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A tau approach for solution of the space fractional diffusion equation. (English) Zbl 1228.65203
Summary: Fractional differentials provide more accurate models of systems under consideration. In this paper, approximation techniques based on the shifted Legendre-tau idea are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Legendre polynomials. Using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use.

MSC:
65M99Numerical methods for IVP of PDE
35R11Fractional partial differential equations
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References:
[1] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006) · Zbl 1092.45003
[2] Dalir, M.; Bashour, M.: Applications of fractional calculus, Appl. math. Sci. 4, 1021-1032 (2010) · Zbl 1195.26011 · http://www.m-hikari.com/ams/ams-2010/ams-21-24-2010/index.html
[3] Su, L.; Wang, W.; Xu, Q.: Finite difference methods for fractional dispersion equations, Appl. math. Comput. 216, 3329-3334 (2010) · Zbl 1193.65158 · doi:10.1016/j.amc.2010.04.060
[4] Oldham, K. B.; Spanier, J.: The fractional calculus, (1974) · Zbl 0292.26011
[5] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993) · Zbl 0789.26002
[6] Diethelm, K.; Ford, N. J.; Freed, A. D.; Luchko, Yu.: Algorithms for the fractional calculus: a selection of numerical methods, Comput. methods appl. Mech. engrg. 194, 743-773 (2005) · Zbl 1119.65352 · doi:10.1016/j.cma.2004.06.006
[7] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[8] 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. 345, 476-484 (2008) · Zbl 1146.35304 · doi:10.1016/j.jmaa.2008.04.007
[9] Momanim, S.; Odibat, Z.: Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. lett. A 355, 271-279 (2006) · Zbl 05675858
[10] Momani, S.; Shawagfeh, N. T.: Decomposition method for solving fractional Riccati differential equations, Appl. math. Comput. 182, 1083-1092 (2006) · Zbl 1107.65121 · doi:10.1016/j.amc.2006.05.008
[11] Ray, S. S.: Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method, Commun. nonlinear sci. Numer. simul. 14, 1295-1306 (2009) · Zbl 1221.65284 · doi:10.1016/j.cnsns.2008.01.010
[12] Dehghan, M.; Manafian, J.; Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. methods partial differential equations 26, 448-479 (2010) · Zbl 1185.65187 · doi:10.1002/num.20460
[13] Dehghan, M.; Manafian, J.; Saadatmandi, A.: The solution of the linear fractional partial differential equations using the homotopy analysis method, Z. naturforsch. A 65, 935-945 (2010) · Zbl 1185.65187
[14] Tadjeran, C.; Meerschaert, M. M.; Scheffler, H. P.: A second-order accurate numerical approximation for the fractional diffusion equation, J. comput. Phys. 213, 205-213 (2006) · Zbl 1089.65089 · doi:10.1016/j.jcp.2005.08.008
[15] Kumar, P.; Agrawal, O. P.: An approximate method for numerical solution of fractional differential equations, Signal process. 86, 2602-2610 (2006) · Zbl 1172.94436 · doi:10.1016/j.sigpro.2006.02.007
[16] Yuste, S. B.: Weighted average finite difference methods for fractional diffusion equations, J. comput. Phys. 216, 264-274 (2006) · Zbl 1094.65085 · doi:10.1016/j.jcp.2005.12.006
[17] Saadatmandi, A.; Dehghan, M.: A new operational matrix for solving fractional-order differential equations, Comput. math. Appl. 59, 1326-1336 (2010) · Zbl 1189.65151 · doi:10.1016/j.camwa.2009.07.006
[18] Khader, M. M.: On the numerical solutions for the fractional diffusion equation, Commun. nonlinear sci. Numer. simul. 16, 2535-2542 (2011) · Zbl 1221.65263 · doi:10.1016/j.cnsns.2010.09.007
[19] Dehghan, M.; Yousefi, S. A.; Lotfi, A.: The use of he’s variational iteration method for solving the telegraph and fractional telegraph equations, Internat. J. Numer. methods biomed. Eng. 27, 219-231 (2011) · Zbl 1210.65173 · doi:10.1002/cnm.1293
[20] Lotfi, A.; Dehghan, M.; Yousefi, S. A.: A numerical technique for solving fractional optimal control problems, Comput. math. Appl. 62, No. 3, 1055-1067 (2011) · Zbl 1228.65109 · doi:10.1016/j.camwa.2011.03.044
[21] Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. comput. Simulation 71, 16-30 (2006) · Zbl 1089.65085 · doi:10.1016/j.matcom.2005.10.001
[22] Lanczos, C.: Trigonometric interpolation of empirical and analytic functions, J. math. Phys. 17, 123-199 (1938) · Zbl 0020.01301
[23] Ortiz, E. L.: The tau method, SIAM J. Numer. anal. Optim. 12, 480-492 (1969) · Zbl 0195.45701 · doi:10.1137/0706044
[24] Ortiz, E. L.; Samara, H.: Numerical solution of partial differential equations with variable coefficients with an operational approach to the tau method, Comput. math. Appl. 10, No. 4, 5-13 (1984) · Zbl 0575.65118 · doi:10.1016/0898-1221(84)90081-6
[25] Abadi, M. H. A.; Ortiz, E. L.: The algebraic kernel method for the numerical solution of partial differential equations, J. numer. Funct. anal. Optim. 12, 339-360 (1991) · Zbl 0787.65080 · doi:10.1080/01630569108816433
[26] Saadatmandi, A.; Dehghan, M.: Numerical solution of the one-dimensional wave equation with an integral condition, Numer. methods partial differential equations 23, 282-292 (2007) · Zbl 1112.65097 · doi:10.1002/num.20177
[27] Saadatmandi, A.; Dehghan, M.: A tau method for the one-dimensional parabolic inverse problem subject to temperature overspecification, Comput. math. Appl. 52, 933-940 (2006) · Zbl 1125.65340 · doi:10.1016/j.camwa.2006.04.017
[28] M.R. Eslahchi, M. Dehghan, Application of Taylor series in obtaining the orthogonal operational matrix, Comput. Math. Appl., in press (doi:10.1016/j.camwa.2011.03.004). · Zbl 1221.33016
[29] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A.: Spectral methods in fluid dynamic, (1988) · Zbl 0658.76001
[30] Canuto, C.; Quarteroni, A.; Hussaini, M. Y.; Zang, T. A.: Spectral methods: fundamentals in single domains, (2006) · Zbl 1093.76002