Functional Separable Solutions to Nonlinear Diffusion Equations by Group Foliation Method

We consider the functional separation of variables to the nonlinear diffusion equation with source and convection term： ut = （A（x）D（u）ux）x ＋ B（x）Q（u）, Ax ≠ 0. The functional separation of variables to this equation is studied by using the group foliation method. A classification is carried out for the equations which admit the function separable solutions. As a consequence, some solutions to the resulting equations are obtained.

Variable Separation and Derivative-Dependent Functional Separable Solutions to Generalized Nonlinear Wave Equations

Using the generalized conditional symmetry approach, we obtain a number of new generalized (1+1)-dimensional nonlinear wave equations that admit derivative-dependent functional separable solutions.

Classification and Functional Separable Solutions to Extended Nonlinear Wave Equations

The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained and exact separable solutions to some of the resulting equations are constructed.

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Furthermore, blow up and behavior of those exact solutions are also investigated.

Functional Separable Solutions of Nonlinear Heat Equations in Non-Newtonian Fluids

We study the functional separation of variables to the nonlinear heat equation： ut = （A（x）D（u）ux^n）x＋ B（x）Q（u）, Ax≠0. Such equation arises from non-Newtonian fluids. Its functional separation of variables is studied by using the group foliation method. A classification of the equation which admits the functional separable solutions is performed. As a consequence, some solutions to the resulting equations are obtained.

Effects of Catalysis and Separator Functionalization on High-Energy Lithium–Sulfur Batteries:A Complete Review

Lithium–sulfur(Li-S)batteries have the advantages of high theoretical specific capacity(1675 mAh g^(−1)),rich sulfur resources,low production cost,and friendly environment,which makes it one of the most promising next-generation rechargeable energy storage devices.However,the“shuttle effect”of polysulfide results in the passivation of metal lithium anode,the decrease of battery capacity and coulombic efficiency,and the deterioration of cycle stability.To realize the commercialization of Li-S batteries,its serious“shuttle effect”needs to be suppress.The commercial separators are ineffective to suppress this effect because of its large pore size.Therefore,it is an effective strategy to modify the separator surface and introduce functional modified layer.In addition to the blocking strategy,the catalysis of polysulfide conversion reaction is also an important factor hindering the migration of polysulfides.In this review,the principles of separator modification,functionalization,and catalysis in Li-S batteries are reviewed.Furthermore,the research trend of separator functionalization and polysulfide catalysis in the future is prospected.

Three-in-one LaNiO_(3) functionalized separator boosting electrochemical stability and redox kinetics for high-performance Li-S battery

The lithium-sulfur(Li-S)battery,as one of the energy storage devices,has been in the limelight due to its high theoretical energy density.However,the poor redox kinetics and the"shuttle effect"of polysulfides severely restrict the use of Li-S batteries in practical applications.Herein,a novel bimetallic LaNiO_(3) functional material with high electrical conductivity and catalytic property is prepared to act as a high-efficiency polysulfide shuttling stopper.The three LaNiO_(3) samples with different physical/chemical characteristics are obtained by controlling the calcination temperature.In conjunction with the high electrical conductivity and excellent catalytic properties of the as-prepared materials,the appropriate chemisorption toward polysulfides offers great potential to enhance electrochemical stability for highperformance Li-S batteries.Particularly,the Li-S cell with the separator modified by such functional material gives a specific capacity of 658 mA h g^(-1) after 500 cycles at a high current density of 2 C.Even with high sulfur loading of 6.05 mg cm^(-2),the Li-S battery still exhibits an areal specific capacity of 2.81 m A h cm^(-2)after 150 cycles.This work paves a new avenue for the rational design of materials for separator modification in high-performance Li-S batteries.

Understanding Dual-Polar Group Functionalized COFs for Accelerating Li-Ion Transport and Dendrite-Free Deposition in Lithium Metal Anodes

Lithium metal batteries(LMBs)have attracted wide attentions because of their high theoretical specific capacity and low electrochemical potential.However,the growth of lithium dendrites seriously affects the practical application of LMBs.Thus,the lithium-philic carbonyl and carboxy dualgroup-modified covalent organic framework(COF-COOH)is designed to coat the polypropylene(PP)separator(COF-COOH@PP separator),realizing the regulation of ion transport and uniform lithium deposition.The plentiful and negative charge sites in the COF-COOH can suppress the diffusion of the freely movable lithium salt anion by the electrostatic interaction.Density functional theory(DFT)calculations demonstrate that the COF-COOH possesses the function of anchoring anion and desolvation.Consequently,the Li^(+)transference number(0.7),ion conductivity(0.64 mS cm^(-1)),and desolvating of Li^(+)are obviously improved by using the COF-COOH@PP separator.The modified Li-Li symmetric battery delivers stable cycle for more than 1000 h and lower voltage hysteresis(0.02 V).This dendrite-free deposition strategy holds great promise for practical application of Li metal anodes.

Restraining migration and dissolution of transition-metal-ions via functionalized separator for Li-rich Mn-based cathode with high-energy-density

Lithium-rich manganese-based materials(LRMs) are promising cathode for high-energy-density lithiumion batteries due to their high capacity,low toxicity,and low cost.However,LRMs suffer from serious voltage decay and capacity fade due to continual migration and dissolution of transition metal ions(TMs) during cycling process.Herein,a novel strategy is proposed to inhibit the TMs migration of LRMs through a modified separator by means of functionalized carbon coating layer,which depends on the chemical constraint of the abundant functional groups in the modified super P.In addition,it has been found that the dissolution of TMs can be restrained based on the Le Chatelier's principle.Moreover,the modified separator owns good wettability toward the electrolyte.As a result,the LRMs cathode with the modified separator delivers a high discharge capacity of 329.93 mA h g-1 at 0.1 C,and achieves good cyclic performance,the enhanced reaction kinetics and low voltage decay.Therefore,this work provides a new idea to promote the comprehensive electrochemical performances of Li-ion batteries with LRMs cathode through a strategy of separator modification.

Regulating the non-effective carriers transport for high-performance lithium metal batteries

The absence of control over carriers transport during electrochemical cycling,accompanied by the deterioration of the solid electrolyte interphase(SEI)and the growth of lithium dendrites,has hindered the development of lithium metal batteries.Herein,a separator complexion consisting of polyacrylonitrile(PAN)nanofiber and MIL-101(Cr)particles prepared by electrospinning is proposed to bind the anions from the electrolyte utilizing abundant effective open metal sites in the MIL-101(Cr)particles to modulate the transport of non-effective carriers.The binding effect of the PANM separator promotes uniform lithium metal deposition and enhances the stability of the SEI layer and long cycling stability of ultra-high nickel layered oxide cathodes.Taking PANM as the Li||NCM96 separator enables high-voltage cycling stability,maintaining 72%capacity retention after 800 cycles at a charging and discharging rate of 0.2 C at a cut-off voltage of 4.5 V and 0°C.Meanwhile,the excellent high-rate performance delivers a specific capacity of 156.3 mA h g^(-1) at 10 C.In addition,outstanding cycling performance is realized from−20 to 60°C.The separator engineering facilitates the electrochemical performance of lithium metal batteries and enlightens a facile and promising strategy to develop fast charge/discharge over a wide range of temperatures.

Extended Group Foliation Method and Functional Separation of Variables to Nonlinear Wave Equations

Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.

Functional Variable Separation for Extended Nonlinear Elliptic Equations

This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the generalized conditional symmetry, we obtain complete classification of those equations which admit functional separable solutions （FSSs） and construct some exact FSSs to the resulting equations.

2D spinel ZnCo_(2)O_(4) microsheet-coated functional separator for promoted redox kinetics and inhibited polysulfide dissolution

Lithium-sulfur(Li-S)batteries are receiving increasing attention as one of the potential next-generation batteries,owing to their high energy densities and low cost.However,practical Li-S batteries with high energy densities are extremely hindered by the sulfur loss,low Coulombic efficiency,and short cycling life originating from the polysulfide(LiPS)shuttle.In this study,two-dimensional(2D)ZnCo_(2)O_(4) microsheets fabricated by a facile hydrothermal process are employed to modify the separator,for improving the electrochemical performances of Li-S cells.The resulting 2D Zn Co_(2)O_(4)-coated separator features a coating thickness of approximately 10 lm,high ionic conductivity of 1.8 m S/cm,and low mass loading of 0.2 mg/cm^(2).This 2D ZnCo_(2)O_(4)-coated separator effectively inhibits Li PS shuttle by a strong chemical interaction with Li PS as well as promotes the redox kinetics by Zn CO2O4-coated layers,as determined by X-ray photoelectron spectroscopy analysis,self-discharge,time-dependent permeation test,Li symmetric cell test,and Li2S nucleation analyses.Consequently,the Li-S batteries based on the 2D Zn Co_(2)O_(4)-coated separator exhibit a high initial discharge capacity of 1292.2 m Ah/g at 0.1 C.Moreover,they exhibit excellent long cycle stability at 1 and 2 C with capacity retention of 84%and 86%even after800 cycles,corresponding to a capacity fading rate of 0.020%and 0.016%per cycle,respectively.Effectively,these Li-S cells with a high sulfur loading at 5.3 mg/cm^(2) and low electrolyte concentration of 9 l L/mg deliver a high discharge capacity of 4.99 m Ah/cm^(2) after 200 cycles at 0.1 C.

The derivative-dependent functional variable separation for the evolution equations

This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A（u, ux）uxx＋B（u, ux, ut） which admits the derivative- dependent functional separable solutions （DDFSSs）. We also extend the concept of the DDFSS to cover other variable separation approaches.

Functional lithiophilic polymer modified separator for dendrite-free and pulverization-free lithium metal batteries

Severe performance drop and fire risk due to the uneven lithium(Li) dendrite formation and growth during charge/discharge process has been considered as the major obstacle to the practical application of Li metal batteries.So inhibiting dendrite growth and producing a stable and robust solid electrolyte interface(SEI) layer are essential to enable the use of Li metal anodes.In this work,a functional lithiophilic polymer composed of chitosan(CTS),polyethylene oxide(PEO),and poly(triethylene glycol dimethacrylate)(PTEGDMA),was homogeneously deposited on a commercial Celgard separator by combining electrospraying and polymer photopolymerization techniques.The lithiophilic environment offered by the CTS-PEO-PTEGDMA layer enables uniform Li deposition and facilitates the formation of a robust homogeneous SEI layer,thus prevent the formation and growth of Li dendrites.As a result,both Li/Li symmetric cells and LiFePO4/Li full cells deliver significantly enhanced electrochemical performance and cycle life.Even after 1000 cycles,the specific capacity of the modified full cell could be maintained at65.8 mAh g^(-1), twice which of the unmodified cell(32.8 mAh g^(-1)).The long-term cycling stability in Li/Li symmetric cells,dendrite-free anodes in SEM images and XPS analysis suggest that the pulverization of the Li anode was effectively suppressed by the lithiophilic polymer layer.

Approximate derivative-dependent functional variable separation for quasi-linear diffusion equations with a weak source

By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approxi- mate solutions to the resulting perturbed equations are constructed via examples.

The Separation of Powers from Functions and the Establishment of a New Structure of Management and Operation of State-owned Assets

The development of the socialist market economy demands the furtherintensification of the reform of the management operating system (MOS)of the state-owned assets, the strengthening of the efficiency in the man-agement of state-owned assets,and the improvement of the operating bene-

Novel 2D bifunctional layered rare-earth hydroxides@GO catalyst as a functional interlayer for improved liquid-solid conversion of polysulfides in lithium-sulfur batteries

The detrimental“shuttle effect”of lithium polysulfides(LiPSs)together with sluggish multi-order reaction kinetics are the main drawbacks hindering lithium-sulfur(Li-S)batteries from commercial success.Here,we first propose the implementability of layered rare-earth hydroxides(LREHs)in Li-S batteries to optimize electrochemical performance.In this work,a two-dimensional(2D)rare-earth-based composite constructed by the layered gadolinium hydroxy chloride[Gd_(2)(OH)_(5)(H_(2)O)_(n)]Cl nanoplates(LGdH NPs)and graphene oxide(GO)was designed as a sulfur immobilizer for Li-S batteries.Combining the experimental results and density functional theory(DFT)calculations,it is revealed that the LGdH@GO composite not only provides a strong anchoring of the intermediates during cycling,but also acts as an effective catalyst to accelerate the liquid-solid conversion of polysulfides.The Li-S batteries assembled by LGdH@GO modified separators delivered a superior rate performance with a specific capacity of 605.34 mAh/g at 5 C,as well as excellent cycle stability with a decay rate of 0.087%over 500 cycles at 2 C.This study provided a deep understanding of the mechanism to suppress the“shuttle effect”by the LREHs,and a guide to design effective functional interlayers for high-performance Li-S batteries with excellent electrocatalytic activity.

CONVERGENCE ANALYSIS AND MRALLEL IMPLEMENTION FOR THE DIRECTEDGRAPH -ALGORITHM

In this paper we discuss the convergence of the directed graph-algorithm for solving a kind of optimization problems where the objective and subjective functions are all separable, and the parallel implementation process for the directed graph -algorithm is introduced.

Regulating the growth of lithium dendrite by coating an ultra-thin layer of gold on separator for improving the fast-charging ability of graphite anode

With the ever-growing application of lithium-ion batteries(LIBs), their fast-charging technology has attracted great interests of scientists. However, growth of lithium dendrites during fast charge of the bat teries with high energy density may pose great threats to the operation and cause serious safety issues Herein, we prepared a functional separator with an ultra-thin(20 nm) layer of Au nanoparticles deposited by evaporation coating method which could regulate growth direction and morphology of the lithium dendrites, owing to nearly zero overpotential of lithium meal nucleation on lithiated Au. Once the Li den drites are about to form on the graphite anode during fast charging(or lithiation), they plate predomi nantly on the Au deposited separator rather than on the graphite. Such selective deposition does no compromise the electrochemical performance of batteries under normal cycling. More importantly, i enables the better cycling stability of batteries at fast charge condition. The Li/Graphite cells with Au nanoparticles coated separator could cycle stably with a high areal capacity retention of 90.5% over 95 cycles at the current density of 0.72 m A cm^(-2). The functional separator provides an effective strategy to adjust lithium plating position at fast charge to ensure high safety of batteries without a compromise on the energy density of LIBs.