Wave Equations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 1, 2022, pp. 116-32, doi:10.31801/cfsuasmas.884025.
Vancouver Ekici M, Ünal M. Application of the rational (G’ /G)-expansion method for solving some coupled and combined wave equations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(1):116-32.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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