An element-free Galerkin (EFG) method for generalized Fisher equations (GFE)

doi:10.1088/1674-1056/22/6/060210

中国物理B ›› 2013, Vol. 22 ›› Issue (6): 60210-060210.doi: 10.1088/1674-1056/22/6/060210

An element-free Galerkin (EFG) method for generalized Fisher equations (GFE)

时婷玉^a, 程荣军^b, 葛红霞^a

a Faculty of Science, Ningbo University, Ningbo 315211, China;
b Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China

收稿日期:2012-11-07 修回日期:2012-12-15 出版日期:2013-05-01 发布日期:2013-05-01
基金资助:
Project supported by the National Natural Science Foundation of China (Grant No. 11072117), the Natural Science Foundation of Ningbo City (Grant Nos. 2012A610038 and 2012A610152), the Scientific Research Fund of Education Department of Zhejiang Province, China (Grant No. Z201119278), and K.C. Wong Magna Fund in Ningbo University.

An element-free Galerkin (EFG) method for generalized Fisher equations (GFE)

Shi Ting-Yu (时婷玉)^a, Cheng Rong-Jun (程荣军)^b, Ge Hong-Xia (葛红霞)^a

a Faculty of Science, Ningbo University, Ningbo 315211, China;
b Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China

Received:2012-11-07 Revised:2012-12-15 Online:2013-05-01 Published:2013-05-01
Contact: Ge Hong-Xia E-mail:gehongxia@nbu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant No. 11072117), the Natural Science Foundation of Ningbo City (Grant Nos. 2012A610038 and 2012A610152), the Scientific Research Fund of Education Department of Zhejiang Province, China (Grant No. Z201119278), and K.C. Wong Magna Fund in Ningbo University.

摘要/Abstract

摘要： A generalized Fisher equation (GFE) relates the time derivative of the average of the intrinsic rate of growth to its variance. The exact mathematical result of GFE has been widely used in population dynamics and genetics, where it originated. Many researchers have studied the numerical solutions of GFE, up to now. In this paper, we introduce an element-free Galerkin (EFG) method based on the moving least-square approximation to approximate positive solutions of the GFE from population dynamics. Compared with other numerical methods, the EFG method for GFE needs only scattered nodes instead of meshing the domain of the problem. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. In comparison with the traditional method, numerical solutions show that the new method has higher accuracy and better convergence. Several numerical examples are presented to demonstrate the effectiveness of the method.

关键词: element-free Galerkin (EFG) method, meshless method, generalized Fisher equations (GFE)

Abstract: A generalized Fisher equation (GFE) relates the time derivative of the average of the intrinsic rate of growth to its variance. The exact mathematical result of GFE has been widely used in population dynamics and genetics, where it originated. Many researchers have studied the numerical solutions of GFE, up to now. In this paper, we introduce an element-free Galerkin (EFG) method based on the moving least-square approximation to approximate positive solutions of the GFE from population dynamics. Compared with other numerical methods, the EFG method for GFE needs only scattered nodes instead of meshing the domain of the problem. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. In comparison with the traditional method, numerical solutions show that the new method has higher accuracy and better convergence. Several numerical examples are presented to demonstrate the effectiveness of the method.

Key words: element-free Galerkin (EFG) method, meshless method, generalized Fisher equations (GFE)

中图分类号: (Ordinary and partial differential equations; boundary value problems)

02.60.Lj

03.65.Ge (Solutions of wave equations: bound states)

时婷玉, 程荣军, 葛红霞. An element-free Galerkin (EFG) method for generalized Fisher equations (GFE)[J]. 中国物理B, 2013, 22(6): 60210-060210.

Shi Ting-Yu (时婷玉), Cheng Rong-Jun (程荣军), Ge Hong-Xia (葛红霞). An element-free Galerkin (EFG) method for generalized Fisher equations (GFE)[J]. Chin. Phys. B, 2013, 22(6): 60210-060210.

参考文献 33

[1]	Fisher R A 1930 The General Theory of Natural Selection (Oxford: Oxford University Press) pp. 22-47
[2]	Vlad M O, Szedlacsek S E, Pourmand N, Cavalli-Sforza L L, Oefner P and Ross J 2005 Proc. Natl. Acad. Sci. USA 102 9848
[3]	Ross J 2010 Proc. Natl. Acad. Sci. USA 107 12777
[4]	Gazdag J and Canosa J 1974 J. Appl. Probab. 11 445
[5]	Qiu Y and Sloan D M 1998 J. Comput. Phys. 146 726
[6]	Wazwaz A M and Gorguis A 2004 Appl. Math. Comput. 154 609
[7]	Tang S and Weber R O 1991 J. Austr. Math. Soc. Sci. B 33 27
[8]	Al-Khaled K 2001 J. Comput. Appl. Math. 137 245
[9]	Mickens R E 1994 Numer. Meth. Part. Differ. Eqns. 10 581
[10]	El-Azab M S 2007 Appl. Math. Comput. 186 579
[11]	Mittal R C and Ceeta A 2010 Int. J. Comput. Math. 87 3039
[12]	Sahin A, Dag I and Saka B 2008 Kybernetes 37 326
[13]	Macías-Díaz J E and Puri A 2012 J. Comput. Appl. Math. 218 5829
[14]	Zhang R P and Zhang L W 2012 Chin. Phys. B 21 090206
[15]	Alfaro M and Coville J 2012 Appl. Math. Lett. 25 2095
[16]	Belytschko T, Lu Y Y and Gu L 1994 Int. J. Numer. Methods Engng. 37 229
[17]	Wang J F, Sun F X and Cheng R J 2010 Chin. Phys. B 19 060201
[18]	Cheng R J and Ge H X 2009 Chin.Phys. B 18 4059
[19]	Du C 2000 Comput. Methods Appl. Mesh. Engng. 182 89
[20]	Kryl P and Belytschko T 1995 Comput. Mech. 17 26
[21]	Bai F N, Li D M, Wang J F and Cheng Y M 2012 Chin. Phys. B 21 020204
[22]	Cheng Y M, Wang J F and Bai F N 2012 Chin. Phys B. 21 090203
[23]	Wang J F, Sun F X and Cheng Y M 2012 Chin. Phys B 21 090204
[24]	Cheng Y M, Li R X and Peng M J 2012 Chin. Phys B 21 090205
[25]	Cheng Y M and Peng M J 2005 Sci. China G: Phys. Mech. & Astron. 35 435 in Chinese
[26]	Cheng Y M and Li J H 2005 Acta Phys. Sin. 54 4463 (in Chinese)
[27]	Qin Y X and Cheng Y M 2006 Acta Phys. Sin. 55 3215 (in Chinese)
[28]	Cheng R J and Cheng Y M 2007 Acta Phys. Sin. 56 5569 (in Chinese)
[29]	Dai B D and Cheng Y M 2007 Acta Phys. Sin. 56 597 (in Chinese)
[30]	Cheng R J and Cheng Y M 2008 Acta Phys. Sin. 57 6037 (in Chinese)
[31]	Ge H X, Liu Y Q and Cheng R J 2010 Chin. Phys. B 21 010206
[32]	Cheng R J and Cheng Y M 2011 Chin. Phys. B 20 070206
[33]	Xiao Y M and Wu Y J 2011 J. Lanzhou University 47 2 (in Chinese)

An element-free Galerkin (EFG) method for generalized Fisher equations (GFE)

An element-free Galerkin (EFG) method for generalized Fisher equations (GFE)

PDF (PC)

赞

可视化

摘要/Abstract

引用本文

使用本文

参考文献 33

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	马吉超, 魏高峰, 刘丹丹. Improved reproducing kernel particle method for piezoelectric materials[J]. 中国物理B, 2018, 27(1): 10201-010201.
[2]	吴意, 马永其, 冯伟, 程玉民. Topology optimization using the improved element-free Galerkin method for elasticity[J]. 中国物理B, 2017, 26(8): 80203-080203.
[3]	唐耀宗, 李小林. Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems[J]. 中国物理B, 2017, 26(3): 30203-030203.
[4]	陈莘莘, 王娟, 李庆华. Two-dimensional fracture analysis of piezoelectric material based on the scaled boundary node method[J]. 中国物理B, 2016, 25(4): 40203-040203.
[5]	程荣军, 程玉民. Solving unsteady Schrödinger equation using the improved element-free Galerkin method[J]. 中国物理B, 2016, 25(2): 20203-020203.
[6]	马永其, 周延凯. Hybrid natural element method for large deformation elastoplasticity problems[J]. 中国物理B, 2015, 24(3): 30204-030204.
[7]	程玉民, 刘超, 白福浓, 彭妙娟. Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method[J]. 中国物理B, 2015, 24(10): 100202-100202.
[8]	周延凯, 马永其, 董轶, 冯伟. Hybrid natural element method for viscoelasticity problems[J]. , 2015, 24(1): 10204-010204.
[9]	王延冲, 李小林. A meshless algorithm with moving least square approximations for elliptic Signorini problems[J]. , 2014, 23(9): 90202-090202.
[10]	葛红霞, 程荣军. A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation[J]. 中国物理B, 2014, 23(4): 40203-040203.
[11]	翁云杰, 程玉民. Analysis of variable coefficient advection–diffusion problems via complex variable reproducing kernel particle method[J]. 中国物理B, 2013, 22(9): 90204-090204.
[12]	李小林, 李淑玲. A meshless Galerkin method with moving least square approximations for infinite elastic solids[J]. 中国物理B, 2013, 22(8): 80204-080204.
[13]	王启防, 戴保东, 栗振锋. A complex variable meshless local Petrov-Galerkin method for transient heat conduction problems[J]. 中国物理B, 2013, 22(8): 80203-080203.
[14]	Cheng Rong-Jun, Wei Qi. Analysis of the generalized Camassa and Holm equation with the improved element-free Galerkin method[J]. 中国物理B, 2013, 22(6): 60209-060209.
[15]	王健菲, 程玉民. A new complex variable meshless method for advection–diffusion problems[J]. 中国物理B, 2013, 22(3): 30208-030208.