An Efficient Boubakar-Chebyshev Polynomial Algorithm for High-Order Non-Linear Volterra-Fredholm Integro-Differential Equations

This study addresses the challenge of solving high-order nonlinear Volterra-Fredholm integro-differential equations (VFIDEs), which are essential in modeling complex physical and engineering systems but often pose significant computational difficulties due to their nonlinearity and mixed boundary conditions. While existing numerical methods struggle with accuracy and efficiency, this research bridges the gap by introducing a novel Boubaker-Chebyshev polynomial algorithm that leverages Chebyshev-Gauss-Radau collocation points for enhanced precision. The aim is to develop a highly accurate and computationally efficient method for solving high-order VFIDEs, addressing the shortcomings of traditional approaches. Implemented in Maple 18, the proposed algorithm is rigorously tested, demonstrating superior error estimation and computational optimization compared to conventional polynomial methods, particularly for problems with mixed boundary conditions. The results reveal marked improvements in both solution accuracy and computational performance, highlighting the effectiveness of combining Boubaker and Chebyshev polynomials. These findings have significant implications for scientific computing, offering a robust tool for tackling complex integro-differential problems. For future investigations, it is suggested to expand this method to fractional-order equations, fine-tune the selection of collocation points, and utilize it in practical nonlinear systems to further support and improve its usability. This research not only propels numerical methods for VFIDEs forward but also introduces new possibilities for high-precision computational techniques in the fields of applied mathematics and engineering.