一个简单的有限元方法用于带有Riemann-Liouville导数的边界值问题
A simple finite element method for the boundary value problem with a Riemann-Liouville derivative
AI总结：
本文提出一种简单方法，通过将问题转化为具有非局部低阶项的二阶两点边界值问题，以解决Riemann-Liouville导数的边界值问题，提供L^2(D)误差估计及特征值近似误差估计。
摘要：
我们考虑一个涉及在单位区间(0,1)上Riemann-Liouville分数导数阶数α∈(3/2,2)的边界值问题。由于解表示中的奇异性项x^{α-1}，标准Galerkin有限元近似收敛缓慢。本文开发了一种简单技术，将其转换为具有非局部低阶项的二阶两点边界值问题，其解可直接重构原问题的解。分析了变分公式的稳定性及解的最优正则性。开发了一种新的Galerkin有限元方法，使用分段线性或二次有限元，并提供了L^2(D)误差估计。然后将该方法应用于相应的分数Sturm-Liouville问题，给出了特征值近似误差估计。大量数值结果完全证实了我们的理论研究。
英文摘要：
We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order $α\in (3/2,2)$ on the unit interval $(0,1)$. The standard Galerkin finite element approximation converges slowly due to the presence of singularity term $x^{α-1}$ in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and $L^2(D)$ error estimates are provided. The approach is then applied to the corresponding fractional Sturm-Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study.
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。
