Time-dependent fractional advection-diffusion equations by an implicit MLS meshless method

International Journal for Numerical Methods in Engineering

Volumn 88, Issue 13, 2011, Pages 1346-1362

  Zhuang, P ^a   Gu, Y T ^b   Liu, F ^b   Turner, I ^b   Yarlagadda, P K D V ^b  

^a XIAMEN UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

Author keywords

Fractional advection diffusion equation; Fractional partial differential equation; Implicit numerical schemes; Meshfree method; Meshless method; Moving least squares

Indexed keywords

ADVECTION DIFFUSION EQUATION; MESH-FREE METHOD; MESH-LESS METHODS; MOVING LEAST SQUARES; NUMERICAL SCHEME;

ADVECTION; COMPUTER SIMULATION; DIFFERENTIATION (CALCULUS); DIFFUSION; FINITE DIFFERENCE METHOD; LEAST SQUARES APPROXIMATIONS; NUMERICAL METHODS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 81955168106     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.3223     Document Type: Article

References (41)

1. Agrawal OP, Machado JAT, Sabatier J, Introduction (Preface). Nonlinear Dynamics 2004; 38: 1-2.
2. Benson DA, Wheatcraft SW, Meerschaert MM, Application of a fractional advection-dispertion equation. Water Resources Research 2000; 36(6): 1403-1412.
3. Butzer PL, Westphal U, An Introduction to Fractional Calculus. World Scientific: Singapore, 1-85.
4. Miller KS, Ross B, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley: New York, 1993.
5. Metzler R, Barkai E, Klafter J, Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck approach. Physics Review Letter 1999; 82: 3563-3567.
6. Podlubny I, Fractional Differential Equations. Academic Press: New York, 1999.
7. Chechkin A, Gonchar VY, Kflafter J, Metzler R, Fundamentals of Lévy fight processes. Advances in Chemical Physics 2006; 133: 436-496.
8. Metzler R, Klafter J, The random walk's guide to anomalous diffusion a fractional dynamics approach. Physics Reports 2000; 339(1): 1-77.
9. Sokolov I, Klafter J, Blumen A, Fractional kinetic equations. Physics Today 2002; 55: 48-54.
10. Zaslavsky G, Chao, fractional kinetics, and anomalous transport. Physics Reports 2002; 371(1): 461-580.
11. Metzler R, Klafter J, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A: Mathematical and General 2004; 37(2): R161-R208.
12. Meerschaert M, Tadjeran C, Finite difference approximations for fractional advection-dispersion flow equations. Journal of Computational and Applied Mathematics 2004; 172: 65-77.
13. Liu F, Zhuang P, Anh V, Turner I, Burrage K, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Applied Mathematics and Computation 2007; 191: 12-20.
14. Zhuang P, Liu F, Anh V, Turner IW, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM Journal on Numerical Analysis 2009; 47(3): 1760-1781.
15. Liu Q, Liu F, Turner IW, Anh V, Numerical simulation for the 3D seepage flow with fractional derivatives in porous media. IMA Journal of Applied Mathematics 2008; 74(2): 201-229.
16. Yang Q, Novel analytical and numerical methods for solving fractional dynamical systems. Ph.D. Thesis, Queensland University of Technology, 2010.
17. Ervin VJ, Roop JP, Variational formulation for the stationary fractional advection-dispersion equation. Numerical Methods Partial Differential Equations 2004; 22: 558-576.
18. Ervin VJ, Heuer N, Roop JP, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation. SIAM Journal on Numerical Analysis 2007; 45: 572-591.
19. 2. Journal of Computational and Applied Mathematics 2006; 193(1): 243-268.
20. Lin Y, Xu C, Finite difference/spectral approximations for the time-fractional diffusion equation. Journal of Computational Physics 2007; 225(2): 1533-1552.
21. Belytschko T, Krongauz Y, Organ D, Meshless methods: overview and recent developments. Computer Methods in Applied Mechanics and Engineering 1996; 139: 3-47.
22. Onate E, Idelsohn S, Zienkiewicz OC, Taylor RL, Sacco C, A finite method in computational mechanics: applications to convective transport and fluid flow. International Journal for Numerical Methods in Engineering 1996; 39: 3839-3866.
23. Gingold RA, Moraghan JJ, Smooth particle hydrodynamics: theory and applications to non spherical stars. Monthly Notices of the Royal Astronomical Society 1977; 181: 375-389.
24. Kansa EJ, Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics. Computers and Mathematics with Applications 1990; 19(8/9): 127-145.
25. Nayroles B, Touzot G, Villon P, Generalizing the finite element method: diffuse approximation and diffuse elements. Computational Mechanics 1992; 10(5): 307-318.
26. Liu WK, Jun S, Zhang Y, Reproducing kernel particle methods. International Journal for Numerical Methods in Engineering 1995; 20: 1081-1106.
27. Liu GR, Gu YT, A point interpolation method for two-dimensional solids. International Journal for Numerical Methods in Engineering 2001; 50(4): 937-951.
28. Liu GR, Gu YT, Assessment and applications of point interpolation methods for computational mechanics. International Journal for Numerical Methods in Engineering 2004; 59(10): 1373-1397.
29. Atluri SN, Zhu T, A new meshfree local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics 1998; 22: 117-127.
30. Liu GR, Gu YT, A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids. Journal of Sound and Vibration 2001; 246(1): 29-46.
31. Liu GR, Gu YT, Comparisons of two meshfree local point interpolation methods for structural analyses. Computational Mechanics 2002; 29(2): 107-121.
32. Mukherjee YX, Mukherjee S, Boundary node method for potential problems. International Journal for Numerical Methods in Engineering 1997; 40: 797-815.
33. Gu YT, Liu GR, A boundary point interpolation method for stress analysis of solids. Computational Mechanics 2002; 28: 47-54.
34. Gu YT, Liu GR, A boundary radial point interpolation method (BRPIM) for 2-D structural analyses. Structural Engineering and Mechanics 2003; 15(5): 535-550.
35. Liu GR, Gu YT, An Introduction to Meshfree Methods and their Programming. Springer: Berlin, 2005.
36. Chen W, Ye L, Sun H, Fractional diffusion equations by the Kansa method. Computers and Mathematics with Applications 2010; 59(5): 1614-1620.
37. Gu YT, Zhuang PH, Liu FW, An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation. CMES-Computer Modeling in Engineering and Sciences 2010; 56(3): 303-333.
38. Zhu T, Atluri SN, A Modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method. Computational Mechanics 1998; 21: 211-222.
39. Atluri SN, Kim HG, Cho JY, A critical assessment of the truly meshless local Petrov-Galerkin (MLPG), and local boundary integral equation (LBIE) methods. Computational Mechanics 1999; 24: 348-372.
40. Belytschko T, Krongauz Y, Organ D, Liu WK, Smoothing and accelerated computations in the element free Galerkin method. Journal of Computational and Applied Mathematics 1996; 74: 111-126.
41. Liu GR, Mesh Free Methods: Moving Beyond the Finite Element Method (2nd edn). CRC Press: U.S.A., 2009.