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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.

##### 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
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