论文标题
分数微分方程的替换方法
Substitution Method for Fractional Differential Equations
论文作者
论文摘要
具有分数衍生物的数值求解微分方程需要消除分数导数的标准定义固有的奇异性。众所周知。它允许求解一些方程，但增加了方程的顺序，有时会导致错误的数值结果或不稳定。 我们建议另一种方法：通过替代消除奇异性。它不会增加方程的顺序，其数值实现为将分数导数定义为离散限制提供了机会。 我们为取代生成的差异近似提供了足够的条件。 我们证明了如何使用此方法充分置信地证明了如何在近似值至少二阶的近似值中准确地求解某些方程式。
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.
