On the numerical solutions for the fractional diffusion equation.

*(English)*Zbl 1221.65263Summary: 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.

##### MSC:

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 |

##### Keywords:

finite difference method; fractional diffusion equation; Chebyshev polynomials; Caputo derivative
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\textit{M. M. Khader}, Commun. Nonlinear Sci. Numer. Simul. 16, No. 6, 2535--2542 (2011; Zbl 1221.65263)

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