In this paper, we focus on designing a well-conditioned Galerkin spectral methods for solving a two-sided fractional diffusion equations with drift in which the fractional operators are defined neither in Riemann–Liouville nor Caputo sense, and its physical meaning is clear. Based on the image spaces of Riemann–Liouville fractional integral operators on L_p([a, b]) space discussed in our previous work, after a step by step deduction, three kinds of Galerkin spectral formulations are proposed, the final obtained corresponding scheme of which shows to be well-conditioned—the condition number of the stiff matrix can be reduced from O(N^2α) to O(N^α), where N is the degree of the polynomials used in the approximation. Another point is that the obtained schemes can also be applied successfully to approximate fractional Laplacian with generalized homogeneous boundary conditions, whose fractional order α ∈ (0, 2), not only having to be limited to α ∈ (1, 2). Several numerical experiments demonstrate the effectiveness of the derived schemes. Besides, based on the numerical results, we can observe the behavior of mean first exit time, an interesting quantity that can provide us with a further understanding about the mechanism of abnormal diffusion.

中文翻译：

---

带漂移和分数拉普拉斯算子的双边分数扩散方程的良条件伽辽金谱法

在本文中，我们重点设计了一种条件良好的Galerkin谱方法，用于求解具有漂移的双边分数扩散方程，其中分数算子既不是黎曼-刘维尔意义上的也不是卡普托意义上的定义，其物理意义是明确的。基于我们前人工作中讨论的L _p ([ a ,  b ])空间上的黎曼-刘维尔分数积分算子的图像空间，经过逐步推导，提出了三种Galerkin谱公式，最终得到对应方案其中表明是良条件的——刚性矩阵的条件数可以从O ( N ^{2 α} ) 减少到O (N ^α )，其中N是近似中使用的多项式的次数。另一点是，所获得的方案也可以成功地应用于具有广义齐次边界条件的分数拉普拉斯算子，其分数阶α  ∈ (0, 2)，而不仅限于α  ∈ (1, 2)。几个数值实验证明了导出方案的有效性。此外，根据数值结果，我们可以观察到平均首次退出时间的行为，这是一个有趣的量，可以让我们进一步了解异常扩散的机制。