Hybrid Multistep Block Method for Solving Neutral Volterra Integro-Differential Equation with Proportional and Mixed Delays

Abstract

The neutral Volterra integro-differential equation with proportional and mixed delays (NDVIDE) is being solved by a newly proposed technique in numerical method, namely, the two-point one off-point block multistep method (1OBM3). The method is also known as a hybrid multistep block method. Subsequently, Lagrange interpolating polynomial is utilized in order to develop the hybrid block method. The foundation of the technique is taken from predictor and corrector formulae. The proposed method will solve NDVIDE in two steps simultaneously, with three predictor formulae including one off-point. The NDVIDE problems are solved via the constant step size technique. In order to solve the integral and differential parts of the problems, two alternative numerical approaches are applied. The differentiation part is approximated by deriving the divided difference formula, while the integration part is interpolated using composite Simpson’s rule. Note that the proposed method has been analysed thoroughly regarding its order, consistency, zero stability and convergence of the method. The stability region for 1OBM3 has been constructed based on the stability polynomial obtained. Consequently, numerical results are presented to demonstrate the effectiveness of the proposed method, 1OBM3.