Error estimates of piecewise Hermite collocation method for highly oscillatory Volterra integral equation with Bessel kernel

This work concerns the convergence of the piecewise Hermite collocation method for highly oscillatory integral equations. The collocation method is constructed by calculating highly oscillatory integrals efficiently. To study the convergence with respect to frequency, some estimates about the highly oscillatory integrals are studied. Then, we obtain the asymptotic order of the method in a different way and get a sharper result compared with the existing study. Besides, we also analyze the convergence with respect to step length in detail. The last part provides some examples to confirm the theoretical estimate.