Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind

Abstract

This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel ϕ(t, s) = (t − s)^−µ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233: 938–950], the error analysis for this approach is carried out for 0 < µ < 1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., µ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-type but also establish the error estimates under a more general regularity assumption on the exact solution.

Keywords

Jacobi spectral collocation method / Abel-Volterra integral equation / convergence analysis

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