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On the numerical solutions for the fractional diffusion equation. (English) Zbl 1221.65263
Summary: Fractional differential equations have recently been applied in various area of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional diffusion equation (FDE) is considered. The fractional derivative is described in the Caputo sense. The method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FDE to a system of ordinary differential equations, which solved by the finite difference method. Numerical simulation of FDE is presented and the results are compared with the exact solution and other methods.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
35K20 Initial-boundary value problems for second-order parabolic equations
45K05 Integro-partial differential equations
Full Text: DOI
[1] Bagley, R.L.; Torvik, P.J., On the appearance of the fractional derivative in the behavior of real materials, J appl mech, 51, 294-298, (1984) · Zbl 1203.74022
[2] Das, S., Functional fractional calculus for system identification and controls, (2008), Springer New York · Zbl 1154.26007
[3] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electron trans numer anal, 5, 1-6, (1997) · Zbl 0890.65071
[4] Hashim, I.; Abdulaziz, O.; Momani, S., Homotopy analysis method for fractional ivps, Commun nonlinear sci numer simul, 14, 674-684, (2009) · Zbl 1221.65277
[5] He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput meth appl mech eng, 167, 1-2, 57-68, (1998) · Zbl 0942.76077
[6] Inc, M., The approximate and exact solutions of the space- and time-fractional burger’s equations with initial conditions by variational iteration method, J math anal appl, 345, 476-484, (2008) · Zbl 1146.35304
[7] Jafari, H.; Daftardar-Gejji, V., Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition, Appl math comput, 180, 488-497, (2006) · Zbl 1102.65135
[8] Lubich, Ch., Discretized fractional calculus, SIAM J math anal, 17, 704-719, (1986) · Zbl 0624.65015
[9] Meerschaert, M.M.; Tadjeran, C., Finite difference approximations for fractional advection – dispersion flow equations, J comput appl math, 172, 1, 65-77, (2004) · Zbl 1126.76346
[10] Meerschaert, M.M.; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl numer math, 56, 80-90, (2006) · Zbl 1086.65087
[11] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[12] Rawashdeh, E.A., Numerical solution of fractional integro-differential equations by collocation method, Appl math comput, 176, 1-6, (2006) · Zbl 1100.65126
[13] Samko, S.; Kilbas, A.; Marichev, O., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach London · Zbl 0818.26003
[14] Snyder, M.A., Chebyshev methods in numerical approximation, (1966), Prentice-Hall Inc. Englewood Cliffs, N.J · Zbl 0173.44102
[15] Sweilam, N.H.; Khader, M.M.; Al-Bar, R.F., Numerical studies for a multi-order fractional differential equation, Phys lett A, 371, 26-33, (2007) · Zbl 1209.65116
[16] Tadjeran, C.; Meerschaert, M.M., A second-order accurate numerical method for the two dimensional fractional diffusion equation, J comput phys, 220, 813-823, (2007) · Zbl 1113.65124
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