Research Article
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Application of the GKM to some nonlinear partial equations

Year 2024, Volume: 73 Issue: 1, 274 - 284, 16.03.2024

Şeyma Tülüce Demiray Uğur Bayrakcı Vehpi Yıldırım

https://doi.org/10.31801/cfsuasmas.1313970

Abstract

In this manuscript, the strain wave equation, which plays an important role in describing different types of wave propagation in microstructured solids and the (2+1) dimensional Bogoyavlensky Konopelchenko equation, is defined in fluid mechanics as the interaction of a Riemann wave propagating along the $y$-axis and a long wave propagating along the $x$-axis, were studied. The generalized Kudryashov method (GKM), which is one of the solution methods of partial differential equations, was applied to these equations for the first time. Thus, a series of solutions of these equations were obtained. These found solutions were compared with other solutions. It was seen that these solutions were not shown before and were presented for the first time in this study. The new solutions of these equations might have been useful in understanding the phenomena in which waves are governed by these equations. In addition, 2D and 3D graphs of these solutions were constructed by assigning certain values and ranges to them.

Keywords

Generalized Kudryashov method strain wave equation (2+1)-dimensional Bogoyavlensky-Konopelchenko equation

References

  • Ahmad, H., Seadawy, A. R., Khan, T. A., Thounthong P., Analytic approximate solutions for some nonlinear parabolic dynamical wave equations, Journal of Taibah University for Science, 14(1) (2020), 346–358. https://doi.org/10.1080/16583655.2020.1741943
  • Arshed, S., New soliton solutions to the perturbed nonlinear Schrödinger equation by $exp(-\Phi(\xi))$-expansion method, Optik-International Journal for Light and Electron Optics, 220(165123) (2020), 1–12. https://doi.org/10.1016/j.ijleo.2020.165123
  • Dusunceli, F., Celik, E., Askin, M., Bulut, H., New exact solutions for the doubly dispersive equation using the improved Bernoulli sub-equation function method, Indian Journal of Physics, 95(2) (2021), 309–314. https://doi.org/10.1007/s12648-020-01707-5
  • Ekici, M., Unal, 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, 71(1) (2022), 116–132. https://doi.org/10.31801/cfsuasmas.884025
  • Rahman, M., Habiba, U., Salam, M., Datta, M., The traveling wave solutions of space-time fractional partial differential equations by modified Kudryashov method., Journal of Applied Mathematics and Physics, 8(11) (2020), 2683–2690. https://doi.org/10.4236/jamp.2020.811198
  • Taşbozan, O., Kurt, A., The new travelling wave solutions of time fractional Fitzhugh-Nagumo equation with Sine-Gordon expansion method, Adıyaman University Journal of Science, 10(1) (2020), 256–263. https://doi.org/10.37094/adyujsci.515011
  • Arnous, A. H., Zhou, Q., Biswas, A., Guggilla, P., Khan, S., Yıldırım, Y., Alshomrani, A. S., Alshehri, H. M., Optical solitons in fiber Bragg gratings with cubic–quartic dispersive reflectivity by enhanced Kudryashov’s approach, Physics Letters A, 422 (2022), 127797. https://doi.org/10.1016/j.physleta.2021.127797
  • Zayed, E. M. E., Gepreel, K. A., Shohib, R. M. A., Alngar, M. E. M., Yıldırım, Y., Optical solitons for the perturbed Biswas-Milovic equation with Kudryashov’s law of refractive index by the unified auxiliary equation method, Optik, 230 (2021), 166286. https://doi.org/10.1016/j.ijleo.2021.166286
  • Yıldırım, Y., Topkara, E., Biswas, A., Triki, H., Ekici, M., Guggilla, P., Khan, S., Belic, M. R., Cubic–quartic optical soliton perturbation with Lakshmanan–Porsezian–Daniel model by sine-Gordon equation approach, Journal of Optics, 50 (2021), 322–329. https://doi.org/10.1007/s12596-021-00685-z
  • Yıldırım, Y., Biswas, A., Asma, M., Ekici, M., Ntsime, B. P., Zayed, E. M. E., Moshokoa, S. P., Alzahrani, A. K., Belic, M. R., Optical soliton perturbation with Chen–Lee–Liu equation, Optik, 220 (2020), 165177. https://doi.org/10.1016/j.ijleo.2020.165177
  • Seadawy, A. R., Arshad, M., Lu, D., Dispersive optical solitary wave solutions of strain wave equation in micro-structured solids and its applications, Physica A: Statistical Mechanics and its Applications, 540 (2020), 1–13. https://doi.org/10.1016/j.physa.2019.123122
  • Ayati, Z., Hosseini, K., Mirzazadeh, M., Application of Kudryashov and functional variable methods to the strain wave equation in microstructured solids, Nonlinear Engineering, 6(1) (2017), 25–29. https://doi.org/10.1515/nleng-2016-0020
  • Arshad, M., Seadawy, A. R., Lu, D., Study of bright–dark solitons of strain wave equation in micro-structured solids and its applications, Modern Physics Letters B, 33(33) (2019), 1–12. https://doi.org/10.1142/S0217984919504177
  • Gao, W., Silambarasan, R., Baskonus, H. M., Anand, R. V., Rezazadeh, H., Periodic waves of the non dissipative double dispersive micro strain wave in the micro structured solids, Physica A: Statistical Mechanics and its Applications, 545 (2020), 1–30. https://doi.org/10.1016/j.physa.2019.123772
  • Irshad, A., Ahmed, N., Nazir, A., Khan, U., Mohyud-Din, S. T., Novel exact double periodic soliton solutions to strain wave equation in micro structured solids, Physica A: Statistical Mechanics and its Applications, 550 (2020), 1–15. https://doi.org/10.1016/j.physa.2019.124077
  • Kumar, S., Kumar, A., Wazwaz, A. M., New exact solitary wave solutions of the strain wave equation in microstructured solids via the generalized exponential rational function method, The European Physical Journal Plus, 135(870) (2020), 1–17. https://doi.org/10.1140/epjp/s13360-020-00883-x
  • Joseph, S. P., New traveling wave rational form exact solutions for strain wave equation in micro structured solids, IOP SciNotes, 2(1) (2021), 1–7. https://doi.org/10.1088/2633-1357/abec2a
  • Ray, S. S., Lie symmetry analysis and reduction for exact solution of (2+1)-dimensional Bogoyavlensky–Konopelchenko equation by geometric approach, Modern Physics Letters B, 32(11) (2018), 1–9. https://doi.org/10.1142/S0217984918501270
  • Yan, H., Tian, S. F., Feng, L. L., Zhang, T. T., Quasi-periodic wave solutions, soliton solutions, and integrability to a (2+1)-dimensional generalized Bogoyavlensky- Konopelchenko equation, Waves in Random and Complex Media, 26(4) (2016), 1–14. https://doi.org/10.1080/17455030.2016.1166289
  • Xiang-Peng, X., Xi-Qiang, L., Lin-Lin, Z., Explicit solutions of the Bogoyavlensky–Konoplechenko equation, Applied Mathematics and Computation, 215(10) (2010), 3669–3673. https://doi.org/10.1016/j.amc.2009.11.005
  • Zhou, X. M., Tian, S. F., Zhang, L. D., Zhang, T. T., General high-order breather, lump, and semi-rational solutions to the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation. Modern Physics Letters B, 35(3) (2021), 1–12. https://doi.org/10.1142/S0217984921500573
  • Ray, S. S., On conservation laws by Lie symmetry analysis for (2+1)-dimensional Bogoyavlensky–Konopelchenko equation in wave propagation, Computers and Mathematics with Applications, 74(6) (2017), 1158–1165. https://doi.org/10.1016/j.camwa.2017.06.007
  • Chen, S. T., Ma, W. X., Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation, Frontiers of Mathematics in China, 13(3) (2018), 525–534. https://doi.org/10.1007/s11464-018-0694-z
  • Tuluce Demiray, S., New solutions of Biswas-Arshed equation with beta time derivative, Optik-International Journal for Light and Electron Optics, 222(165405) (2020a), 1–5. https://doi.org/10.1016/j.ijleo.2020.165405
  • Tuluce Demiray, S, Bayrakci, U., Soliton solutions of generalized third-order nonlinear Schrödinger equation by using GKM, Journal of the Institute of Science and Technology, 11(2) (2021), 1481–1488. https://doi.org/10.21597/jist.861864
  • Tuluce Demiray, S, Bayrakci, U., Soliton solutions for space-time fractional Heisenberg ferromagnetic spin chain equation by generalized Kudryashov method and modified $exp(-\Omega(\eta))$ -expansion function method, Revista Mexicana de Fisica, 67(3) (2021), 393–402. https://doi.org/10.31349/RevMexFis.67.393
  • Gurefe, Y., The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative, Revista Mexicana de Fisica, 66(6) (2020), 771–781. https://doi.org/10.31349/RevMexFis.66.771

Year 2024, Volume: 73 Issue: 1, 274 - 284, 16.03.2024

Şeyma Tülüce Demiray Uğur Bayrakcı Vehpi Yıldırım

https://doi.org/10.31801/cfsuasmas.1313970

Abstract

References

  • Ahmad, H., Seadawy, A. R., Khan, T. A., Thounthong P., Analytic approximate solutions for some nonlinear parabolic dynamical wave equations, Journal of Taibah University for Science, 14(1) (2020), 346–358. https://doi.org/10.1080/16583655.2020.1741943
  • Arshed, S., New soliton solutions to the perturbed nonlinear Schrödinger equation by $exp(-\Phi(\xi))$-expansion method, Optik-International Journal for Light and Electron Optics, 220(165123) (2020), 1–12. https://doi.org/10.1016/j.ijleo.2020.165123
  • Dusunceli, F., Celik, E., Askin, M., Bulut, H., New exact solutions for the doubly dispersive equation using the improved Bernoulli sub-equation function method, Indian Journal of Physics, 95(2) (2021), 309–314. https://doi.org/10.1007/s12648-020-01707-5
  • Ekici, M., Unal, 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, 71(1) (2022), 116–132. https://doi.org/10.31801/cfsuasmas.884025
  • Rahman, M., Habiba, U., Salam, M., Datta, M., The traveling wave solutions of space-time fractional partial differential equations by modified Kudryashov method., Journal of Applied Mathematics and Physics, 8(11) (2020), 2683–2690. https://doi.org/10.4236/jamp.2020.811198
  • Taşbozan, O., Kurt, A., The new travelling wave solutions of time fractional Fitzhugh-Nagumo equation with Sine-Gordon expansion method, Adıyaman University Journal of Science, 10(1) (2020), 256–263. https://doi.org/10.37094/adyujsci.515011
  • Arnous, A. H., Zhou, Q., Biswas, A., Guggilla, P., Khan, S., Yıldırım, Y., Alshomrani, A. S., Alshehri, H. M., Optical solitons in fiber Bragg gratings with cubic–quartic dispersive reflectivity by enhanced Kudryashov’s approach, Physics Letters A, 422 (2022), 127797. https://doi.org/10.1016/j.physleta.2021.127797
  • Zayed, E. M. E., Gepreel, K. A., Shohib, R. M. A., Alngar, M. E. M., Yıldırım, Y., Optical solitons for the perturbed Biswas-Milovic equation with Kudryashov’s law of refractive index by the unified auxiliary equation method, Optik, 230 (2021), 166286. https://doi.org/10.1016/j.ijleo.2021.166286
  • Yıldırım, Y., Topkara, E., Biswas, A., Triki, H., Ekici, M., Guggilla, P., Khan, S., Belic, M. R., Cubic–quartic optical soliton perturbation with Lakshmanan–Porsezian–Daniel model by sine-Gordon equation approach, Journal of Optics, 50 (2021), 322–329. https://doi.org/10.1007/s12596-021-00685-z
  • Yıldırım, Y., Biswas, A., Asma, M., Ekici, M., Ntsime, B. P., Zayed, E. M. E., Moshokoa, S. P., Alzahrani, A. K., Belic, M. R., Optical soliton perturbation with Chen–Lee–Liu equation, Optik, 220 (2020), 165177. https://doi.org/10.1016/j.ijleo.2020.165177
  • Seadawy, A. R., Arshad, M., Lu, D., Dispersive optical solitary wave solutions of strain wave equation in micro-structured solids and its applications, Physica A: Statistical Mechanics and its Applications, 540 (2020), 1–13. https://doi.org/10.1016/j.physa.2019.123122
  • Ayati, Z., Hosseini, K., Mirzazadeh, M., Application of Kudryashov and functional variable methods to the strain wave equation in microstructured solids, Nonlinear Engineering, 6(1) (2017), 25–29. https://doi.org/10.1515/nleng-2016-0020
  • Arshad, M., Seadawy, A. R., Lu, D., Study of bright–dark solitons of strain wave equation in micro-structured solids and its applications, Modern Physics Letters B, 33(33) (2019), 1–12. https://doi.org/10.1142/S0217984919504177
  • Gao, W., Silambarasan, R., Baskonus, H. M., Anand, R. V., Rezazadeh, H., Periodic waves of the non dissipative double dispersive micro strain wave in the micro structured solids, Physica A: Statistical Mechanics and its Applications, 545 (2020), 1–30. https://doi.org/10.1016/j.physa.2019.123772
  • Irshad, A., Ahmed, N., Nazir, A., Khan, U., Mohyud-Din, S. T., Novel exact double periodic soliton solutions to strain wave equation in micro structured solids, Physica A: Statistical Mechanics and its Applications, 550 (2020), 1–15. https://doi.org/10.1016/j.physa.2019.124077
  • Kumar, S., Kumar, A., Wazwaz, A. M., New exact solitary wave solutions of the strain wave equation in microstructured solids via the generalized exponential rational function method, The European Physical Journal Plus, 135(870) (2020), 1–17. https://doi.org/10.1140/epjp/s13360-020-00883-x
  • Joseph, S. P., New traveling wave rational form exact solutions for strain wave equation in micro structured solids, IOP SciNotes, 2(1) (2021), 1–7. https://doi.org/10.1088/2633-1357/abec2a
  • Ray, S. S., Lie symmetry analysis and reduction for exact solution of (2+1)-dimensional Bogoyavlensky–Konopelchenko equation by geometric approach, Modern Physics Letters B, 32(11) (2018), 1–9. https://doi.org/10.1142/S0217984918501270
  • Yan, H., Tian, S. F., Feng, L. L., Zhang, T. T., Quasi-periodic wave solutions, soliton solutions, and integrability to a (2+1)-dimensional generalized Bogoyavlensky- Konopelchenko equation, Waves in Random and Complex Media, 26(4) (2016), 1–14. https://doi.org/10.1080/17455030.2016.1166289
  • Xiang-Peng, X., Xi-Qiang, L., Lin-Lin, Z., Explicit solutions of the Bogoyavlensky–Konoplechenko equation, Applied Mathematics and Computation, 215(10) (2010), 3669–3673. https://doi.org/10.1016/j.amc.2009.11.005
  • Zhou, X. M., Tian, S. F., Zhang, L. D., Zhang, T. T., General high-order breather, lump, and semi-rational solutions to the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation. Modern Physics Letters B, 35(3) (2021), 1–12. https://doi.org/10.1142/S0217984921500573
  • Ray, S. S., On conservation laws by Lie symmetry analysis for (2+1)-dimensional Bogoyavlensky–Konopelchenko equation in wave propagation, Computers and Mathematics with Applications, 74(6) (2017), 1158–1165. https://doi.org/10.1016/j.camwa.2017.06.007
  • Chen, S. T., Ma, W. X., Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation, Frontiers of Mathematics in China, 13(3) (2018), 525–534. https://doi.org/10.1007/s11464-018-0694-z
  • Tuluce Demiray, S., New solutions of Biswas-Arshed equation with beta time derivative, Optik-International Journal for Light and Electron Optics, 222(165405) (2020a), 1–5. https://doi.org/10.1016/j.ijleo.2020.165405
  • Tuluce Demiray, S, Bayrakci, U., Soliton solutions of generalized third-order nonlinear Schrödinger equation by using GKM, Journal of the Institute of Science and Technology, 11(2) (2021), 1481–1488. https://doi.org/10.21597/jist.861864
  • Tuluce Demiray, S, Bayrakci, U., Soliton solutions for space-time fractional Heisenberg ferromagnetic spin chain equation by generalized Kudryashov method and modified $exp(-\Omega(\eta))$ -expansion function method, Revista Mexicana de Fisica, 67(3) (2021), 393–402. https://doi.org/10.31349/RevMexFis.67.393
  • Gurefe, Y., The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative, Revista Mexicana de Fisica, 66(6) (2020), 771–781. https://doi.org/10.31349/RevMexFis.66.771
There are 27 citations in total.

Details

Primary Language English
Subjects Partial Differential Equations
Journal Section Research Articles
Authors

Şeyma Tülüce Demiray 0000-0002-8027-7290

Uğur Bayrakcı 0000-0002-1765-2318

Vehpi Yıldırım 0000-0003-3837-4756

Publication Date March 16, 2024
Submission Date June 13, 2023
Acceptance Date October 31, 2023
Published in Issue Year 2024 Volume: 73 Issue: 1

Cite

APA Tülüce Demiray, Ş., Bayrakcı, U., & Yıldırım, V. (2024). Application of the GKM to some nonlinear partial equations. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 73(1), 274-284. https://doi.org/10.31801/cfsuasmas.1313970
AMA Tülüce Demiray Ş, Bayrakcı U, Yıldırım V. Application of the GKM to some nonlinear partial equations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. March 2024;73(1):274-284. doi:10.31801/cfsuasmas.1313970
Chicago Tülüce Demiray, Şeyma, Uğur Bayrakcı, and Vehpi Yıldırım. “Application of the GKM to Some Nonlinear Partial Equations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73, no. 1 (March 2024): 274-84. https://doi.org/10.31801/cfsuasmas.1313970.
EndNote Tülüce Demiray Ş, Bayrakcı U, Yıldırım V (March 1, 2024) Application of the GKM to some nonlinear partial equations. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73 1 274–284.
IEEE Ş. Tülüce Demiray, U. Bayrakcı, and V. Yıldırım, “Application of the GKM to some nonlinear partial equations”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 73, no. 1, pp. 274–284, 2024, doi: 10.31801/cfsuasmas.1313970.
ISNAD Tülüce Demiray, Şeyma et al. “Application of the GKM to Some Nonlinear Partial Equations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73/1 (March 2024), 274-284. https://doi.org/10.31801/cfsuasmas.1313970.
JAMA Tülüce Demiray Ş, Bayrakcı U, Yıldırım V. Application of the GKM to some nonlinear partial equations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73:274–284.
MLA Tülüce Demiray, Şeyma et al. “Application of the GKM to Some Nonlinear Partial Equations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 73, no. 1, 2024, pp. 274-8, doi:10.31801/cfsuasmas.1313970.
Vancouver Tülüce Demiray Ş, Bayrakcı U, Yıldırım V. Application of the GKM to some nonlinear partial equations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73(1):274-8.

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

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