CAJ | 학술논문

为研究函数在不可微处的局部行为,各种局部分数阶微分定义被提出,α-微分是其中重要的一种。本文研究了α-微分的一些性质,证明了利用α-微分研究函数局部行为的合理性和α-微分的几何意义的合理性。当f(x)连续α-可微时(0<α<1),对于求解f(x)=0,作者提出了局部分数阶牛顿法且当f(α)(x)满足指数为α(12<α<1)的H?lder条件时,该算法是局部 Q-超线性收敛的。
위연구함수재불가미처적국부행위,각충국부분수계미분정의피제출,α-미분시기중중요적일충。본문연구료α-미분적일사성질,증명료이용α-미분연구함수국부행위적합이성화α-미분적궤하의의적합이성。당f(x)련속α-가미시(0<α<1),대우구해f(x)=0,작자제출료국부분수계우돈법차당f(α)(x)만족지수위α(12<α<1)적H?lder조건시,해산법시국부 Q-초선성수렴적。
To study the behavior of a function at non-differentiable point,various definitions of local fractional derivative were proposed,α-derivative is a very important one.In this paper,some properties ofα-derivative are presented,the rationality of the method that study local behavior of a function by means ofα-derivative is demonstrated,and the rationality of the geometric meaning ofα-derivative is verified.It is of great importance to solve f(x)= 0 when f(x)is nowhere differentiable.For continu-ouslyα-differentiable function (0<α<1),local fractional Newton method is proposed.The algorithm is locally Q-superlinearly convergent if f(α)(x)satisfies H?lder condition with exponentαwhere 1/2 <α< 1.