Research Article
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Year 2018, Volume: 6 Issue: 1, 19 - 28, 27.04.2018
https://doi.org/10.36753/mathenot.421751

Abstract

References

  • [1] Aksoy, E., Kaplan, M., Bekir A., Exponential rational function method for space−time fractional differential equations, Waves in Random Media 26 (2016), no.2, 142-151.
  • [2] Alzaidy, J. F. , Fractional Sub-Equation Method and its Applications to the Space Time Fractional Differential Equations in Mathematical Physics, Br. J. of Maths. Comp. Sci. 2 (2013), no.3, 152-163.
  • [3] Baleanu, D., Machado, J. A. T., Luo, A. C. J., Fractional Dynamics and Control, Springer, (2012), 49-57.
  • [4] Bekir, A. and Guner, O., The (G0/G)-expansion method using modified Riemann–Liouville derivative for some space-time fractional differential equations, Ain Shams Engin. J. 5 (2014), no.3, 959-965.
  • [5] Bekir, A. and Aksoy, E., Exact solutions of shallow water wave equations by using (G0/G)-expansion method, Waves in Random Complex Media, 22 (2012), no.3, 317-331.
  • [6] Boudjehem, B., Boudjehem, D., Parameter tuning of a fractional-order PI Controller using the ITAE Criteria, Fractional Dynamics Control, (2011), 49-57.
  • [7] Bulut, H., Pandir, Y. and Demiray, S. T., Exact Solutions of Time-Fractional KdV Equations by Using Generalized Kudryashov Method, Int. J. Model. Opt. 4 (2014), no.4, 315-320.
  • [8] Bulut, H., Baskonus, H. M. and Pandir, Y., The modified trial equation method for fractional wave equation and time fractional generalized burgers equation, Abst. Applied Analy. (2013), 1-8.
  • [9] Ege, S. M. and Misirli, E., The modified Kudryashov method for solving some fractional-order nonlinear equations, Advances in Difference Equations, 135 (2014), 1-13.
  • [10] Ege, S. M. and Misirli, E., Solutions of the space-time fractional foam-drainage equation and the fractional Klein-Gordon equation by use of modified Kudryashov method,Int. J. of Research Adv. Tech. 2321(2014), no.9637 384-388.
  • [11] Ege, S. M., On semianalytical solutions of some nonlinear physical evolution equations with polynomial type auxilary equation,PhD Thesis, Ege University (2015).
  • [12] Guner, O., Bekir, A. and Bilgil, H. , A note on exp-function method combined with complex transform method applied to fractional differential equations, Advances in Nonlinear Analysis 4 (2015), no.3, 201-208.
  • [13] Guoa, S., Meia, Y., Lia, Y. and Sunb, Y., The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics,Phys. Letters A. 376 (2012), 407-411.
  • [14] He, J. H., Li. Z. B., Converting fractional differential equations into partial differential equations,Thermal Science, 16 (2012), no.2, 331-337.
  • [15] Jumarie, G. , Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Compt. Math. Appl., 51 (2006), 1367-1376.
  • [16] Jumarie, G. , Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution, J. Appl. Math. Compt., 24, (2007), 31-48.
  • [17] Kudryashov, N. A. , One method for finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci., 17 (2012), 2248–2253.
  • [18] Martinez, H. Y., Sosa, I. O. and Reyes, J. M. , Feng’s First Integral Method Applied to the ZKBBM and the Generalized Fisher Space-Time Fractional Equations, J. Appl. Math. (2015), 1-5.
  • [19] Mohamed, M. S., Al-Malki, F. and Gepreel, K. A., Approximate solution for fractional Zakharov-Kuznetsov equation using the fractional complex transform, AIP Conf. Proc. 1558 (2013), no.1, 1989.
  • [20] Meng, F., A New Approach for Solving Fractional Partial Differential Equations,J. Appl. Math. (2013), 1-5.
  • [21] Miller, K. S. and Ross, B, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York, (1993).
  • [22] Odabasi M. and Misirli, E., On the solutions of the nonlinear fractional differential equations via the modified trial equation method,Math. Methods Appl. Sci. 2015, 1-8.
  • [23] Pandir, Y., Symmetric Fibonacci Function Solutions of some Nonlinear Partial Differential Equations,Appl. Math. Inf. Sci. 8 (2014), no.5, 2237-2241.
  • [24] Pandir, Y., Gurefe, Y., New exact solutions of the generalized fractional Zakharov-Kuznetsov equations, Life Sci. J. 10 (2013), no.2, 2701-2705.
  • [25] Podlubny, I., Fractional Differential Equations, Academic Press, California, (1999).
  • [26] Ryabov, P. N. , Sinelshchikov, D. I., Kochanov, M. B., Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations, Applied Mathematics and Computation, 218 (1999), no.1, 3965-3971.
  • [27] Zayed, E. M. E., Sonmezoglu, A. and Ekici, M., A new fractional sub-equation method for solving the space-time fractional differential equations in mathematical physics, Computational Methods for Differential Equations, 2 (2014), no.3, 153-170.
  • [28] Zayed, E. M. E., Alurrfi, K. A. E. , The modified Kudryashov method for solving some seventh order nonlinear PDEs in mathematical physics, World Journal of Modelling and Simulation, 11 (2015), no.4, 308-319.
  • [29] Zheng, B., Exp−function method for solving fractional partial differential equations, Sci. World J. (2013), 1-8.
  • [30] Zheng, B., Wen, C. , Exact solutions for fractional partial differential equations by a new fractional sub-equation method, Advances in Difference Equations, 199 (2013), 1-12.

Extended Kudryashov Method for Fractional Nonlinear Differential Equations

Year 2018, Volume: 6 Issue: 1, 19 - 28, 27.04.2018
https://doi.org/10.36753/mathenot.421751

Abstract

In this study, we have propesed the extended Kudryashov method to obtain the exact solutions of
nonlinear fractional differential equations. Definiton of modified Riemann Liouville sense fractional
derivative is used and the proposed method is applied to two nonlinear fractional differential equations.
Analytical solutions including hyperbolic functions are obtained.

References

  • [1] Aksoy, E., Kaplan, M., Bekir A., Exponential rational function method for space−time fractional differential equations, Waves in Random Media 26 (2016), no.2, 142-151.
  • [2] Alzaidy, J. F. , Fractional Sub-Equation Method and its Applications to the Space Time Fractional Differential Equations in Mathematical Physics, Br. J. of Maths. Comp. Sci. 2 (2013), no.3, 152-163.
  • [3] Baleanu, D., Machado, J. A. T., Luo, A. C. J., Fractional Dynamics and Control, Springer, (2012), 49-57.
  • [4] Bekir, A. and Guner, O., The (G0/G)-expansion method using modified Riemann–Liouville derivative for some space-time fractional differential equations, Ain Shams Engin. J. 5 (2014), no.3, 959-965.
  • [5] Bekir, A. and Aksoy, E., Exact solutions of shallow water wave equations by using (G0/G)-expansion method, Waves in Random Complex Media, 22 (2012), no.3, 317-331.
  • [6] Boudjehem, B., Boudjehem, D., Parameter tuning of a fractional-order PI Controller using the ITAE Criteria, Fractional Dynamics Control, (2011), 49-57.
  • [7] Bulut, H., Pandir, Y. and Demiray, S. T., Exact Solutions of Time-Fractional KdV Equations by Using Generalized Kudryashov Method, Int. J. Model. Opt. 4 (2014), no.4, 315-320.
  • [8] Bulut, H., Baskonus, H. M. and Pandir, Y., The modified trial equation method for fractional wave equation and time fractional generalized burgers equation, Abst. Applied Analy. (2013), 1-8.
  • [9] Ege, S. M. and Misirli, E., The modified Kudryashov method for solving some fractional-order nonlinear equations, Advances in Difference Equations, 135 (2014), 1-13.
  • [10] Ege, S. M. and Misirli, E., Solutions of the space-time fractional foam-drainage equation and the fractional Klein-Gordon equation by use of modified Kudryashov method,Int. J. of Research Adv. Tech. 2321(2014), no.9637 384-388.
  • [11] Ege, S. M., On semianalytical solutions of some nonlinear physical evolution equations with polynomial type auxilary equation,PhD Thesis, Ege University (2015).
  • [12] Guner, O., Bekir, A. and Bilgil, H. , A note on exp-function method combined with complex transform method applied to fractional differential equations, Advances in Nonlinear Analysis 4 (2015), no.3, 201-208.
  • [13] Guoa, S., Meia, Y., Lia, Y. and Sunb, Y., The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics,Phys. Letters A. 376 (2012), 407-411.
  • [14] He, J. H., Li. Z. B., Converting fractional differential equations into partial differential equations,Thermal Science, 16 (2012), no.2, 331-337.
  • [15] Jumarie, G. , Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Compt. Math. Appl., 51 (2006), 1367-1376.
  • [16] Jumarie, G. , Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution, J. Appl. Math. Compt., 24, (2007), 31-48.
  • [17] Kudryashov, N. A. , One method for finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci., 17 (2012), 2248–2253.
  • [18] Martinez, H. Y., Sosa, I. O. and Reyes, J. M. , Feng’s First Integral Method Applied to the ZKBBM and the Generalized Fisher Space-Time Fractional Equations, J. Appl. Math. (2015), 1-5.
  • [19] Mohamed, M. S., Al-Malki, F. and Gepreel, K. A., Approximate solution for fractional Zakharov-Kuznetsov equation using the fractional complex transform, AIP Conf. Proc. 1558 (2013), no.1, 1989.
  • [20] Meng, F., A New Approach for Solving Fractional Partial Differential Equations,J. Appl. Math. (2013), 1-5.
  • [21] Miller, K. S. and Ross, B, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York, (1993).
  • [22] Odabasi M. and Misirli, E., On the solutions of the nonlinear fractional differential equations via the modified trial equation method,Math. Methods Appl. Sci. 2015, 1-8.
  • [23] Pandir, Y., Symmetric Fibonacci Function Solutions of some Nonlinear Partial Differential Equations,Appl. Math. Inf. Sci. 8 (2014), no.5, 2237-2241.
  • [24] Pandir, Y., Gurefe, Y., New exact solutions of the generalized fractional Zakharov-Kuznetsov equations, Life Sci. J. 10 (2013), no.2, 2701-2705.
  • [25] Podlubny, I., Fractional Differential Equations, Academic Press, California, (1999).
  • [26] Ryabov, P. N. , Sinelshchikov, D. I., Kochanov, M. B., Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations, Applied Mathematics and Computation, 218 (1999), no.1, 3965-3971.
  • [27] Zayed, E. M. E., Sonmezoglu, A. and Ekici, M., A new fractional sub-equation method for solving the space-time fractional differential equations in mathematical physics, Computational Methods for Differential Equations, 2 (2014), no.3, 153-170.
  • [28] Zayed, E. M. E., Alurrfi, K. A. E. , The modified Kudryashov method for solving some seventh order nonlinear PDEs in mathematical physics, World Journal of Modelling and Simulation, 11 (2015), no.4, 308-319.
  • [29] Zheng, B., Exp−function method for solving fractional partial differential equations, Sci. World J. (2013), 1-8.
  • [30] Zheng, B., Wen, C. , Exact solutions for fractional partial differential equations by a new fractional sub-equation method, Advances in Difference Equations, 199 (2013), 1-12.
There are 30 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Serife Muge Ege This is me

Emine Misirli

Publication Date April 27, 2018
Submission Date March 29, 2016
Published in Issue Year 2018 Volume: 6 Issue: 1

Cite

APA Ege, S. M., & Misirli, E. (2018). Extended Kudryashov Method for Fractional Nonlinear Differential Equations. Mathematical Sciences and Applications E-Notes, 6(1), 19-28. https://doi.org/10.36753/mathenot.421751
AMA Ege SM, Misirli E. Extended Kudryashov Method for Fractional Nonlinear Differential Equations. Math. Sci. Appl. E-Notes. April 2018;6(1):19-28. doi:10.36753/mathenot.421751
Chicago Ege, Serife Muge, and Emine Misirli. “Extended Kudryashov Method for Fractional Nonlinear Differential Equations”. Mathematical Sciences and Applications E-Notes 6, no. 1 (April 2018): 19-28. https://doi.org/10.36753/mathenot.421751.
EndNote Ege SM, Misirli E (April 1, 2018) Extended Kudryashov Method for Fractional Nonlinear Differential Equations. Mathematical Sciences and Applications E-Notes 6 1 19–28.
IEEE S. M. Ege and E. Misirli, “Extended Kudryashov Method for Fractional Nonlinear Differential Equations”, Math. Sci. Appl. E-Notes, vol. 6, no. 1, pp. 19–28, 2018, doi: 10.36753/mathenot.421751.
ISNAD Ege, Serife Muge - Misirli, Emine. “Extended Kudryashov Method for Fractional Nonlinear Differential Equations”. Mathematical Sciences and Applications E-Notes 6/1 (April 2018), 19-28. https://doi.org/10.36753/mathenot.421751.
JAMA Ege SM, Misirli E. Extended Kudryashov Method for Fractional Nonlinear Differential Equations. Math. Sci. Appl. E-Notes. 2018;6:19–28.
MLA Ege, Serife Muge and Emine Misirli. “Extended Kudryashov Method for Fractional Nonlinear Differential Equations”. Mathematical Sciences and Applications E-Notes, vol. 6, no. 1, 2018, pp. 19-28, doi:10.36753/mathenot.421751.
Vancouver Ege SM, Misirli E. Extended Kudryashov Method for Fractional Nonlinear Differential Equations. Math. Sci. Appl. E-Notes. 2018;6(1):19-28.

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