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Yıl 2021, Cilt: 9 Sayı: 3, 108 - 123, 30.09.2021

Handan Yaslan

https://doi.org/10.36753/mathenot.734019

Öz

Kaynakça

  • [1] Bagley, R.L., Torvik, P.J.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294-298 (1984). https://doi.org/10.1115/1.3167615
  • [2] Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calculus. App. Anal. 5, 367 - 386 (2002).
  • [3] Mainardi,F.:Fractionalcalculus:Somebasicproblemsincontinuumandstatisticalmechanics,In:A.Carpinteri, F. Mainardi, Editors, Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag, New York (1997).
  • [4] Thomas, M.D., Bamforth,P.B.: Modelling chloride diffusion in concrete: Effect of fly ash and slag. Cem. Concr. Res. 29 (4), 487-495 (1999).
  • [5] Khitab, A., Lorente, S., Ollivier, J.P.: Predictive model for chloride penetration through concrete. Mag. Concr. Res. 57 (9), 511-520 (2005). https://doi.org/10.1680/macr.2005.57.9.511
  • [6] Sahoo, S., Saha Ray S.: New travelling wave and anti-kink wave solutions of space-time fractional (3+1)-Dimensional Jimbo-Miwa equation. Chin. J. Phys. 67, 79-85 (2020). https://doi.org/10.1016/j.cjph.2020.04.016.
  • [7] Kim, H., Sakthivel, R., Debbouchecd, A., Torres, Delfim F.M.: Traveling wave solutions of some important Wick-type fractional stochastic nonlinear partial differential equation. Chaos soliton fract. 131, 109542, (2020). https://doi.org/10.1016/j.chaos.2019.109542.
  • [8] Zulfiqar, A., Ahmad, J.: Exact solitary wave solutions of fractional modified Camassa-Holm equation using an efficient method. Alexandria Eng. J. 59 (5), 3565-3574 (2020). https://doi.org/10.1016/j.aej.2020.06.002.
  • [9] Sulaimana, T. A., Bulut, H.: Boussinesq equations: M-fractional solitary wave solutions and convergence analysis. JOES 4 (1), 1-6 (2019). https://doi.org/10.1016/j.joes.2018.12.001.
  • [10] Dianchen, L. Y., Arshad, L. M., Xu, X.: New Exact Traveling Wave Solutions of the Unstable Nonlinear Schrödinger Equations and their Applications. Optik 165386 (2020). https://doi.org/10.1016/j.ijleo.2020.165386.
  • [11] Guo, S., Mei, L., Zhou, Y.: The compound (G′/G)-expansion method and double non-traveling wave solu- tions of (2 + 1)-dimensional nonlinear partial differential equations. Comput. Math. Appl. 69, 804-816 (2015). https://doi.org/10.1016/j.camwa.2015.02.016.
  • [12] Bekir,A., Uygun, F.: Exact travelling wave solutions of nonlinear evolution equations by using (G′/G)-expansion method. Arab J. Math. Sci. 18, 73-85 (2012). https://doi.org/10.1016/j.ajmsc.2011.08.002.
  • [13] Dai, C., Zhang, J.: Chaotic behaviors in the (2 + 1)-dimensional breaking soliton system. Chaos soliton fract. 39, 889-894 (2009). https://doi.org/10.1016/j.chaos.2007.01.063.
  • [14] Ping, Z.: New exact solutions to breaking soliton equations and Whitham-Broer-Kaup equations. Appl. Math. Comput. 217, 1688-1696 (2010). https://doi.org/10.1016/j.amc.2009.09.062.
  • [15] Tascan, F., Bekir, A.: Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method. Appl. Math. Comput. 215, 3134-3139 (2009). https://doi.org/10.1016/j.amc.2009.09.027.
  • [16] Zhang, S.: A further improved extended Fan sub-equation method for (2+1)-dimensional breaking soliton equations. Appl. Math. Comput. 199, 259-267 (2008). https://doi.org/10.1016/j.amc.2007.09.052.
  • [17] Xia,T., Xiong, S.: Exact solutions of (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation with symbolic computation. Comput. Math. Appl. 60, 919-923 (2010). https://doi.org/10.1016/j.camwa.2010.05.037.
  • [18] Zhao, Z., Dai, Z., Mu, G.: The breather-type and periodic-type soliton solutions for the (2 + 1)-dimensional breaking soliton equation. Comput. Math. Appl. 61, 2048-2052 (2011). https://doi.org/10.1016/j.camwa.2010.08.065.
  • [19] Zheng, B.: A new fractional Jacobi elliptic equation method for solving fractional partial differential equations. Adv. Differ. Equ. 228, 1-11 (2014). https://doi.org/10.1186/1687-1847-2014-228.
  • [20] Choi, J. H., Kim, H.: Soliton solutions for the space-time nonlinear partial differential equations with fractional-orders. Chinese J. Phys. 55, 556-565 (2017). https://doi.org/10.1016/j.cjph.2016.10.019.
  • [21] Kaplan, M., Akbulut, A., Bekir, A.: Solving Space-Time Fractional Differential Equations by Using Modified Simple Equation Method. Commun. Theor. Phys. 65, 563-568 (2016). https://doi.org/10.1088/0253-6102/65/5/563.
  • [22] Li,C., Zhao, M.: Analytical solutions of the (2 + 1)-dimensional space-time fractional Bogoyavlenskii’s breaking soliton equation. Appl. Math. Lett. 84, 13-18 (2018). https://doi.org/10.1016/j.aml.2018.04.011.
  • [23] Mohyud-Din, S. T., Bibi, S.: Exact solutions for nonlinear fractional differential equations using exponential rational function method. Opt. Quant. Electron. 49, 1-12 (2017). https://doi.org/10.1007/s11082-017-0895-9
  • [24] Shang,N.,Zheng,B.:ExactSolutionsforThreeFractionalPartialDifferentialEquationsbythe(G′/G)Method.IJAM 43, 1-6 (2013).
  • [25] Salas, A. H., Gomez, A.: Exact Solutions for a Third-Order KdV Equation with Variable Coefficients and Forcing Term. Math. Probl. Eng. 2009, 1-13 (2009). https://doi.org/10.1155/2009/737928.
  • [26] Wazzan, L.: Exact solutions for the family of third order Korteweg de-Vries equations. Commun. Appl. Anal. 2016, 108-117 (2016). https://doi.org/10.5899/2016/cna-00242.
  • [27] Seadawy, A. R., Sayed, A.: Traveling Wave Solutions of the Benjamin-Bona-Mahony Water Wave Equations. Abstr. Appl. Anal. 2014, 1-7 (2014). https://doi.org/10.1155/2014/926838.
  • [28] Raslan, K. R., EL-Danaf, T. S., Ali, K. K.: New numerical treatment for solving the KDV equation. JACM 1, 1-12 (2017).
  • [29] Demiray, S. T., Pandir, Y., Bulut, H.: Generalized Kudryashov Method for Time-Fractional Differential Equations. Abstr. Appl. Anal. 2014 1-13 (2014). https://doi.org/10.1155/2014/901540.
  • [30] Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9, 225-236 (1951). https://doi.org/10.1090/qam/42889.
  • [31] Xiea,S.,Heo,S.,Kimc,S.,Woo,G.,Yi,S.:Numericalsolutionofone-dimensionalBurgers’equationusingreproducing kernel function. J. Comput. Appl. Math. 214, 417-434 (2008). https://doi.org/10.1016/j.cam.2007.03.010.
  • [32] AL-Jawary, M. A., Azeez, M. M., Radhi, G. H.: Analytical and numerical solutions for the nonlinear Burgers and advection-diffusion equations by using a semi-analytical iterative method. Comput. Math. Appl. 76, 155-171 (2018). https://doi.org/10.1016/j.camwa.2018.04.010.
  • [33] Huda,M.A.,Akbar,M.A.,Shanta,S.S.:ThenewtypesofwavesolutionsoftheBurger’sequationandtheBenjamin- Bona-Mahony equation. JOES 3, 1-10 (2018). https://doi.org/10.1016/j.joes.2017.11.002.
  • [34] Liu, H., Zhang, T.: A note on the improved tan(φ(ξ)/2)-expansion method. Optik 131, 273-278 (2017). https://doi.org/10.1016/j.ijleo.2016.11.029.
  • [35] Khalil, R., Horani, M. A., Yousef, A., Sababheh, M.: A new defnition of fractional derivative. J. Comput. Appl. Math. 264 , 65-70 (2014). https://doi.org/10.1016/j.cam.2014.01.002.
  • [36] Bogoyavlenskii, O.I.: Breaking solitons in (2 + 1)-dimensional integrable equations. Russian Math. Surveys 45, 1-86 (1990). https://doi.org/10.1070/RM1990v045n04ABEH002377.

SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger's Equations

Yıl 2021, Cilt: 9 Sayı: 3, 108 - 123, 30.09.2021

Handan Yaslan

https://doi.org/10.36753/mathenot.734019

Öz

In the present paper, new analytical solutions for the conformable space-time fractional (2+1)-dimensional breaking soliton, third-order KdV and Burger's equations are obtained by using the simplified tan(ϕ(ξ)2)tan⁡(ϕ(ξ)2)-expansion method (SITEM). Here, fractional derivatives are described in conformable sense. The obtained traveling wave solutions are expressed by the trigonometric, hyperbolic, exponential and rational functions. Simulation of the obtained solutions are given at the end of the paper.

Anahtar Kelimeler

Space-time fractional (2+1)-dimensional breaking soliton equations Space-time fractional third-order KdV equation Space-time fractional Burger's equation Simplified $\tan(\frac{\phi (\xi) }{2})$-expansion method (SITEM)

Kaynakça

  • [1] Bagley, R.L., Torvik, P.J.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294-298 (1984). https://doi.org/10.1115/1.3167615
  • [2] Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calculus. App. Anal. 5, 367 - 386 (2002).
  • [3] Mainardi,F.:Fractionalcalculus:Somebasicproblemsincontinuumandstatisticalmechanics,In:A.Carpinteri, F. Mainardi, Editors, Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag, New York (1997).
  • [4] Thomas, M.D., Bamforth,P.B.: Modelling chloride diffusion in concrete: Effect of fly ash and slag. Cem. Concr. Res. 29 (4), 487-495 (1999).
  • [5] Khitab, A., Lorente, S., Ollivier, J.P.: Predictive model for chloride penetration through concrete. Mag. Concr. Res. 57 (9), 511-520 (2005). https://doi.org/10.1680/macr.2005.57.9.511
  • [6] Sahoo, S., Saha Ray S.: New travelling wave and anti-kink wave solutions of space-time fractional (3+1)-Dimensional Jimbo-Miwa equation. Chin. J. Phys. 67, 79-85 (2020). https://doi.org/10.1016/j.cjph.2020.04.016.
  • [7] Kim, H., Sakthivel, R., Debbouchecd, A., Torres, Delfim F.M.: Traveling wave solutions of some important Wick-type fractional stochastic nonlinear partial differential equation. Chaos soliton fract. 131, 109542, (2020). https://doi.org/10.1016/j.chaos.2019.109542.
  • [8] Zulfiqar, A., Ahmad, J.: Exact solitary wave solutions of fractional modified Camassa-Holm equation using an efficient method. Alexandria Eng. J. 59 (5), 3565-3574 (2020). https://doi.org/10.1016/j.aej.2020.06.002.
  • [9] Sulaimana, T. A., Bulut, H.: Boussinesq equations: M-fractional solitary wave solutions and convergence analysis. JOES 4 (1), 1-6 (2019). https://doi.org/10.1016/j.joes.2018.12.001.
  • [10] Dianchen, L. Y., Arshad, L. M., Xu, X.: New Exact Traveling Wave Solutions of the Unstable Nonlinear Schrödinger Equations and their Applications. Optik 165386 (2020). https://doi.org/10.1016/j.ijleo.2020.165386.
  • [11] Guo, S., Mei, L., Zhou, Y.: The compound (G′/G)-expansion method and double non-traveling wave solu- tions of (2 + 1)-dimensional nonlinear partial differential equations. Comput. Math. Appl. 69, 804-816 (2015). https://doi.org/10.1016/j.camwa.2015.02.016.
  • [12] Bekir,A., Uygun, F.: Exact travelling wave solutions of nonlinear evolution equations by using (G′/G)-expansion method. Arab J. Math. Sci. 18, 73-85 (2012). https://doi.org/10.1016/j.ajmsc.2011.08.002.
  • [13] Dai, C., Zhang, J.: Chaotic behaviors in the (2 + 1)-dimensional breaking soliton system. Chaos soliton fract. 39, 889-894 (2009). https://doi.org/10.1016/j.chaos.2007.01.063.
  • [14] Ping, Z.: New exact solutions to breaking soliton equations and Whitham-Broer-Kaup equations. Appl. Math. Comput. 217, 1688-1696 (2010). https://doi.org/10.1016/j.amc.2009.09.062.
  • [15] Tascan, F., Bekir, A.: Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method. Appl. Math. Comput. 215, 3134-3139 (2009). https://doi.org/10.1016/j.amc.2009.09.027.
  • [16] Zhang, S.: A further improved extended Fan sub-equation method for (2+1)-dimensional breaking soliton equations. Appl. Math. Comput. 199, 259-267 (2008). https://doi.org/10.1016/j.amc.2007.09.052.
  • [17] Xia,T., Xiong, S.: Exact solutions of (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation with symbolic computation. Comput. Math. Appl. 60, 919-923 (2010). https://doi.org/10.1016/j.camwa.2010.05.037.
  • [18] Zhao, Z., Dai, Z., Mu, G.: The breather-type and periodic-type soliton solutions for the (2 + 1)-dimensional breaking soliton equation. Comput. Math. Appl. 61, 2048-2052 (2011). https://doi.org/10.1016/j.camwa.2010.08.065.
  • [19] Zheng, B.: A new fractional Jacobi elliptic equation method for solving fractional partial differential equations. Adv. Differ. Equ. 228, 1-11 (2014). https://doi.org/10.1186/1687-1847-2014-228.
  • [20] Choi, J. H., Kim, H.: Soliton solutions for the space-time nonlinear partial differential equations with fractional-orders. Chinese J. Phys. 55, 556-565 (2017). https://doi.org/10.1016/j.cjph.2016.10.019.
  • [21] Kaplan, M., Akbulut, A., Bekir, A.: Solving Space-Time Fractional Differential Equations by Using Modified Simple Equation Method. Commun. Theor. Phys. 65, 563-568 (2016). https://doi.org/10.1088/0253-6102/65/5/563.
  • [22] Li,C., Zhao, M.: Analytical solutions of the (2 + 1)-dimensional space-time fractional Bogoyavlenskii’s breaking soliton equation. Appl. Math. Lett. 84, 13-18 (2018). https://doi.org/10.1016/j.aml.2018.04.011.
  • [23] Mohyud-Din, S. T., Bibi, S.: Exact solutions for nonlinear fractional differential equations using exponential rational function method. Opt. Quant. Electron. 49, 1-12 (2017). https://doi.org/10.1007/s11082-017-0895-9
  • [24] Shang,N.,Zheng,B.:ExactSolutionsforThreeFractionalPartialDifferentialEquationsbythe(G′/G)Method.IJAM 43, 1-6 (2013).
  • [25] Salas, A. H., Gomez, A.: Exact Solutions for a Third-Order KdV Equation with Variable Coefficients and Forcing Term. Math. Probl. Eng. 2009, 1-13 (2009). https://doi.org/10.1155/2009/737928.
  • [26] Wazzan, L.: Exact solutions for the family of third order Korteweg de-Vries equations. Commun. Appl. Anal. 2016, 108-117 (2016). https://doi.org/10.5899/2016/cna-00242.
  • [27] Seadawy, A. R., Sayed, A.: Traveling Wave Solutions of the Benjamin-Bona-Mahony Water Wave Equations. Abstr. Appl. Anal. 2014, 1-7 (2014). https://doi.org/10.1155/2014/926838.
  • [28] Raslan, K. R., EL-Danaf, T. S., Ali, K. K.: New numerical treatment for solving the KDV equation. JACM 1, 1-12 (2017).
  • [29] Demiray, S. T., Pandir, Y., Bulut, H.: Generalized Kudryashov Method for Time-Fractional Differential Equations. Abstr. Appl. Anal. 2014 1-13 (2014). https://doi.org/10.1155/2014/901540.
  • [30] Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9, 225-236 (1951). https://doi.org/10.1090/qam/42889.
  • [31] Xiea,S.,Heo,S.,Kimc,S.,Woo,G.,Yi,S.:Numericalsolutionofone-dimensionalBurgers’equationusingreproducing kernel function. J. Comput. Appl. Math. 214, 417-434 (2008). https://doi.org/10.1016/j.cam.2007.03.010.
  • [32] AL-Jawary, M. A., Azeez, M. M., Radhi, G. H.: Analytical and numerical solutions for the nonlinear Burgers and advection-diffusion equations by using a semi-analytical iterative method. Comput. Math. Appl. 76, 155-171 (2018). https://doi.org/10.1016/j.camwa.2018.04.010.
  • [33] Huda,M.A.,Akbar,M.A.,Shanta,S.S.:ThenewtypesofwavesolutionsoftheBurger’sequationandtheBenjamin- Bona-Mahony equation. JOES 3, 1-10 (2018). https://doi.org/10.1016/j.joes.2017.11.002.
  • [34] Liu, H., Zhang, T.: A note on the improved tan(φ(ξ)/2)-expansion method. Optik 131, 273-278 (2017). https://doi.org/10.1016/j.ijleo.2016.11.029.
  • [35] Khalil, R., Horani, M. A., Yousef, A., Sababheh, M.: A new defnition of fractional derivative. J. Comput. Appl. Math. 264 , 65-70 (2014). https://doi.org/10.1016/j.cam.2014.01.002.
  • [36] Bogoyavlenskii, O.I.: Breaking solitons in (2 + 1)-dimensional integrable equations. Russian Math. Surveys 45, 1-86 (1990). https://doi.org/10.1070/RM1990v045n04ABEH002377.
Toplam 36 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Articles
Yazarlar

Handan Yaslan 0000-0002-3243-3703

Yayımlanma Tarihi 30 Eylül 2021
Gönderilme Tarihi 8 Mayıs 2020
Kabul Tarihi 26 Ekim 2020
Yayımlandığı Sayı Yıl 2021 Cilt: 9 Sayı: 3

Kaynak Göster

APA Yaslan, H. (2021). SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger’s Equations. Mathematical Sciences and Applications E-Notes, 9(3), 108-123. https://doi.org/10.36753/mathenot.734019
AMA Yaslan H. SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger’s Equations. Math. Sci. Appl. E-Notes. Eylül 2021;9(3):108-123. doi:10.36753/mathenot.734019
Chicago Yaslan, Handan. “SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger’s Equations”. Mathematical Sciences and Applications E-Notes 9, sy. 3 (Eylül 2021): 108-23. https://doi.org/10.36753/mathenot.734019.
EndNote Yaslan H (01 Eylül 2021) SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger’s Equations. Mathematical Sciences and Applications E-Notes 9 3 108–123.
IEEE H. Yaslan, “SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger’s Equations”, Math. Sci. Appl. E-Notes, c. 9, sy. 3, ss. 108–123, 2021, doi: 10.36753/mathenot.734019.
ISNAD Yaslan, Handan. “SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger’s Equations”. Mathematical Sciences and Applications E-Notes 9/3 (Eylül 2021), 108-123. https://doi.org/10.36753/mathenot.734019.
JAMA Yaslan H. SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger’s Equations. Math. Sci. Appl. E-Notes. 2021;9:108–123.
MLA Yaslan, Handan. “SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger’s Equations”. Mathematical Sciences and Applications E-Notes, c. 9, sy. 3, 2021, ss. 108-23, doi:10.36753/mathenot.734019.
Vancouver Yaslan H. SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger’s Equations. Math. Sci. Appl. E-Notes. 2021;9(3):108-23.
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