Direct and conjugate Neumann-Ulam schemes for solving non-linear integral equations

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Abstract

ELEMENTARY unbiased estimates are constructed for a linear functional of the solution of a non-linear integral equation of fairly general type. The method of constructing the estimates, which is based on the "equivalence" of the initial equation to an infinite system of linear equations, makes it possible to transfer to the so-called conjugate Neumann-Ulam scheme, which can prove more advantageous when solving physical problems. We shall discuss the problem of estimating the functional I=∝φ{symbol}hdμ0 of the solution φ{symbol} of a non-linear integral equation of fairly general type. In accordance with the analogue of the Neumann-Ulam scheme, constructed for this case in [1], the functional is estimated on trajectories of a branched Markov process, which proves to be connected in a natural way with the iterations of the non-linear equation. Below we analyze a different approach to the Neumann-Ulam scheme, based on the equivalence of the initial equation to an infinite system of linear equations. It leads to somewhat more general Markov processes, the most natural of which prove to be branched, and it also enables us to consider the question of the so-called conjugate Neumann-Ulam scheme. Throughout, only elementary estimates of the "absorption" type will be constructed, though it is obviously possible to construct different types of estimate. In addition, we make no explicit stipulation about the absolute convergence of integrals and series, about the σ-finiteness of measures, or about the measurability of functions etc.; such conditions carry over almost automatically from the linear case examined in [2].

Original languageEnglish
Pages (from-to)39-45
Number of pages7
JournalUSSR Computational Mathematics and Mathematical Physics
Volume14
Issue number6
DOIs
StatePublished - 1974

Scopus subject areas

  • Engineering(all)

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