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.

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.

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.

THE FACTORIZATION METHOD FOR AN OPEN ARC

We consider the inverse scattering problem of determining the shape of a thin dielectric infinite cylinder having an open arc as cross section. Assuming that the electric field is polarized in the TM mode, this leads to a mixed boundary value problem for the Helmholtz equation defined in the exterior of an open arc in R2. We suppose that the arc has mixed Dirichlet impedance boundary condition, and try to recover the shape of the arc through the far field pattern by using the factorization method. However, we are not able to apply the basic theorem introduced by Kirsch to treat the far field operator F, and some auxiliary operators have to be considered. The theoretical validation of the factorization method to our problem is given in this paper, and some numerical results are presented to show the viability of our method.

APPLICATION OF THE FACTORIZATION METHOD TO RECOVER CUTS WITH OBLIQUE DERIVATIVE BOUNDARY CONDITION

Direct and inverse problems for the scattering of cracks with mixed oblique derivative boundary conditions from the incident plane wave are considered,which describe the scattering phenomenons such as the scattering of tidal waves by spits or reefs.The solvability of the direct scattering problem is proven by using the boundary integral equation method.In order to show the equivalent boundary integral system is Fredholm of index zero,some relationships concerning the tangential potential operator is used.Due to the mixed oblique derivative boundary conditions,we cannot employ the factorization method in a usual manner to reconstruct the cracks.An alternative technique is used in the theoretical analysis such that the far field operator can be factorized in an appropriate form and fulfills the range identity theorem.Finally,we present some numerical examples to demonstrate the feasibility and effectiveness of the factorization method.

Amplitude Factorization Method in the Atmospheric Gravity-Wave Equation

A novel amplitude factorization method is applied to solve a discrete buoyancy wave equation with arbitrary wind and temperature height distribution.The solution is given in the form of a cumulative product of complex factors,which are computed by a nonlinear,inhomogeneous,two-member recurrence formula,initiated from a radiative condition on top.Singularities of the wave equation due to evanescent winds are eliminated by turbulent friction.The method provides an estimation of the minimal vertical resolution,required to attain a stable accurate solution.The areas of application of the developed numerical scheme are high resolution modelling of orographic waves for arbitrary orography in general atmospheric stratification conditions,and testing of adiabatic kernels of numerical weather prediction models.

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.

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.

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.

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.

A method to calculate displacement factors using SVM

In order to improve the precision of mining subsidence prediction, a mathematical model using Support Vector Machine （SVM） was established to calculate the displacement factor. The study is based on a comprehensive analysis of factors affecting the displacement factor, such as mechanical properties of the cover rock, the ratio of mining depth to seam thickness, dip angle of the coal seam and the thickness of loose layer. Data of 63 typical observation stations were used as a training and testing sample set. A SVM regression model of the displacement factor and the factors affecting it was established with a kernel function, an insensitive loss factor and a properly selected penalty factor. Given an accurate calculation algorithm for testing and analysis, the results show that an SVM regression model can calcu- late displacement factor precisely and reliable precision can be obtained which meets engineering requirements. The experimental results show that the method to calculation of the displacement factor, based on the SVM method, is feasible. The many factors affecting the displacement factor can be consid- ered with this method. The research provides an efficient and accurate approach for the calculation of displacement in mining subsidence orediction.

Research on Design Method of the Full Form Ship with Minimum Thrust Deduction Factor

In the preliminary design stage of the full form ships, in order to obtain a hull form with low resistance and maximum propulsion efficiency, an optimization design program for a full form ship with the minimum thrust deduction factor has been developed, which combined the potential flow theory and boundary layer theory with the optimization technique. In the optimization process, the Sequential Unconstrained Minimization Technique（SUMT） interior point method of Nonlinear Programming（NLP） was proposed with the minimum thrust deduction factor as the objective function. An appropriate displacement is a basic constraint condition, and the boundary layer separation is an additional one. The parameters of the hull form modification function are used as design variables. At last, the numerical optimization example for lines of after-body of 50000 DWT product oil tanker was provided, which indicated that the propulsion efficiency was improved distinctly by this optimal design method.

Source Apportionment of Ambient PM_(10) in the Urban Area of Longyan City,China:a Comparative Study Based on Chemical Mass Balance Model and Factor Analysis Method

In order to identify the day and night pollution sources of PM10 in ambient air in Longyan City,the authors analyzed the elemental composition of respirable particulate matters in the day and night ambient air samples and various pollution sources which were collected in January 2010 in Longyan with inductivity coupled plasma-mass spectrometry（ICP-MS）.Then chemical mass balance（CMB） model and factor analysis（FA） method were applied to comparatively study the inorganic components in the sources and receptor samples.The results of factor analysis show that the major sources were road dust,waste incineration and mixed sources which contained automobile exhaust,soil dust/secondary dust and coal dust during the daytime in Longyan City,China.There are two major sources of pollution which are soil dust and mixture sources of automobile exhaust and secondary dust during the night in Longyan.The results of CMB show that the major sources are secondary dust,automobile exhaust and road dust during the daytime in Longyan.The major sources are secondary dust,soil dust and automobile exhaust during the night in Longyan.The results of the two methods are similar to each other and the results will guide us to plan to control the PM10 pollution sources in Longyan.

Application of scaled boundary finite element method in static and dynamic fracture problems

The scaled boundary finite element method （SBFEM） is a recently developed numerical method combining advantages of both finite element methods （FEM） and boundary element methods （BEM） and with its own special features as well. One of the most prominent advantages is its capability of calculating stress intensity factors （SIFs） directly from the stress solutions whose singularities at crack tips are analytically represented. This advantage is taken in this study to model static and dynamic fracture problems. For static problems, a remeshing algorithm as simple as used in the BEM is developed while retaining the generality and flexibility of the FEM. Fully-automatic modelling of the mixed-mode crack propagation is then realised by combining the remeshing algorithm with a propagation criterion. For dynamic fracture problems, a newly developed series-increasing solution to the SBFEM governing equations in the frequency domain is applied to calculate dynamic SIFs. Three plane problems are modelled. The numerical results show that the SBFEM can accurately predict static and dynamic SIFs, cracking paths and load-displacement curves, using only a fraction of degrees of freedom generally needed by the traditional finite element methods.

Dynamic Crack Propagation Analysis Using Scaled Boundary Finite Element Method

The prediction of dynamic crack propagation in brittle materials is still an important issue in many engineering fields. The remeshing technique based on scaled boundary finite element method(SBFEM) is extended to predict the dynamic crack propagation in brittle materials. The structure is firstly divided into a number of superelements, only the boundaries of which need to be discretized with line elements. In the SBFEM formulation, the stiffness and mass matrices of the super-elements can be coupled seamlessly with standard finite elements, thus the advantages of versatility and flexibility of the FEM are well maintained. The transient response of the structure can be calculated directly in the time domain using a standard time-integration scheme. Then the dynamic stress intensity factor(DSIF) during crack propagation can be solved analytically due to the semi-analytical nature of SBFEM. Only the fine mesh discretization for the crack-tip super-element is needed to ensure the required accuracy for the determination of stress intensity factor(SIF). According to the predicted crack-tip position, a simple remeshing algorithm with the minimum mesh changes is suggested to simulate the dynamic crack propagation. Numerical examples indicate that the proposed method can be effectively used to deal with the dynamic crack propagation in a finite sized rectangular plate including a central crack. Comparison is made with the results available in the literature, which shows good agreement between each other.

A Research on Competitiveness of Guangxi City——Based on System Clustering Method and Principal Component Analysis Method

A total of 10 indices of regional economic development in Guangxi are selected.According to the relevant economic data,regional economic development in Guangxi is analyzed by using System Clustering Method and Principal Component Analysis Method.Result shows that System Clustering Method and Principal Component Analysis Method have revealed similar results analysis of economic development level.Overall economic strength of Guangxi is weak and Nanning has relatively high scores of factors due to its advantage of the political,economic and cultural center.Comprehensive scores of other regions are all lower than 1,which has big gap with the development of Nanning.Overall development strategy points out that Guangxi should accelerate the construction of the Ring Northern Bay Economic Zone,create a strong logistics system having strategic significance to national development,use the unique location advantage and rely on the modern transportation system to establish a logistics center and business center connecting the hinterland and the Asean Market.Based on the problems of unbalanced regional economic development in Guangxi,we should speed up the development of service industry in Nanning,construct the circular economy system of industrial city,and accelerate the industrialization process of tourism city in order to realize balanced development of regional economy in Guangxi,China.

Comprehensive Evaluation on the Regional Economy of Commodity Grain Base in Heilongjiang Province Based on Factor Analysis Method

Taking a total of 13 areas in Heilongjiang commodity grain base as the research objects,9 indices are selected,which are regional GDP(X1),per capita GDP(X2),total value of tertiary industry(X3),financial revenue(X4),urban fixed assets investment(X5),average salary(X6),gross industrial output value(X7),total output value of farming,forestry,husbandry and fishing(X8),and retail sales of social consumer goods(X9).Based on this,evaluation index system of regional economy is established.According to the 2006-2008 Heilongjiang Statistical Yearbook,average values within 3 years are used as analytical data.Factor Analysis Method is adopted to establish regression model and to carry out comprehensive analysis.Result shows that Heilongjiang commodity grain base has extremely uneven regional economic development in different areas.According to the score order and actual situation,the 13 areas are divided into 4 types.The first and second types are Harbin and Daqing,respectively.The third type is Qiqihaer,Suihua,Mudanjiang and Jiamusi.And the forth type is Jixi,Shuangyashan,Heihe,Yichun,Qitaihe,Hegang and Daxinganling.Suggestions for the development of these areas are put forward.