Abstract
The [Formula: see text]-fractional differential equation usually describes the physics process imposed on the time scale set [Formula: see text]. In this paper, we first propose a difference formula for discretizing the fractional [Formula: see text]-derivative [Formula: see text] on the time scale set [Formula: see text] with order [Formula: see text] and scale index [Formula: see text]. We establish a rigours truncation error boundness and prove that this difference formula is unconditionally stable. Then, we consider the difference method for solving the initial value problem of [Formula: see text]-fractional differential equation: [Formula: see text] on the time scale set. We prove the unique existence and stability of the difference solution and give the convergence analysis. Numerical experiments show the effectiveness and high accuracy of the proposed difference method.
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