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In this paper, a novel compact operator is derived for the approximation of the Riesz derivative with order α ∈ (1, 2]. The compact operator is proved with fourth-order accuracy. Combining the compact operator in space discretization, a linearized difference scheme is proposed for a two-dimensional nonlinear space fractional Schrödinger equation. It is proved that the difference scheme is uniquely solvable, stable, and convergent with order O(τ 2 + h 4 ), where τ is the time step size, h = max{h 1 , h 2 }, and h 1 , h 2 are space grid sizes in the x direction and the y direction, respectively. Based on the linearized difference scheme, a compact alternating direction implicit scheme is presented and analyzed. Numerical results demonstrate that the compact operator does not bring in extra computational cost but improves the accuracy of the scheme greatly. Introduction.Fractional quantum mechanics is a theory used to discuss quantum phenomena in fractal environments. In quantum physics the first successful attempt to apply the fractality concept was the Feynman path integral approach to quantum mechanics. Feynman and Hibbs reformulated the nonrelativistic quantum mechanics as a path integral over Brownian paths [1,2]. Thus the Feynman-Hibbs fractional background leads to standard (nonfractional) quantum mechanics. Laskin [3] extended the fractality concept in quantum physics to construct a fractional path integral and formulate the fractional quantum mechanics as a path integral over the paths of the Lévy flights. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Lévy paths leads to fractional quantum mechanics and fractional statistical mechanics, namely, if the path integral over Brownian trajectories leads to the well-known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation.The space fractional Schrödinger equation includes a space fractional derivative of order α (1 < α < 2) instead of the Laplacian in the standard Schrödinger equation [4]. Assumption that an anomalous relation between energy and the angular frequency is of fractional Planck quantum energy relation leads to a time fractional Schrödinger equation [5], which replaces iu t in the classical time dependent Schrödinger equation by i α ∂ α u ∂t α , where i 2 = −1 and α is the order of fractional derivative.

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a b s t r a c tThis article is devoted to the study of high order difference methods for the fractional diffusion-wave equation. The time fractional derivatives are described in the Caputo's sense. A compact difference scheme is presented and analyzed. It is shown that the difference scheme is unconditionally convergent and stable in L 1 -norm. The convergence order is Oðs 3Àa þ h 4 Þ.Two numerical examples are also given to demonstrate the theoretical results.

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