ABSTRACT This project focuses on developing and implementing two numerical methods namely; Successive Approximation and Power Series to solve Volterra Integral Equations of the Second Kind. These equations are characterized by their integral form, which often requires numerical methods for solution. The Successive Approximation method is an iterative technique that refines the solution through repeated applications of an integral operator. The method starts with an initial guess and iteratively improves the solution until convergence is achieved. The theoretical framework for this method involves establishing the conditions for convergence and determining the rate of convergence. The Power Series method represents the solution as an infinite series of powers. This method is particularly useful for solving integral equations with polynomial or analytic kernels. The theoretical framework for this method involves determining the coefficients of the power series and establishing the radius of convergence. The results show that both methods can effectively solve Volterra Integral Equations of the Second Kind. However, the Successive Approximation method exhibits faster convergence for certain types of kernels, while the Power Series method provides a more accurate solution for equations with smooth kernels, but each method yields an exact solution to the Volterra equation of the Second Kind. A comparative analysis of the two methods reveals that the Successive Approximation method is more suitable for equations with singular kernels, while the Power Series method is more effective for equations with analytic kernels. Keywords: Volterra Integral Equation, Successive Approximation, Power series.
Jeremiah Adeniyi (Wed,) studied this question.