Exponential Time Differencing Method for a Reaction-Diffusion System with Free Boundary

For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.

Spatiotemporal Evolution of Ecosystem Service Value in Caohai National Nature Reserve

Wetland ecosystems are important regulators of global climate change.Studying the spatiotemporal changes and driving mechanisms of their ecosystem service values(ESV)is beneficial for the sustainable development of wetlands.This paper uses the equivalent factor method,based on land use changes,to reveal the spatiotemporal evolution of the ecosystem service value in the Caohai National Nature Reserve(CNNR).The results show the following:①from 2000 to 2020,there was a significant decrease in the core zone s arable land area,with an increase in forest and water areas.Construction land mainly increased in the experimental area,and the grassland area showed a fluctuating change of first increasing and then decreasing;②in 2000,2010,and 2020,the ecosystem service value of the study area was 302 million,296 million,and 325 million yuan,respectively,showing a trend of fluctuating growth,with the value of wetland ecosystems playing a dominant role;③regulatory services are the main contributors to the ecosystem service value in the study area,with a contribution rate of 60%.Hydrological regulation is the ecosystem function with the highest value in wetland ecosystem services,contributing more than 35%to the ESV in all three periods;④in terms of spatial distribution,the core zone s ecosystem service value is dominant.Looking at the total ecosystem service value of the region,the core zone>the experimental area>the buffer zone.In terms of ESV per unit area,the core zone(89000 yuan/hm^(2))is significantly higher than the buffer zone(39100 yuan/hm^(2))and the experimental area(15800 yuan/hm^(2)).The study can provide a basis for research and spatial management of ecosystem services in wetland ecosystems and nature reserves.

The Factorization Method to Solve a Class of Inverse Potential Scattering Problems for Schrdinger Equations

This paper is concerned with the inverse scattering problems for Schrdinger equations with compactly supported potentials.For purpose of reconstructing the support of the potential,we derive a factorization of the scattering amplitude operator A and prove that the ranges of （A＊ A） ^1/4 and G which maps more general incident fields than plane waves into the scattering amplitude coincide.As an application we characterize the support of the potential using only the spectral data of the operator A.

Compact implicit integration factor methods for some complex-valued nonlinear equations

The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger （NLS） equation and the complex Ginzburg-Landau （GL） equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.

Factorization method for inverse obstacle scattering problem in three-dimensional planar acoustic waveguides

In this paper, we consider the inverse scattering problem of reconstructing a bounded obstacle in a three-dimensional planar waveguide from the scattered near-field data measured on a finite cylindrical surface containing the obstacle and corresponding to infinitely many incident point sources also placed on the measurement surface. The obstacle is allowed to be an impenetrable scatterer or a penetrable scatterer. We establish the validity of the factorization method with the nearfield data to characterize the obstacle in the planar waveguide by constructing an outgoing-to-incoming operator which is an integral operator defined on the measurement surface with the kernel given in terms of an infinite series.

Barut-Girardello and Gilmore-Perelomov coherent states for pseudoharmonic oscillators and their nonclassical properties: Factorization method

In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrrdinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is much easier to work with, in comparison to the functional Hilbert space. The SU（1, 1） dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut-Girardello and Gilmore-Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.

Relativistic Schrodinger Wave Equation for Hydrogen Atom Using Factorization Method

In this investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.

INVERSE CONDUCTIVE MEDIUM SCATTERING WITH UNKNOWN BURIED OBJECTS

This paper is concerned with inverse acoustic scattering in an inhomogeneous medium with a conductive boundary condition and the unknown buried impenetrable objects inside.Using a variational approach,we establish the well-posedness of the direct problem.For the inverse problem,we shall numerically reconstruct the inhomogeneous medium from the far-field data for different kinds of cases.For the case when a Dirichlet boundary condition is imposed on the buried object,the classical factorization method proposed in[1]is justified as valid for reconstructing the inhomogeneous medium from the far-field data.For the case when a Neumann boundary condition is imposed on the buried object,the classical factorization method of[1]cannot be applied directly,since the middle operator of the factorization of the far-field operator is only compact.In this case,we develop a modified factorization method to locate the inhomogeneous medium with a conductive boundary condition and the unknown buried objects.Some numerical experiments are provided to demonstrate the practicability of the inversion algorithms developed.

Safe mining technology of undersea metal mine

Xinli district of Sanshandao Gold Mine is the first subsea metal mine in China.To achieve 6 kt/d production capacity under the premise of safe mining,high-intensity mining might destroy the in-situ stress filed and the stability of rockmass.According to sampling and testing of ore-rock and backfill and in-situ stress field measurement,safety factor method calculation model based on stress-strain strength reduction at arbitrary points and Mohr-Coulomb yield criterion was established and limit displacement subsidence values under the safety factor of different limit stoping steps were calculated.The results from three years in-situ mining and strata movement monitoring using multi-point displacements meter showed that the lower settlement frame stope hierarchical level filling mining method,mining sequence are reasonable and rockmass stability evaluation using safety factor method,in-situ real-time monitoring can provide the technical foundation for the safety of seabed mining.

Arrhenius relationship and two-step scheme in AF hyperdynamics simulation of diffusion of Mg/Zn interface

The accelerating factor （AF） method is a simple and appropriate way to investigate the atomic long-time deep diffusion at solid-solid interface. In the framework of AF hyperdynamics （HD） simulation, the relationship between the diffusion coefficient along the direction of z-axis which is normal to the Mg/Zn interface and temperature was investigated, and the AF＇s impact on the diffusion constant （D0） and activation energy （Q^＊） was studied. Then, two steps were taken to simulate the atomic diffusion process and the formation of new phases： one for acceleration and the other for equilibration. The results show that： the Arrhenius equation works well for the description of Dz with different accelerating factors; the AF has no effect on the diffusion constant Do in the case of no phase transition; and the relationship between Q＊ and Q conforms to Q^＊=Q/A. Then, the new Arrhenius equation for AFHD is successfully constructed as Dz=Doexp[-Q/（ART）]. Meanwhile, the authentic equilibrium conformations at any dynamic moment can only be reproduced by the equilibration simulation of the HD-simulated configurations. Key words： accelerating factor method; Arrhenius equation; two-steps scheme; Mg/Zn interface; hyperdynamic simulation

Quantum Chemical Studies on Proteins in the Reaction Center of Rhodobacter sphaeroides

The electronic structure of protein chains L and M in photosynthetic reaction center (PRC) of Rhodobacter sphaeroides (Van Niel) Imhoff, Truper et Pfennig) was studied by using the Overlapping Dimer Approximation method and the Extended Negative Factor Counter method at ab initio level. The result indicated that: (1) Amino acid residues, the molecular orbitals of which composed the main components of frontier orbitals of protein chain L (M), are located at the random coil areas of chain L (alpha helix areas of chain M). Since the random coil is flexible and more easy to change its conformation in the electron transfer process and to reduce the energy of the system, and the structure of the alpha helix is reletively stable, this difference might be one of the causes for the electron transfer in photosynthetic reaction center (PRC) only takes place along the L branch. (2) The His residues which axially coordinated to the 'special pair' P and accessory chlorophyll molecules (ABChls) are essentially important for the E-LUMO levels of P and ABChl. But, the corresponding molecular orbitals of these His residues do not appear in the composition of frontier orbitals of protein chains. It means that the interaction between pigment molecules and protein chains do not influence the contribution to the frontier orbitals of protein chains explicitly, but influences the corresponding E-LUMO levels significantly.

Evaluation, prediction, and protection of water quality in Danjiangkou Reservoir, China

The water quality in the Danjiangkou Reservoir has attracted considerable attention from the Chinese public and government since the announcement of the Middle Route of the South to North Water Diversion Project （SNWDP）, which commenced transferring water in 2014. Integrated research on the evaluation, prediction, and protection of water quality in the Danjiangkou Reservoir was carried out in this study in order to improve environmental management. Based on 120 water samples, wherein 17 water quality indices were measured at 20 monitoring sites, a single factor evaluation method was used to evaluate the current status of water quality. The results show that the main indices influencing the water quality in the Danjiangkou Reservoir are total phosphorus （TP）, permanganate index （CODM,）, dissolved oxygen （DO）, and five-day biochemical oxygen demand （BODs）, and the concentrations of TP, BODs, ammonia nitrogen （NH3--N）, CODM,, DO, and anionic surfactant （Surfa） do not reach the specified standard levels in the tributaries. Seasonal Mann--Kendall tests indicated that the CODMn concentration shows a highly significant increasing trend, and the TP concentration shows a significant increasing trend in the Danjiangkou Reservoir. The distribution of the main water quality indices in the Danjiangkou Reservoir was predicted using a two-dimensional water quality numerical model, and showed that the sphere of influence from the tributaries can spread across half of the Han Reservoir if the pollutants are not controlled. Cluster analysis （CA） results suggest that the Shending River is heavily polluted, that the Jianghe, Sihe, and Jianhe rivers are moderately polluted, and that they should be the focus of environmental remediation.

Optimal choice of parameters for particle swarm optimization

The constriction factor method (CFM) is a new variation of the basic particle swarm optimization (PSO), which has relatively better convergent nature. The effects of the major parameters on CFM were systematically investigated based on some benchmark functions. The constriction factor, velocity constraint, and population size all have significant impact on the per- formance of CFM for PSO. The constriction factor and velocity constraint have optimal values in practical application, and im- proper choice of these factors will lead to bad results. Increasing population size can improve the solution quality, although the computing time will be longer. The characteristics of CFM parameters are described and guidelines for determining parameter values are given in this paper.

Rutting influencing factors and prediction model for asphalt pavements based on the factor analysis method

To clarify the importance of various influencing factors on asphalt pavement rutting deformation and determine a screening method of model indicators,the data of the RIOHTrack full-scale track were examined using the factor analysis method(FAM).Taking the standard test pavement structure of RIOHTrack as an example,four rutting influencing factors from different aspects were determined through statistical analysis.Furthermore,the common influencing factors among the rutting influencing factors were studied based on FAM.Results show that the common factor can well characterize accumulative ESALs,center-point deflection,and temperature,besides humidity,which indicates that these three influencing factors can have an important impact on rutting.Moreover,an empirical rutting prediction model was established based on the selected influencing factors,which proved to exhibit high prediction accuracy.These analysis results demonstrate that the FAM is an effective screening method for rutting prediction model indicators,which provides a reference for the selection of independent model indicators in other rutting prediction model research when used in other areas and is of great significance for the prediction and control of rutting distress.

DYNAMIC CHARACTERISTIC ANALYSIS OF FUZZY-STOCHASTIC TRUSS STRUCTURES BASED ON FUZZY FACTOR METHOD AND RANDOM FACTOR METHOD

A new fuzzy stochastic finite element method based on the fuzzy factor method and random factor method is given and the analysis of structural dynamic characteristic for fuzzy stochastic truss structures is presented. Considering the fuzzy randomness of the structural physical parameters and geometric dimensions simultaneously, the structural stiffness and mass matrices axe constructed based on the fuzzy factor method and random factor method; from the Rayleigh＇s quotient of structural vibration, the structural fuzzy random dynamic characteristic is obtained by means of the interval arithmetic; the fuzzy numeric characteristics of dynamic characteristic axe then derived by using the random variable＇s moment function method and algebra synthesis method. Two examples axe used to illustrate the validity and rationality of the method given. The advantage of this method is that the effect of the fuzzy randomness of one of the structural parameters on the fuzzy randomness of the dynamic characteristic can be reflected expediently and objectively.

Analytical analysis for vibration of longitudinally moving plate submerged in infinite liquid domain

The vibration of a longitudinally moving rectangular plate submersed in an infinite liquid domain is studied analytically with the Rayleigh-Ritz method. The liquid is assumed to be incompressible, inviscid, and irrotational, and the velocity potential is used to describe the fluid velocity in the whole liquid field. The classical thin plate theory is used to derive mechanical energies of the traveling plate. As derivative of transverse displacement with respect to time in the compatibility condition equation exists, an exponential function is introduced to depict the dynamic deformation of the moving plate. It is shown that this exponential function works well with the Rayleigh- Ritz method. A convergence study shows a quick convergence speed for the immersed moving plate. Furthermore, the parametric study is carried out to demonstrate the effect of system parameters including the moving speed, the plate location, the liquid depth, the plate-liquid ratio, and the boundary condition. Results show that the above system parameters have significant influence on the vibration characteristics of the immersed moving plate. To extend the study, the method of added virtual mass incremental （AVMI） factor is used. The results show good agreement with those from the Rayleigh-Ritz method.

ANALYSIS OF THE LARGE DEFLECTION DYNAMIC PLASTIC RESPONSE OF SIMPLY-SUPPORTED CIRCULAR PLATES BY THE“MEMBRANE FACTOR METHOD”

Based on energy equilibrium,a new procedure called the Membrane Factor Method is devel- oped to analyze the dynamic plastic response of plates with deflections in the range where both bending mo- ments and membrane forces are important.The final deflection of a simply -supported circular rigid-plastic plate loaded by a uniformly distributed impulse is obtained.In comparison with other approximate solutions, the present results are found to be simpler and in better agreement with the corresponding experimental values reoorded by Florence.

Exact solutions of the Klein-Gordon equation with Makarov potential and a recurrence relation

In this paper, the Klein-Gordon equation with equal scalar and vector Makaxov potentials is studied by the factorization method. The energy equation and the normalized bound state solutions are obtained, a recurrence relation between the different principal quantum number n corresponding to a certain angular quantum number l is established and some special cases of Makarov potential axe discussed.

Anti-plane analysis of semi-infinite crack in piezoelectric strip

Using the complex variable function method and the technique of the conformal mapping, the fracture problem of a semi-infinite crack in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load. The analytic solutions of the field intensity factors and the mechanical strain energy release rate are presented under the assumption that the surface of the crack is electrically impermeable. When the height of the strip tends to infinity, the analytic solutions of an infinitely large piezoelectric solid with a semi-infinite crack are obtained. Moreover, the present results can be reduced to the well-known solutions for a purely elastic material in the absence of the electric loading. In addition, numerical examples are given to show the influences of the loaded crack length, the height of the strip, and the applied mechanical/electric loads on the mechanical strain energy release rate.

Analyzing the Urban Hierarchical Structure Based on Multiple Indicators of Economy and Industry: An Econometric Study in China

For a city,analyzing its advantages,disadvantages and the level of economic development in a country is important,especially for the cities in China developing at flying speed.The corresponding literatures for the cities in China have not considered the indicators of economy and industry in detail.In this paper,based on multiple indicators of economy and industry,the urban hierarchical structure of 285 cities above the prefecture level in China is investigated.The indicators from the economy,industry,infrastructure,medical care,population,education,culture,and employment levels are selected to establish a new indicator system for analyzing urban hierarchical structure.The factor analysis method is used to investigate the relationship between the variables of selected indicators and obtain the score of each common factor and comprehensive scores and rankings for 285 cities above the prefecture level in China.According to the comprehensive scores,285 cities above the prefecture level are clustered into 15 levels by using K-means clustering algorithm.Then,the hierarchical structure system of the cities above the prefecture level in China is obtained and corresponding policy implications are proposed.The results and implications can not only be applied to the urban planning and development in China but also offer a reference on other developing countries.The methodologies used in this paper can also be applied to study the urban hierarchical structure in other countries.