Approximations of Parabolic Equations at the Vicinity of Hyperbolic Equilibrium Point

This article is devoted to the numerical analysis of the abstract semilinear parabolic problem u′(t) = Au(t) + f(u(t)), u(0) = u ⁰, in a Banach space E. We are developing a general approach to establish a discrete dichotomy in a very general setting and prove shadowing theorems that compare solutions of the continuous problem with those of discrete approximations in space and time. In [3 W.-J. Beyn and S. Piskarev ( 2008 ). Shadowing for discrete approximations of abstract parabolic equations . Discrete Contin. Dyn. Syst., B 10 : 19 – 42 .[Crossref], [Web of Science ®] , [Google Scholar]] the discretization in space was constructed under the assumption of compactness of the resolvent. It is a well-known fact (see [10 S. Larsson ( 1999 ). Numerical analysis of semilinear parabolic problems . In: The Graduate Student's Guide to Numerical Analysis ’98 ( M. Ainsworth , ed.). Lecture Notes from the 8th EPSRC Summer School in Numerical Analysis . Leicester , UK , July 5–17. 1998. Ser. Comput. Math. 26:83–117; Springer, Berlin .[Crossref] , [Google Scholar], 11 S. Larsson and J. M. Sanz-Serna (1994). The behavior of finite element solutions of semilinear parabolic problems near stationary points. SIAM J. Numer. Anal. 31:1000–1018.[Crossref], [Web of Science ®] , [Google Scholar]]) that the phase space in the neighborhood of the hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced to initial value problems with exponential bounded solutions on the corresponding subspaces. We show that such a decomposition of the flow persists under rather general approximation schemes, utilizing a uniform condensing property. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite elements as well as finite differences methods.     

The exponential dichotomy on general approximation scheme

This paper is devoted to the numerical analysis of abstract parabolic problems u′(t) = Au(t), u(0) = u ⁰ with hyperbolic generator A. We develop a general approach to establish a discrete dichotomy in a very general setting in the case of discrete approximation in space and time. It is a well-known fact that the phase space in the neighborhood of the hyperbolic equilibrium can be split in such a way that the original initial value problem is reduced to initial value problems with exponentially decaying solutions in opposite time directions. We use the theory of compact approximation principle and collectively condensing approximation to show that such a decomposition of the flow persists under rather general approximation schemes. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for the finite element method as well as finite difference methods.     

Mechanism of water luminescence upon radiolysis under the effect of background radiation

The luminescence of pure water and Fenton reagent is studied. Photons of the red spectral region characterizing the dimer of singlet oxygen are recorded. It is shown that water luminescence can be explained by the accumulation of water radiolysis products under the influence of background radiation.     

On the approximation of fractional resolution families

We study the approximation of fractional resolution families with respect to space and time. In the approach, we use the notions of general approximation scheme including the finite-difference method, the finite-element method, and projection methods.     

Formation of active species in spark corona discharge at a liquid electrode

The formation of chemically active species upon a spark corona discharge on a liquid electrode at the liquid surface. Test liquids were 0.1 M and 0.005 M oxalic acid, 0.1 M KI and 3% KCl solutions, and tap water. It was found that the reaction yield due to the active species (which might include hydroxyl radicals) was substantially higher in the case of liquid electrode, depending on the electrode-liquid composition.__________Translated from Khimiya Vysokikh Energii, Vol. 39, No. 3, 2005, pp. 228–231.Original Russian Text Copyright © 2005 by Piskarev, Solovev, Karelin, Selemir, Spirov.     

Hydrogen Peroxide Formation in Aqueous Solutions under UV-C Radiation

The formation of hydrogen peroxide in bidistilled water under the influence of UV-C radiation from a DKB-9 low-pressure mercury lamp has been studied. The yield of hydrogen peroxide was (1 ± 0.2) × 10^?7 mol (L s)^?1. The wavelengths of radiation under the influence of which the formation of H₂O₂ is possible have been estimated. It has been assumed that the intermediate product of the reaction is the HO₂^./O₂^.-radical. To identify it, oxidation–reduction reactions in aqueous solutions containing Fe²⁺, Fe³⁺, and I^? ions at pH values from 0.8 to 8.1 have been studied. The quantum yield of HO₂^. radicals in an acidic medium under the influence of radiation from the mercury lamp is 0.015 ± 0.005.     

Redox processes induced in methemoglobin solution by mercury lamp UV light and spark-discharge plasma radiation

The buildup of–SH groups in methemoglobin by irradiation with emission of spark discharge plasma in air and UV light from a low-pressure mercury lamp (λ = 253.7 nm) has been studied. The concentration of–SH groups increases the during spark discharge treatment, with the methionine amino acid residue in the methemoglobin molecule being reduced by nitrous and peroxynitrous acids; the bonds in the methemoglobin molecule are destroyed within 20 minutes of the spark treatment. By the action of UV radiation, methionine is reduced through chromophores (tryptophan, tyrosine) and methemoglobin is reduced by O₂^.- radical ions to oxy- and deoxyhemoglobin.