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In this work, the one-step Adams-Bashforth-Moulton (ABM) predictor-corrector method is introduced to solve the initial value problems (IVPs) of nonlinear fractional delay differential equations (FDDEs). Moreover, this method is extended to solve the variable-order FDDEs, where the derivative is defined in the Caputo variable-order fractional sense. Special attention is given to study the error estimate of the proposed method. Numerical test examples are presented to demonstrate utility of the method, one of these examples is the variable-order fractional delay version of the Chen system. All...