In this paper, we study a time-fractional diffusion equation with a time-invariant type variable fractional order. We propose an implicit finite difference scheme to approximate the variable-order Caputo fractional derivative, while the central difference method is employed to discretize the spatial differential operator. A novel decomposition of the temporal discretization coefficients is adopted to overcome their loss of monotonicity due to the impact of the variable order and thus to support the proof of the convergence and unconditionally stability of the numerical scheme. Numerical examples are presented to verify the effectiveness of the proposed method.

中文翻译：

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时不变类型变量阶的时间分数阶扩散方程的隐式差分格式

在本文中，我们研究了具有时不变类型可变分数阶的时间分数阶扩散方程。我们提出了一个隐式有限差分方案来近似变阶Caputo分数阶导数，而采用中心差分方法来离散化空间微分算子。通过对时间离散系数进行新的分解来克服它们由于变量阶数的影响而导致的单调性损失，从而支持数值格式的收敛性和无条件稳定性的证明。数值算例验证了所提方法的有效性。