非均匀生长域上扩散输运的精确解
Exact solutions for diffusive transport on heterogeneous growing domains
AI总结：
本文针对空间非均匀生长域上的扩散输运模型，通过变换、卷积和叠加技术求出精确解，经格子随机游走模拟验证，揭示了扩散率与生长速率等参数对存活和分裂概率等统计量的影响。
摘要：
从最小的生物系统到最大的宇宙结构，空间域都会经历扩张和收缩。在这些生长域内，扩散输运是一种常见现象。数学模型已被广泛用于研究生长域上的扩散过程。然而，一个标准假设是域的生长在空间上是均匀的。在许多相关实例中情况并非如此，例如神经嵴细胞对生长中肠道组织的定植。因此，理清非均匀生长与扩散输运各自的作用并非易事。在此，我们提出在经历空间非均匀生长的域上扩散输运模型的精确解。这些精确解是通过结合变换、卷积和叠加技术获得的。我们通过与相应的基于格子的随机游走模拟进行比较，验证了这些解的准确性。我们探索了多种域生长函数，包括线性生长、指数生长与收缩，以及振荡生长。我们发现，只要域大小保持为正，推导出的解就是有效的。这些精确解揭示了模型参数（如扩散率、域生长的类型与速率）与关键统计量（如存活概率和分裂概率）之间的关系。
英文摘要：
From the smallest biological systems to the largest cosmological structures, spatial domains undergo expansion and contraction. Within these growing domains, diffusive transport is a common phenomenon. Mathematical models have been widely employed to investigate diffusive processes on growing domains. However, a standard assumption is that the domain growth is spatially uniform. There are many relevant examples where this is not the case, such as the colonisation of growing gut tissue by neural crest cells. As such, it is not straightforward to disentangle the individual roles of heterogeneous growth and diffusive transport. Here we present exact solutions to models of diffusive transport on domains undergoing spatially non-uniform growth. The exact solutions are obtained via a combination of transformation, convolution and superposition techniques. We verify the accuracy of these solutions via comparison with simulations of a corresponding lattice-based random walk. We explore various domain growth functions, including linear growth, exponential growth and contraction, and oscillatory growth. Provided the domain size remains positive, we find that the derived solutions are valid. The exact solutions reveal the relationship between model parameters, such as the diffusivity and the type and rate of domain growth, and key statistics, such as the survival and splitting probabilities.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
