Finite differences and collocation methods for the solution of the two‐dimensional heat equation

In this article, we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two‐dimensional heat equation. We employ, respectively, second‐order and fourth‐order schemes for the spatial derivatives, and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is nonsingular. Numerical experiments carried out on serial computers show the unconditional stability of the proposed method and the high accuracy achieved by the fourth‐order scheme. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 54–63, 2001     

A dynamic injection operator in a multigrid solution of convection-diffusion equations

A multigrid method is studied for the solution of a linear system resulting from the high-order nine-point discretization of the convection-diffusion equations. The residual injection operator is used as a substitute for the usual full-weighting in the multigrid process. A heuristic analysis is given to obtain a dynamic injection operator that is cost-effective for both diffusion- and convection-dominated problems. Numerical experiments are employed to test the stability and efficiency of the proposed method. © 1998 John Wiley & Sons, Ltd.     

An overview of recent progress in multiple dual-mode counter-current chromatography

Counter-current chromatography is a chromatographic separation and purification technique being developed. The development of different elution modes has significantly contributed to this field. Multiple dual-mode elution is a method developed based on dual-mode elution, which consists of a series of changing cycles of the phase role and the direction by switching between normal and reverse elution modes of counter-current chromatography. This dual-mode elution method takes full advantage of the liquid nature of stationary and mobile phases of counter-current chromatography and effectively improves the separation efficiency. So, this unique elution mode has gained extensive attention for separating complex samples. This review mainly describes and summarizes in detail its development, applications, and characteristics in recent years. Meanwhile, its advantages, limitations, and future outlook also have been discussed in this paper.     

Dynamic Phosphorus: Thiolate Exchange Cascades with Higher Phosphorothioates

In this study, we introduce phosphorus, a pnictogen, as an exchange center for dynamic covalent chemistry. Cascade exchange of neutral phosphorotri- and -tetrathioates with thiolates is demonstrated in organic solvents, aqueous micellar systems, and in living cells. Exchange rates increase with the pH value, electrophilicity of the exchange center, and nucleophilicity of the exchangers. Molecular walking of the dynamic phosphorus center along Hammett gradients is simulated by the sequential addition of thiolate exchangers. Compared to phosphorotrithioates, tetrathioates are better electrophiles with higher exchange rates. Dynamic phosphorotri- and -tetrathioates are non-toxic to HeLa Kyoto cells and participate in the dynamic networks that account for thiol-mediated uptake into living cells.