Computation of Volterra and Fredholm Integro-Differential Nonlinear Boundary Value Problems by a Modified Regula Falsi-Bisection Shooting Approach

Abstract

A modified Regula Falsi shooting method approach is deployed to solve a set of Volterra and Fredholm integro- nonlinear boundary value problems. Efforts to solve this class of problems with traditional shooting methods have generally failed and the outcomes from most domain based approaches are often plagued by ill conditioning. Our modification is based on an exponential series technique embedded in a shooting bracketing method. For the purposes of validation, we initially solved problems with known closed form solutions before considering those that do not come with this property but are singular, genuinely nonlinear, and are of practical interest. Although in most of these tests, convergence was found to be super linear, the errors decreased monotonically after few iterations. This suggests that the method is robust and can be trusted to yield faithful results and by far surpasses in simplicity various other techniques that have been applied to solve similar problems. In order to buttress the effectiveness and utility of this approach, we display both the graphical and error analysis outcomes. And for each test case, the method can be seen to demonstrate the closeness of numerically generated results to the analytical solutions.