Numerical Solution of Volterra Integro-Differential Equations by Hybrid Block with Quadrature Rules Method

In this paper, the implementation of one-step hybrid block method with quadrature rules will be proposed for solving linear and non-linear first order Volterra Integro-Differential Equations (VIDEs) of the second kind. VIDEs have important applications in many branches of sciences and engineering, such as analysing rhythmic biological data can be conducted by utilizing a curve fitting technique based on solutions of the VIDEs. The formulation of the hybrid block method is based on the Lagrange interpolation polynomial. The approximation of the integral part in the VIDEs will be estimated using the quadrature rules. The proposed hybrid block method of order five will compute the numerical solutions at two points simultaneously at each integration steps. The stability analysis such as order of the method, consistency, zero stable and stability region of the method are deliberated. The fixed step size is used to generate the results and the code is written in C language. Numerical simulations are presented to show the efficiency and accuracy of the hybrid method when compared to the Runge-Kutta of order four and five in terms of accuracy, total steps, and total function calls.