Integrating factors and conservation theorems of constrained Birkhoffian systems

In this paper the conservation theorems of the constrained Birkhoffian systems are studied by using the method of integrating factors. The differential equations of motion of the system are written. The definition of integrating factors is given for the system. The necessary conditions for the existence of the conserved quantity for the system are studied. The conservation theorem and its inverse for the system are established. Finally, an example is given to illustrate the application of the results.

ON THE INTEGRATING FACTOR OF ABEL EQUATION

The necessary and sufficient condition of a sort of rational faction integrating factor of Abel equation is derived from this paper.

POLYNOMIAL INVERSE INTEGRATING FACTORS

Let (P, Q) be a C1 vector field defined in an open subset U IR2. We call inverse integrating factor a C1 solution V(x, y) of the equation . In previous works it has been shown that this function plays an important role in the problem of the center and in the determination of limit cycles. In this paper we obtain necessary conditions for a polynomial vector field (P, Q) to have a polynomial inverse integrating factor.

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.

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.

The uniqueness of the integration factor associated with the exchanged heat in thermodynamics

State functions play important roles in thermodynamics.Different from the process function,such as the exchanged heatδQ and the applied workδW,the change of the state function can be expressed as an exact differential.We prove here that,for a generic thermodynamic system,only the inverse of the temperature,namely 1/T,can serve as the integration factor for the exchanged heatδQ.The uniqueness of the integration factor invalidates any attempt to define other state functions associated with the exchanged heat,and in turn,reveals the incorrectness of defining the entransy E_(vh)=CVT^(2)/2 as a state function by treating T as an integration factor.We further show the errors in the derivation of entransy by treating the heat capacity C_(V)as a temperature-independent constant.

To Explore the Chinese Medicine Syndrome Types and Integrative Therapy from Clinical Relative Factors of Patients with Advanced Prostate Cancer

Current Situation and Problems of the Treatment in Advanced Prostate Cancer In recent years,the incidence of prostate cancer shows a rising trend in China with an increase of 70%and has been the first place in the growth rate of malignant tumor in the male reproductive system. Prostate cancer has become a serious threat to male senior’s health.Because of the application of

Implicit integration factor method for the nonlinear Dirac equation

A high-order accuracy time discretization method is developed in this paper to solve the one-dimensional nonlinear Dirac(NLD)equation.Based on the implicit integration factor(IIF)method,two schemes are proposed.Central differences are applied to the spatial discretization.The semi-discrete scheme keeps the conservation of the charge and energy.For the temporal discretization,second-order IIF method and fourth-order IIF method are applied respectively to the nonlinear system arising from the spatial discretization.Numerical experiments are given to validate the accuracy of these schemes and to discuss the interaction dynamics of the NLD solitary waves.

Weighted Interior Penalty Methodwith Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation

Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh.There are three weights including the arithmetic,the harmonic,and the geometric weight in the weighted discontinuous Galerkin scheme.For the time discretization,we treat the nonlinear diffusion coefficients explicitly,and apply the semiimplicit integration factormethod to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization.The semi-implicit integration factor method can not only avoid severe timestep limits,but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method.Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.

Investigation of new solutions for an extended(2+1)-dimensional Calogero-Bogoyavlenskii-Schif equation

We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended(2+1)-dimensional Calogero-Bogoyavlenskii-Schif(eCBS)equation using the commutator table which results in a system of nonlinear ordinary differential equations(ODEs)which can be manually solved.Through two stages of Lie symmetry reductions,the eCBS equation is reduced to non-solvable nonlinear ODEs using different combinations of optimal Lie vectors.Using the integration method and the Riccati and Bernoulli equation methods,we investigate new analytical solutions to those ODEs.Back substituting to the original variables generates new solutions to the eCBS equation.These results are simulated through three-and two-dimensional plots.

Weierstrass Integrability of Complex Differential Equations

We characterize the complex differential equations of the form dy/dx=a_(n)(x)y^)n_+a_(n-1)(x)y^(n-1)+…+a_(1)(x)y+a_(0)(x) where a_(j)(x) are meromorphic functions in the variable x for j = 0,..., n that admit either a Weierstrass first integral or a Weierstrass inverse integrating factor.

Complete Study on a Bi-Center Problem for the Z2-Equivariant Cubic Vector Fields

For the planar Z2-equivariant cubic systems having twoelementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Liapunov constants are completely discussed. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z2-equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme （6 II 6） is proved.

A conservative local discontinuous Galerkin method for the solution of nonlinear Schrdinger equation in two dimensions

In this study, we present a conservative local discontinuous Galerkin(LDG) method for numerically solving the two-dimensional nonlinear Schrdinger(NLS) equation. The NLS equation is rewritten as a firstorder system and then we construct the LDG formulation with appropriate numerical flux. The mass and energy conserving laws for the semi-discrete formulation can be proved based on different choices of numerical fluxes such as the central, alternative and upwind-based flux. We will propose two kinds of time discretization methods for the semi-discrete formulation. One is based on Crank-Nicolson method and can be proved to preserve the discrete mass and energy conservation. The other one is Krylov implicit integration factor(IIF) method which demands much less computational effort. Various numerical experiments are presented to demonstrate the conservation law of mass and energy, the optimal rates of convergence, and the blow-up phenomenon.

The Growth of China’s Private Sector: A Case Study of Zhejiang Province

Private capital is one of the main driving forces in China's initiatives towards stimulating the market economy. The development of private economy in China has always been based on integrating industrial and corporate structures with product composition and market structures. This paper explores the development of the private economy and how it integrates different industries with specific markets by analyzing the leading private sector in Zhejiang province. It also examines the trends of industrial cluster, the formation of the agglomerative economy and their effects on private economy development. Finally, the paper explains why Zhejiang people have profited much from the Wenzhou model and discusses some existing problems and future possibilities for development of the Wenzhou model.

Evaluation of heavy metal contamination hazards in nuisance dust particles, in Kurdistan Province, western Iran

The effects of natural andgeochemical factors depending on heavy metal contamination in nuisance dust particles were evaluated. The nuisance dust particles were sampled using passive deposit gauge method for one year from April2010 to March2011 and the obtained samples were measured for the total contents and the contamination levels of Fe, Mn, Cu and As usinggeo-accumulation index （Igeo ）, enrichment factor （EF） and the integrated pollution index （IPI）. The results showed that, the contamination levels of Fe and Mn based on Igeo values, were uncontaminated （Igeo 〈 0） （variations of the Igeo index was from -3.11 to -1.751 for Fe, from -0.630 to -1.925 for Mn）, while the values of Cu and As were demonstrated to have moderate contamination based on Igeo values （variations of Igeo index was from -1.125 to 0.848 for Cu, and from -2.002 to 1.249 for As）. The analysis of EF also revealed minor to moderate enrichment for Mn （1.215-4.214）, minor to moderately severe enrichment for Cu （2.791-6.484）, and As （1.370-8.462）, respectively. The variation of the IPI index also showed low to moderate level of heavy metal pollution in nuisance dust particulates （0.511-1.829）. The analysis of the results also approved that the natural processes andgeochemical variables （the changing meteorological parameters） can significantly affect the availability of heavy metals in nuisance dust particles in Western Iran.

A conservative numerical method for the fractional nonlinear Schrodinger equation in two dimensions

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference(WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schr?dinger(FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations(ODEs) in matrix formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson(CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor(c IIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ~2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.

NUMERICAL SOLUTIONS FOR A NEARLY CIRCULAR CRACK WITH DEVELOPING CUSPS UNDER SHEAR LOADING

In this paper, we study the behavior of the solution at the crack edges for a nearly circular crack with developing cusps subject to shear loading. The problem of finding the resulting force can be written in the form of a hypersingular integral equation. The equation is then trans-formed into a similar equation over a circular region using conformal mapping. The equation is solved numerically for the unknown coefficients, which will later be used in finding the stress intensity factors. The sliding and tearing mode stress intensity factors are evaluated for cracks and displayed graphically. Our results seem to agree with the existing asymptotic solution.

Study on Integral Dissolution Model Based on Biological Potency for Compound Chinese Materia Medica

Objective To investigate the integral dissolution model based on biological potency in order to evaluate the dissolution of Compound Chinese materia medica（CCMM） in vitro. Methods The contents of paeoniflorin, phillyrin, ginsenoside Rg1, and adenosine of ten batches of Compound Biejia Ruangan Tablet（CBRT） were determined at different times. The self-defined weighting coefficient based on the contents has been created to establish the integral dissolution model. In addition, the biological potency of CBRT was measured by MTT assay. Then, the f2 similar factor was used to evaluate the similarity of the batches. Results Compared with batch a, some batches’ f2 values of paeoniflorin and adenosine were less than 50, while f2 values of ginsenoside Rg1, phillyrin, and integral component were more than 50. Likewise, ginsenoside Rg1, phillyrin, and integral component were all in good correlation with biological dissolution. Conclusion The results of the integral dissolution based on biological test of CBRT demonstrate that the bioassay method may be a promising supplement for its quality evaluation.