Spectral Approaches with Alternative Polynomial Bases for Quadratic Integral Systems: An Expanded Investigation

Abstract

This work presents a detailed exploration of spectral methods tailored to systems of quadratic integral equations (SQIEs). While conventional approaches often rely on Legendre, Jacobi, or the classical Chebyshev families, here we emphasize three less conventional polynomial bases: the Chebyshev polynomials of the eighth kind (CP8K), the Boubaker sequence, and Bernoulli polynomials. We establish a rigorous operator framework, analyze conditions for existence and uniqueness, and provide stability bounds. The paper further elaborates on discretization strategies—collocation, Galerkin, and Tau. Also we discuss quadrature adjustments for both smooth and weakly singular kernels. Error estimates, conditioning concerns, and preconditioning remedies are also studied. Finally, we present algorithmic templates and conceptual numerical experiments to illustrate comparative performance. The emphasis is on providing not only thoretical assurances but also practical insights that make these alternative bases viable in real computations.