A connection between the(G'/G)-expansion method and the truncated Painlevé expansion method and its application to the mKdV equation

Recently the （G′/G）-expansion method was proposed to find the traveling wave solutions of nonlinear evolution equations. This paper shows that the （G′/G）-expansion method is a special form of the truncated Painlev＇e expansion method by introducing an intermediate expansion method. Then the generalized （G′/G）-（G/G′） expansion method is naturally derived from the standpoint of the nonstandard truncated Painlev＇e expansion. The application of the generalized method to the mKdV equation shows that it extends the range of exact solutions obtained by using the （ G′/ G）-expansion method.

A new expansion method of first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term and its application to mBBM model

Based on a first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term which is extended from a type of elliptic equation, and by converting it into a new expansion form, this paper proposes a new algebraic method to construct exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to modified Benjamin-Bona-Mahony （mBBM） model, and some new exact solutions to the system are obtained. The algorithm is of important significance in exploring exact solutions for other nonlinear evolution equations.

TAYLOR EXPANSION METHOD FOR NONLINEAR EVOLUTION EQUATIONS

A new numerical method of integrating the nonlinear evolution equations, namely the Taylor expansion method, was presented. The standard Galerkin method can be viewed as the 0_th order Taylor expansion method; while the nonlinear Galerkin method can be viewed as the 1_st order modified Taylor expansion method. Moreover, the existence of the numerical solution and its convergence rate were proven. Finally, a concrete example, namely, the two_dimensional Navier_Stokes equations with a non slip boundary condition,was provided. The result is that the higher order Taylor expansion method is of the higher convergence rate under some assumptions about the regularity of the solution.

Eigenfunction expansion method and its application to two-dimensional elasticity problems based on stress formulation

This paper proposes an eigenfunction expansion method to solve twodimensional （2D） elasticity problems based on stress formulation. By introducing appropriate state functions, the fundamental system of partial differential equations of the above 2D problems is rewritten as an upper triangular differential system. For the associated operator matrix, the existence and the completeness of two normed orthogonal eigenfunction systems in some space are obtained, which belong to the two block operators arising in the operator matrix. Moreover, the general solution to the above 2D problem is given by the eigenfunction expansion method.

Exact Traveling Wave Solutions for Generalized Camassa-Holm Equation by Polynomial Expansion Methods

We formulate efficient polynomial expansion methods and obtain the exact traveling wave solutions for the generalized Camassa-Holm Equation. By the methods, we obtain three types traveling wave solutions for the generalized Camassa-Holm Equation: hyperbolic function traveling wave solutions, trigonometric function traveling wave solutions, and rational function traveling wave solutions. At the same time, we have shown graphical behavior of the traveling wave solutions.

NEW TRUNCATED EXPANSION METHOD AND SOLITON-LIKE SOLUTION OF VARIABLE COEFFICIENT KdV-MKdV EQUATION WITH THREE ARBITRARY FUNCTIONS

The truncated expansion method for finding explicit and exact soliton-like solution of variable coefficient nonlinear evolution equation was described. The crucial idea of the method was first the assumption that coefficients of the truncated expansion formal solution are functions of time satisfying a set of algebraic equations, and then a set of ordinary different equations of undetermined functions that can be easily integrated were obtained. The simplicity and effectiveness of the method by application to a general variable coefficient KdV-MKdV equation with three arbitrary functions of time is illustrated.

THE CONVERGENCE FOR NODAL EXPANSION METHOD

In this paper, we prove the convergence of the nodal expansion method, a new numerical method for partial differential equations and provide the error estimates of approximation solution.

Application of Fourier Series Expansion Method with PMLs to the Microcavities on Two-Dimensional Photonic Crystals

By using a Fourier series expansion method combined with Chew＇s perfectly matched layers （PMLs）, we analyze the frequency and quality factor of a micro-cavity on a two-dimensional photonic crystal is analyzed. Compared with the results by the method without PML and finite-difference time-domain （FDTD） based on supercell approximation, it can be shown that by the present method with PMLs, the resonant frequency and the quality factor values can be calculated satisfyingly and the characteristics of the micro-cavity can be obtained by changing the size and permittivity of the point defect in the micro-cavity.

Eigenfunction expansion method of upper triangular operator matrixand application to two-dimensional elasticity problems based onstress formulation

This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper tri angular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler com plete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and conve nient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.

DEFECT AND IMPROVEMENT OF TWO-TIME EXPANSION METHOD IN NONLINEAR OSCILLATIONS

In this paper, the defect of the Two-Time Expansion method is indicated and an improvement of this method is suggested. Certain examples.in which the present method is used, are given. Moreover, the paper shows the equivalence of the improved Two-Time Expansion Method and the method of KBM(Kryloy-Bogoliuboy-Mitropolski).

Sensitivity Analysis of Electromagnetic Scattering from Dielectric Targets with Polynomial Chaos Expansion and Method of Moments

In this paper,an adaptive polynomial chaos expansion method(PCE)based on the method of moments(MoM)is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis.The MoM is applied to accurately solve the electric field integral equation(EFIE)of electromagnetic scattering from homogeneous dielectric targets.Within the bistatic radar cross section(RCS)as the research object,the adaptive PCE algorithm is devoted to selecting the appropriate order to construct the multivariate surrogate model.The corresponding sensitivity results are given by the further derivative operation,which is compared with those of the finite difference method(FDM).Several examples are provided to demonstrate the effectiveness of the proposed algorithm for sensitivity analysis of electromagnetic scattering from homogeneous dielectric targets.

AN OPTIMIZATION OF EIGENFUNCTION EXPANSION METHOD FOR THE INTERACTION OF WATER WAVES WITH AN ELASTIC PLATE

The hydroelastic interaction of an incident wave with a semi-infinite horizontal elastic plate floating on a homogenous fluid of finite depth is analyzed using the eigenfunction expansion method. The fluid is assumed to be inviscid and incompressible and the wave amplitudes are assumed to be small. A two-dimensional problem is formulated within the framework of linear potential theory. The fluid domain is divided into two regions, namely an open water region and a plate-covered region. In this paper, the orthogonality property of eigenfunctions in the open water region is used to obtain the set of simultaneous equations for the expansion coefficients of the velocity potentials and the edge conditions are included as a part of the equation system. The results indicate that the thickness and the density of plate have almost no influence on the reflection and transmission coefficients. Numerical analysis shows that the method proposed here is effective and has higher convergence than the previous results.

Singularity analysis for single gimbal control moment gyroscope system using space expansion method

Control Moment Gyroscope（CMG） is an effective candidate for agile satellites and large spacecraft attitude control because of its powerful torque amplification capability. The most serious situation, however, in using CMG is the inherent geometric singularity problem, where there＇s no torque output along a particular direction. Space expansion method has been proposed in this work for the singularity analysis. Based on inverse mapping transformation, an expanded Jacobian matrix which is a full rank square matrix is obtained. The singular angle sets of the 3-parallel cluster and pyramid cluster are distinguished using space expansion method. An effective hybrid steering strategy, able to deal with the elliptic singularity, is further proposed. Simulation results demonstrate the excellent performance of the proposed steering logic compared to the generalized singular robust logic and pseudo inverse logic in terms of energy consumption and torque error.

A polynomial chaos expansion method for the uncertain acoustic field in shallow water

To obtain a universal model solving the uncertain acoustic field in shallow water, a non-intrusive model coupled polynomial chaos expansion （PCE） method with Helmholtz equa- tion is established, in which the polynomial coefficients are solved by probabilistic collocation method （PCM）. For the cases of Pekeris waveguide which have uncertainties in depth of water column, in both sound speed profile and depth of water column, and for the case of thermocline with lower limit depth uncertain, probability density functions （PDF） of transmission loss （TL） are calculated. The results show that the proposed model is universal for the acoustic propa- gation codes with high computational efficiency and accuracy, and can be applied to study the uncertainty of acoustic propagation in the shallow water en^-ironment with multiple parameters uncertain.

THE EXTENDED JACOBIAN ELLIPTIC FUNCTION EXPANSION METHOD AND ITS APPLICATIONS IN WEAKLY NONLINEAR WAVE EQUATIONS

The extended Jacobian elliptic function expansion method is introduced and applied to solve the coupled ZK equations and the coupled KP equations describing two weakly long nonlinear wave models in fluid system. Many types of doubly periodic traveling wave solutions are obtained. Under limiting conditions these solutions are reduced into solitary wave solutions.

Relanon between singularity expansion method(SEM) and resonance scattering theory(RST)

In recent years there are two theories for the acoustic scattering, one is the Singularity Expansion Method (SEM) , the other is the Resonance Scattering Theory (RST). In this paper, relation between these two theories was established. For the examples of the acoustic scattering from the solid elastic cylinder and sphere immersed in water, we prove that the RST can be directly derived from the SEM, so that these two theories are equivalent. By use of the Mittag- Leffler theorem we expand the pure elastic scattering wave, which is extracted by isolating the rigid background from the total scattering wave, in an exact resonance expansion. We specially prove that the reradiation efficiency and the resonance width are nearly proportional to the imaginary part of the corresponding pole for most solid objects immersed in water. This shows that the resonance scattering behavious can be entirely determined by the complex frequency poles. For the cases of an aluminum cylinder and a tungsten carbide sphere immersed in water, we calculate the partial-wave form functions by using the new resonance formulae. The results agree with the exact calculation well.

A Comparative Study on Polynomial Expansion Method and Polynomial Method of Particular Solutions

In this study,the polynomial expansion method(PEM)and the polynomial method of particular solutions(PMPS)are applied to solve a class of linear elliptic partial differential equations(PDEs)in two dimensions with constant coefficients.In the solution procedure,the sought solution is approximated by the Pascal polynomials and their particular solutions for the PEM and PMPS,respectively.The multiple-scale technique is applied to improve the conditioning of the resulted linear equations and the accuracy of numerical results for both of the PEM and PMPS.Some mathematical statements are provided to demonstrate the equivalence of the PEM and PMPS bases as they are both bases of a certain polynomial vector space.Then,some numerical experiments were conducted to validate the implementation of the PEM and PMPS.Numerical results demonstrated that the PEM is more accurate and well-conditioned than the PMPS and the multiple-scale technique is essential in these polynomial methods.

Calculation of dissociation temperature of quarkonium using Gaussian Expansion Method

The dissociation temperatures of quarkonium states in a thermal medium are obtained in the framework of the quark model with the help of the Gaussian Expansion Method（GEM）.This is the first time this method has been applied to the dissociation problem of mesons.The temperature-dependent potential is obtained by fitting the lattice results.Solving the Schr¨odinger equation with the GEM,the binding energies and corresponding wave functions of the ground states and the excited states are obtained at the same time.The accuracy and efficiency of the GEM provide a great advantage for the dissociation problem of mesons.The results show that the ground states1^1S（0 ）and 1^3S（1 ）have much higher dissociation temperatures than other states,and the spin-dependent interaction has a significant effect on the dissociation temperatures of 1^3S（1 ）and 1^1S0.We also suggest using the radius of the bound state as a criterion of quarkonium dissociation.This can help to avoid the inaccuracy caused by the long tail of quarkonium binding energy dependence on temperature.

Study of various few-body systems using Gaussian expansion method （GEM）

We review our calculation method, Gaussian expansion method （GEM）, to solve accurately the Schrodinger equations for bound, resonant and scattering states of few-body systems. Use is made of the Rayleigh-Ritz variational method for bound states, the complex-scaling method for resonant states and the Kohn-type variational principle to S-matrix for scattering states. GEM was proposed 30 years ago and has been applied to a variety of subjects in few-body （3- to 5-body） systems, such as 1） few-nucleon systems, 2） few-body structure of hypernuelei, 3） clustering structure of light nuclei and unstable nuclei, 4） exotic atoms/molecules, 5） cold atoms, 6） nuclear astrophysics and 7） structure of exotic hadrons. Showing examples in our published papers, we explain i） high accuracy of GEM calculations and its reason, ii） wide applicability of GEM to various few-body systems, iii） successful predictions by GEM calculations before measurements. The total bound-state wave function is expanded in terms of few-body Gaussian basis functions spanned over all the sets of rearrangement Jacobi coordinates. Gaussians with ranges in geometric progression work very well both for short- range and long-range behavior of the few-body wave functions. Use of Gaussians with complex ranges gives much more accurate solution than in the case of real-range Gaussians, especially, when the wave function has many nodes （oscillations）. These basis functions can well be applied to calculations using the complex-scaling method for resonances. For the few-body scattering states, the amplitude of the interaction region is expanded in terms of those few-body Gaussian basis functions.

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

In this paper,a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations,which mainly originate from parameters dispersions and measurement errors.Taking the perturbations into account,the perturbed system is regarded as a modification of the nominal system.By combining the perturbation series expansion method with the deterministic linear Hamiltonian system,the solution to the perturbed system is expressed in the form of asymptotic series by introducing a small parameter and a series of Hamiltonian canonical equations to predict the dynamic response are derived.Eventually,the response of the perturbed system can be obtained successfully by solving these Hamiltonian canonical equations using the symplectic difference schemes.The symplectic conservation of the proposed method is demonstrated mathematically indicating that the proposed method can preserve the characteristic property of the system.The performance of the proposed method is evaluated by three examples compared with the Runge-Kutta algorithm.Numerical examples illustrate the superiority of the proposed method in accuracy and stability,especially symplectic conservation for solving linear Hamiltonian systems with perturbations and the applicability in structural dynamic response estimation.