In this paper, we introduce a new family of interpolants, called piecewise fractional interpolants (PFIs). The proposed interpolants are used to construct a new class of predictor–corrector methods for solving nonlinear fractional differential equations (FDEs). To construct the PFIs, it is essential to know the regularity behavior of the exact solution of an FDE. To do so, we present a pre-algorithm to provide the singular indexes of the exact solution at the initial time. Also, an error analysis with a rigorous proof of the convergence order of the proposed method is given. Some numerical examples are carried out to support the theoretical analysis.

中文翻译：

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分段分数插值及其在分数阶微分方程中的应用

在本文中，我们介绍了一个新的插值族，称为分段分数插值（PFI）。提出的插值用于构造一类新的预测器-校正器方法，用于求解非线性分数阶微分方程（FDE）。要构造PFI，必须了解FDE精确解的规则性行为。为此，我们提出了一个预算法，以在初始时间提供精确解的奇异索引。此外，给出了误差分析，并严格证明了所提出方法的收敛顺序。进行了一些数值算例以支持理论分析。