18 research outputs found
Complexes of Polyvinylpyrrolidone and Polyethylene Glycol with Palladium(II) Ions: Characterization and Catalytic Activity
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
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
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 , 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
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