Fractional diffusion equations by the Kansa method

Computers and Mathematics with Applications

Volumn 59, Issue 5, 2010, Pages 1614-1620

  Chen, Wen ^a   Ye, Linjuan ^a   Sun, Hongguang ^a  

^a HOHAI UNIVERSITY   (China)

Author keywords

Anomalous diffusion; Kansa method; Radial basis function; Time fractional derivative

Indexed keywords

ANALYTICAL SOLUTIONS; ANOMALOUS DIFFUSION; DISCRETIZATIONS; FINITE DIFFERENCE SCHEME; FRACTIONAL DERIVATIVES; FRACTIONAL DIFFUSION EQUATION; MULTIQUADRICS; NUMERICAL SOLUTION; RADIAL BASIS FUNCTIONS; SPATIAL DERIVATIVES; THIN PLATE SPLINE; TIME FRACTIONAL DIFFUSION EQUATION;

ATTITUDE CONTROL; NUMERICAL ANALYSIS; PARTIAL DIFFERENTIAL EQUATIONS; RADIAL BASIS FUNCTION NETWORKS;

DIFFUSION;

EID: 76449113714     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2009.08.004     Document Type: Article

References (29)

1. Scher H., and Montroll E.W. Anomalous transit-time dispersion in amorphous solids. Physical Review B 12 6 (1975) 2455-2477
2. Szabo T.L., and Wu J. A model for longitudinal and shear wave propagation in viscoelastic media. Journal of the Acoustical Society of America 107 5 (2000) 2437-2446
3. Gorenflo R., Mainardi F., Moretti D., Pagnini G., and Paradisi P. Discrete random walk models for space-time fractional diffusion. Chemical Physics 284 (2002) 521-541
4. Del-Castillo-Negrete D., Carreras B.A., and Lynch V.E. Front dynamics in reaction-diffusion systems with Lévy flights: A fractional diffusion approach. Physical Review Letters 91 1 (2003) 018301-018304
5. Sokolov I.M., Klafter J., and Blumen A. Ballistic versus diffusive pair-dispersion in the Richardson regime. Physical Review E 61 3 (2000) 2717-2722
6. Chen W. A speculative study of 2/3-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures. Chaos 16 (2006) 023126
7. Hilfer R. Applications of Fractional Calculus in Physics (2000), World Scientific, Singapore
8. Chen W. Time-space fabric underlying anomalous diffusion. Soliton, Fractal & Chaos 28 4 (2006) 923-929
9. Witten T.A. Insights from soft condensed matter. Reviews of Modern Physics 71 2 (1999) 367-373
10. Chang F.X., Chen J., and Huang W. Anomalous diffusion and fractional advection-diffusion equation. Acta Physica Sinica 54 3 (2005) 1113-1117 (in Chinese)
11. Momani S. An algorithm for solving the fractional convection-diffusion equation with nonlinear source term. Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1283-1290
12. Chen C.M., Liu F., and Burrage K. Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation. Applied Mathematics and Computation 198 2 (2008) 754-769
13. Wyss W. The fractional diffusion equation. Journal of Mathematical Physics 27 (1986) 2782-2785
14. Podlubny I. Fractional Differential Equations (1999), Academic Press, San Diego
15. Agrawal O.P. Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dynamics 29 (2002) 145-155
16. Gorenflo R., Mainardi F., Moretti D., and Paradisi P. Time fractional diffusion: A discrete random walk approach. Nonlinear Dynamics 29 (2002) 129-143
17. Meerschaert M., and Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. Journal of Computer and Applied Mathematics 172 1 (2004) 65-77
18. Meerschaert M., and Tadjeran C. Finite difference approximations for two-sided space-fractional partial differential equations. Applied Numerical Mathematics 56 1 (2006) 80-90
19. Liu F., Anh V., Turner I., and Zhuang P. Time fractional advection dispersion equation. Journal of Applied Mathematics and Computation 13 (2003) 233-245
20. Lin Y., and Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation. Journal of Computational Physics 225 (2007) 1533-1552
21. Murio D.A. Implicit finite difference approximation for time fractional diffusion equations. Computers and Mathematics with Applications 56 (2008) 1138-1145
22. Diethelm K., Ford N.J., Freed A.D., and Luchko Y. Algorithms for the fractional calculus: A selection of numerical methods. Computer Methods in Applied Mechanics and Engineering 194 (2005) 743-773
23. Kansa E.J. Multiquadrics-A scattered data approximation scheme with application to computation fluid dynamics, I. Surface approximations and partial derivative estimates. Computers & Mathematics with Applications 19 (1990) 127-145
24. Kansa E.J. Multiquadrics-A scattered data approximation scheme with application to computation fluid dynamics, II. Solutions to hyperbolic, parabolic, and elliptic partial differential equations. Computers & Mathematics with Applications 19 (1990) 149-161
25. Zerroukat M., Power H., and Chen C.S. A numerical method for heat transfer problems using collocation and radial basis functions. International Journal for Numerical Methods in Engineering 42 (1998) 1263-1278
26. Zerroukat M., Djidjeli K., and Charafi A. Explicit and implicit meshless methods for linear advection-diffusion-type partial differential equations. International Journal for Numerical Methods in Engineering 48 (2000) 19-35
27. Chen J.H. An implicit approximation for the Caputo fractional reaction-dispersion equation. Journal of Xiamen University (Natural Science) 46 5 (2007) 617-619 (in Chinese)
28. Zhang X., Song K.Z., Lu M.W., and Liu X. Meshless methods based on collocation with radial basis functions. Computational Mechanics 26 (2000) 333-343
29. Tadjeran C., Meerschaert M.M., and Scheffler H.P. A second-order accurate numerical approximation for the fractional diffusion equation. Journal of Computational Physics 213 (2006) 205-213