基于全局帕德逼近的米塔-列夫勒函数及其导数的数值算法

Numerical algorithm of Mittag-Leffler functions and its derivatives based on global Padé approximation

米塔-列夫勒函数类在分数阶微积分中起着非常重要的作用,是应用非常广泛的一类特殊函数。针对米塔-列夫勒函数及其导数的高精度计算问题,提出一种基于全局帕德逼近的数值算法。该算法从泰勒级数和渐进级数出发,构造有理多项式分式,实现双参数米塔-列夫勒函数E_(α,β)(x)(x≤0)及其任意阶导数d^(s)E_(α,β)(x)/d(x);(s∈N^(*))的逼近。通过调节逼近阶数,获得最佳的稳定性和精度。将数值解与解析解做对比,通过Matlab仿真实验证明了算法的运算有效性和可行性,数值求解结果稳定可靠,逼近性能优越。

The class of Mittag-Leffler functions plays a very important role in fractional calculus and is a widely used class of special functions.Aiming at the high precision calculation of Mittag-Leffler functions and their derivatives,a numerical algorithm based on global Padéapproximation is proposed.Starting from Taylor series and asymptotic series,the algorithm constructs rational fractions to approximate the two-parametric Mittag-Leffler function E_(α,β)(x)(x≤0)and its arbitrary derivatives d^(s)E_(α,β)(x)/d(x);(s∈N^(*)).The approximation order can be adjusted to obtain the best stability and accuracy.Comparing the numerical solution with the analytic solution,the Matlab simulation results show that the algorithm is effective and feasible,the numerical solutions are stable and reliable,and the approximation performance is superior.