Two-point diagonally implicit multistep block method for solving volterra integro-differential equation of second kind

The first part of the thesis focuses on solving Volterra integro-differential equation (VIDE) of the second kind with the multistep block method. The two points diagonally implicit multistep block (2PDIB) method is formulated for the numerical solution of the second kind of VIDE. The derivation of the 2PDIB method can be obtained using Lagrange interpolating polynomial. The numerical solution of the second kind of VIDE computed at two points simultaneously in block form using the proposed method using constant step size. These numerical solutions are executed in the predictor-corrector mode. Since an integral part of VIDE cannot be solved explicitly and analytically, the idea to approximate the solution of the integral part is discussed and the appropriate order of numerical integration formulae is chosen to approximate the solution of the integral part of VIDE which include trapezoidal rule, Simpson’s rule and Boole’s rule. Regarding the general form of VIDE, there are two cases of the kernel which are K(x; s) = 1 and K(x; s) ≠= 1. Two different procedures are developed to obtain the solution for these cases. The stability region is discussed based on the stability polynomial of the 2PDIB method paired with the appropriate quadrature rule. Linear and nonlinear problems of VIDE have been solved numerically using the 2PDIB method. Six tested problems are presented in order to study the performance and efficiency of the 2PDIB method in terms of maximum error, total function calls, total steps taken and the execution time taken. Numerical results showed that the efficiency of 2PDIB method when solving VIDE compared to the existing methods.

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