Zanariah Abdul Majid

One-step block method for solving linear Volterra integro-differential equations (VIDEs) is presented in this paper. In VIDEs, the unknown function appears in the form of derivative and under the integral sign. The popular methods for solving VIDEs are the method of quadrature or quadrature method combined with numerical method. The proposed block method will solve the ordinary differential equations (ODEs) part and Newton-Cotes quadrature rule is applied to calculate the integral part of VIDEs. Numerical problems are presented to illustrate the performance of the proposed method.

Two-point block method with variable step-size strategy is presented to obtain the solutions for boundary value problems directly. Dirichlet type and Neumann type of boundary conditions are studied in this paper. Multiple shooting techniques adapted with the three-step iterative method are employed for generating the guessing value. Six boundary value problems are solved using the proposed method, and the numerical results are compared to the existing methods. The results suggest a significant improvement in the efficiency of the proposed methods in terms of the number of steps, execution time, and accuracy.

The three-point fully implicit block methods are developed for solving large systems of ordinary differential equations using variable step size on a parallel shared memory computer. The methods calculate the numerical solution at three points simultaneously and are suitable for parallelization across the method. The methods are in a simple form as Adams Moulton method with the specific aim of gaining efficiency. For large problems, the parallel implementation produced a good speed-up with respect to the sequential timing and hence better efficiency for the methods developed.