On the basis of product integration techniques a discrete version of a piecewise polynomial collocation method for the numerical solution of initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels is constructed. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms.
Pedas, A., & Tamme, E. (2011). Product integration for weakly singular integro-differential equations. Mathematical Modelling and Analysis, 16(1), 153-172. https://doi.org/10.3846/13926292.2011.564771
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