Fractional backward spectral approximation theory for weakly singular adjoint integral equations

Abstract

We introduce a new class of fractional backward orthogonal functions designed for the spectral approximation of weakly singular adjoint Volterra integral equations. These basis functions generate an approximation space that naturally reflects the terminal-endpoint singular behaviour produced by weakly singular kernels. We develop the basic approximation theory for the proposed backward orthogonal basis, including weighted projection estimates, Gauss-type interpolation estimates, inverse inequalities, and stability bounds for the associated weakly singular adjoint integral operator. The error analysis and numerical results show that the proposed backward Jacobi method is particularly suitable for solutions with terminal-endpoint weak singularities and can recover high-order convergence rates that are typically lost when usual polynomial approximations are applied directly to such weakly regular solutions.