Numerical approximation of Lévy–Feller diffusion equation and its probability interpretation

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.