Year 2018,
, 19 - 28, 27.04.2018
Serife Muge Ege
Emine Misirli
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,
, 19 - 28, 27.04.2018
Serife Muge Ege
Emine Misirli
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.