Finite element method for space-time fractional diffusion equation
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In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order β (1 < β ≤ 2), and the first order time derivative by a Caputo fractional derivative of order γ (0 < γ ≤ 1). For the 0 < γ < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(τ 2−γ + h 2) and O(τ 2 + h 2), respectively, in which τ is the time step size and h is the space step size. And for the case γ = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(τ 2 + h 2) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.
KeywordsFinite element method Space-time fractional diffusion equation Riesz derivative Caputo derivative Riemann-Liouville derivative
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