Recent studies highlight that diffusion processes in highly heterogeneous, fractal-like media can exhibit anomalous transport phenomena, which motivates us to consider the use of generalised transport models based on fractional operators. In this work, we harness the properties of the distributed-order space fractional potential to provide a new perspective on dealing with boundary conditions for nonlocal operators on finite domains. Firstly, we consider a homogeneous space distributed-order model with Beta distribution weight. An a priori estimate based on the $L_2$ norm is presented. Secondly, utilising finite Fourier and Laplace transform techniques, the analytical solution to the model is derived in terms of Kummer’s confluent hypergeometric function. Moreover, the finite volume method combined with Jacobi-Gauss quadrature is applied to derive the numerical solution, which demonstrates high accuracy even when the weight function shows near singularity. Finally, a one-dimensional two-layered problem involving the use of the fractional Laplacian operator and space distributed-order operator is developed and the correct form of boundary conditions to impose is analysed. Utilising the idea of ‘geometric reconstruction’, we introduce a ‘transition layer’ whereby the fractional operator index varies from fractional order to integer order across a fine layer at the boundary of the domain when transitioning from the complex internal structure to the external conditions exposed to the medium. An important observation is that for a fractional dominated case, the diffusion behaviour in the main layer is similar to fractional diffusion, while near the boundary the behaviour transitions to the case of classical diffusion.

最近的研究强调，高度异质的分形介质中的扩散过程会表现出异常的传输现象，这促使我们考虑使用基于分数算子的广义传输模型。在这项工作中，我们利用分布阶空间分数势的特性，为处理有限域上的非局部算子的边界条件提供了一个新视角。首先，我们考虑一个具有 Beta 分布权重的齐次空间分布阶模型。基于先验估计${\mathrm{大号}}_2$提出了规范。其次，利用有限傅里叶和拉普拉斯变换技术，根据Kummer的汇合超几何函数推导出模型的解析解。此外，采用有限体积法结合雅可比-高斯求积法推导出数值解，即使在权函数接近奇异的情况下也表现出较高的精度。最后，开发了涉及使用分数拉普拉斯算子和空间分布阶算子的一维二维问题，并分析了施加的边界条件的正确形式。利用“几何重构”的思想，我们引入了一个“过渡层”，当从复杂的内部结构过渡到暴露于介质的外部条件时，分数算子指数在域边界的精细层上从分数阶变为整数阶。一个重要的观察结果是，对于分数主导的情况，主层中的扩散行为类似于分数扩散，而在边界附近，行为转变为经典扩散的情况。