Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation

Sevil Çulha Ünal; Ayşegül Daşcıoğlu; Dilek Varol Bayram

doi:10.36753/mathenot.688493

Research Article

Year 2021, Volume: 9 Issue: 2, 53 - 63, 01.06.2021

Sevil Çulha Ünal Ayşegül Daşcıoğlu Dilek Varol Bayram

https://doi.org/10.36753/mathenot.688493

Cited By: 10

Abstract

References

[1] Seyler, C. E., Fenstermacher, D. L.: A symmetric regularized-long-wave equation. Physics of Fluids. 27 (1), (1984).
[2] Ahmadian, S., Darvishi, M. T.: New exact traveling wave solutions for space-time fractional (1+1)-dimensional SRLW equation. Optik. 127, 10697–10704 (2010).
[3] Wang, T., Zhang, L., Chen, F.: Conservative schemes for the symmetric regularized long wave equations. Appl. Math. Comput. 190, 1063–1080 (2007).
[4] Xu, F.: Application of Exp-function method to symmetric regularized long wave (SRLW) equation. Phys. Lett. A. 372, 252–257 (2008).
[5] Abazari, R.: Application of $(G'/G)$-expansion method to traveling wave solutions of three nonlinear evolution equations. Comput. Fluids., 39, 1957–1963 (2010).
[6] Jafari, H., Kadkhoda, N., Khalique, C.M.: Travelling wave solutions of nonlinear evolution equations using the simplest equation method. Comput. Math. Appl., 64, 2084–2088 (2012).
[7] Hu, J., Zheng, K., Zheng, M.: Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation. Appl. Math. Modelling., 38, 5573–5581 (2014).
[8] Ugurlu, Y., Kaya, D., Inan, I. E.: Generalized Jacobi Elliptic Function Method for Periodic Wave Solutions of SRLW Equation and (1+1)-Dimensional Dispersive Long Wave Equation. Çankaya Univ. J. Sci. Eng., 8(2), 205-223 (2011).
[9] Yimnet, S., Wongsaijai, B., Rojsiraphisal, T. and Poochinapan, K.: Numerical implementation for solving the symmetric regularized long wave equation. Appl. Math. Comput. 273, 809–825 (2016).
[10] Alzaidy, J. F.: The fractional sub-equation method and exact analytical solutions for some nonlinear fractional PDEs. Amer. J. Math. Anal. 1 (1), 14-19 (2013).
[11] Guner, O., Eser, D.: Exact solutions of the space time fractional symmetric regularized long wave equation using different methods. Adv. Math. Physics. 2014, Article ID 456804 (2014).
[12] Shakeel M., Mohyud-Din, S. T.: A novel $(G'/G)$-expansion method and its application to the space-time fractional symmetric regularized long wave (SRLW) equation. Adv. Trends Math., 2, 1-16 (2015).
[13] Zhouzheng, K.: Infinite sequence solutions for space-time fractional symmetric regularized long wave equation. J. Partial Differential Equations, 29, 48-58 (2016).
[14] Islam, T., Akbar, M. A., Azad, A. K.: Travelling Wave Solutions to Some Nonlinear Fractional Partial Differential Equations Through the Rational $(G'/G)$-expansion Method. J. Ocean Eng. Sci., 3, 76-81 (2018).
[15] Yaro, D., Seadawy, A. R., Lu, D., ApeantiW. O., Akuamoah, S.W.: Dispersive Wave Solutions of the Nonlinear fractional Zakhorov-Kuznetsov-Benjamin-Bono-Mahony Equation and Fractional Symmetric Regularized Long Wave Equation. Results Phys. 12, 1971-1979 (2019).
[16] Sonmezoglu, A.: Exact Solutions for Some Fractional Differential Equations. Adv. Math. Phys. 2015, Article ID 567842 (2015).
[17] Akbulut, A., Kaplan M., Bekir, A.: Auxiliary equation method for fractional differential equations with modified Riemann-Liouville derivative. Int. J. Nonlin. Sci. Num. (2016). DOI: 10.1515/ijnsns-2016-0023.
[18] Senol, M.: New analytical solutions of fractional symmetric regularized-long-wave equation. Rev. Mex. Fis. E, 66(3), 297–307 (2020).
[19] Ala, V., Demirbilek, U., Mamedov, K. R.: An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation. AIMS Math., 5 (4), 3751–3761 (2020).
[20] Ali, K. K., Nuruddeen, R. I., Raslan, K. R.: New structures for the space-time fractional simplified MCH and SRLW equations. Chaos Solitons Fractals. 106, 304-309 (2018).
[21] Jebreen, H. B.: Some Nonlinear Fractional PDEs Involving -Derivative by Using Rational $exp(-\Omega(\nu))$-Expansion Method. Complexity. 2020, Article ID 9179826 (2020). https://doi.org/10.1155/2020/9179826
[22] Hanif, M., Habib, M. A.: Exact solitary wave solutions for a system of some nonlinear space–time fractional differential equations. Pramana - J. Phys. 94, 7 (2020).
[23] Manafian, J., Ilhan, O. A., Avazpour, L.: The extended auxiliary equation mapping method to determine novel exact solitary wave solutions of the nonlinear fractional PDEs. (2020). https://doi.org/10.1515/ijnsns-2019-0279.
[24] Alhamdan, W. M., Wazzan, L.: Exact Solutions for the Space-Time Fractional SRLW and STO Equations by the $(D^\alpha G)/G$-Expansion Method. American J. Appl. Math. Stat., 4(3), 87-93 (2016).
[25] Ege, S. M., Mısırlı, E.: Traveling wave solutions of some fractional differential equations. Romanian J. Math. Comput. Sci., 6(1), 106-115 (2016).
[26] Zayed, E. M. E., Amer, Y. A., Shohib, R. M. A.: The fractional $(D^{\alpha}_{\xi}/G)$-expansion method and its applications for solving four nonlinear space-time fractional PDES in mathematical physics. Ital. J. Pure Appl. Math., 34, 463-482 (2015).
[27] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Higher Transcendental Functions, vol. 2. McGraw-Hill. New York (1953).
[28] Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical tables. Dover. Newyork (1972).
[29] Khalil, R., Horani, M. A., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65-70 (2014).
[30] Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57-66 (2015).
[31] Kovacic, I., Brennan, M. J.: The Duffing Equation: Nonlinear Oscillators and their Behaviour. JohnWiley and Sons. United Kingdom (2011).
[32] Daşcıoglu, A., Çulha, S., Varol Bayram, D.: New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions. New Trends Math. Sci. 5(4), 232-241 (2017).
[33] Çulha S., Dascıoglu, A.: Analytic solutions of the space–time conformable fractional Klein–Gordon equation in general form.Wave Random Complex 29(4), 775-790 (2019).
[34] Çulha Ünal, S., Daşcıoglu, A., Varol Bayram, D.: New exact solutions of space and time fractional modified Kawahara equation. Phys. A, 551, 124550 (2020).
[35] Dascıoglu, A., Çulha Ünal, S., Varol Bayram, D.: New analytical solutions for space and time Fractional Phi-4 equation. Naturengs, MTU J. Eng. Nat. Sci., 1(1), 30-46 (2020).
[36] Dascıoglu, A., Çulha Ünal, S.: New exact solutions for the space-time fractional Kawahara equation. Appl. Math. Model., 89, 952-965 (2021).

Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation

Year 2021, Volume: 9 Issue: 2, 53 - 63, 01.06.2021

Sevil Çulha Ünal Ayşegül Daşcıoğlu Dilek Varol Bayram

https://doi.org/10.36753/mathenot.688493

Cited By: 10

Abstract

In this paper, by using a direct method based on the Jacobi elliptic functions, the exact solutions of the space-time fractional symmetric regularized long wave (SRLW) equation have been obtained. The elliptic function solutions of a nonlinear ordinary differential (auxiliary) equation $\left({dF}/{d \xi}\right) ^{2} = PF^{4} (\xi)+QF^{2} (\xi) + R$ have also been examined. Besides, the solutions have been found in general form including rational, trigonometric and hyperbolic functions. Moreover, the complex valued solutions, periodic solutions, and soliton solutions, have also been gained. Some solutions have been illustrated by the graphics.

Keywords

SRLW equation, Jacobi elliptic function, Nonlinear differential equation, Fractional partial differential equation

References

[1] Seyler, C. E., Fenstermacher, D. L.: A symmetric regularized-long-wave equation. Physics of Fluids. 27 (1), (1984).
[2] Ahmadian, S., Darvishi, M. T.: New exact traveling wave solutions for space-time fractional (1+1)-dimensional SRLW equation. Optik. 127, 10697–10704 (2010).
[3] Wang, T., Zhang, L., Chen, F.: Conservative schemes for the symmetric regularized long wave equations. Appl. Math. Comput. 190, 1063–1080 (2007).
[4] Xu, F.: Application of Exp-function method to symmetric regularized long wave (SRLW) equation. Phys. Lett. A. 372, 252–257 (2008).
[5] Abazari, R.: Application of $(G'/G)$-expansion method to traveling wave solutions of three nonlinear evolution equations. Comput. Fluids., 39, 1957–1963 (2010).
[6] Jafari, H., Kadkhoda, N., Khalique, C.M.: Travelling wave solutions of nonlinear evolution equations using the simplest equation method. Comput. Math. Appl., 64, 2084–2088 (2012).
[7] Hu, J., Zheng, K., Zheng, M.: Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation. Appl. Math. Modelling., 38, 5573–5581 (2014).
[8] Ugurlu, Y., Kaya, D., Inan, I. E.: Generalized Jacobi Elliptic Function Method for Periodic Wave Solutions of SRLW Equation and (1+1)-Dimensional Dispersive Long Wave Equation. Çankaya Univ. J. Sci. Eng., 8(2), 205-223 (2011).
[9] Yimnet, S., Wongsaijai, B., Rojsiraphisal, T. and Poochinapan, K.: Numerical implementation for solving the symmetric regularized long wave equation. Appl. Math. Comput. 273, 809–825 (2016).
[10] Alzaidy, J. F.: The fractional sub-equation method and exact analytical solutions for some nonlinear fractional PDEs. Amer. J. Math. Anal. 1 (1), 14-19 (2013).
[11] Guner, O., Eser, D.: Exact solutions of the space time fractional symmetric regularized long wave equation using different methods. Adv. Math. Physics. 2014, Article ID 456804 (2014).
[12] Shakeel M., Mohyud-Din, S. T.: A novel $(G'/G)$-expansion method and its application to the space-time fractional symmetric regularized long wave (SRLW) equation. Adv. Trends Math., 2, 1-16 (2015).
[13] Zhouzheng, K.: Infinite sequence solutions for space-time fractional symmetric regularized long wave equation. J. Partial Differential Equations, 29, 48-58 (2016).
[14] Islam, T., Akbar, M. A., Azad, A. K.: Travelling Wave Solutions to Some Nonlinear Fractional Partial Differential Equations Through the Rational $(G'/G)$-expansion Method. J. Ocean Eng. Sci., 3, 76-81 (2018).
[15] Yaro, D., Seadawy, A. R., Lu, D., ApeantiW. O., Akuamoah, S.W.: Dispersive Wave Solutions of the Nonlinear fractional Zakhorov-Kuznetsov-Benjamin-Bono-Mahony Equation and Fractional Symmetric Regularized Long Wave Equation. Results Phys. 12, 1971-1979 (2019).
[16] Sonmezoglu, A.: Exact Solutions for Some Fractional Differential Equations. Adv. Math. Phys. 2015, Article ID 567842 (2015).
[17] Akbulut, A., Kaplan M., Bekir, A.: Auxiliary equation method for fractional differential equations with modified Riemann-Liouville derivative. Int. J. Nonlin. Sci. Num. (2016). DOI: 10.1515/ijnsns-2016-0023.
[18] Senol, M.: New analytical solutions of fractional symmetric regularized-long-wave equation. Rev. Mex. Fis. E, 66(3), 297–307 (2020).
[19] Ala, V., Demirbilek, U., Mamedov, K. R.: An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation. AIMS Math., 5 (4), 3751–3761 (2020).
[20] Ali, K. K., Nuruddeen, R. I., Raslan, K. R.: New structures for the space-time fractional simplified MCH and SRLW equations. Chaos Solitons Fractals. 106, 304-309 (2018).
[21] Jebreen, H. B.: Some Nonlinear Fractional PDEs Involving -Derivative by Using Rational $exp(-\Omega(\nu))$-Expansion Method. Complexity. 2020, Article ID 9179826 (2020). https://doi.org/10.1155/2020/9179826
[22] Hanif, M., Habib, M. A.: Exact solitary wave solutions for a system of some nonlinear space–time fractional differential equations. Pramana - J. Phys. 94, 7 (2020).
[23] Manafian, J., Ilhan, O. A., Avazpour, L.: The extended auxiliary equation mapping method to determine novel exact solitary wave solutions of the nonlinear fractional PDEs. (2020). https://doi.org/10.1515/ijnsns-2019-0279.
[24] Alhamdan, W. M., Wazzan, L.: Exact Solutions for the Space-Time Fractional SRLW and STO Equations by the $(D^\alpha G)/G$-Expansion Method. American J. Appl. Math. Stat., 4(3), 87-93 (2016).
[25] Ege, S. M., Mısırlı, E.: Traveling wave solutions of some fractional differential equations. Romanian J. Math. Comput. Sci., 6(1), 106-115 (2016).
[26] Zayed, E. M. E., Amer, Y. A., Shohib, R. M. A.: The fractional $(D^{\alpha}_{\xi}/G)$-expansion method and its applications for solving four nonlinear space-time fractional PDES in mathematical physics. Ital. J. Pure Appl. Math., 34, 463-482 (2015).
[27] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Higher Transcendental Functions, vol. 2. McGraw-Hill. New York (1953).
[28] Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical tables. Dover. Newyork (1972).
[29] Khalil, R., Horani, M. A., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65-70 (2014).
[30] Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57-66 (2015).
[31] Kovacic, I., Brennan, M. J.: The Duffing Equation: Nonlinear Oscillators and their Behaviour. JohnWiley and Sons. United Kingdom (2011).
[32] Daşcıoglu, A., Çulha, S., Varol Bayram, D.: New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions. New Trends Math. Sci. 5(4), 232-241 (2017).
[33] Çulha S., Dascıoglu, A.: Analytic solutions of the space–time conformable fractional Klein–Gordon equation in general form.Wave Random Complex 29(4), 775-790 (2019).
[34] Çulha Ünal, S., Daşcıoglu, A., Varol Bayram, D.: New exact solutions of space and time fractional modified Kawahara equation. Phys. A, 551, 124550 (2020).
[35] Dascıoglu, A., Çulha Ünal, S., Varol Bayram, D.: New analytical solutions for space and time Fractional Phi-4 equation. Naturengs, MTU J. Eng. Nat. Sci., 1(1), 30-46 (2020).
[36] Dascıoglu, A., Çulha Ünal, S.: New exact solutions for the space-time fractional Kawahara equation. Appl. Math. Model., 89, 952-965 (2021).

There are 36 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Articles
Authors	Sevil Çulha Ünal 0000-0001-7447-9219 Ayşegül Daşcıoğlu 0000-0001-8931-6930 Dilek Varol Bayram 0000-0002-5158-5614
Publication Date	June 1, 2021
Submission Date	February 12, 2020
Acceptance Date	October 30, 2020
Published in Issue	Year 2021 Volume: 9 Issue: 2

Cite

APA	Çulha Ünal, S., Daşcıoğlu, A., & Varol Bayram, D. (2021). Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation. Mathematical Sciences and Applications E-Notes, 9(2), 53-63. https://doi.org/10.36753/mathenot.688493
AMA	Çulha Ünal S, Daşcıoğlu A, Varol Bayram D. Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation. Math. Sci. Appl. E-Notes. June 2021;9(2):53-63. doi:10.36753/mathenot.688493
Chicago	Çulha Ünal, Sevil, Ayşegül Daşcıoğlu, and Dilek Varol Bayram. “Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation”. Mathematical Sciences and Applications E-Notes 9, no. 2 (June 2021): 53-63. https://doi.org/10.36753/mathenot.688493.
EndNote	Çulha Ünal S, Daşcıoğlu A, Varol Bayram D (June 1, 2021) Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation. Mathematical Sciences and Applications E-Notes 9 2 53–63.
IEEE	S. Çulha Ünal, A. Daşcıoğlu, and D. Varol Bayram, “Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation”, Math. Sci. Appl. E-Notes, vol. 9, no. 2, pp. 53–63, 2021, doi: 10.36753/mathenot.688493.
ISNAD	Çulha Ünal, Sevil et al. “Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation”. Mathematical Sciences and Applications E-Notes 9/2 (June 2021), 53-63. https://doi.org/10.36753/mathenot.688493.
JAMA	Çulha Ünal S, Daşcıoğlu A, Varol Bayram D. Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation. Math. Sci. Appl. E-Notes. 2021;9:53–63.
MLA	Çulha Ünal, Sevil et al. “Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 2, 2021, pp. 53-63, doi:10.36753/mathenot.688493.
Vancouver	Çulha Ünal S, Daşcıoğlu A, Varol Bayram D. Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation. Math. Sci. Appl. E-Notes. 2021;9(2):53-6.