The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations

The cutoff method, which cuts off the values of a function less than a given number, is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations. A convergence analysis is given for a broad class of finite difference methods combined with cutoff for linear parabolic equations. Two applications are investigated, linear anisotropic diffusion problems satisfying the setting of the convergence analysis and nonlinear lubrication-type equations for which it is unclear if the convergence analysis applies. The numerical results are shown to be consistent with the theory and in good agreement with existing results in the literature. The convergence analysis and applications demonstrate that the cutoff method is an effective tool for use in the computation of nonnegative solutions. Cutoff can also be used with other discretization methods such as collocation, finite volume, finite element, and spectral methods and for the computation of positive solutions.Comment: 19 pages, 41 figure

Structural evolution of palladium nanoparticles and their electrocatalytic activity toward ethanol oxidation in alkaline solution

A series of palladium nanoparticles with sizes ranging from 3 to 10 nm were synthesized with ethylene glycol as the solvent and reducing agent. Based on X-ray diffraction, transmission electron microscopy and high-resolution transmission electron microscopy characterization, the as-prepared palladium nanoparticles were well-crystalline and uniformly dispersed. Electrochemical measurements showed that the smallest (∼3 nm) palladium nanoparticles exhibited the highest current density and the best cycling stability and durability among the four kinds of palladium nanoparticles for the electrocatalytic oxidation of ethanol. In addition, small palladium nanoparticles were spherical while large nanoparticles with a certain decline in electrocatalytic activity showed distinct shape. The formation and shape evolution of the palladium nanoparticles as well as their structure-property relationship have been studied and analyzed based on the experimental data

Solvothermal Synthesis of Hierarchical Colloidal Nanocrystal Assemblies of ZnFe2O4 and Their Application in Water Treatment

Hierarchical colloidal nanocrystal assemblies (CNAs) of ZnFe2O4 have been synthesized controllably by a solvothermal method. Hollow ZnFe2O4 spheres can be formed with the volume ratios of ethylene glycol to ethanol of 1:4 in the starting systems, while solid ZnFe2O4 CNAs are obtained by adjusting the volume proportion of ethylene glycol to ethanol from 1:2 to 2:1. Magnetometric measurement data showed that the ZnFe2O4 CNAs obtained with the volume ratios of 1:2 and 1:1 exhibited weak ferromagnetic behavior with high saturation magnetization values of 60.4 and 60.3 emu·g−1, respectively. However, hollow spheres showed a saturation magnetization value of 52.0 emu·g−1, but the highest coercivity among all the samples. It was found that hollow spheres displayed the best ability to adsorb Congo red dye among all the CNAs. The formation mechanisms of ZnFe2O4 CNAs, as well as the relationship between their structure, crystallite size, and properties were discussed based on the experimental results