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An Application of Improved Bernoulli Sub-Equation Function Method to The Nonlinear Time-Fractional Burgers Equation

Year 2016, Volume: 5 , 1 - 7, 30.12.2016

Hasan Bulut Gülnur Yel Hacı Mehmet Başkonuş

Abstract

In this work, we study on the improved Bernoulli sub-equation function method. We apply this method to the nonlinear time-fractional Burgers equation. We obtain new analytical solutions to this model for values of n, m and M. Numerical simulation were depicted for di erent values of alpha.

Keywords

Improved Bernoulli sub-equation function method time-fractional Burgers equation exponential function solution

References

  • Atangana, A., On the new fractional derivative and application to nonlinear Fisher’s reaction-di ffusion equation, Applied Mathematics and Computation, 273(2016), 948-956. 2.1
  • Atangana, A., Baleanu, D., New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 1(2016), 763–769. 2.1, 2.2, 2.4
  • Atangana, A., Koca, I., Chaos in a simple nonlinear system with Atangana- Baleanu derivatives with fractionalorder, Chaos, Solitons and Fractals, 1(2016), 447–454. 2.3
  • Baskonus, H. M., Bulut, H., An e ective scheme for solving some nonlinear partial di erential equation arising in nonlinear physics, Open Physics, 13(2015), 280–289. 3
  • 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, 25(2015), 720–728. 3
  • Baskonus, H. M., Bulut, H., Exponential prototype structures for (2+1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics, Waves in Random and Complex Media, 26(2016), 201–208. 3
  • Baskonus, H. M., Bulut, H., Regarding on the prototype solutions for the nonlinear fractional-order biological population model, AIP Conference Proceedings 1738(2016), 290004. 3
  • Baskonus, H. M., Koc¸, D. A., Bulut, H., Dark and new travelling wave solutions to the nonlinear evolution equation, Optic-International Journal for Light and Electron Optics, 127(2016), 8043–8055. 3
  • Bekir, A., Guner, O., Exact solutions of nonlinear fractional di erential equations by (G’/G)-expansion method, Chinese Physics B, 22(2013), 110202. 1, 4
  • Burgers, J. M., A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, 1(1948), 171–199. 1
  • Doha, E. H., Bhrawy, A. H., Abdelkawy, M. A., Hafez, R. M., A Jacobi collocation approximation for nonlinear coupled viscous Burgers equation, Central European Journal of Physics, 12(2014), 111–122. 1
  • Gepreel, K. A., Omran, S., Exact solutions for nonlinear partial fractional di erential equations, Chinese Physics B, 21(2012), 110204. 1
  • Guo, S., Mei, L., Li, Y., Sun, Y., The improved fractional sub-equation method and its applications to the space-time fractional di erential equations in fluid mechanics, Physics Letters A, 376(2012), 407. 3
  • Hammouch, Z., Mekkaoui, T., Travelling-wave solutions for some fractional partial di erential equation by means of generalized trigonometry functions, International Journal of Applied Mathematical Research, 1(2012), 206–212. 1
  • Hammouch, Z., Mekkaoui, T., Chaos synchronization of a fractional nonautonomous system, Nonautonomous Dynamical Systems, 1(2014), 61–71. 1
  • Harris, P. A., Garra, R., Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method, Nonlinear Studies, (2013), 471–481. 1
  • He, J., Elagan, S. K., Li, Z. B., Geometrical explanation of the fractional complex transform and derivative Chain rule for fractional calculus, Physics Letters A, 376(2012) 257–259. 3
  • Hussein, A., Selim, M. M., New soliton solutions for some important nonlinear partial di erential equations using a generalized Bernoulli method, International Journal of Mathematical Analysis and Applications, 1(2014), 1–8. 3
  • Kurulay, M., The approximate and exact solutions of the space- and time- fractional Burgers equations, IJRRAS, 3(2010), 257. 1
  • Momani, S., Non-perturbative analytical solutions of the space- and time- fractional Burgers equations, Chaos, Solitons and Fractals, 28(2006), 930–937. 1
  • Moslem, W. M., Sabry, R., Zakharov-Kuznetsov-Burgers equation for Dust Ion acoustic waves, Chaos, Solitons and Fractals, 36(2008), 628. 1
  • Rashidi, M. M., Erfani, E., New analytical method for solving Burgers and nonlinear heat transfer equations and comparison with HAM, Computer Physics Communications, 180(2009), 1539. 1
  • Saad, M., Elagan, S. K., Hamed, Y. S., Sayed, M., Using a complex transformation to get an exact solutions for fractional generalized coupled MKDV and KDV equations, International Journal of Basic and Applied Sciences, 13(2013), 23–25. 3
  • Sugimoto, N., Burgers equation with a fractional derivative: Hereditary e ects on nonlinear acoustic waves, Journal of Fluid Mechanics, 225(1991), 631. 1
  • Yildirim, A., Mohyud-Din, S. T., Analytical approach to space- and time- fractional Burgers equations, Chinese Physics Letters, 27(2010), 9. 1
  • Zhang, S., Hang, H. Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters A, 375(2011), 1069. 1
  • Zhang, S., Zong, Q. A., Liu, D., Gao, Q., A Generalized exp-function method for fractional Riccati di erential equations, The Communications in Fractional Calculus, 1(2010), 48–51. 1
  • Zhen, B. H., (G’/G)- expansion method for solving fractional partial di erential equations in the theory of mathematical physics, Communication Theory Physics, 58(2012), 623. 1

Year 2016, Volume: 5 , 1 - 7, 30.12.2016

Hasan Bulut Gülnur Yel Hacı Mehmet Başkonuş

Abstract

References

  • Atangana, A., On the new fractional derivative and application to nonlinear Fisher’s reaction-di ffusion equation, Applied Mathematics and Computation, 273(2016), 948-956. 2.1
  • Atangana, A., Baleanu, D., New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 1(2016), 763–769. 2.1, 2.2, 2.4
  • Atangana, A., Koca, I., Chaos in a simple nonlinear system with Atangana- Baleanu derivatives with fractionalorder, Chaos, Solitons and Fractals, 1(2016), 447–454. 2.3
  • Baskonus, H. M., Bulut, H., An e ective scheme for solving some nonlinear partial di erential equation arising in nonlinear physics, Open Physics, 13(2015), 280–289. 3
  • 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, 25(2015), 720–728. 3
  • Baskonus, H. M., Bulut, H., Exponential prototype structures for (2+1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics, Waves in Random and Complex Media, 26(2016), 201–208. 3
  • Baskonus, H. M., Bulut, H., Regarding on the prototype solutions for the nonlinear fractional-order biological population model, AIP Conference Proceedings 1738(2016), 290004. 3
  • Baskonus, H. M., Koc¸, D. A., Bulut, H., Dark and new travelling wave solutions to the nonlinear evolution equation, Optic-International Journal for Light and Electron Optics, 127(2016), 8043–8055. 3
  • Bekir, A., Guner, O., Exact solutions of nonlinear fractional di erential equations by (G’/G)-expansion method, Chinese Physics B, 22(2013), 110202. 1, 4
  • Burgers, J. M., A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, 1(1948), 171–199. 1
  • Doha, E. H., Bhrawy, A. H., Abdelkawy, M. A., Hafez, R. M., A Jacobi collocation approximation for nonlinear coupled viscous Burgers equation, Central European Journal of Physics, 12(2014), 111–122. 1
  • Gepreel, K. A., Omran, S., Exact solutions for nonlinear partial fractional di erential equations, Chinese Physics B, 21(2012), 110204. 1
  • Guo, S., Mei, L., Li, Y., Sun, Y., The improved fractional sub-equation method and its applications to the space-time fractional di erential equations in fluid mechanics, Physics Letters A, 376(2012), 407. 3
  • Hammouch, Z., Mekkaoui, T., Travelling-wave solutions for some fractional partial di erential equation by means of generalized trigonometry functions, International Journal of Applied Mathematical Research, 1(2012), 206–212. 1
  • Hammouch, Z., Mekkaoui, T., Chaos synchronization of a fractional nonautonomous system, Nonautonomous Dynamical Systems, 1(2014), 61–71. 1
  • Harris, P. A., Garra, R., Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method, Nonlinear Studies, (2013), 471–481. 1
  • He, J., Elagan, S. K., Li, Z. B., Geometrical explanation of the fractional complex transform and derivative Chain rule for fractional calculus, Physics Letters A, 376(2012) 257–259. 3
  • Hussein, A., Selim, M. M., New soliton solutions for some important nonlinear partial di erential equations using a generalized Bernoulli method, International Journal of Mathematical Analysis and Applications, 1(2014), 1–8. 3
  • Kurulay, M., The approximate and exact solutions of the space- and time- fractional Burgers equations, IJRRAS, 3(2010), 257. 1
  • Momani, S., Non-perturbative analytical solutions of the space- and time- fractional Burgers equations, Chaos, Solitons and Fractals, 28(2006), 930–937. 1
  • Moslem, W. M., Sabry, R., Zakharov-Kuznetsov-Burgers equation for Dust Ion acoustic waves, Chaos, Solitons and Fractals, 36(2008), 628. 1
  • Rashidi, M. M., Erfani, E., New analytical method for solving Burgers and nonlinear heat transfer equations and comparison with HAM, Computer Physics Communications, 180(2009), 1539. 1
  • Saad, M., Elagan, S. K., Hamed, Y. S., Sayed, M., Using a complex transformation to get an exact solutions for fractional generalized coupled MKDV and KDV equations, International Journal of Basic and Applied Sciences, 13(2013), 23–25. 3
  • Sugimoto, N., Burgers equation with a fractional derivative: Hereditary e ects on nonlinear acoustic waves, Journal of Fluid Mechanics, 225(1991), 631. 1
  • Yildirim, A., Mohyud-Din, S. T., Analytical approach to space- and time- fractional Burgers equations, Chinese Physics Letters, 27(2010), 9. 1
  • Zhang, S., Hang, H. Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters A, 375(2011), 1069. 1
  • Zhang, S., Zong, Q. A., Liu, D., Gao, Q., A Generalized exp-function method for fractional Riccati di erential equations, The Communications in Fractional Calculus, 1(2010), 48–51. 1
  • Zhen, B. H., (G’/G)- expansion method for solving fractional partial di erential equations in the theory of mathematical physics, Communication Theory Physics, 58(2012), 623. 1
There are 28 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Hasan Bulut

Gülnur Yel

Hacı Mehmet Başkonuş

Publication Date December 30, 2016
Published in Issue Year 2016 Volume: 5

Cite

APA Bulut, H., Yel, G., & Başkonuş, H. M. (2016). An Application of Improved Bernoulli Sub-Equation Function Method to The Nonlinear Time-Fractional Burgers Equation. Turkish Journal of Mathematics and Computer Science, 5, 1-7.
AMA Bulut H, Yel G, Başkonuş HM. An Application of Improved Bernoulli Sub-Equation Function Method to The Nonlinear Time-Fractional Burgers Equation. TJMCS. December 2016;5:1-7.
Chicago Bulut, Hasan, Gülnur Yel, and Hacı Mehmet Başkonuş. “An Application of Improved Bernoulli Sub-Equation Function Method to The Nonlinear Time-Fractional Burgers Equation”. Turkish Journal of Mathematics and Computer Science 5, December (December 2016): 1-7.
EndNote Bulut H, Yel G, Başkonuş HM (December 1, 2016) An Application of Improved Bernoulli Sub-Equation Function Method to The Nonlinear Time-Fractional Burgers Equation. Turkish Journal of Mathematics and Computer Science 5 1–7.
IEEE H. Bulut, G. Yel, and H. M. Başkonuş, “An Application of Improved Bernoulli Sub-Equation Function Method to The Nonlinear Time-Fractional Burgers Equation”, TJMCS, vol. 5, pp. 1–7, 2016.
ISNAD Bulut, Hasan et al. “An Application of Improved Bernoulli Sub-Equation Function Method to The Nonlinear Time-Fractional Burgers Equation”. Turkish Journal of Mathematics and Computer Science 5 (December 2016), 1-7.
JAMA Bulut H, Yel G, Başkonuş HM. An Application of Improved Bernoulli Sub-Equation Function Method to The Nonlinear Time-Fractional Burgers Equation. TJMCS. 2016;5:1–7.
MLA Bulut, Hasan et al. “An Application of Improved Bernoulli Sub-Equation Function Method to The Nonlinear Time-Fractional Burgers Equation”. Turkish Journal of Mathematics and Computer Science, vol. 5, 2016, pp. 1-7.
Vancouver Bulut H, Yel G, Başkonuş HM. An Application of Improved Bernoulli Sub-Equation Function Method to The Nonlinear Time-Fractional Burgers Equation. TJMCS. 2016;5:1-7.