Fractional complex transform and exp-function methods for fractional differential equations

Abstract and Applied Analysis

Volumn 2013, Issue , 2013, Pages

  Bekir, Ahmet ^a   Güner, Özkan ^b   Cevikel, Adem C ^c  

^a ESKISEHIR OSMANGAZI UNIVERSITY   (Turkey)

^b DUMLUPINAR UNIVERSITY   (Turkey)

^c YILDIZ TECHNICAL UNIVERSITY   (Turkey)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84880174334     PISSN: 10853375     EISSN: 16870409     Source Type: Journal    
DOI: 10.1155/2013/426462     Document Type: Article

References (35)

1. Oldham K. B., Spanier J., The Fractional Calculus 1974 New York, NY, USA Akademic Press MR0361633 ZBL0292.26011
2. Miller K. S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 New York, NY, USA John Wiley & Sons A Wiley-Interscience Publication MR1219954 ZBL0943.82582
3. Podlubny I., Fractional differential equations 1999 198 San Diego, CA, USA Academic Press Mathematics in Science and Engineering MR1658022 ZBL1056.93542
4. Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations 2006 Amsterdam, The Netherlands Elsevier
5. Zhang S., Zong Q.-A., Liu D., Gao Q., A generalized exp-function method for fractional riccati differential equations. Communications in Fractional Calculus 2010 1 1 48 51
6. Zhang S., Zhang H.-Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs. Physics Letters A 2011 375 7 1069 1073 10.1016/j.physleta.2011.01.029 MR2765013 ZBL1242.35217
7. Tang B., He Y., Wei L., Zhang X., A generalized fractional sub-equation method for fractional differential equations with variable coefficients. Physics Letters A 2012 376 38-39 2588 2590 10.1016/j.physleta.2012.07.018 MR2961121
8. Guo S., Mei L., Li Y., Sun Y., The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics. Physics Letters A 2012 376 4 407 411 10.1016/j.physleta. 2011.10.056 MR2877750 ZBL1255.37022
9. Zheng B., (G′/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Communications in Theoretical Physics 2012 58 5 623 630
10. Gepreel K. A., Omran S., Exact solutions for nonlinear partial fractional differential equations. Chinese Physics B 2012 21 11 110204
11. Lu B., The first integral method for some time fractional differential equations. Journal of Mathematical Analysis and Applications 2012 395 2 684 693 10.1016/j.jmaa.2012.05.066 MR2948259 ZBL1246.35202
12. He J.-H., Elagan S. K., Li Z. B., Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Physics Letters A 2012 376 4 257 259 10.1016/j.physleta.2011.11.030 MR2877722 ZBL1255.26002
13. Ibrahim R. W., Fractional complex transforms for fractional differential equations. Advances in Difference Equations 2012 2012 article 192
14. He J.-H., Wu X.-H., Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals 2006 30 3 700 708 10.1016/j.chaos.2006.03.020 MR2238695 ZBL1141.35448 (Pubitemid 43903182)
15. Zhang S., Application of Exp-function method to high-dimensional nonlinear evolution equation. Chaos, Solitons and Fractals 2008 38 1 270 276 10.1016/j.chaos.2006.11.014 MR2417662 ZBL1142.35593 (Pubitemid 351503992)
16. Bekir A., Cevikel A. C., New solitons and periodic solutions for nonlinear physical models in mathematical physics. Nonlinear Analysis. Real World Applications 2010 11 4 3275 3285 10.1016/j.nonrwa.2009.10.015 MR2661988 ZBL1196.35178
17. El-Wakil S. A., Madkour M. A., Abdou M. A., Application of Exp-function method for nonlinear evolution equations with variable coefficients. Physics Letters A 2007 369 1-2 62 69 2-s2.0-34548246435 10.1016/j.physleta.2007.04.075
18. Zhu S. D., Exp-function method for the Hybrid-Lattice system. International Journal of Nonlinear Sciences and Numerical Simulation 2007 8 3 461 464 2-s2.0-34548577315
19. Bekir A., Application of the exp-function method for nonlinear differential-difference equations. Applied Mathematics and Computation 2010 215 11 4049 4053 10.1016/j.amc.2009.12.003 MR2578871 ZBL1185.35312
20. Dai C. Q., Chen J. L., New analytic solutions of stochastic coupled KdV equations. Chaos, Solitons & Fractals 2009 42 4 2200 2207 10.1016/j.chaos.2009.03.157 MR2559882 ZBL1198.35292
21. Caputo M., Linear models of dissipation whose Q is almost frequency independent II. Geophysical Journal International 1967 13 5 529 539
22. Samko S. G., Kilbas A. A., Marichev O. I., Fractional integrals and derivatives 1993 Switzerland Gordon and Breach Science Publishers MR1347689
23. Jumarie G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications 2006 51 9-10 1367 1376 10.1016/j.camwa.2006.02.001 MR2237634 ZBL1137.65001
24. Jumarie G., Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. Applied Mathematics Letters 2009 22 3 378 385 10.1016/j.aml.2008.06.003 MR2483503 ZBL1171.26305
25. Li Z.-B., He J.-H., Fractional complex transform for fractional differential equations. Mathematical & Computational Applications 2010 15 5 970 973 MR2777702 ZBL1215.35164
26. Li Z.-B., He J.-H., Application of the fractional complex transform to fractional differential equations. Nonlinear Science Letters A 2011 2 121 126
27. He J.-H., Abdou M. A., New periodic solutions for nonlinear evolution equations using exp-function method. Chaos, Solitons & Fractals 2007 34 5 1421 1429 10.1016/j.chaos.2006.05.072 MR2335393 ZBL1152.35441 (Pubitemid 46907583)
28. Ebaid A., Exact solitary wave solutions for some nonlinear evolution equations via exp-function method. Physics Letters A 2007 365 3 213 219 10.1016/j.physleta.2007.01.009 MR2308767 ZBL1203.35213
29. Bekir A., The exp-function method for Ostrovsky equation. International Journal of Nonlinear Sciences and Numerical Simulation 2009 10 6 735 739 2-s2.0-69149084662
30. El-Sayed A. M. A., Rida S. Z., Arafa A. A. M., Exact solutions of fractional-order biological population model. Communications in Theoretical Physics 2009 52 6 992 996 10.1088/0253-6102/52/6/04 MR2683018 ZBL1184.92038
31. Lu B., Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations. Physics Letters A 2012 376 28-29 2045 2048 10.1016/j.physleta.2012.05.013 MR2929392
32. Inc M., The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method. Journal of Mathematical Analysis and Applications 2008 345 1 476 484 10.1016/j.jmaa.2008.04.007 MR2422665 ZBL1146.35304
33. Jafari H., Tajadodi H., Kadkhoda N., Baleanu D., Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations. Abstract and Applied Analysis 2013 2013 5 587179
34. Choo S. M., Chung S. K., Lee Y. J., A conservative difference scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient. Applied Numerical Mathematics 2004 51 2-3 207 219 10.1016/j.apnum.2004.02.006 MR2091400 ZBL1112.65078
35. Kim J., A numerical method for the Cahn-Hilliard equation with a variable mobility. Communications in Nonlinear Science and Numerical Simulation 2007 12 8 1560 1571 10.1016/j.cnsns.2006.02.010 MR2332646 ZBL1118.35049 (Pubitemid 46838592)