Abstract We introduce a new generalized Caputo-type fractional derivative which generalizes Caputo fractional derivative. Some characteristics were derived to display the new generalized derivative features. Then, we present an adaptive predictor corrector method for the numerical solution of generalized Caputo-type initial value problems. The proposed algorithm can be considered as a fractional extension of the classical Adams-Bashforth-Moulton method. Dynamic behaviors of some fractional derivative models are numerically discussed. We believe that the presented generalized Caputo-type fractional derivative and the proposed algorithm are expected to be further used to formulate and simulate many generalized Caputo type fractional models.

中文翻译：

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广义Caputo型分数阶导数初值问题的数值模拟

摘要 我们介绍了一种新的广义 Caputo 型分数导数，它推广了 Caputo 分数阶导数。导出了一些特征以显示新的广义衍生特征。然后，我们提出了一种用于广义 Caputo 型初值问题数值解的自适应预测器校正方法。所提出的算法可以被认为是经典的 Adams-Bashforth-Moulton 方法的分数扩展。数值讨论了一些分数阶导数模型的动力学行为。我们相信，所提出的广义 Caputo 型分数阶导数和所提出的算法有望进一步用于制定和模拟许多广义 Caputo 型分数模型。