Lévy flights are generalised random walk processes where the independent stationary increments are drawn from a long-tailed α-stable jump length distribution. We consider the formulation of Lévy flights, for 0 < α < 1, in terms of a space-fractional diffusion equation which fundamental solutions are the probability density functions. First, we present how to obtain the governing equation of Lévy motion from the Fourier transform of the jump distribution. Then, we derive a family of implicit numerical methods to determine the numerical solutions and we study their consistency and stability. Although numerical algorithms for the case 1 < α < 2 have been widely discussed, very few works paid attention to the case we discuss here. We present numerical experiments to show the performance of the numerical methods and to highlight the advantages and disadvantages of the different approaches. In the end we determine the numerical solutions of an initial value problem, that considers an approximation of the Dirac delta function as the initial condition, in order to obtain approximations of the probability density functions.

Lévy飞行是广义的随机游走过程，其中从长尾α稳定跳跃长度分布得出独立的固定增量。我们根据空间分数扩散方程来考虑对于0 < α <1的Lévy飞行的公式，其基本解是概率密度函数。首先，我们介绍如何从跳跃分布的傅立叶变换中获得李维运动的控制方程。然后，我们派生出一系列隐式数值方法来确定数值解，并研究它们的一致性和稳定性。尽管情况1 < α的数值算法<2已被广泛讨论，很少有人关注我们在此讨论的案例。我们目前进行数值实验，以显示数值方法的性能，并强调不同方法的优缺点。最后，我们确定初始值问题的数值解，该解将Dirac德尔塔函数的近似值作为初始条件，以便获得概率密度函数的近似值。