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New analytical solutions for space and time fractional Phi-4 equation

Year 2020, Volume: 1 Issue: 1, 30 - 46, 01.06.2020

Ayşegül Daşcıoğlu Sevil Çulha Ünal Dilek Varol Bayram

Abstract

In this paper, by utilizing an analytical method based on the Jacobi elliptic functions, the largest set in the literature of time, space and space-time conformable fractional Phi-4 equations is found. These solutions are obtained in general form including rational, trigonometric, hyperbolic and complex functions. Four problems are presented to illustrate the application of the method and some of the solutions are also demonstrated by the graphics.

Keywords

Jacobi elliptic functions Space-time fractional Phi-4 equation Conformable fractional derivative Analytic method

References

  • [1] Lundquist, S., March, N. H. and Tosi, M. P., (1988), Order and Chaos in Nonlinear Physical systems, Springer, New York.
  • [2] Calogero, F. and Degasperis, A., (1982), Spectral Transform and Solitons: Tools to solve and Investigate Nonlinear Evolution Equations, New York: North-Holland.
  • [3] Wazwaz, A. M., (2007), Analytic study on nonlinear variants of the RLW and the PHI-four equations, Communications in Nonlinear Science and Numerical Simulation 12, 314–327.
  • [4] Wazwaz, A. M., (2005), Generalized forms of the phi-four equation with compactons, solitons and periodic solutions, Mathematics and Computers in Simulation, 69, 580–588.
  • [5] Alofi, A. S., (2013), Exact and Explicit Traveling Wave Solutions for the Nonlinear Partial Differntial Equations, World Applied Sciences Journal, 21, 62-67.
  • [6] Soliman, A. A., (2007), Exact travelling wave solution of nonlinear variants of the RLW and the PHI-four equations, Physics Letters A, 368, 383–390.
  • [7] Deng, X., Zhao, M. and Li, X., (2009), Travelling wave solutions for a nonlinear variant of the PHI-four equation, Mathematical and Computer Modelling, 49, 617–622.
  • [8] Xiang, C., (2012), A note on exact travelling wave solutions for nonlinear PHI-four equation, Applied Mechanics and Materials, 4569-4572.
  • [9] Najafi, M., (2012), Using He’s Variational Method to Seek the Traveling Wave Solution of PHI-Four Equation, International Journal of Applied Mathematical Research, 1 (4), 659-665.
  • [10] Younis, M. and Zafar, A., (2013), The modified simple equation method for solving nonlinear Phi-Four equation, International Journal of Innovation and Applied Studies, 2, 661-664.
  • [11] Akter, J. and Akbar, M. A., (2015), Exact solutions to the Benney–Luke equation and the Phi-4 equations by using modified simple equation method, Results in Physics, 5, 125-130.
  • [12] Ehsani, F., Ehsani, F., Hadi, A. and Hadi, A., (2013), Analytical Solution of Phi-Four Equation, Technical Journal of Engineering and Applied Sciences, 3 (14), 1378-1388.
  • [13] Akbulut, A., Kaplan, M. and Tascan, F., (2016), Conservation laws and Exact Solutions of Phi-Four (Phi-4) Equation via the (G^'⁄G,1⁄G )- Expansion Method, Naturforsch, 71 (5), 439-446.
  • [14] Zahra, W. K., (2017), Trigonometric B-Spline Collocation Method for Solving PHI-Four and Allen–Cahn Equations, Mediterranean Journal of Mathematics, 14, 122-141.
  • [15] Mahmud, F., Samsuzzoha, M. and Akbar, M. A., (2017), A generalized Kudrashov method to obtain exact travelling wave solutions of the PHI-four equation and the Fisher equation, Results in Physics, 7, 4296-4302.
  • [16] Islam, M. S., Khan, K. and Akbar, M. A., (2017), Application of the improved F-expansion method with Riccati equation to find the exact solution of the nonlinear evolution equations, Journal of the Egyptian Mathematical Society, 25, 13–18.
  • [17] Tariq, H. and Akram, G., (2017), New approach for exact solutions of time fractional Cahn–Allen equation and time fractional Phi-4 equation, Physica A, 473, 352–362.
  • [18] Alquran, M., Jaradat, H. M. and Syam, M. I., (2017), Analytical solution of the time-fractional Phi-4 equation by using modified residual power series method, Nonlinear Dyn, 90:2525–2529.
  • [19] Akram, G., Batool, F. and Riaz, A., (2018), Two reliable techniques for the analytical study of conformable time-fractional Phi-4 equation, Opt Quant Electron, 50:22.
  • [20] Rezazadeh, H., Tariq, H., Eslami, M., Mirzazadeh, M., and Zhou, Q., (2018), New exact solutions of nonlinear conformable time-fractional Phi-4 equation, Chinese Journal of Physics, 2, in press.
  • [21] Korpınar, Z., (2019), Some analytical solutions by mapping methods for non-linear conformable time-fractıonal Phi-4 equation, Thermal Science, 23.
  • [22] Sirisubtawee, S., Koonprasert, S., Sungnul, S. and Leekparn, T., (2019), Exact traveling wave solutions of the space–time fractional complex Ginzburg–Landau equation and the space-time fractional Phi-4 equation using reliable methods, Advances in Difference Equations, Doi: 10.1186/s13662-019-2154-9.
  • [23] Khalil, R., Horani, M. A., Yousef, A. and Sababheh, M., (2014), A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65–70.
  • [24] Abdeljawad, T., (2015), On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279, 57–66.
  • [25] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., (1993), Higher Transcendental Functions, vol. 2, McGraw-Hill: New York.
  • [26] Abramowitz, M. and Stegun, I. A., (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Government Printing Office: Washington, D.C.
  • [27] Ali, A. T., (2011), New generalized Jacobi elliptic function rational expansion method, Journal of Computational and Applied Mathematics, 235, 4117–4127.
  • [28] Dascioglu, A., Culha, S., and Varol Bayram, D., (2017), New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions, New Trends in Mathematical Sciences, 5, No. 4, 232-241.
  • [29] Çulha, S. and Daşcıoğlu, A., (2019), Analytic solutions of the space–time conformable fractional Klein–Gordon equation in general form, Waves in Random and complex Media, 29, No. 4, 775-790.
  • [30] Remoissenet, M., (1993), Waves Called Solitons: Concepts and Experiments, Springer-Verlag, Berlin.

Year 2020, Volume: 1 Issue: 1, 30 - 46, 01.06.2020

Ayşegül Daşcıoğlu Sevil Çulha Ünal Dilek Varol Bayram

Abstract

References

  • [1] Lundquist, S., March, N. H. and Tosi, M. P., (1988), Order and Chaos in Nonlinear Physical systems, Springer, New York.
  • [2] Calogero, F. and Degasperis, A., (1982), Spectral Transform and Solitons: Tools to solve and Investigate Nonlinear Evolution Equations, New York: North-Holland.
  • [3] Wazwaz, A. M., (2007), Analytic study on nonlinear variants of the RLW and the PHI-four equations, Communications in Nonlinear Science and Numerical Simulation 12, 314–327.
  • [4] Wazwaz, A. M., (2005), Generalized forms of the phi-four equation with compactons, solitons and periodic solutions, Mathematics and Computers in Simulation, 69, 580–588.
  • [5] Alofi, A. S., (2013), Exact and Explicit Traveling Wave Solutions for the Nonlinear Partial Differntial Equations, World Applied Sciences Journal, 21, 62-67.
  • [6] Soliman, A. A., (2007), Exact travelling wave solution of nonlinear variants of the RLW and the PHI-four equations, Physics Letters A, 368, 383–390.
  • [7] Deng, X., Zhao, M. and Li, X., (2009), Travelling wave solutions for a nonlinear variant of the PHI-four equation, Mathematical and Computer Modelling, 49, 617–622.
  • [8] Xiang, C., (2012), A note on exact travelling wave solutions for nonlinear PHI-four equation, Applied Mechanics and Materials, 4569-4572.
  • [9] Najafi, M., (2012), Using He’s Variational Method to Seek the Traveling Wave Solution of PHI-Four Equation, International Journal of Applied Mathematical Research, 1 (4), 659-665.
  • [10] Younis, M. and Zafar, A., (2013), The modified simple equation method for solving nonlinear Phi-Four equation, International Journal of Innovation and Applied Studies, 2, 661-664.
  • [11] Akter, J. and Akbar, M. A., (2015), Exact solutions to the Benney–Luke equation and the Phi-4 equations by using modified simple equation method, Results in Physics, 5, 125-130.
  • [12] Ehsani, F., Ehsani, F., Hadi, A. and Hadi, A., (2013), Analytical Solution of Phi-Four Equation, Technical Journal of Engineering and Applied Sciences, 3 (14), 1378-1388.
  • [13] Akbulut, A., Kaplan, M. and Tascan, F., (2016), Conservation laws and Exact Solutions of Phi-Four (Phi-4) Equation via the (G^'⁄G,1⁄G )- Expansion Method, Naturforsch, 71 (5), 439-446.
  • [14] Zahra, W. K., (2017), Trigonometric B-Spline Collocation Method for Solving PHI-Four and Allen–Cahn Equations, Mediterranean Journal of Mathematics, 14, 122-141.
  • [15] Mahmud, F., Samsuzzoha, M. and Akbar, M. A., (2017), A generalized Kudrashov method to obtain exact travelling wave solutions of the PHI-four equation and the Fisher equation, Results in Physics, 7, 4296-4302.
  • [16] Islam, M. S., Khan, K. and Akbar, M. A., (2017), Application of the improved F-expansion method with Riccati equation to find the exact solution of the nonlinear evolution equations, Journal of the Egyptian Mathematical Society, 25, 13–18.
  • [17] Tariq, H. and Akram, G., (2017), New approach for exact solutions of time fractional Cahn–Allen equation and time fractional Phi-4 equation, Physica A, 473, 352–362.
  • [18] Alquran, M., Jaradat, H. M. and Syam, M. I., (2017), Analytical solution of the time-fractional Phi-4 equation by using modified residual power series method, Nonlinear Dyn, 90:2525–2529.
  • [19] Akram, G., Batool, F. and Riaz, A., (2018), Two reliable techniques for the analytical study of conformable time-fractional Phi-4 equation, Opt Quant Electron, 50:22.
  • [20] Rezazadeh, H., Tariq, H., Eslami, M., Mirzazadeh, M., and Zhou, Q., (2018), New exact solutions of nonlinear conformable time-fractional Phi-4 equation, Chinese Journal of Physics, 2, in press.
  • [21] Korpınar, Z., (2019), Some analytical solutions by mapping methods for non-linear conformable time-fractıonal Phi-4 equation, Thermal Science, 23.
  • [22] Sirisubtawee, S., Koonprasert, S., Sungnul, S. and Leekparn, T., (2019), Exact traveling wave solutions of the space–time fractional complex Ginzburg–Landau equation and the space-time fractional Phi-4 equation using reliable methods, Advances in Difference Equations, Doi: 10.1186/s13662-019-2154-9.
  • [23] Khalil, R., Horani, M. A., Yousef, A. and Sababheh, M., (2014), A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65–70.
  • [24] Abdeljawad, T., (2015), On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279, 57–66.
  • [25] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., (1993), Higher Transcendental Functions, vol. 2, McGraw-Hill: New York.
  • [26] Abramowitz, M. and Stegun, I. A., (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Government Printing Office: Washington, D.C.
  • [27] Ali, A. T., (2011), New generalized Jacobi elliptic function rational expansion method, Journal of Computational and Applied Mathematics, 235, 4117–4127.
  • [28] Dascioglu, A., Culha, S., and Varol Bayram, D., (2017), New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions, New Trends in Mathematical Sciences, 5, No. 4, 232-241.
  • [29] Çulha, S. and Daşcıoğlu, A., (2019), Analytic solutions of the space–time conformable fractional Klein–Gordon equation in general form, Waves in Random and complex Media, 29, No. 4, 775-790.
  • [30] Remoissenet, M., (1993), Waves Called Solitons: Concepts and Experiments, Springer-Verlag, Berlin.
There are 30 citations in total.

Details

Primary Language English
Journal Section Research Articles
Authors

Ayşegül Daşcıoğlu 0000-0001-8931-6930

Sevil Çulha Ünal 0000-0001-7447-9219

Dilek Varol Bayram 0000-0002-5158-5614

Publication Date June 1, 2020
Submission Date March 3, 2020
Acceptance Date May 18, 2020
Published in Issue Year 2020 Volume: 1 Issue: 1

Cite

APA Daşcıoğlu, A., Çulha Ünal, S., & Varol Bayram, D. (2020). New analytical solutions for space and time fractional Phi-4 equation. NATURENGS, 1(1), 30-46.
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