Periodic B-spline-Heaviside collocation for Fredholm integral equations with piecewise H\"older data

Abstract

We propose and analyze a periodic B-spline-Heaviside collocation method for Fredholm integral equations of the second kind on a smooth closed contour with piecewise H\"older data and finitely many prescribed jumps. Since the regular integral term is globally continuous, the solution jumps coincide with those of the data. The approximation space combines one global periodic B-spline component with cut-compensated Heaviside generators carrying exactly the prescribed physical jumps, which yields a triangular algorithm: the jump amplitudes are reconstructed first, and only the continuous component is obtained from an $n_B\times n_B$ collocation system. For B-spline orders $m=2,3,4$, exact collocation is uniformly stable and converges in $PH_\beta$, $0<\beta<\alpha<\mu\le1$, with order $O(h^{\alpha-\beta})$. If the parametrized kernel density is H\"older continuous of exponent $\rho$ in the integration variable, a panelwise $q$-point Gauss rule with $q\ge\lceil m/2\rceil$ produces an $O(h^\rho)$ quadrature perturbation on the discrete spline space and preserves the exact-collocation rate whenever $\rho\ge\alpha-\beta$. The exact-operator one-step B-spline--Heaviside extension satisfies the stronger estimate $O(h^\alpha)$ in $H_\mu$. We also treat sampled lateral data and a nonvanishing piecewise H\"older leading coefficient. Numerical experiments are consistent with the predicted stability and convergence, show the substantial error reduction produced by iteration, and indicate competitive accuracy relative to established discontinuity-adapted collocation and Nystr\"om alternatives, while the proposed representation introduces no mesh-dependent artificial jumps.