Numerous anomalous diffusion processes are characterized by crossovers of the scaling exponent in the mean squared displacement at some correlations time. The bi-fractional diffusion equation containing two time-fractional derivatives is a versatile mathematical tool describing specifically retarded subdiffusive transport, in which the scaling exponents acquires a smaller value, i.e., the diffusion becomes even slower after the crossover. We here derive closed-form multi-dimensional solutions for this integro-differential equation in $n$ spatial dimensions by generalizing the classical Schneider-Wyss solution of the fractional diffusion equation with a single fractional derivative. In the two-dimensional case we develop a limiting approach based on the solution of the space-time fractional diffusion equation. The probabilistic interpretation in higher dimensions is discussed. The asymptotic long- and short-time behaviors are derived. It is shown that the solution of the bi-fractional diffusion equation can be interpreted in terms of the Fox $H$-transform of the Gaussian distribution.

许多异常扩散过程的特征在于在某些相关时间均方位移中标度指数的交叉。包含两个时间分数阶导数的双分数阶扩散方程是一种通用数学工具，用于描述特定延迟的亚扩散传输，其中标度指数获得较小的值，即，在交叉后扩散变得更慢。我们在这里推导出这个积分微分方程的封闭形式的多维解$n$通过用单个分数阶导数概括分数阶扩散方程的经典 Schneider-Wyss 解来获得空间维度。在二维情况下，我们开发了一种基于时空分数扩散方程解的限制方法。讨论了更高维度的概率解释。导出渐近的长时和短时行为。结果表明，双分数扩散方程的解可以解释为 Fox$H$- 高斯分布的变换。