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Multilevel correction for collocation solutions of Volterra integral equations with proportional delays. (English) Zbl 1274.65348

Summary: In this paper, we propose a convergence acceleration method for collocation solutions of the linear second-kind Volterra integral equations with proportional delay \(qt(0<q<1)\). This convergence acceleration method called multilevel correction method is based on a kind of hybrid mesh, which can be viewed as a combination between the geometric meshes and the uniform meshes. It will be shown that, when the collocation solutions are continuous piecewise polynomials whose degrees are less than or equal to \(m(m\leqslant 2)\), the global accuracy of \(k\) level corrected approximation is \(O(N^{-(2m(k+1)-\varepsilon)})\), where \(N\) is the number of the nodes, and \(\varepsilon\) is an arbitrary small positive number.

MSC:

65R20 Numerical methods for integral equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
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