A High-Accuracy Hybrid Method for Linear Fredholm Integral Systems Using Bernoulli Polynomials Coupled with Enhanced Block-Pulse Functions

This paper proposes a novel mixed numerical scheme for approximating linear Fredholm integral equation systems (LFISs) using a combination of Bernoulli polynomials (BPs) and enhanced block-pulse functions (EBPFs). This suggested representation makes use of both the local support nature and computation efficiency of the (EBPFs) as well as the high-order approximating nature of BPs. Using the operational matrices, the system of coupled integrals can be transformed into a finite-dimensional algebraic system (AS) of expansion coefficients. A theoretical analysis is established to investigate the solvability, convergence, stability, and approximation error of the resulting scheme. In addition, the effects of polynomial degree and partition refinement on the numerical accuracy are examined. Several test problems are considered, and the obtained results demonstrate that the proposed BEBPF approach provides highly accurate approximations while requiring relatively small basis dimensions. Comparisons with previously reported numerical techniques further illustrate their computational effectiveness and accuracy.