Numerical Approximation of Volterra Integro-Differential Equations of the Second Kind Using Boole’s Quadrature Rule Method

This article presents the numerical approximation of Volterra integro-differential equations (VIDEs) of the second kind using the quadrature rule in the modified block method. The new implementation of new block method which considers the closest point to approximate two solutions of y(xn+1 ) and y(xn+2 ) concurrently was taken into account. This method is said to have an advantage in reducing the number of total steps and function evaluations compared to the classical multistep method. The techniques of quadrature rule which consist of the trapezoidal rule, Simpson’s 1/3 rule, Simpson’s 3/8 rule and Boole’s quadrature rule have been used to approximate the integral parts of Kernel function, zn for n = 1, 2, 3, 4, 5 for the case of K(x, s) = 1. The analysis of the order, error constant, consistency and convergence of VIDEs in the proposed method has also been presented. The stability analysis is derived using the specified linear test equation for both approximate solutions until obtained the stability polynomial. To validate the efficiency of the developed method, some of the numerical results are presented and compared with the existing method. It is shown that the modified block method has given better accuracy and efficiency in terms of maximum error and number of steps and function calls.