Research Article
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Exact Solutions of Nonlinear Time Fractional Schrödinger Equation with Beta-Derivative

Year 2023, Volume: 4 Issue: 1, 1 - 8, 31.01.2023
https://doi.org/10.54974/fcmathsci.1083724

Abstract

This article consists of Improved Bernoulli Sub-Equation Function Method (IBSEFM) to get the new solutions of nonlinear fractional Schrödinger equation described by beta-derivative. Foremost, it is dealt with derivative of Atangana. Secondly, basic properties of the IBSEFM are given. Finally, the proposed method has been applicated to the considered equation to get its new solutions. Moreover, the graphs of the obtained solutions are plotted via Mathematica. It is inferred from the results that IBSEFM is effectual technique for new solutions of nonlinear equations containing conformable derivatives.

References

  • Atangana A., Derivative with a New Parameter: Theory, Methods and Applications, Academic Publish Press, 2015.
  • Atangana A., Alqahtani R.T., Modelling the spread of river blindness disease via the Caputo fractional derivative and the beta-derivative, Entropy, 18(2), 40, 2016.
  • Atangana A., Alkahtani B.S.T., Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter, Complexity, 21(6), 442-451, 2016.
  • Atangana A., Baleanu D., New fractional derivatives with nonlocal and nonsingular kernel: Theory and application to heat transfer model, Thermal Science, 20(2), 763-769, 2016.
  • Ala V., Demirbilek U., Mamedov Kh.R., An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation, AIMS Mathematics, 5(4), 3751- 3761, 2020.
  • Ala V., Demirbilek U., Mamedov Kh.R., On the exact solutions to conformable equal width wave equation by improved Bernoulli sub-equation function method, Bulletin of the SUSU, Series Math. Mech. Phy., 13(3), 5-13, 2021.
  • Atangana A., Baleanu D., Alsaedi A., Analysis of time-fractional Hunter-Saxton equation: A model of neumatic liquid crystal, Open Physics 14, 145-149, 2016.
  • Baskonus H.M., Bulut H., On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method, Waves in Random and Complex Media, 66(3), 720-728, 2015.
  • Caputo M., Fabrizio M., A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1(2), 73-85, 2015.
  • Cenesiz Y., Kurt A., The solution of time fractional heat equation with new fractional derivative definition, 8th International Conference on Applied Mathematics, Simulation and Modelling, 195, 2014.
  • Gürefe Y., The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative, Revista Mexicana de Fisica, 66(6), 771-781, 2020.
  • Gomez-Aguilar J.F., Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel, Physica A: Statistical Mechanics and its Applications, 465(1), 562-572, 2017.
  • Jumarie G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Computers and Mathematics with Applications, 51(9-10), 1367-1376, 2006.
  • Khalil R., Al Horani M., Yousef A., Sababheh M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65-70, 2014.
  • Kumar D., Singh J., Baleanu D., A hybrid computational approach for Klein-Gordon equations on Cantor sets, Nonlinear Dynamics, 87, 511-517, 2017.
  • Owolabi K.M., Atangana A., Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction diffusion systems, Computational and Applied Mathematics, 37, 2166–2189, 2018.
  • Owolabi K.M., Atangana A., Numerical simulation of noninteger order system in subdiffusive, diffusive and super diffusive scenarios, Journal of Computational and Nonlinear Dynamics, 12(3), 031010, 2017.
  • Pandir Y., Gurefe Y., Misirli E., New exact solutions of the time-fractional nonlinear dispersive KdV equation, International Journal of Modeling and Optimization, 3(4), 349-352, 2013.
  • Podlubny I., Fractional Differential Equations, Academic Press, 1999.
  • Senol M., New analytical solutions of fractional symmetric regularized-long-wave equation, Revista Mexicana de Fisica, 66(3), 297-307, 2020.
  • Srivastava H.M., Kumar D., Singh J., An efficient analytical technique for fractional model of vibration equation, Applied Mathematical Modelling, 45, 192-204, 2017.
  • Yepez-Martinez H., Gomez-Aguilar J.F., Atangana A., First integral method for nonlinear differential equations with conformable derivative, Mathematical Modelling of Natural Phenomena, 13(1), 14, 2018.
  • Yepez-Martinez H., Gomez-Aguilar J.F., Fractional subequation method for Hirota-Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative, Waves in Random and Complex Media, 29(4), 678-693, 2019.
  • Yepez-Martinez H., Gomez-Aguilar J.F., Atangana A., Optical solitons solution of resonance nonlinear Schrödinger type equation with Atangana’s-conformable derivative using sub-equation method, Waves in Random and Complex Media, 31(3), 573-596, 2021.
Year 2023, Volume: 4 Issue: 1, 1 - 8, 31.01.2023
https://doi.org/10.54974/fcmathsci.1083724

Abstract

References

  • Atangana A., Derivative with a New Parameter: Theory, Methods and Applications, Academic Publish Press, 2015.
  • Atangana A., Alqahtani R.T., Modelling the spread of river blindness disease via the Caputo fractional derivative and the beta-derivative, Entropy, 18(2), 40, 2016.
  • Atangana A., Alkahtani B.S.T., Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter, Complexity, 21(6), 442-451, 2016.
  • Atangana A., Baleanu D., New fractional derivatives with nonlocal and nonsingular kernel: Theory and application to heat transfer model, Thermal Science, 20(2), 763-769, 2016.
  • Ala V., Demirbilek U., Mamedov Kh.R., An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation, AIMS Mathematics, 5(4), 3751- 3761, 2020.
  • Ala V., Demirbilek U., Mamedov Kh.R., On the exact solutions to conformable equal width wave equation by improved Bernoulli sub-equation function method, Bulletin of the SUSU, Series Math. Mech. Phy., 13(3), 5-13, 2021.
  • Atangana A., Baleanu D., Alsaedi A., Analysis of time-fractional Hunter-Saxton equation: A model of neumatic liquid crystal, Open Physics 14, 145-149, 2016.
  • Baskonus H.M., Bulut H., On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method, Waves in Random and Complex Media, 66(3), 720-728, 2015.
  • Caputo M., Fabrizio M., A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1(2), 73-85, 2015.
  • Cenesiz Y., Kurt A., The solution of time fractional heat equation with new fractional derivative definition, 8th International Conference on Applied Mathematics, Simulation and Modelling, 195, 2014.
  • Gürefe Y., The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative, Revista Mexicana de Fisica, 66(6), 771-781, 2020.
  • Gomez-Aguilar J.F., Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel, Physica A: Statistical Mechanics and its Applications, 465(1), 562-572, 2017.
  • Jumarie G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Computers and Mathematics with Applications, 51(9-10), 1367-1376, 2006.
  • Khalil R., Al Horani M., Yousef A., Sababheh M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65-70, 2014.
  • Kumar D., Singh J., Baleanu D., A hybrid computational approach for Klein-Gordon equations on Cantor sets, Nonlinear Dynamics, 87, 511-517, 2017.
  • Owolabi K.M., Atangana A., Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction diffusion systems, Computational and Applied Mathematics, 37, 2166–2189, 2018.
  • Owolabi K.M., Atangana A., Numerical simulation of noninteger order system in subdiffusive, diffusive and super diffusive scenarios, Journal of Computational and Nonlinear Dynamics, 12(3), 031010, 2017.
  • Pandir Y., Gurefe Y., Misirli E., New exact solutions of the time-fractional nonlinear dispersive KdV equation, International Journal of Modeling and Optimization, 3(4), 349-352, 2013.
  • Podlubny I., Fractional Differential Equations, Academic Press, 1999.
  • Senol M., New analytical solutions of fractional symmetric regularized-long-wave equation, Revista Mexicana de Fisica, 66(3), 297-307, 2020.
  • Srivastava H.M., Kumar D., Singh J., An efficient analytical technique for fractional model of vibration equation, Applied Mathematical Modelling, 45, 192-204, 2017.
  • Yepez-Martinez H., Gomez-Aguilar J.F., Atangana A., First integral method for nonlinear differential equations with conformable derivative, Mathematical Modelling of Natural Phenomena, 13(1), 14, 2018.
  • Yepez-Martinez H., Gomez-Aguilar J.F., Fractional subequation method for Hirota-Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative, Waves in Random and Complex Media, 29(4), 678-693, 2019.
  • Yepez-Martinez H., Gomez-Aguilar J.F., Atangana A., Optical solitons solution of resonance nonlinear Schrödinger type equation with Atangana’s-conformable derivative using sub-equation method, Waves in Random and Complex Media, 31(3), 573-596, 2021.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Volkan Ala 0000-0002-8499-9979

Publication Date January 31, 2023
Published in Issue Year 2023 Volume: 4 Issue: 1

Cite

19113 FCMS is licensed under the Creative Commons Attribution 4.0 International Public License.