时间分数阶扩散方程的布朗映射有效数值解
Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart
摘要：
给出了$0<α<1$阶Caputo时间分数阶扩散方程的解表示为相应的整数阶扩散方程的解。我们证明了这些解决方案之间的线性时间映射，允许加速计算的分数阶问题的解决方案。在$N$ -点有限差分时间离散化的上下文中，在给定$O\left(N^{1+α}\ln N\right)$的预计算的情况下，映射允许从$O\left(N^2\right)$到$O\left(N^α\right)$的时间计算复杂度的改进。该映射成功地应用于最小二乘拟合的分数平流扩散模型的当前在飞行时间的实验，导致在一个到三个数量级的范围内为现实的问题大小的计算速度。
英文摘要：
The solution of a Caputo time fractional diffusion equation of order $0<α<1$ is expressed in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an $N$-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from $O\left(N^2\right)$ to $O\left(N^α\right)$, given a precomputation of $O\left(N^{1+α}\ln N\right)$. The mapping is applied successfully to the least-squares fitting of a fractional advection diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes.
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。
