Considering the large number of fractional operators that exist, and since it does not seem that their number will stop increasing soon at the time of writing this paper, it is presented for the first time, as far as the authors know, a simplified and compact way to work the fractional calculus through the classification of fractional operators using sets. This new way of working with fractional operators, which may be called as fractional calculus of sets, allows to generalize objects of the conventional calculus such as tensor operators, the diffusion equation, the heat equation, the Taylor series of a vector-valued function, and the fixed point method in several variables which allows to generate the method known as the fractional fixed point method. It is also shown that each fractional fixed point method that generates a convergent sequence has the ability to generate an uncountable family of fractional fixed point methods that generate convergent sequences. So, it is shown one way to estimate numerically the mean order of convergence of any fractional fixed point method in a region $\Omega$ through the problem of determining the critical points of a scalar function, and it is shown how to construct a hybrid fractional iterative method to determine the critical points of a scalar function.
翻译:考虑到现有的分数操作员数量众多,而且由于在撰写本文件时它们的数目似乎不会很快停止增加,因此,就作者所知,这是第一次提出一种简化和紧凑的方法,通过对使用组数的分数操作员进行分微计算。这种与分数操作员合作的新方式,可称为分数计算器,可被称作组数分微微微分计算器,从而可以将常规微分计算器的物体,如推线操作员、扩散方程、热等方程、矢量估函数的泰勒系列以及若干变量中的固定点方法,这些变量能够产生被称为分数固定点方法的方法。还表明,产生一个相趋同序列的每个分数固定点方法都有能力产生无法计算的分数固定点方法组合,产生相趋同序列。因此,可以用一种方法从数字角度估计某一区域任何分数固定点方法的平均趋同顺序,通过确定一个矢量值函数的临界点的问题来计算,计算出一个矢数级函数的固定点的固定点的固定点。