Variable-order space-fractional diffusion equations provide very competitive modeling capabilities of challenging phenomena, including anomalously superdiffusive transport of solutes in heterogeneous porous media, long-range spatial interactions and other applications, as well as eliminating the nonphysical boundary layers of the solutions to their constant-order analogues. In this paper, we prove the uniqueness of determining the variable fractional order of the homogeneous Dirichlet boundary-value problem of the one-sided linear variable-order space-fractional diffusion equation with some observed values of the unknown solutions near the boundary of the spatial domain. We base on the analysis to develop a spectral-Galerkin Levenberg–Marquardt method and a finite difference Levenberg–Marquardt method to numerically invert the variable order. We carry out numerical experiments to investigate the numerical performance of these methods.

中文翻译：

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将具有可变扩散率的可变阶空间分数扩散方程中的可变分数阶求逆：分析和模拟

可变阶空间分数扩散方程提供了具有挑战性的现象的非常有竞争力的建模能力，包括非均质多孔介质中溶质的异常超扩散传输，远距离空间相互作用和其他应用，以及消除了其解的非物理边界层常数阶类似物。在本文中，我们证明了在空间边界附近具有未知解的一些观测值的情况下，确定单线性线性变量级空间分数分数扩散方程的齐次Dirichlet边值问题的变量分数阶的唯一性领域。我们在分析的基础上发展了频谱加勒金·勒芬贝格—马夸特方法和有限差分勒芬贝格—马夸特方法，以数值方式对变量阶进行了求逆。