A general collocation analysis for weakly singular Volterra integral equations with variable exponent

Abstract

Abstract Piecewise polynomial collocation of weakly singular Volterra integral equations (VIEs) of the second kind has been extensively studied in the literature, where integral kernels of the form $(t-s)^{-\alpha }$ for some constant $\alpha \in (0,1)$ are considered. Variable-order fractional-derivative differential equations currently attract much research interest, and in Zheng & Wang (2020, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J. Numer. Anal., 58, 330–352), such a problem is transformed to a weakly singular VIE whose kernel has the above form with variable $\alpha = \alpha (t)$, then solved numerically by piecewise linear collocation, but it is unclear whether this analysis could be extended to more general problems or to polynomials of higher degree. In the present paper, the general theory (existence, uniqueness, regularity of solutions) of variable-exponent weakly singular VIEs is developed using novel techniques. These results then underpin an error analysis of collocation methods where piecewise polynomials of any degree can be used. This error analysis is also novel—it makes no use of the usual resolvent representation, which is a key technique in the error analysis of collocation methods for VIEs in the current research literature. Furthermore, all the above analysis for a scalar VIE can be extended to certain nonlinear VIEs and to systems of VIEs. The sharpness of the theoretical error bounds obtained for the collocation methods is demonstrated by numerical examples.