A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

doi:10.1088/1674-1056/23/4/040203

Abstract Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging interpolation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.

Keywords: meshless method moving Kriging interpolation time-fractional diffusion equation

Received: 10 August 2013
Revised: 08 October 2013
Accepted manuscript online:

PACS:	02.60.Lj	(Ordinary and partial differential equations; boundary value problems)
	03.65.Ge	(Solutions of wave equations: bound states)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11072117), the Natural Science Foundation of Ningbo City, China (Grant No. 2013A610103), the Natural Science Foundation of Zhejiang Province, China (Grant No. Y6090131), the DisciplinaryProject of Ningbo City, China (Grant No. SZXL1067), and the K. C. Wong Magna Fund in Ningbo University, China.

Corresponding Authors: Ge Hong-Xia
E-mail: gehongxia@nbu.edu.cn

About author: 02.60.Lj; 03.65.Ge

Cite this article:

Ge Hong-Xia (葛红霞), Cheng Rong-Jun (程荣军) A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation 2014 Chin. Phys. B 23 040203

[1]	Liu F, Shen S, Anh V and Turner I 2005 ANZIAM J. 46 488
[2]	Metzler R and Klafter J 2005 Phys. Rep. 339 1
[3]	Podlubny I 1999 Fractional Differential Equation (New York: Academic Press) p. 21
[4]	Debnath L and Bhatta D 2004 Frac. Calc. Appl. Anal. 7 21
[5]	Sweilam N H, Khader M M and Al-Bar R F 2007 Phys. Lett. A 371 26
[6]	Murio A 2008 Comput. Math. Appl. 56 1138
[7]	Khader M M 2011 Commun. Nonlinear Sci. Numer. Simul. 16 2535
[8]	Sweilam N H, Khader M M and Mahdy A M S 2012 J. Fract. Calc. Appl. 2 1
[9]	Bagley R L and Torvik P J 1984 J. Appl. Mech. 51 294
[10]	Diethelm K 1997 Electron. Trans. Numer. Anal. 5 1
[11]	Enelund M and Josefson B L 1997 AIAA J. 35 1630
[12]	He J H 1998 Comput. Methods Appl. Mech. Eng. 167 57
[13]	Tadjeran C and Meerschaert M M 2007 J. Comput. Phys. 220 813
[14]	Ge H X, Liu Y Q and Cheng R J 2012 Chin. Phys. B 21 010206
[15]	Zhang Y 2009 Appl. Math. Comput. 215 524
[16]	Sweilama N H, Khader M M and Nagyb A M 2011 J. Comput. Appl. Math. 235 2832
[17]	Zhuang P and Liu F 2007 J. Alg. Comput. Technol. 1 1
[18]	Meerschaert M M, Scheffer H P and Tadjeran C 2006 J. Comput. Phys. 11 249
[19]	West B J, Bolognab M and Grigolini P 2003 Physics of Fractal Operators (New York: Springer) p. 12
[20]	Yuste S B 2006 J. Comput. Phys. 216 264
[21]	Yuste B and Acedo L 2005 SIAMJ Numer. Anal. 42 1862
[22]	Gingold R A and Moraghan J J 1977 Mon. Not. R. Astr. Soc. 181 375
[23]	Belytschko T, Lu YY and Gu L 1994 Int. J. Numer. Methods Eng. 37 229
[24]	Nayroles B, Touzot G and Villon P 1992 Comput. Mech. 10 307
[25]	Liu W K, Jun S and Zhang Y F 2001 Int. J. Numer. Methods Fluids 20 1081
[26]	Arluri S N and Zhu T L 1998 Comput. Mech. 22 117
[27]	Liszka T J, Duarte C A M and Tworzydlo W W 1996 Comput. Methods Appl. Mech. Eng. 139 263
[28]	Cheng R J and Liew K M 2012 Comput. Method Appl. Mech. Eng. 245 132
[29]	Lim C W and Liew K M 1994 J. Sound Vib. 173 343
[30]	Liew K M and Feng Z C 2001 Int. J. Mech. Sci. 43 2613
[31]	Peng M J and Cheng Y M 2009 Eng. Anal. Bound Elem. 33 77
[32]	Cheng Y M and Peng M J 2005 Sci. Chin. Ser. G: Phys. Mech. Astron. 48 641
[33]	Liew K M and Cheng Y M 2009 Comput. Methods Appl. Mech. Eng. 198 3925
[34]	Cheng R J and Cheng Y M 2007 Acta Phys. Sin. 56 5569 (in Chinese)
[35]	Dai B D and Cheng Y M 2007 Acta Phys. Sin. 56 597 (in Chinese)
[36]	Cheng R J and Cheng Y M 2008 Acta Phys. Sin. 57 6037 (in Chinese)
[37]	Cheng R J and Ge H X 2009 Chin. Phys. B 18 4059
[38]	Wang J F, Sun F X and Cheng R J 2010 Chin. Phys. B 19 060201
[39]	Cheng R J and Ge H X 2010 Chin. Phys. B 19 090201
[40]	Cheng Y M and Li J H 2006 Sci. China Ser. G: Phys. Mech. Astron. 49 46
[41]	Wang J F and Cheng Y M 2011 Chin. Phys. B 20 030206
[42]	Cheng R J and Cheng Y M 2011 Chin. Phys. B 20 070206
[43]	Cheng R J and Cheng Y M 2011 Acta Phys. Sin. 60 070206 (in Chinese)
[44]	Cheng R J and Ge H X 2012 Chin. Phys. B 21 040203
[45]	Cheng R J and Ge H X 2012 Chin. Phys. B 21 100209
[46]	Cheng R J and Liew K M 2014 Appl. Math. Comput. 227 274
[47]	Cheng R J and Wei Q 2013 Chin. Phys. B 22 060209
[48]	Shi T Y, Cheng R J and Ge H X 2013 Chin. Phys. B 22 060210
[49]	Krige D G 1976 Advanced Geostatistics in the Mining Industry 23 279
[50]	Gu L 2003 Int. J. Numer. Methods Eng. 56 1

[1]	Improved reproducing kernel particle method for piezoelectric materials Ji-Chao Ma(马吉超), Gao-Feng Wei(魏高峰), Dan-Dan Liu(刘丹丹). Chin. Phys. B, 2018, 27(1): 010201.
[2]	Topology optimization using the improved element-free Galerkin method for elasticity Yi Wu(吴意), Yong-Qi Ma(马永其), Wei Feng(冯伟), Yu-Min Cheng(程玉民). Chin. Phys. B, 2017, 26(8): 080203.
[3]	Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems Yao-Zong Tang(唐耀宗), Xiao-Lin Li(李小林). Chin. Phys. B, 2017, 26(3): 030203.
[4]	Two-dimensional fracture analysis of piezoelectric material based on the scaled boundary node method Shen-Shen Chen(陈莘莘), Juan Wang(王娟), Qing-Hua Li(李庆华). Chin. Phys. B, 2016, 25(4): 040203.
[5]	Solving unsteady Schrödinger equation using the improved element-free Galerkin method Rong-Jun Cheng(程荣军) and Yu-Min Cheng(程玉民). Chin. Phys. B, 2016, 25(2): 020203.
[6]	Hybrid natural element method for large deformation elastoplasticity problems Ma Yong-Qi (马永其), Zhou Yan-Kai (周延凯). Chin. Phys. B, 2015, 24(3): 030204.
[7]	Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method Cheng Yu-Min (程玉民), Liu Chao (刘超), Bai Fu-Nong (白福浓), Peng Miao-Juan (彭妙娟). Chin. Phys. B, 2015, 24(10): 100202.
[8]	Hybrid natural element method for viscoelasticity problems Zhou Yan-Kai (周延凯), Ma Yong-Qi (马永其), Dong Yi (董轶), Feng Wei (冯伟). Chin. Phys. B, 2015, 24(1): 010204.
[9]	A meshless algorithm with moving least square approximations for elliptic Signorini problems Wang Yan-Chong (王延冲), Li Xiao-Lin (李小林). Chin. Phys. B, 2014, 23(9): 090202.
[10]	Analysis of variable coefficient advection–diffusion problems via complex variable reproducing kernel particle method Weng Yun-Jie (翁云杰), Cheng Yu-Min (程玉民). Chin. Phys. B, 2013, 22(9): 090204.
[11]	A complex variable meshless local Petrov-Galerkin method for transient heat conduction problems Wang Qi-Fang (王启防), Dai Bao-Dong (戴保东), Li Zhen-Feng (栗振锋). Chin. Phys. B, 2013, 22(8): 080203.
[12]	A meshless Galerkin method with moving least square approximations for infinite elastic solids Li Xiao-Lin (李小林), Li Shu-Ling (李淑玲). Chin. Phys. B, 2013, 22(8): 080204.
[13]	Analysis of the generalized Camassa and Holm equation with the improved element-free Galerkin method Cheng Rong-Jun, Wei Qi. Chin. Phys. B, 2013, 22(6): 060209.
[14]	An element-free Galerkin (EFG) method for generalized Fisher equations (GFE) Shi Ting-Yu (时婷玉), Cheng Rong-Jun (程荣军), Ge Hong-Xia (葛红霞). Chin. Phys. B, 2013, 22(6): 060210.
[15]	A new complex variable meshless method for advection–diffusion problems Wang Jian-Fei (王健菲), Cheng Yu-Min (程玉民). Chin. Phys. B, 2013, 22(3): 030208.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Cite this article:

Online attention