The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients

Communications in Nonlinear Science and Numerical Simulation

Volumn 17, Issue 11, 2012, Pages 4125-4136

  Saadatmandi, Abbas ^a   Dehghan, Mehdi ^b   Azizi, Mohammad Reza ^c  

^a UNIVERSITY OF KASHAN   (Iran)

^b AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

^c Shariaty Technical College   (Iran)

Author keywords

Caputo derivative; Collocation method; Fractional diffusion equation; Legendre polynomials; Sinc functions

Indexed keywords

DIFFUSION IN LIQUIDS; HEAT CONVECTION; LINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIALS;

CAPUTO DERIVATIVES; COLLOCATION METHOD; CONVECTION-DIFFUSION EQUATIONS; FRACTIONAL DIFFUSION EQUATION; LEGENDRE POLYNOMIALS; LEGENDRE-COLLOCATION METHOD; SHIFTED LEGENDRE POLYNOMIALS; SINC FUNCTION; UNKNOWN COEFFICIENTS; VARIABLE COEFFICIENTS;

NUMERICAL METHODS;

EID: 84861716404     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2012.03.003     Document Type: Article

References (46)

1. Podlubny I. Fractional differential equations 1999, Academic Press.
2. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and applications of fractional differential equations 2006, Elsevier, San Diego.
3. Miller K.S., Ross B. An introduction to the fractional calculus and fractional differential equations 1993, Wiley, New York.
4. Machado J.T., Kiryakova V., Mainardi F. Recent history of fractional calculus. Commun Nonlinear Sci Numer Simul 2011, 16:1140-1153.
5. Podlubny I., Chechkin A., Skovranek T., Chen Y.Q., Jara B.M.V. Matrix approach to discrete fractional calculus II: Partial fractional differential equations. J Comput Phys 2009, 228:3137-3153.
6. Diethelm K. The analysis of fractional differential equations 2010, Springer-Verlag, Berlin.
7. Luchko Y., Gorenflo R. An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math Vietnam 1999, 24:207-233.
8. Su L., Wang W., Xu Q. Finite difference methods for fractional dispersion equations. Appl Math Comput 2010, 216:3329-3334.
9. Tadjeran C., Meerschaert M.M., Scheffler H.P. A second-order accurate numerical approximation for the fractional diffusion equation. J Comput Phys 2006, 213:205-213.
10. Odibat Z., Momani S. A generalized differential transform method for linear partial differential equations of fractional order. Appl Math Lett 2008, 21:194-199.
11. Rida S.Z., El-Sayed A.M.A., Arafa A.A.M. On the solutions of time-fractional reaction-diffusion equations. Nonlinear Sci Numer Simulat 2010, 15:3847-3854.
12. Chen Y., Wu Y., Cui Y., Wang Z., Jin D. Wavelet method for a class of fractional convection-diffusion equation with variable coefficients. J Comput Sci 2010, 1:146-149.
13. Ray S.S. Analytical solution for the space fractional diffusion equation by two-step adomian decomposition method. Commun Nonlinear Sci Numer Simul 2009, 14:1295-1306.
14. Inc M. The approximate and exact solutions of the space and time-fractional Burgers equations with initial conditions by variational iteration method. J Math Anal Appl 2008, 345:476-484.
15. Dehghan M., Yousefi S.A., Lotfi A. The use of He's variational iteration method for solving the telegraph and fractional telegraph equations. Int J Numer Methods Biomed Eng 2011, 27:219-231.
16. Dehghan M., Manafian J., Saadatmandi A. Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer Methods Partial Differ Equ 2010, 26:448-479.
17. Momani S., Odibat Z. Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations. Comput Math Appl 2007, 54:910-919.
18. Karimi Vanani S., Aminataei A. Tau approximate solution of fractional partial differential equations. Comput Math Appl 2011, 62:1075-1085.
19. Saadatmandi A., Dehghan M. A tau approach for solution of the space fractional diffusion equation. Comput Math Appl 2011, 62:1135-1142.
20. Saadatmandi A., Dehghan M. A new operational matrix for solving fractional-order differential equations. Comput Math Appl 2010, 59:1326-1336.
21. Saadatmandi A., Dehghan M. A Legendre collocation method for fractional integro-differential equations. J Vib Control 2011, 17:2050-2058.
22. Esmaeili S., Shamsi M. A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations. Commun Nonlinear Sci Numer Simul 2011, 16:3646-3654.
23. Esmaeili S., Shamsi M., Luchko Y. Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials. Comput Math Appl 2011, 62:918-929.
24. Lin Y., Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation. J Comput Phys 2007, 255:1533-1552.
25. Dehghan M., Manafian J., Saadatmandi A. The Solution of the linear fractional partial differential equations using the homotopy analysis method. Z. Naturforsch. 2010, 65a:935-949.
26. Agrawal O.P. Solution for a fractional diffusion-wave equation defined in a bounded domain. J Nonlinear Dyn 2002, 29:145-155.
27. Uddin M, Haq S. RBFs approximation method for time fractional partial differential equations. Commun. Nonlinear Sci Numer Simulat, http://dx.doi.org/10.1016/j.cnsns.2011.03.021.
28. Jiang Y., Ma J. High-order finite element methods for time-fractional partial differential equations. J Comput Appl Math 2011, 235:3285-3290.
29. Murio D.A. Implicit finite difference approximation for time fractional diffusion equations. Comput Math Appl 2008, 56:1138-1145.
30. Li X., Xu C. A spacetime spectral method for the time fractional diffusion equation. SIAM J Numer Anal 2009, 45:2108-2131.
31. Stenger F. Numerical methods based on sinc and analytic functions 1993, Springer.
32. Lund J., Bowers K. Sinc methods for quadrature and differential equations 1992, SIAM, Philadelphia.
33. Dehghan M., Saadatmandi A. The numerical solution of a nonlinear system of second-order boundary value problems using the Sinc-collocation method. Math Comput Model 2007, 46:1434-1441.
34. Saadatmandi A., Razzaghi M. The numerical solution of third-order boundary value problems using Sinc-collocation method. Commun Numer Meth Eng 2007, 23:681-689.
35. Parand K., Dehghan M., Pirkhedri A. Sinc-collocation method for solving the Blasius equation. Phys Lett A 2009, 373:4060-4065.
36. Rashidinia J., Zarebnia M. The numerical solution of integro-differential equation by means of the Sinc method. Appl Math Comput 2007, 188:1124-1130.
37. Saadatmandi A., Dehghan M. Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications. Int J Comput Math 2010, 87:997-1008.
38. Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A. Spectral methods in fluid dynamic 1988, Prentice-Hall, Englewood Cliffs, NJ.
39. Molliq R.Y., Molliq M.S.M., Noorani M.S.M., Hashim I. Variational iteration method for fractional heat- and wave-like equations. Nonlinear Anal RWA 2009, 10:1854-1869.
40. Momani S. Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method. Appl Math Comput 2005, 165:459-472.
41. Schumer R., Meerschaert M.M., Baeumer B. Fractional advection-dispersion equations for modeling transport at the Earth surface. J Geophys Res 2009, 114:F00A07. 10.1029/2008JF001246.
42. Tatari M., Dehghan M. On the solution of the non-local parabolic partial differential equations via radial basis functions. Appl Math Model 2009, 33:1729-1738.
43. Dehghan M., Shokri A. A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions. Math Comput Simulat 2008, 79:700-715.
44. Dehghan M. Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simulat 2006, 71:16-30.
45. Dehghan M. On the numerical solution of the one-dimensional convection-diffusion equation. Math Prob Eng 2005, 2005:61-74.
46. Dehghan M. Weighted finite difference techniques for the one-dimensional advection-diffusion equation. Appl Math Comput 2004, 147:307-319.