Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbationmethod and the approximate direct method.The similarity reduction solutions of different orders are obtainedfor both methods, series reduction solutions are consequently derived.Higher order similarity reduction equations arelinear variable coefficients ordinary differential equations.By comparison, it is find that the results generated from theapproximate direct method are more general than the results generated from the approximate symmetry perturbationmethod.

Direct perturbation method for perturbed complex Burgers equation

So far, Lou＇s direct perturbation method has been applied successfully to solve the nonlinear SchrSdinger equation（NLSE） hierarchy, such as the NLSE, the coupled NLSE, the critical NLSE, and the derivative NLSE. But to our knowledge, this method for other types of perturbed nonlinear evolution equations has still been lacking. In this paper, Lou＇s direct perturbation method is applied to the study of perturbed complex Burgers equation. By this method, we calculate not only the zero-order adiabatic solution, but also the first order modification.

DIAGONALIZATION METHOD FOR A SINGULARLY PERTURBED VECTOR ROBIN PROBLEM

In this paper the method and technique of the diagonalization are employed to transform a vector second-order nonlinear system into two first-order approximate diagonalized systems. The existence and the asymptotic behavior of the solutions are obtained for a vector second-order nonlinear Robin problem of singular perturbation type.

Generalized nth-Order Perturbation Method Based on Loop Subdivision Surface Boundary Element Method for Three-Dimensional Broadband Structural Acoustic Uncertainty Analysis

In this paper,a generalized nth-order perturbation method based on the isogeometric boundary element method is proposed for the uncertainty analysis of broadband structural acoustic scattering problems.The Burton-Miller method is employed to solve the problem of non-unique solutions that may be encountered in the external acoustic field,and the nth-order discretization formulation of the boundary integral equation is derived.In addition,the computation of loop subdivision surfaces and the subdivision rules are introduced.In order to confirm the effectiveness of the algorithm,the computed results are contrasted and analyzed with the results under Monte Carlo simulations(MCs)through several numerical examples.

SECOND-ORDER ACCURATE DIFFERENCE METHOD FOR THE SINGULARLY PERTURBED PROBLEM OF FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.

Measurement of the g Factor of the 3.1232 MeV 19/2^(-) Level in ^(43)Sc by Perturbed Angular Distribution Method

The g-factor hence the magnetic moment,of the isomeric state 43Sc（|9/2-,3.1232 MeV）has been measured by the time differential perturbed angular distributionmethod.The measured values are g=0.3279（19）and μ=3.108（18）nm.

PERTURBATION METHODS IN SEMILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY-SOBOLEV EXPONENT

In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.

Method of Semi Analytic Perturbation Weighted Residuals for Nonlinear Bending of Shallow Shells

The semi? analytic perturbation weighted residuals method was used to solve the nonlinear bending problem of shallow shells, and the fifth order B spline was taken as trial function to seek an efficient method for nonlinear bending problem of shallow shells. The results from the present method are in good agreement with those derived from other methods. The present method is of higher accuracy, lower computing time and wider adaptability. In addition, the design of computer program is simple and it is easy to be programmed.

Perturbation method of studying the EI Nifio oscillation with two parameters by using the delay sea-air oscillator model

The EI Nino-southern oscillation （ENSO） is an interannual phenomenon involved in tropical Pacific ocean- atmosphere interactions. In this paper, we develop an asymptotic method of solving the nonlinear equation using the ENSO model. Based on a class of the oscillator of the ENSO model, a approximate solution of the corresponding problem is studied employing the perturbation method.

Precise integration method for a class of singular two-point boundary value problems

In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.

Solving ground eigenvalue and eigenfunction of spheroidal wave equation at low frequency by supersymmetric quantum mechanics method

The spheroidal wave functions are found to have extensive applications in many branches of physics and mathematics. We use the perturbation method in supersymmetric quantum mechanics to obtain the analytic ground eigenvalue and the ground eigenfunction of the angular spheroidal wave equation at low frequency in a series form. Using this approach, the numerical determinations of the ground eigenvalue and the ground eigenfunction for small complex frequencies are also obtained.

Distributed Capacitance of Coaxial Line With Perturbed Walls

Perturbation method of boundary geometry(PMOBG) used in Lapiacian problems is dealt with and the three--term perturbation expression of distributed capacitance of a coaxial line with perturbed walls is obtained. As an example,four-order expression of distributed capacitance of a elliptic coaxial line with small eccentricity is given.

Ground-state energy of beryllium atom with parameter perturbation method

We present a perturbation study of the ground-state energy of the beryllium atom by incorporating double parameters in the atom＇s Hamiltonian. The eigenvalue of the Hamiltonian is then solved with a double-fold perturbation scheme,where the spin-spin interaction of electrons from different shells of the atom is also considered. Calculations show that the obtained ground-state energy is in satisfactory agreement with experiment. It is found that the Coulomb repulsion of the inner-shell electrons enhances the effective nuclear charge seen by the outer-shell electrons, and the shielding effect of the outer-shell electrons to the nucleus is also notable compared with that of the inner-shell electrons.

IMPROVEMENTS ON THE ARC-LENGTH-TYPE METHOD

Arc-length-type and energy-type methods are two main strategies used in structural nonlinear tracing analysis, but the former is widely used due to the explicitness and clarity in conception, as well as the convenience and reliability in calculation. It is very important to trace the complete load-deflection path in order to know comprehensively the characteristics of structures subjected to loads. Unfortunately, the nonlinear analysis techniques are only workable for tracing the limit-point-type equilibrium path. For the bifurcation-point-type path, most of them cannot secure a satisfactory result. In this paper, main arc-length-type methods are reviewed and compared, and the possible reasons of failures in tracing analysis are briefly discussed. Some improvements are proposed, a displacement perturbation method and a force perturbation method are presented for tracing the bifurcation-point-type paths. Finally, two examples are analyzed to verify the ideas, and some conclusions are drawn with respect to the arc-length-type methods.

Thermoelastic Analysis of Non-uniform Pressurized Functionally Graded Cylinder with Variable Thickness Using First Order Shear Deformation Theory(FSDT) and Perturbation Method

Recently application of functionally graded materials（FGMs） have attracted a great deal of interest. These materials are composed of various materials with different micro-structures which can vary spatially in FGMs. Such composites with varying thickness and non-uniform pressure can be used in the aerospace engineering. Therefore, analysis of such composite is of high importance in engineering problems. Thermoelastic analysis of functionally graded cylinder with variable thickness under non-uniform pressure is considered. First order shear deformation theory and total potential energy approach is applied to obtain the governing equations of non-homogeneous cylinder. Considering the inner and outer solutions, perturbation series are applied to solve the governing equations. Outer solution for out of boundaries and more sensitive variable in inner solution at the boundaries are considered. Combining of inner and outer solution for near and far points from boundaries leads to high accurate displacement field distribution. The main aim of this paper is to show the capability of matched asymptotic solution for different non-homogeneous cylinders with different shapes and different non-uniform pressures. The results can be used to design the optimum thickness of the cylinder and also some properties such as high temperature residence by applying non-homogeneous material.

POINCARE-LIGHTHILL-KUO METHOD AND SYMBOLIC COMPUTATION

This paper elucidates the effectiveness of combining the Poincare-Lighthill-Kuo method (PLK method, for short) and symbolic computation. Firstly, the idea and history of the PLK method are briefly introduced. Then, the difficulty of intermediate expression swell, often encountered in symbolic computation, is outlined. For overcoming the difficulty, a semi-inverse algorithm was proposed by the author, with which the lengthy ports of intermediate expressions are first frozen in the form of symbols till the Fnal stage of seeking perturbation solutions. Tn discuss the applications of the above algorithm, the related work of the author and his research group on nonlinear oscillations and waves is concisely reviewed. The computer-extended perturbation solution of the Duffing equation shows that the asymptotic solution obtained with the PLK method possesses the convergence radius of 1 and thus the range of validity of the solution is considerably enlarged. The studies on internal solitary waves in stratified fluid and on the head-on collision between two solitary waves in a hyperelastic rod indicate that by means of the presented methods, very complicated manipulation, unconceivable in hand calculation, can be conducted and thus result in higher-order evolution equations and asymptotic solutions. The examples illustrate that the algorithm helps to realize the symbolic computation on micro-commputers. Finally, it is concluded that,vith the aid of symbolic computation, the vitality of the PLK method is greatly. Strengthened and at least for the solutions to conservative systems of oscillations and waves, it is a powerful tool.

INTERPOLATION PERTURBATION METHOD FORSOLVING NONLINEAR PROBLEMS

In this paper, using the interpolation perturbation method. the author seeks tosolve several nonlinear problems. Numerical examples show that the method Df thispaper has good accuracy.

PERTURBATIONAL FINITE VOLUME METHOD FOR THE SOLUTION OF 2-D NAVIER-STOKES EQUATIONS ON UNSTRUCTURED AND STRUCTURED COLOCATED MESHES

Based on the first-order upwind and second-order central type of finite volume (UFV and CFV) scheme, upwind and central type of perturbation finite volume (UPFV and CPFV) schemes of the Navier-Stokes equations were developed. In PFV method, the mass fluxes of across the cell faces of the control volume (CV) were expanded into power series of the grid spacing and the coefficients of the power series were determined by means of the conservation equation itself. The UPFV and CPFV scheme respectively uses the same nodes and expressions as those of the normal first-order upwind and second-order central scheme, which is apt to programming. The results of numerical experiments about the flow in a lid-driven cavity and the problem of transport of a scalar quantity in a known velocity field show that compared to the first-order UFV and second-order CFV schemes, upwind PFV scheme is higher accuracy and resolution, especially better robustness. The numerical computation to flow in a lid-driven cavity shows that the under-relaxation factor can be arbitrarily selected ranging from (0.3) to (0.8) and convergence perform excellent with Reynolds number variation from 10~2 to 10~4.

A New Approximation to the Linear Matrix Equation AX = B by Modification of He’s Homotopy Perturbation Method

It is well known that the matrix equations play a significant role in engineering and applicable sciences. In this research article, a new modification of the homotopy perturbation method (HPM) will be proposed to obtain the approximated solution of the matrix equation in the form AX = B. Moreover, the conditions are deduced to check the convergence of the homotopy series. Numerical implementations are adapted to illustrate the properties of the modified method.

A METHOD OF DETERMINING BUCKLED STATES OF THIN PLATES AT A DOUBLE EIGENVALUE

A method of determining bifurcation directions at a double eigenvalue is presented by combining the finite element method with the perturbation method. By using the present method, the buckled states of rectangular plates at a double eigenvalue are numerically analyzed. The results show that this method is effective.