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. 2010 Mathematics Subject Classifications: 45Dxx; 65R20; 41A15.