Diagonally implicit multistep block method for solving delay volterra integro-differential equation

In this study, two points diagonally implicit multistep block (2DIMB) methods are constructed for the numerical solution of the first and second order delay Volterra integro-differential equation (DVIDE). The second order of DVIDE is solved directly without reducing the problem in the system of the first order of DVIDE. Two distinct types of DVIDE are solved, namely unbounded and bounded time lag cases. Furthermore, the constant and pantograph delay types indicate that the delay conditions for DVIDE are also considered in this study. The strategy of the constant step size is implemented for finding the numerical solution to DVIDE. When finding the approximate solution to DVIDE, three components must be considered: the initial value problem of DVIDE, the delay solution, and the integral part. The 2DIMB method is formulated for the numerical solution of initial value problem of DVIDE and computed two solutions simultaneously in block form. This method is built on a predictor-corrector formula. The previously calculated solutions are used to obtain the delay solution for the constant delay type. Meanwhile, Lagrange interpolation polynomial is implemented to approximate the delay solution for the pantograph delay type. Since an integral part of DVIDE cannot be solved explicitly and analytically, the idea of approximating the solution is discussed. The appropriate order of the numerical integration method is chosen to approximate the solution of the integral part of DVIDE, which include trapezoidal rule, Simpson’s rule, and Boole’s rule. Analysis on order, error constants, consistency, zero-stability, and convergence of the proposed method are given in this study. Moreover, the stability region is discussed based on the stability polynomial of the 2DIMB method paired with the appropriate numerical integration method. All the computational procedures were undertaken using the C programming language in a CODE::BLOCKS platform. Numerical results showed that where the proposed methods are reliable and suitable for solving the unbounded and bounded time lag of the DVIDE for the constant and pantograph delay types. Three advantages in terms of the total steps taken, function evaluations and the execution time taken by these methods have been identified when comparing the numerical results with the Runge-Kutta and Adam-Bashforth- Moulton methods.

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