分数阶光滑函数线性和二次插值公式余项估计

THE REMAINDER ESTIMATIONS FOR LINEAR AND QUADRATIC INTERPOLATIONS OF FRACTIONAL SMOOTH FUNCTIONS

本文在局部分数阶导数定义的基础上给出了高阶局部分数阶导数定义,并据此得到了一般形式的分数阶Taylor公式.用该公式给出了分数阶光滑函数线性和二次插值公式余项的表达式,并进一步导出了分段线性插值的收敛阶估计.针对分数阶导数临界阶计算困难的问题,本文利用线性插值余项设计了一种外推算法,能够比较准确地求出函数在某点的局部分数阶导数的临界阶.最后通过编写算法的Mathematica程序,验证了理论分析的正确性,并用实例说明了算法的有效性.

This paper presents a definition for high order local fractional derivatives based on the local fractional derivative and from which a general form of the fractional Taylor＇s expansion is derived. The formula is used to derive the remainder expansions of linear and quadratic interpolations for fractional smooth functions. Further, the convergence order of piecewise linear interpolation for fractional smooth functions is obtained. For the problem of how to compute the critical orders efficiently, this paper designs an extrapolation algorithm to accurately evaluate the critical orders of local fractional derivative at the point where the function is not sufficiently smooth by using the remainder of linear interpolation. Finally, the correctness of the theoretical analysis is verified by implementing Mathematica program. Numerical examples also show that the method is effective.