分数阶微分方程：非恒定系数、仿真与模型降阶
Fractional differential equations: non-constant coefficients, simulation and model reduction
AI总结：
该研究针对带非恒定系数的Riemann-Liouville分数阶边值问题，推导变分公式证明适定性，分析基于贪心算法的降基法收敛性与分数阶的关联，并通过数值结果验证结论。
摘要：
我们研究阶数为$s\in(1,2)$的Riemann-Liouville（黎曼-刘维尔）分数阶导数的边值问题，该问题具有非恒定扩散与反应系数。我们推导并分析了一种变分公式，证明了连续问题及其有限元离散化的适定性。随后，分析了针对参数化扩散与反应系数、基于贪心算法的Reduced Basis Method（降基法）。其收敛特性，尤其是Kolmogorov $n$-宽度的衰减特性，被证明依赖于分数阶数$s$。最后，我们给出了验证上述结论的数值结果。
英文摘要：
We consider boundary value problems with Riemann-Liouville fractional derivatives of order $s\in (1, 2)$ with non-constant diffusion and reaction coefficients. A variational formulation is derived and analyzed leading to the well-posedness of the continuous problem and its Finite Element discretization. Then, the Reduced Basis Method through a greedy algorithm for parametric diffusion and reaction coefficients is analyzed. Its convergence properties, and in particular the decay of the Kolmogorov $n$-width, are seen to depend on the fractional order $s$. Finally, numerical results confirming our findings are presented.
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