18 research outputs found

    Complexes of Polyvinylpyrrolidone and Polyethylene Glycol with Palladium(II) Ions: Characterization and Catalytic Activity

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    Received: 21.06.23. Revised: 13.07.23. Accepted: 19.07.23. Available online: 24.07.23.The composition of the complex compounds was determined by potentiometric and conductometric methods.IR spectroscopy and SEM confirmed the coordination of polymeric ligand to palladium and allowed evaluating the morphology of the complex surface.The catalytic activity of the complexes in the oxidation of octene-1 by inorganic oxidizers under mild conditions was evaluated.In this work, we obtained complexes by mixing aqueous solution of palladium(II) chloride with polyvinylpyrrolidone and polyethylene glycol. The composition of the complex compounds was determined by potentiometric and conductometric titration. IR spectroscopy and scanning electron microscopy (SEM) confirmed the coordination of polymeric ligand to palladium and allowed evaluating the morphology and features of the complex surface. The catalytic activity of the synthesized compounds in the oxidation of octene-1 by inorganic oxidizers (NaBrO3, K2S2O8) in aqueous-organic media in dimethyl sulfoxide (DMSO) under mild conditions was calculated. The reaction product was octanone-2, obtained in good yield (62–98%). Quantitative analysis of octanone-2 was made by the gas-chromatographic method. Mass spectrometry confirms the formation of octanone-2. The complexes are able to participate in five consecutive catalytic cycles without significant loss of catalytic efficiency. Oxidation of octene-1 proceeds by the oxidation-reduction mechanism and consists of two key stages

    Symmetry-preserving discrete schemes for some heat transfer equations

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    Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their difference counterparts.Comment: 21 pages, 4 ps figure

    Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

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    Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation ut=Δu+upu_t= \Delta u+ u^p, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure

    Continuous Symmetries of Difference Equations

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    Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl
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