RBFs approximation method for time fractional partial differential equations

Communications in Nonlinear Science and Numerical Simulation

Volumn 16, Issue 11, 2011, Pages 4208-4214

  Uddin, Marjan ^a,c   Haq, Sirajul ^b  

^a Illinois Institute of Technology   (United States)

^b GIK INSTITUTE OF ENGINEERING SCIENCES AND TECHNOLOGY   (Pakistan)

^c KIU   (Pakistan)

Author keywords

Interpolation method; Radial basis function (RBFs); Time fractional partial differential equations (PDEs)

Indexed keywords

ADVECTION; APPROXIMATION THEORY; DIFFERENTIATION (CALCULUS); FUNCTIONS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; RADIAL BASIS FUNCTION NETWORKS;

ADVECTION-DIFFUSION EQUATION; APPROXIMATION METHODS; BASE FUNCTION; FRACTIONAL ADVECTION-DIFFUSIONS; FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS; FUNCTIONS APPROXIMATIONS; INTERPOLATION METHOD; RADIAL BASE FUNCTION; RADIAL BASIS; TIME FRACTIONAL PARTIAL DIFFERENTIAL EQUATION;

INTERPOLATION;

EID: 79957926033     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2011.03.021     Document Type: Article

References (28)

1. Babuska I., Banerjee U., Osborn J.E. Survey of meshless and generalized finite element methods: a unified approach. Acta Numerica 2003, 12:1-125.
2. Brown D., Ling L., Kansa E., Levesley J. On approximate cardinal preconditioning methods for solving PDEs with radial basis functions. Eng Anal Bound Elem 2005, 29:343-353.
3. Chen Change-ming, Lui F., Burrage K. Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation. Appl Math Comput 2008, 198:754-769.
4. Debnath L., Bhatta D. Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics. Frac Calc Appl Anal 2004, 7:21-36.
5. Diego, Murio A. Implicite finite difference approximation for time fractional diffusion equations. Comput Math Appl 2008, 56:1138-1145.
6. Fasshauer G.E., Khaliq A.Q.M., Voss D.A. Using meshfree approximation for multi-asset American option problems. J Chines Inst Eng 2004, 27:563-571.
7. Fasshauer G.E. Meshfree aprooximation methods with matlab 2007, Word Scientific Publishing Co. Pte. Ltd.
8. Fix G.J., Roop J.P. Least squares finite element solution of a fractional order two-point boundary value problem. Comput Math Appl 2004, 48:1017-1033.
9. Fornberg B., Larsson E., Wright G. A new class of oscillatory radial basis functions. Comput Math Appl 2004, 51:1209-1222.
10. Franke C. Scattered data interpolation: tests of some methods. Math Comput 1982, 38:181-200.
11. Franke C., Schaback R. Convergence order estimates of meshless collocation methods using radial basis functions. Adv Comput Math 1998, 8:381-399.
12. Hon Y.C., Mao X.Z. An efficient numerical scheme for burgers equation. Appl Math Comput 1998, 95:37-50.
13. Hon Y.C., Schaback R. On nonsymmetric collocation by radial basis functions. Appl Math Comput 2001, 119:177-186.
14. Dag Idris, Dereli Yilmaz Numerical solutions of KdV equation using radial basis functions. Appl Math Model 2008, 32:535-546.
15. Podlubny Igor, Chechkin A., skovranek T., Chen Y., Jara B.M. Matrix approach to discrete fractional calculas II: partial fractional differential equation. J Comput Phys 2009, 228:3137-3153.
16. Kansa E.J. Multiquadrics scattered data approximation scheme with applications to computational fluid-dynamics, II. Solutions to hyperbolic, parabolic, and elliptic partial differential equations. Comput Math Appl 1990, 19:149-161.
17. Kansa E.J., Hon Y.C. Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Comput Math Appl 2000, 39:123-137.
18. 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. Appl Math Comput 2007, 191:12-20.
19. Madych W.R., Nelson S.A. Multivariate interpolation and conditionally positive definite functions II. Math Comput 1990, 54:211-230.
20. Uddin Marjan, Haq Sirajul, Siraj-ul-islam A mesh-free numerical method for solution of the Kuramoto-Sivashinsky equations. Appl Math Comput 2009, 212:458-469.
21. Dehghan Mehdi, Tatari Mehdi Determination of control parameter in a one-dimensional parabolic equation using the method of radial basis functions. Math Comput Model 2006, 44:1160-1168.
22. Micchelli C.A. Interpolation of scatterted data: distance matrix and conditionally positive definite functions. Construct Approx 1986, 2:11-22.
23. Miller K.S., Ross B. An Introduction to the fractional calculus and fractional differential equations 1974, Academic Press, New York and London.
24. Oldham K.B., Spanier J. The fractional calculas 1974, Academic Press, New York and London.
25. Podulbny I. Fractional differential equations 1999, Academic Press, New York.
26. Samko S.G., Kilbas A.A., Marichev O.I. Fractional integrals and derivatives: theory and applications 1993, Springer, USA.
27. Zheng Yunying, Li Changpin, Zhao Zhengang A note on the finite element method for the space-fractional advection diffusion equation. Comput Math Appl 2010, 59:1718-1726.
28. Odibat Zaid, Momani Shaher The Variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput Math Appl 2009, 58:2199-2208.