非线性时间分数阶微分方程的exp(-Φ(ξ))解法
The Exp(-Φ(ξ)) Method for the Nonlinear Time Fractional Differential Equations
DOI: 10.12677/AAM.2017.64062, PDF, HTML, XML, 下载: 2,107  浏览: 5,264  国家自然科学基金支持
作者: 王亚东, 张新东*：新疆师范大学数学科学学院，新疆 乌鲁木齐
关键词: 时间分数阶KdV -ZK方程；分数阶导数；精确解；exp(-Φ(ξ))方法；Time Fractional KdV-ZK Equation； Fractional Derivative； Exact Solutions； Exp(-Φ(x)) Method
摘要: 本文主要研究(3+1)维Korteweg-de Vries Zakharov (KdV-ZK)方程的exp(-Φ(ξ))解法。利用exp(-Φ(ξ))方法获得所研究方程的近似解析解。数值算例表明，该方法在求解非线性分数阶微分方程的近似解析解时非常有效。
Abstract: In this paper, the exp(-Φ(ξ)) method is used to construct the approximate analytical solution of Korteweg-de Vries Zakharov Kuznetsov (KdV-ZK) equation via the (3 + 1)-dimensional. The results of numerical example show that the method is very useful in solving nonlinear time fractional differential equations.
文章引用：王亚东, 张新东. 非线性时间分数阶微分方程的exp(-Φ(ξ))解法[J]. 应用数学进展, 2017, 6(4): 515-522. https://doi.org/10.12677/AAM.2017.64062

参考文献

[1] Sahoo, S. and Ray, S.S. (2015) Improved Fractional Sub-Equation Method for (3+1)-Dimensional Generalized Fractional KdV-Zakharov-Kuznetsov Equations. Computers and Mathematics with Applications, 70, 158-166.
https://doi.org/10.1016/j.camwa.2015.05.002
[2] Ray, S.S. and Gupta, A.K. (2014) An Approach with Haar Wavelet Collocation Method for Numerical Simulations of Modified KDV and Modified Burgers Equations. Computer Modeling in Engineering and Sciences, 103, 315-341.
[3] Cairns, R.A., Mamun, A.A., Bingham, R., Dendy, R.O., Bostrom, R., Shukla, P.K. and Nairn, C.M.C. (1995) Electrostatic Solitary Structures in Non-Thermal Plasmas. Geophysical Research Letters, 22, 2709-2712.
https://doi.org/10.1029/95GL02781
[4] Islam, S., Bandyopadhyay, A. and Das, K.P. (2008) Ion-Acoustic Solitary Waves in a Multi-Species Magnetized Plasma Consisting of Non-Thermal and Isothermal Electrons. Journal of Plasma Physics, 74, 765-806.
https://doi.org/10.1017/S0022377808007241
[5] Zakharov, V.E. and Kuznetsov, E.A. (1974) On Three-Dimensional Solitons. Journal of Experimental and Theoretical Physics, 39, 285-286.
[6] Mace, R.L. and Hellberg, M.A. (2001) The Korteweg-de Vries-Zakharov-Kuznetsov Equation for Electron-Acoustic Waves. Physics of Plasmas, 8, 2649-2656.
https://doi.org/10.1063/1.1363665
[7] Zheng, B. (2012) (G'/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics. Communications in Theoretical Physics, 58, 623-630.
https://doi.org/10.1088/0253-6102/58/5/02
[8] Bekir, A., Guner, O. and Cevikel, A.C. (2013) Fractional Complex Transform and Exp-Function Methods for Fractional Differential Equations. Abstract and Applied Analysis, 1, 233-255.
https://doi.org/10.1155/2013/426462
[9] Wang, M.L. (1995) Solitary Wave Solutions for Variant Boussinesq Equations. Physics Letters A, 199, 169-172.
[10] Guner, O. and Cevikel, A.C. (2014) A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations. The Scientific World Journal, 11, 489-495.
https://doi.org/10.1155/2014/489495
[11] Guner, O. and Eser, D. (2014) Exact Solutions of the Space Time Fractional Symmetric Regularized Long Wave Equation Using Different Methods. Advances in Mathematical Physics, 11, 1-8.
https://doi.org/10.1155/2014/456804
[12] Gepreel, K.A. and Omran, S. (2012) Exact Solutions for Nonlinear Partial Fractional Differential Equations. Chinese Physics B, 21, 110-204.
https://doi.org/10.1088/1674-1056/21/11/110204
[13] Bekir, A., Guner, O. and Unsal, O. (2013) Exact Solutions of Nonlinear Fractional Differential Equations by (G'/G)-Expansion Method. Chinese Physics B, 11, 110-202.
https://doi.org/10.1088/1674-1056/22/11/110202
[14] Unsal, O., Guner, O., Bekir, A. and Cevikel, A.C. (2017) The (G'/G)-Expansion Method for the Nonlinear Time Fractional Differential Equations. AIP Conference Proceedings, 1798, Article ID: 020167.
[15] American Institute of Physics (1798) Fractional Differential Equations. American Institute of Physics, Prince George's County, Maryland.
[16] Yang, X.J. (2012) Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York.
[17] Yang, X.J. (2012) A Short Note on Local Fractional Calculus of Function of One Variable. Journal of Applied Library and Information Science, 1, 1-13.
[18] Yang, X.J. (2012) The Zero-Mass Renormalization Group Differential Equations and Limit Cycles in Non-Smooth Initial Value Problems. Prespacetime Journal, 9, 913-923.
[19] Hu, M.S., Baleanu, D. and Yang, X.J. (2013) One-Phase Problems for Discontinuous Heat Transfer in Fractal Media. Mathematical Problems in Engineering, 3, Article ID: 358473.
https://doi.org/10.1155/2013/358473
[20] Feng, Z.S. (2002) The First Integral Method to Study the Burgers-KdV Equation. Journal of Physics A: Mathematical and General, 35, 343-349.
https://doi.org/10.1088/0305-4470/35/2/312
[21] Su, W.H., Yang, X.J., Jafari, H. and Baleanu, D. (2013) Fractional Complex Transform Method for Wave Equations on Cantor Sets within Local Fractional Differential Operator. Advances in Difference Equations, 97, 1-8.
https://doi.org/10.1186/1687-1847-2013-97
[22] Yang, X.J., Baleanu, D. and Srivastava, H.M. (2015) Local Fractional Integral Transforms and Their Applications. Elsevier, 10, 1-12.
[23] He, J.H., Elagan, S.K. and Li, Z.B. (2012) Geometrical Explanation of the Fractional Complex Transform and Derivative Chain Rule for Fractional Calculus. Physics Letters A, 4, 257-259.
https://doi.org/10.1016/j.physleta.2011.11.030

为你推荐