Exact solutions for nonlinear partial fractional differential equations

doi:10.1088/1674-1056/21/11/110204

中国物理B ›› 2012, Vol. 21 ›› Issue (11): 110204-110204.doi: 10.1088/1674-1056/21/11/110204

Exact solutions for nonlinear partial fractional differential equations

Khaled A. Gepreel^{a b}, Saleh Omran^{b c}

a Mathematics Department, Faculty of Science, Zagazig University, Egypt;
b Mathematics Department, Faculty of Science, Taif University, Saudi Arabia;
c Mathematics Department, Faculty of science, South Valley University, Egypt

收稿日期:2012-02-11 修回日期:2012-05-17 出版日期:2012-10-01 发布日期:2012-10-01

Exact solutions for nonlinear partial fractional differential equations

Khaled A. Gepreel^{a b}, Saleh Omran^{b c}

a Mathematics Department, Faculty of Science, Zagazig University, Egypt;
b Mathematics Department, Faculty of Science, Taif University, Saudi Arabia;
c Mathematics Department, Faculty of science, South Valley University, Egypt

Received:2012-02-11 Revised:2012-05-17 Online:2012-10-01 Published:2012-10-01
Contact: Khaled A. Gepreel E-mail:kagepreel@yahoo.com

摘要/Abstract

摘要： In this article, we use the fractional complex transformation to convert the nonlinear partial fractional differential equations to the nonlinear ordinary differential equations. We use the improved (G'/G)-expansion function method to calculate the exact solutions for the time and space fractional derivatives Foam Drainage equation and the time and space fractional derivatives nonlinear KdV equation. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.

关键词: fractional calculus, complex transformation, modified Riemann-Liouville derivative, improved (G'/G)-expansion function method

Abstract: In this article, we use the fractional complex transformation to convert the nonlinear partial fractional differential equations to the nonlinear ordinary differential equations. We use the improved (G'/G)-expansion function method to calculate the exact solutions for the time and space fractional derivatives Foam Drainage equation and the time and space fractional derivatives nonlinear KdV equation. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.

Key words: fractional calculus, complex transformation, modified Riemann-Liouville derivative, improved (G'/G)-expansion function method

中图分类号: (Partial differential equations)

02.30.Jr

Khaled A. Gepreel, Saleh Omran . Exact solutions for nonlinear partial fractional differential equations[J]. 中国物理B, 2012, 21(11): 110204-110204.

Khaled A. Gepreel, Saleh Omran. Exact solutions for nonlinear partial fractional differential equations[J]. Chin. Phys. B, 2012, 21(11): 110204-110204.

参考文献 30

[1]	Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press)
[2]	He J H 2004 Bull. Sci. Technol. 15 86
[3]	Diethelm K and LuchkoY 2008 J. Comput. Anal. Appl. 6 243
[4]	Erturk V S, Momani S and Odibat Z 2008 Commun. Nonlinear Sci. Numer. Simulat. 13 1642
[5]	Daftardar-Gejji V and Bhalekar S 2008 Appl. Math. Comput. 202 113
[6]	Daftardar-Gejji V and Jafari H 2007 Appl. Math. Comput. 189 541
[7]	Sweilam N H, Khader M M and Al-Bar R F 2007 Phys. Lett. A 371 26
[8]	Golbabai A and Sayevand K 2011 Comput. Math. Application 61 2227
[9]	Golbabai A and Sayevand K 2010 Nonlinear Science Lett. A 1 147
[10]	Gepreel K A 2011 Applied Math. Lett. 24 1428
[11]	Deng W H 2010 Nonlinear Analysis: TMA 72 1768
[12]	Deng W H 2007 Journal of Computational Physics 227 1510
[13]	Wang M L, Li X Z and Zhang J L 2008 Phys. Lett. A 372 417
[14]	Zayed E M E and Gepreel K A 2009 J. Math. Phys. 50 013502
[15]	Zhang H 2009 Commun. Nonlinear Sci. Numer. Simulat. 14 3220
[16]	Dahmani Z and Anber A 2010 Inter. J. Nonlinear Sci. 10 39
[17]	Momani S, Odibat Z and Alawneh A 2008 J. Numer. Method. Partial Diff. Equ. 24 262
[18]	Kolwankar K M and Gangal A D 1998 Phys. Rev. Lett. 80 214
[19]	Chen W and Sun H G 2009 Mod. Phys. Lett. B 23 449
[20]	Cresson J 2005 J. Math. Anal. Appl. 307 48
[21]	Jumarie G 2006 Comput. Math. Appl. 51 1367
[22]	Jumarie G 2006 Appl. Math. Lett. 19 873
[23]	Wu G C 2011 Appl. Math. Lett. 24 1046
[24]	Jumarie G 2006 Math. Comput. Modelling 44 231
[25]	Jumarie G 2009 Appl. Math. Lett. 22 1659
[26]	Almeida R, Malinowska A B and Torres D FM 2010 J. Math. Phys. 51 033503
[27]	Wu G C and Lee E W M 2010 Phys. Lett. A 374 2506
[28]	Malinowska A B, Sidi Ammi M R and Torres D F M 2010 Commun. Frac. Calc. 1 32
[29]	Wu G C 2010 Commun. Frac. Calc. 1 23
[30]	Li Z B and He J H 2010 Math. Comput. Applications 15 970

Exact solutions for nonlinear partial fractional differential equations

Exact solutions for nonlinear partial fractional differential equations

PDF (PC)

赞

可视化

摘要/Abstract

引用本文

使用本文

参考文献 30

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	Abderrahmane Abbes, Adel Ouannas, and Nabil Shawagfeh. An incommensurate fractional discrete macroeconomic system: Bifurcation, chaos, and complexity[J]. 中国物理B, 2023, 32(3): 30203-030203.
[2]	Shaobo He(贺少波), Huihai Wang(王会海), and Kehui Sun(孙克辉). Solutions and memory effect of fractional-order chaotic system: A review[J]. 中国物理B, 2022, 31(6): 60501-060501.
[3]	Wu-Yang Zhu(朱伍洋), Yi-Fei Pu(蒲亦非), Bo Liu(刘博), Bo Yu(余波), and Ji-Liu Zhou(周激流). A mathematical analysis: From memristor to fracmemristor[J]. 中国物理B, 2022, 31(6): 60204-060204.
[4]	Xiang Xu(徐翔), Gangquan Si(司刚全), Zhang Guo(郭璋), and Babajide Oluwatosin Oresanya. Modeling and character analyzing of multiple fractional-order memcapacitors in parallel connection[J]. 中国物理B, 2022, 31(1): 18401-018401.
[5]	Mei Li(李梅), Ruo-Xun Zhang(张若洵), and Shi-Ping Yang(杨世平). Adaptive synchronization of a class of fractional-order complex-valued chaotic neural network with time-delay[J]. 中国物理B, 2021, 30(12): 120503-120503.
[6]	Zong-Li Yang(杨宗立), Dong Liang(梁栋), Da-Wei Ding(丁大为), Yong-Bing Hu(胡永兵), and Hao Li(李浩). Transient transition behaviors of fractional-order simplest chaotic circuit with bi-stable locally-active memristor and its ARM-based implementation[J]. 中国物理B, 2021, 30(12): 120515-120515.
[7]	Adel Ouannas, Amina Aicha Khennaoui, Shaher Momani, Viet-Thanh Pham, Reyad El-Khazali. Hidden attractors in a new fractional-order discrete system: Chaos, complexity, entropy, and control[J]. 中国物理B, 2020, 29(5): 50504-050504.
[8]	何秋燕, 余波, 袁晓. Carlson iterating rational approximation and performance analysis of fractional operator with arbitrary order[J]. 中国物理B, 2017, 26(4): 40202-040202.
[9]	张立民, 孙克辉, 刘文浩, 贺少波. A novel color image encryption scheme using fractional-order hyperchaotic system and DNA sequence operations[J]. 中国物理B, 2017, 26(10): 100504-100504.
[10]	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.
[11]	Hossam A. Ghany, Abd-Allah Hyder. Abundant solutions of Wick-type stochastic fractional 2D KdV equations[J]. 中国物理B, 2014, 23(6): 60503-060503.
[12]	Hossam A. Ghany, A. S. Okb El Bab, A. M. Zabel, Abd-Allah Hyder. The fractional coupled KdV equations：Exact solutions and white noise functional approach[J]. 中国物理B, 2013, 22(8): 80501-080501.
[13]	Rajneesh Kumar, Vandana Gupta. Uniqueness, reciprocity theorem, and plane waves in thermoelastic diffusion with fractional order derivative[J]. 中国物理B, 2013, 22(7): 74601-074601.
[14]	王发强, 马西奎. Transfer function modeling and analysis of the open-loop Buck converter using the fractional calculus[J]. 中国物理B, 2013, 22(3): 30506-030506.
[15]	魏含玉, 夏铁成 . A new generalized fractional Dirac soliton hierarchy and its fractional Hamiltonian structure[J]. 中国物理B, 2012, 21(11): 110203-110203.