Numerical solving differential equations with fractional derivatives requires elimination of the singularity which is inherent in the standard definition of fractional derivatives. The method of integration by parts to eliminate this singularity is well known. It allows to solve some equations but increases the order of the equation and sometimes leads to wrong numerical results or instability. We suggest another approach: the elimination of singularity by substitution. It does not increase the order of equation and its numerical implementation provides the opportunity to define fractional derivative as the limit of discretization. We present a sufficient condition for the substitution-generated difference approximation to be well-conditioned. We demonstrate how some equations can be solved using this method with full confidence that the solution is accurate with at least second order of approximation.

翻译：用分数衍生物的数值解析差异方程式要求消除分数衍生物标准定义所固有的独一性。消除这种独一性的各个部分的集成法是众所周知的。它允许解析某些方程式,但增加方程式的顺序,有时导致错误的数字结果或不稳定。我们建议另一种方法:通过替代消除单数。它不提高方程式的顺序,其数字实施为将分数衍生物定义为分化的限度提供了机会。我们为替代产生的差数近似设置了充分的条件,使替代产生的差数近似具备了良好的条件。我们展示了如何在完全有信心的情况下使用这种方法解决某些方程式,即解决方案至少以第二顺序的近似值准确无误。