The Implicit Multistep Block Method with An Off-step Point for Initial Value Problems of Neutral Delay Volterra Integro-differential Equations

The aim of this manuscript is to solve the initial-value problems of neutral delay Volterra integro-differential equations with constant or proportional delays. Hence, a proposed hybrid technique named as an implicit multistep block method with an off-step point (1OBM4) is formulated for the numerical solution of NDVIDE. A LMM associated with an off-point is known as hybrid LMM. The proposed technique, 1OBM4, attempts to solve the problem synchronously in a block manner. Moreover, a Taylor expansion is implemented to develop 1OBM4 in predictor-corrector mode. Two different approaches are presented in order to solve both integral and differential parts of the problem. Some analyses on 1OBM4 are considered in terms of order and convergence of the method. A stability polynomial is also obtained for the stability regions to be constructed. In the last section, some numerical results are demonstrated to show the applicability of 1OBM4 in solving NDVIDE with constant or proportional delays.