ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

15. Numerical Analysis and Scientific Computing

1991 MS Classification: 45, 65 Pereverzev, Sergei, Institute of Mathematics, Kiev, Ukraine: An adaptive discretization of third kind Volterra equations with weakly singular kernels A method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations

p(t)z(t) + ∫_{0}^{t} \frac{h(t, τ)}{(t − τ)^{1−α}} z(τ)dτ = f(t), t ∈ [0, 1], 0 < α < 1,

is proposed. Here p(t) vanishes on some subset of [t_{1}, t_{2}] ⊂ [0, 1] and |p(t)| < δ for t ∈ [t_{1}, t_{2}], where δ is sufficiently small positive number. Proposed method gives the accuracy O(δ^{2ν/(2ν+1)}) with respect to the L_{2}-norm, where ν is the parameter of sourcewise representation of exact solution on [t_{1}, t_{2}], and uses no more than O(δ^{−(2−λ)/α} log^{2+1/α} \frac{1}{δ}) values of Galerkin functionals, where λ ∈ (0, 1/2) is determined in the act of choosing regularization parameter within the framework of Morozov’s discrepancy principle. Part of these results is joint work with S. Prössdorf. Reference: S. V. Pereverzev, and Prössdorf, A discretization of Volterra integral equations of the third kind with weakly singular kernels, Journ. of Inverse and Ill-posed problems (to appear).