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Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach

Yıl 2021, Cilt: 9 Sayı: 2, 64 - 73, 01.06.2021
https://doi.org/10.36753/mathenot.626461

Öz

This study investigates exact analytical solutions of some nonlinear partial differential equations arising in mathematical physics. To this reason, the Kudryashov-Sinelshchikov equation, the ZK-BBM equation and the Gardner equation have been considered. With the implementation of the trial solution algorithm, solitary wave, bright, dark and periodic exact traveling wave solutions of the considered equations have been attained. The solutions have been checked and graphs have been given via package programs to see the behavior of the waves.

Destekleyen Kurum

Ege University, Scientific Research Projects Coordination

Proje Numarası

2017-TKMYO-002

Teşekkür

This research is supported by Ege University, Scientific Research Project (BAP), Project Number: 2017-TKMYO-002.

Kaynakça

  • [1] Ali, A., Iqbal, M.A., Mohyud-Din, S.T.: Solitary wave solutions Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation. J. Egypt. Math. Soc. 24, 44-48 (2016).
  • [2] Baskonus, H.M., Altan Koc, D., Bulut, H.: New travelling wave prototypes to the nonlinear Zakharov-Kuznetsov equation with power law nonlinearity. Nonlinear Sci. Lett. A. 7 (2), 67-76 (2016).
  • [3] Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive system. Philos. Trans. R. Soc. Lond. Ser. A. 272, 47-48 (1972).
  • [4] Daoui, A.K., Triki, H.: Solitary waves, shock waves and singular solitons of Gardner’s equation for shallow water dynamics. Acta Phys. Pol. B. 45 (6), 1135-1145 (2014).
  • [5] Demiray, S.T.: New exact solutions for generalized Gardner equation. Kuwait J. Sci. 44 (1), 1-8 (2017).
  • [6] Du, X.H.: An irrational trial equation method and its applications. Pramana-J. Phys. 75 (3), 415-422 (2010).
  • [7] Gepreel, K.A.: Explicit Jacobi elliptic exact solutions for nonlinear partial fractional differential equations. Adv. Differ. Equ-Ny. 2014:286 (2014).
  • [8] Guner O., Bekir, A., Cevikel, A.C.: Dark soliton and periodic wave solutions of nonlinear evolution equations. Adv. Differ. Equ-Ny. 2013:68 (2013).
  • [9] Hamdi, S., Morse, B., Halphen, B., Schiesser, W.: Analytical solutions of long nonlinear internal waves: Part I. Nat. Hazards. 57, 597-607 (2011).
  • [10] Jawad, A.J.M.: New exact solutions of nonlinear partial differential equations using Tan-Cot function method. Studies in Mathematical Sciences. 5 (2), 13-25 (2012).
  • [11] Kudryashov, N.A., Sinelshchikov, D.I.: Nonlinear wave in bubbly liquids with consideration for viscosity and heat transfer. Phys. Lett. A. 374, 2011-2016 (2010).
  • [12] Kudryashov, N.A., Sinelshchikov, D.I: Nonlinear evolution equation for describing waves in bubbly liquids with viscosity and heat transfer consideration. Appl. Math. Comput. 217, 414-421 (2010).
  • [13] Liu, C.S.: Using trial equation method to solve the exact solutions for two kinds of KdV equations with variable coefficients. Acta Phys. Sin. 54 (10), 4506-4510 (2005).
  • [14] Liu, C.S.: Trial equation method to nonlinear evolution equations with rank inhomogeneous: Mathematical discussions and its applications. Commun. Theor. Phys. 45 (2), 219-223 (2006).
  • [15] Liu, C.S.: A new trial equation method and its applications. Commun. Theor. Phys. 45 (3), 395-397 (2006).
  • [16] Lu, J.: New exact solutions for Kudryashov-Sinelshchikov equation. Adv. Differ. Equ-Ny. 2018:374 (2018).
  • [17] Ma, W.X., Wu, H.Y., He, J.S.: Partial differential equations possessing Frobenius integrable decompositions. Phys. Lett. A. 364, 29-32 (2007).
  • [18] Miura, R.M.: Korteweg-de vries equation and generalizations I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9, 1202-1204 (1968).
  • [19] Odabasi, M., Misirli, E.: Application of the extended trial equation method to the nonlinear evolution equations. Math.Sci. Appl. E-Notes. 2 (1), 28-33 (2014).
  • [20] Odabasi, M., Misirli, E.: A note on the traveling wave solutions of some nonlinear evolution equations. Optik. 142, 394-400 (2017).
  • [21] Odabasi, M., Misirli, E.: On the solutions of the nonlinear fractional differential equations via the modified trial equation method. Math. Method Appl. Sci. 41, 904-911 (2018).
  • [22] Odabasi, M.: Traveling wave solutions of conformable time-fractional Zakharov-Kuznetsov and Zoomeron equations. Chin. J. Phys. 64, 194-202 (2020).
  • [23] Odabasi, M., Pinar, Z., Kocak, H.: Analytical solutions of some nonlinear fractional-order differential equations by different methods. Math Meth Appl Sci. 1-12 (2020). https://doi.org/10.1002/mma.6313
  • [24] Wazwaz, A.M.: A study on KdV and Gardner equations with time-dependent coefficients and forcing terms. Appl. Math. Comput. 217, 2277-2281 (2010).
  • [25] Yel, G., Sulaiman, T.A., Baskonus, H.M.: On the complex solutions to the (3+1)-dimensional conformable fractional modified KdV-Zakharov-Kuznetsov equation. Mod. Phys. Lett. B. 34 (5), 2050069 (2020).
Yıl 2021, Cilt: 9 Sayı: 2, 64 - 73, 01.06.2021
https://doi.org/10.36753/mathenot.626461

Öz

Proje Numarası

2017-TKMYO-002

Kaynakça

  • [1] Ali, A., Iqbal, M.A., Mohyud-Din, S.T.: Solitary wave solutions Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation. J. Egypt. Math. Soc. 24, 44-48 (2016).
  • [2] Baskonus, H.M., Altan Koc, D., Bulut, H.: New travelling wave prototypes to the nonlinear Zakharov-Kuznetsov equation with power law nonlinearity. Nonlinear Sci. Lett. A. 7 (2), 67-76 (2016).
  • [3] Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive system. Philos. Trans. R. Soc. Lond. Ser. A. 272, 47-48 (1972).
  • [4] Daoui, A.K., Triki, H.: Solitary waves, shock waves and singular solitons of Gardner’s equation for shallow water dynamics. Acta Phys. Pol. B. 45 (6), 1135-1145 (2014).
  • [5] Demiray, S.T.: New exact solutions for generalized Gardner equation. Kuwait J. Sci. 44 (1), 1-8 (2017).
  • [6] Du, X.H.: An irrational trial equation method and its applications. Pramana-J. Phys. 75 (3), 415-422 (2010).
  • [7] Gepreel, K.A.: Explicit Jacobi elliptic exact solutions for nonlinear partial fractional differential equations. Adv. Differ. Equ-Ny. 2014:286 (2014).
  • [8] Guner O., Bekir, A., Cevikel, A.C.: Dark soliton and periodic wave solutions of nonlinear evolution equations. Adv. Differ. Equ-Ny. 2013:68 (2013).
  • [9] Hamdi, S., Morse, B., Halphen, B., Schiesser, W.: Analytical solutions of long nonlinear internal waves: Part I. Nat. Hazards. 57, 597-607 (2011).
  • [10] Jawad, A.J.M.: New exact solutions of nonlinear partial differential equations using Tan-Cot function method. Studies in Mathematical Sciences. 5 (2), 13-25 (2012).
  • [11] Kudryashov, N.A., Sinelshchikov, D.I.: Nonlinear wave in bubbly liquids with consideration for viscosity and heat transfer. Phys. Lett. A. 374, 2011-2016 (2010).
  • [12] Kudryashov, N.A., Sinelshchikov, D.I: Nonlinear evolution equation for describing waves in bubbly liquids with viscosity and heat transfer consideration. Appl. Math. Comput. 217, 414-421 (2010).
  • [13] Liu, C.S.: Using trial equation method to solve the exact solutions for two kinds of KdV equations with variable coefficients. Acta Phys. Sin. 54 (10), 4506-4510 (2005).
  • [14] Liu, C.S.: Trial equation method to nonlinear evolution equations with rank inhomogeneous: Mathematical discussions and its applications. Commun. Theor. Phys. 45 (2), 219-223 (2006).
  • [15] Liu, C.S.: A new trial equation method and its applications. Commun. Theor. Phys. 45 (3), 395-397 (2006).
  • [16] Lu, J.: New exact solutions for Kudryashov-Sinelshchikov equation. Adv. Differ. Equ-Ny. 2018:374 (2018).
  • [17] Ma, W.X., Wu, H.Y., He, J.S.: Partial differential equations possessing Frobenius integrable decompositions. Phys. Lett. A. 364, 29-32 (2007).
  • [18] Miura, R.M.: Korteweg-de vries equation and generalizations I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9, 1202-1204 (1968).
  • [19] Odabasi, M., Misirli, E.: Application of the extended trial equation method to the nonlinear evolution equations. Math.Sci. Appl. E-Notes. 2 (1), 28-33 (2014).
  • [20] Odabasi, M., Misirli, E.: A note on the traveling wave solutions of some nonlinear evolution equations. Optik. 142, 394-400 (2017).
  • [21] Odabasi, M., Misirli, E.: On the solutions of the nonlinear fractional differential equations via the modified trial equation method. Math. Method Appl. Sci. 41, 904-911 (2018).
  • [22] Odabasi, M.: Traveling wave solutions of conformable time-fractional Zakharov-Kuznetsov and Zoomeron equations. Chin. J. Phys. 64, 194-202 (2020).
  • [23] Odabasi, M., Pinar, Z., Kocak, H.: Analytical solutions of some nonlinear fractional-order differential equations by different methods. Math Meth Appl Sci. 1-12 (2020). https://doi.org/10.1002/mma.6313
  • [24] Wazwaz, A.M.: A study on KdV and Gardner equations with time-dependent coefficients and forcing terms. Appl. Math. Comput. 217, 2277-2281 (2010).
  • [25] Yel, G., Sulaiman, T.A., Baskonus, H.M.: On the complex solutions to the (3+1)-dimensional conformable fractional modified KdV-Zakharov-Kuznetsov equation. Mod. Phys. Lett. B. 34 (5), 2050069 (2020).
Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Meryem Odabaşı 0000-0002-3025-3063

Proje Numarası 2017-TKMYO-002
Yayımlanma Tarihi 1 Haziran 2021
Gönderilme Tarihi 29 Eylül 2019
Kabul Tarihi 2 Kasım 2020
Yayımlandığı Sayı Yıl 2021 Cilt: 9 Sayı: 2

Kaynak Göster

APA Odabaşı, M. (2021). Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach. Mathematical Sciences and Applications E-Notes, 9(2), 64-73. https://doi.org/10.36753/mathenot.626461
AMA Odabaşı M. Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach. Math. Sci. Appl. E-Notes. Haziran 2021;9(2):64-73. doi:10.36753/mathenot.626461
Chicago Odabaşı, Meryem. “Investigation of Exact Solutions of Some Nonlinear Evolution Equations via an Analytical Approach”. Mathematical Sciences and Applications E-Notes 9, sy. 2 (Haziran 2021): 64-73. https://doi.org/10.36753/mathenot.626461.
EndNote Odabaşı M (01 Haziran 2021) Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach. Mathematical Sciences and Applications E-Notes 9 2 64–73.
IEEE M. Odabaşı, “Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach”, Math. Sci. Appl. E-Notes, c. 9, sy. 2, ss. 64–73, 2021, doi: 10.36753/mathenot.626461.
ISNAD Odabaşı, Meryem. “Investigation of Exact Solutions of Some Nonlinear Evolution Equations via an Analytical Approach”. Mathematical Sciences and Applications E-Notes 9/2 (Haziran 2021), 64-73. https://doi.org/10.36753/mathenot.626461.
JAMA Odabaşı M. Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach. Math. Sci. Appl. E-Notes. 2021;9:64–73.
MLA Odabaşı, Meryem. “Investigation of Exact Solutions of Some Nonlinear Evolution Equations via an Analytical Approach”. Mathematical Sciences and Applications E-Notes, c. 9, sy. 2, 2021, ss. 64-73, doi:10.36753/mathenot.626461.
Vancouver Odabaşı M. Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach. Math. Sci. Appl. E-Notes. 2021;9(2):64-73.

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