A Novel Super-Convergent Numerical Method for Solving Nonlinear Volterra Integral Equations Based on B-Splines

Abstract

We introduce and thoroughly examine a novel approach grounded in B-spline techniques to address the solution of second-kind nonlinear Volterra integral equations. Our method revolves around the application of B-spline interpolation, incorporating innovative end conditions, and delving into the associated existence and error estimation aspects. Notably, we develop this technique separately for even and odd-degree splines, leading to super-convergent approximations, particularly significant when employing even-degree splines. This paper extends its commitment to a comprehensive analysis, delving deeply into the method’s convergence characteristics and providing insightful error bounds. To empirically validate our approach, we present a series of numerical experiments. These experiments underscore the method’s efficacy and practicality, showcasing numerical approximations that closely align with the anticipated theoretical outcomes. Our proposed method thus emerges as a promising and robust tool for addressing the challenging realm of nonlinear Volterra integral equations, bridging the gap between theoretical expectations and practical applications.