Progress in Fractional Differentiation & Applications

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In this paper, we present a numerical method for solving the one-dimensional space fractional Schr¨odinger equation in the case of a particle moving in a potential field. The fractional derivative is defined by the quantum Riesz-Feller fractional derivative. A novel weighted average non-standard finite difference method is presented to solve the underline problem numerically. The stability analysis of the proposed method is given by a recently proposed procedure similar to the standard John von Neumann stability analysis and the truncation error is analyzed. Several numerical examples are introduced for various choices of derivative order a, 1

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