We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the L\'evy-Fokker-Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics.

翻译：我们提出了一种适用于有界域的新的分数阶拉普拉斯算子，该算子表示为守恒律形式，因此特别适合有限体积格式。我们的方法允许直接施加无通量边界条件。首先，我们证明了分数阶热方程的适定性理论。我们还开发了一种数值格式，能够正确捕捉分数阶拉普拉斯算子的作用及其反常扩散效应。我们将Lévy-Fokker-Planck方程的数值解与已知解析解进行了基准测试。最后，我们通过数值方法探讨了这些方程在稳态和长时间渐近行为方面的性质。