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Exact Solutions for Fractional Differential-Difference Equations by (G'/G)-Expansion Method with Modified Riemann-Liouville Derivative

Part of: Real functions General theory in ordinary differential equations Miscellaneous topics - Partial differential equations Random dynamical systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  27 January 2016

Ahmet Bekir ,
Ozkan Guner ,
Burcu Ayhan  and
Adem C. Cevikel
Ahmet Bekir*
Affiliation:
Eskisehir Osmangazi University, Art-Science Faculty, Department of Mathematics-Computer, Eskisehir-Turkey
Ozkan Guner
Affiliation:
Cankiri Karatekin University, Faculty of Economics and Administrative Sciences, Department of International Trade, Cankiri-Turkey
Burcu Ayhan
Affiliation:
Eskisehir Osmangazi University, Art-Science Faculty, Department of Mathematics-Computer, Eskisehir-Turkey
Adem C. Cevikel
Affiliation:
Yildiz Technical University, Education Faculty, Department of Mathematics Education, İstanbul-Turkey
*
*Corresponding author. Email: abekir@ogu.edu.tr (A. Bekir), ozkanguner@karatekin.edu.tr (O. Guner), burcu_ayhan87@hotmail.com (B. Ayhan), acevikel@yildiz.edu.tr (A. C. Cevikel)
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Abstract

In this paper, the (G'/G)-expansion method is suggested to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential difference equation into its differential difference equation of integer order. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.

Keywords

(G'/G)-expansion methodfractional Hybrid Lattice equationfractional modified Korteweg-de Vries equationmodified Riemann-Liouville derivative

MSC classification

Secondary: 34A08: Fractional differential equations 35R11: Fractional partial differential equations 37H10: Generation, random and stochastic difference and differential equations 26A33: Fractional derivatives and integrals 35Q68: PDEs in connection with computer science
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 2 , April 2016 , pp. 293 - 305
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Fermi, E., Pasta, J. and Ulam, S., Collected Papers of Enrico Fermi II, Chicago, IL: University of Chicago Press, 1965.Google Scholar
[2]Marquii, P., Bilbault, J. M. and Rernoissnet, M., Observation of nonlinear localized modes in an electrical lattice, Phys. Rev. E, 51 (1995), pp. 6127.CrossRefGoogle Scholar
[3]Kadalbajoo, M. K. and Sharma, K. K., Numerical treatment for singularly perturbed nonlinear differential difference equations with negative shift, Nonlinear Anal., 63 (2005), pp. 1909.CrossRefGoogle Scholar
[4]Ablowitz, M. J. and Ladik, J., Nonlinear differential-difference equations, J. Math. Phys., 16 (1975), pp. 598.CrossRefGoogle Scholar
[5]Eisenberg, H. S., Silberberg, Y., Morandotti, R., Boyd, A. R., and Aitchison, J. S., Discrete spatial optical solitons in waveguide arrays, Phys. Rev. Lett., 81 (1998), pp. 3383.Google Scholar
[6]Tang, B., He, Y., Wei, L. and Wang, S., Variable-coefficient discrete (G'/G)-expansion method for nonlinear differential-difference equations, Phys. Lett. A, 375(38) (2011), pp. 3355.CrossRefGoogle Scholar
[7]Wang, Z., Discrete tanh method for nonlinear difference-differential equations, Comput. Phys. Commun., 180 (2009), pp. 1104.CrossRefGoogle Scholar
[8]Zhu, S. D., Exp-function method for the discrete mKdV lattice, Int. J. Nonlinear Sci. Num. Simul., 8 (2007), pp. 465.Google Scholar
[9]Wang, Z., and Ma, W. X., Discrete Jacobi sub-equation method for nonlinear differential-difference equations, Math. Methods Appl. Sci., 33(12) (2010), pp. 1463.Google Scholar
[10]Baldwin, D., Goktas, U. and Hereman, W., Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations, Comput. Phys. Commun., 162 (2004), pp. 203.Google Scholar
[11]Zhang, S., Dong, L., Ba, J. and Sun, Y., The (G'/G)-expansion method for nonlinear differential-difference equations, Phys. Lett. A, 373 (2009), pp. 905.Google Scholar
[12]Podlubny, I., Fractional Differential Equations, Academic Press, California, 1999.Google Scholar
[13]Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.Google Scholar
[14]Guner, O. and Cevikel, A. C., A procedure to construct exact solutions of nonlinear fractional differential equations, Scientific World J., 2014 (2014), pp. 489495.CrossRefGoogle ScholarPubMed
[15]Gepreel, K. A. and Omran, S., Exact solutions for nonlinear partial fractional differential equations, China Phys. B, 21 (2012), pp. 110204.Google Scholar
[16]Bekir, A. and Guner, A., Exact solutions of nonlinear fractional differential equations by (G'/G)-expansion method, China Phys. B, 22(11) (2013), pp. 110202.CrossRefGoogle Scholar
[17]Bekir, A., Guner, O. and Cevikel, A. C., Fractional complex transform and exp-function methods for fractional differential equations, Abstract Appl. Anal., 2013 (2013), pp. 426462.CrossRefGoogle Scholar
[18]Zheng, B., Exp-function method for solving fractional partial differential equations, Scientific World J., 2013 (2013), pp. 465723.Google Scholar
[19]Lu, B., The first integral method for some time fractional differential equations, J. Math. Anal. Appl., 395 (2012), pp. 684–693.Google Scholar
[20]Guner, O. and Eser, D., Exact solutions of the space time fractional symmetric regularized long wave equation using different methods, Adv. Math. Phys., 2014 (2014), pp. 456804.Google Scholar
[21]Zhang, S. and Zhang, H. Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A, 375 (2011), pp. 1069–1073.Google Scholar
[22]Liu, W. and Chen, K., The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations, Pramana J. Phys., 81 (2013), pp. 3.CrossRefGoogle Scholar
[23]Bulut, H., Baskonus, H. M. and Pandir, Y., The modified trial equation method for fractional wave equation and time fractional generalized burgers equation, Abstract Appl. Anal., 2013 (2013), pp. 636802.Google Scholar
[24]Li, Z. B. and He, J. H., Fractional complex transform for fractional differential equations, Math. Comput. Appl., 15 (2010), pp. 970–973.Google Scholar
[25]Li, Z. B. and He, J. H., Application of the fractional complex transform to fractional differential equations, Nonlinear Sci. Lett. A Math. Phys. Mech., 2 (2011), pp. 121–126.Google Scholar
[26]Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl., 51 (2006), pp. 1367–1376.CrossRefGoogle Scholar
[27]Jumarie, G., Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution, J. Appl. Maths. Comput., 4(1-2) (2007), pp. 31–48.Google Scholar
[28]Jumarie, G., Table of some basic fractional calculus formulae derived from a modified Riemann-Liouvillie derivative for nondifferentiable functions, Appl. Maths. Lett., 22 (2009), pp. 378–385.Google Scholar
[29]Saad, M., Elagan, S. K., Hamed, Y. S. and Sayed, M., Using a complex transformation to get an exact solutions for fractional generalized coupled MKDV and KDV equations, Int. J. Basic Appl. Sci., 13(01) (2013), pp. 23–25.Google Scholar
[30]Feng, Q. H., Exact solutions for fractional differential-difference equations by an extended riccati SUB-ODE method, Commun. Theor. Phys., 595 (2013), pp. 521–527.CrossRefGoogle Scholar
[31]Bakkyaraj, T. and Sahadevan, R., An approximate solution to some classes of fractional non-linear partial differential difference equation using adomian decomposition method, J. Fractional Calculus Appl., 5(1) (2014), pp. 37–52.Google Scholar