Abstract
In this paper, we consider the Lévy–Feller fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order α ∈ ( 0 , 2 ] ( α ≠ 1 ) and skewness θ ( | θ | ⩽ min { α , 2 - α } ). We construct two new discrete schemes of the Cauchy problem for the above equation with 0 < α < 1 and 1 < α ⩽ 2 , respectively. We investigate their probabilistic interpretation and the domain of attraction of the corresponding stable Lévy distribution. Furthermore, we present a numerical analysis for the Lévy–Feller fractional diffusion equation with 1 < α < 2 in a bounded spatial domain. Finally, we present a numerical example to evaluate our theoretical analysis.
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