New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods

doi:10.1088/1674-1056/25/4/040204

中国物理B ›› 2016, Vol. 25 ›› Issue (4): 40204-040204.doi: 10.1088/1674-1056/25/4/040204

New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods

S Saha Ray

National Institute of Technology, Department of Mathematics, Rourkela-769008, India

收稿日期:2015-08-30 修回日期:2015-11-23 出版日期:2016-04-05 发布日期:2016-04-05
通讯作者: S Saha Ray E-mail:santanusaharay@yahoo.com

New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods

S Saha Ray

National Institute of Technology, Department of Mathematics, Rourkela-769008, India

Received:2015-08-30 Revised:2015-11-23 Online:2016-04-05 Published:2016-04-05
Contact: S Saha Ray E-mail:santanusaharay@yahoo.com

摘要/Abstract

摘要： In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov (KdV-KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.

关键词: KdV-Khokhlov-Zabolotskaya-Kuznetsov equation, Kudryashov method, modified Kudryashov method, fractional complex transform, modified Riemann-Liouville derivative

Abstract: In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov (KdV-KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.

Key words: KdV-Khokhlov-Zabolotskaya-Kuznetsov equation, Kudryashov method, modified Kudryashov method, fractional complex transform, modified Riemann-Liouville derivative

中图分类号: (Computational techniques; simulations)

02.70.-c

S Saha Ray. New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods[J]. 中国物理B, 2016, 25(4): 40204-040204.

S Saha Ray. New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods[J]. Chin. Phys. B, 2016, 25(4): 40204-040204.

参考文献 35

[1]	Podlubny I 1999 Fractional Differential Equations of Mathematics in Science and Engineering (New York: Academic Press) Vol. 198
[2]	Hilfer R 2000 Applications of Fractional Calculus in Physics (Singapore: World Scientific)
[3]	Debnath L, Bhatta D 2006 Integral Transforms and Their Applications (Boca Raton: Chapman and Hall/CRC).
[4]	Sabatier J, Agrawal O P and Tenreiro Machado J A (eds.) 2007 Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (Netherlands: Springer)
[5]	Ortigueira M D 2011 “Fractional Calculus for Scientists and Engineers”, Lecture Notes in Electrical Engineering 84 (Netherlands: Springer)
[6]	Das S 2011 Functional Fractional Calculus (New York: Springer)
[7]	Diethelm K, Ford N J and Freed A D 2002 Nonlinear Dynamics 29 3
[8]	Liu F, Anh V and Turner I 2004 J. Comput. Appl. Math. 166 209
[9]	Saha Ray S 2007 Phys. Scr. 75 article number 008 53
[10]	Ortigueira M D and Tenreiro Machado J A 2006 Signal Processing 86 2503
[11]	Saha Ray S 2008 Appl. Math. Comput. 202 544
[12]	Tong B, He Y, Wei L and Zhang X 2012 Phys. Lett. A 376 2588
[13]	Liu F, Zhuang P and Burrage K 2012 Comput. Math. Appl. 64 2990
[14]	Saha Ray S 2012 Appl. Math. Comput. 218 5239
[15]	Kadem A and Kılıçman A 2012 Abstr. Appl. Anal. 2012 1
[16]	Gupta A K and Saha Ray S 2015 Appl. Math. Model. 39 5121
[17]	Sahoo S and Saha Ray S 2015 Physica A: Statistical Mechanics and Its Applications 434 240
[18]	Saha Ray S and Sahoo S 2015 Comput. Math. Appl. 70 158
[19]	Gan W S 2007 Acoustical Imaging 28 445
[20]	Kudryashov N A 2009 Commun. Nonlinear Sci. Numer. Simul. 14 1891
[21]	Ryabov P N, Sinelshchikov D I and Kochanov M B 2011 Appl. Math. Comput. 218 3965
[22]	Kudryashov N A 2012 Commun. Nonlinear Sci. Numer. Simul. 17 2248
[23]	Kabir M M, Khajeh A, Aghdam E A and Koma A Y 2011 Math. Methods Appl. Sci. 34 244
[24]	Ege S M and Misirli E 2014 Advances in Difference Equation 2014 135
[25]	Taghizadeh N, Mirzazadeh M and Mahmoodirad A 2013 Indian J. Phys. 87 781
[26]	Jumarie G 2005 Appl. Math. Lett. 18 739
[27]	Jumarie G 2006 Comput. Math. Appl. 51 1367
[28]	Jumarie G 2009 Appl. Math. Lett. 22 378
[29]	Jumarie G 2013 Mathematics and Statistics 1 50
[30]	Su W H, Yang X J, Jafari H and Baleanu D 2013 Advances in Difference Equations 2013 1
[31]	Yang X J, Baleanu D and Srivastava H M 2015 Local Fractional Integral Transforms and Their Applications (London: Academic Press)
[32]	He J H, Elagan S K and Li Z B 2012 Phys. Lett. A 376 257
[33]	Güner O, Bekir A and Cevikel A C 2015 Eur. Phys. J. Plus 130 146
[34]	Stakhov A and Rozin B 2005 Chaos, Solitons, and Fractals 23 379
[35]	Pandir Y 2014 Appl. Math. Inform. Sci. 8 2237

New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods

New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods

RichHTML

PDF (PC)

赞

可视化

摘要/Abstract

引用本文

使用本文

参考文献 35

相关文章 2

Metrics

本文评价

推荐阅读 0

[1]	Zillur Rahman, M Zulfikar Ali, and Harun-Or Roshid. Closed form soliton solutions of three nonlinear fractional models through proposed improved Kudryashov method[J]. 中国物理B, 2021, 30(5): 50202-050202.
[2]	Khaled A. Gepreel, Saleh Omran . Exact solutions for nonlinear partial fractional differential equations[J]. 中国物理B, 2012, 21(11): 110204-110204.