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The element-free Galerkin method based on the shifted basis for solving the Kuramoto- Sivashinsky equation

The element-free Galerkin method based on the shifted basis for solving the Kuramoto- Sivashinsky equation

Feng Zhao, Wang Xiao-Dong, Ouyang Jie
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• Abstract

The Kuramoto-Sivashinsky equation is a kind of high-order nonlinear evolution equation which can describe complicated chaotic nature. Due to the existence of high-order derivatives in the equation, the shape functions violate the consistency conditions when using traditional element-free Galerkin method which adopts high-order polynomial basis functions to construct the shape functions. In order to solve the problems encountered in the traditional element-free Galerkin method, a kind of element-free Galerkin method adopting the shifted polynomial basis functions is presented in this paper. Compared with the traditional element-free Galerkin method, the Galerkin principle is still used to discrete the equation in this method, but the shape functions are constructed by moving least squares based on the shifted polynomial basis functions. Numerical results for the Kuramoto-Sivashinsky equation having traveling wave solution and chaotic nature prove the validity of the presented method.

Authors and contacts

• Funds: Project supported by the National Basic Research Program of China (Grant No. 2012CB025903) and the National Natural Science Foundation of China (Grant No. 10871159).

References

 [1] Kuramoto Y, Tsuzuki T 1975 Prog. Theor. Phys. 54 687 [2] Sivashinsky G I 1977 Acta Astrorsautica 4 1177 [3] Hyman J M, Nicolaenko B 1986 Physica D 18 113 [4] Liu B Z, Peng J H 2004 Nonlinear Dynamics (Beijing: Higher Education Press) p120 (in Chinese) [刘秉正, 彭建华 2004 非线性动力学 (北京:高等教育出版社) 第120页] [5] Kuramoto Y, Tsuzuki T 1976 Pron. Theor. Phys. 55 356 [6] Sivashinsky G I, Michelson D M 1980 Pron. Theor. Phys. 63 2112 [7] Sivashinsky G I 1983 Ann. Rev. Fluid Mech. 15 179 [8] Fan E 2000 Phys. Lett. A 277 212 [9] Peng Y Z 2003 Commun. Theor. Phys. 39 641 [10] Nickel J 2007 Chaos, Solitons and Fractals 33 1376 [11] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201 [12] Cheng R J, Cheng Y M 2011 Chin. Phys. B 20 070206 [13] Jin X Z, Li G, Aluru N R 2001 Comput. Modell. Eng. Sci. 2 447 [14] Fernanez-Mendez S, Huerta A 2004 Comput. Methods Appl. Mech. Eng. 193 1257 [15] Uddin M, Haq S, Islam S U 2009 Appl. Math. Comput. 212 458 [16] Abdel-Gawad H I, Abdusalam H A 2001 Chaos, Solitons and Fractals 12 2039

Cited By

•  [1] Kuramoto Y, Tsuzuki T 1975 Prog. Theor. Phys. 54 687 [2] Sivashinsky G I 1977 Acta Astrorsautica 4 1177 [3] Hyman J M, Nicolaenko B 1986 Physica D 18 113 [4] Liu B Z, Peng J H 2004 Nonlinear Dynamics (Beijing: Higher Education Press) p120 (in Chinese) [刘秉正, 彭建华 2004 非线性动力学 (北京:高等教育出版社) 第120页] [5] Kuramoto Y, Tsuzuki T 1976 Pron. Theor. Phys. 55 356 [6] Sivashinsky G I, Michelson D M 1980 Pron. Theor. Phys. 63 2112 [7] Sivashinsky G I 1983 Ann. Rev. Fluid Mech. 15 179 [8] Fan E 2000 Phys. Lett. A 277 212 [9] Peng Y Z 2003 Commun. Theor. Phys. 39 641 [10] Nickel J 2007 Chaos, Solitons and Fractals 33 1376 [11] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201 [12] Cheng R J, Cheng Y M 2011 Chin. Phys. B 20 070206 [13] Jin X Z, Li G, Aluru N R 2001 Comput. Modell. Eng. Sci. 2 447 [14] Fernanez-Mendez S, Huerta A 2004 Comput. Methods Appl. Mech. Eng. 193 1257 [15] Uddin M, Haq S, Islam S U 2009 Appl. Math. Comput. 212 458 [16] Abdel-Gawad H I, Abdusalam H A 2001 Chaos, Solitons and Fractals 12 2039
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• Received Date:  30 March 2012
• Accepted Date:  11 June 2012
• Published Online:  05 December 2012

The element-free Galerkin method based on the shifted basis for solving the Kuramoto- Sivashinsky equation

• 1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, China
Fund Project:  Project supported by the National Basic Research Program of China (Grant No. 2012CB025903) and the National Natural Science Foundation of China (Grant No. 10871159).

Abstract: The Kuramoto-Sivashinsky equation is a kind of high-order nonlinear evolution equation which can describe complicated chaotic nature. Due to the existence of high-order derivatives in the equation, the shape functions violate the consistency conditions when using traditional element-free Galerkin method which adopts high-order polynomial basis functions to construct the shape functions. In order to solve the problems encountered in the traditional element-free Galerkin method, a kind of element-free Galerkin method adopting the shifted polynomial basis functions is presented in this paper. Compared with the traditional element-free Galerkin method, the Galerkin principle is still used to discrete the equation in this method, but the shape functions are constructed by moving least squares based on the shifted polynomial basis functions. Numerical results for the Kuramoto-Sivashinsky equation having traveling wave solution and chaotic nature prove the validity of the presented method.

Reference (16)

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