A hybrid polynomial spectral collocation method for weakly singular Volterra integral equations with variable exponent

Abstract Weakly singular Volterra integral equations (VIEs) of the second kind, where the kernel of the integral term has a factor of the form $(t-s)^{-\mu (t)}$ with $0\le \mu (t) <1$, are studied on the interval $0\le t\le 1$. It is known that typical solutions of these problems lie in the Vainikko space $C^{m,\mu (0)}(0,1],$ so they have a weak singularity at the initial time $t=0$. Estimates for best approximations of functions in $C^{m,\nu }(0,1]$ by polynomials defined on $[t_{1},1]$ are derived in the $L^\infty (t_{1},1)$ norm, where $\nu \in (-\infty ,1)$ and $t_{1}\in [0,1)$ are arbitrary, and these bounds are shown to be sharp in the most singular case $\nu> 1- m/2$; such estimates were not previously known and are of independent interest. To solve these VIEs numerically, an analytical technique of Huang & Stynes (2017, Spectral Galerkin methods for a weakly singular Volterra integral equation of the second kind. IMA J. Numer. Anal., 37, 1411–1436) is used on an initial small interval $[0,t_{1}]$, and then collocation by globally defined polynomials at suitably chosen Jacobi–Gauss collocation points is used on $[t_{1},1]$. Error bounds for the solution of this hybrid method are derived in the $L^\infty (0,1)$ norm. Numerical experiments show that these error bounds are often sharp.