Endpoint-singularity-preserving spectral approximation theory for weakly singular integral equations
For researchers solving weakly singular integral equations and fractional differential equations, this work offers a theoretically grounded spectral method that handles endpoint singularities without mesh refinement.
The paper develops a fractional polynomial basis that preserves endpoint singularities, enabling high-order spectral approximations for weakly singular integral equations. It establishes orthogonality, error estimates, and inverse inequalities, providing a rigorous foundation for solving terminal-value and fractional problems.
We introduce a fractional approximation framework for functions with limited regularity near the terminal point. The proposed basis is constructed by composing classical Jacobi polynomials with an endpoint algebraic mapping, thereby incorporating the terminal singular structure directly into the approximation space. The main structural properties of the fractional polynomials are established, including orthogonality relations, derivative identities, and a singular Sturm--Liouville eigenvalue formulation. We then introduce the associated weighted Sobolev spaces and prove projection and Gauss-type interpolation error estimates in weighted norms. Inverse inequalities and weighted Sobolev embedding estimates are also derived. The resulting theory provides a rigorous foundation for high-order spectral and collocation approximations of endpoint-singular and weakly regular problems, including terminal value problems, fractional differential equations, and weakly singular Volterra integral equations.