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.

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.

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.

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.

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.

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.

A NEW MATRIX PERTURBATION METHOD FOR ANALYTICAL SOLUTION OF THE COMPLEX MODAL EIGENVALUE PROBLEM OF VISCOUSLY DAMPED LINEAR VIBRATION SYSTEMS

A new matrix perturbation analysis method is presented for efficient approximate solution of the complex modal quadratic generalized eigenvalue problem of viscously damped linear vibration systems. First, the damping matrix is decomposed into the sum of a proportional-and a nonproportional-damping parts, and the solutions of the real modal eigenproblem with the proportional dampings are determined, which are a set of initial approximate solutions of the complex modal eigenproblem. Second, by taking the nonproportional-damping part as a small modification to the proportional one and using the matrix perturbation analysis method, a set of approximate solutions of the complex modal eigenvalue problem can be obtained analytically. The result is quite simple. The new method is applicable to the systems with viscous dampings-which do not deviate far away from the proportional-damping case. It is particularly important that the solution technique be also effective to the systems with heavy, but not over, dampings. The solution formulas of complex modal eigenvlaues and eigenvectors are derived up to second-order perturbation terms. The effectiveness of the perturbation algorithm is illustrated by an exemplar numerical problem with heavy dampings. In addition, the practicability of approximately estimating the complex modal eigenvalues, under the proportional-damping hypothesis, of damped vibration systems is discussed by several numerical examples.

DIRECT PERTURBATION METHOD FOR REANALYSIS OF MATRIX SINGULAR VALUE DECOMPOSITION

The perturbational reanalysis technique of matrix singular value decomposition is applicable to many theoretical and practical problems in mathematics, mechanics, control theory, engineering, etc.. An indirect perturbation method has previously been proposed by the author in this journal, and now the direct perturbation method has also been presented in this paper. The second-order perturbation results of non-repeated singular values and the corresponding left and right singular vectors are obtained. The results can meet the general needs of most problems of various practical applications. A numerical example is presented to demonstrate the effectiveness of the direct perturbation method.

Three-Dimensional Static Analysis of Nanoplates and Graphene Sheets by Using Eringen’s Nonlocal Elasticity Theory and the Perturbation Method

A three-dimensional(3D)asymptotic theory is reformulated for the static analysis of simply-supported,isotropic and orthotropic single-layered nanoplates and graphene sheets(GSs),in which Eringen’s nonlocal elasticity theory is used to capture the small length scale effect on the static behaviors of these.The perturbation method is used to expand the 3D nonlocal elasticity problems as a series of two-dimensional(2D)nonlocal plate problems,the governing equations of which for various order problems retain the same differential operators as those of the nonlocal classical plate theory(CST),although with different nonhomogeneous terms.Expanding the primary field variables of each order as the double Fourier series functions in the in-plane directions,we can obtain the Navier solutions of the leading-order problem,and the higher-order modifications can then be determined in a hierarchic and consistent manner.Some benchmark solutions for the static analysis of isotropic and orthotropic nanoplates and GSs subjected to sinusoidally and uniformly distributed loads are given to demonstrate the performance of the 3D nonlocal asymptotic theory.

Investigation on electromagnetic scattering from rough soil surface of layered medium using the small perturbation method

Electromagnetic scattering from a rough surface of layered medium is investigated, and the formulae of the scattering coefficients for different polarizations are derived using the small perturbation method. A rough surface with exponential correlation function is presented for describing a rough soil surface of layered medium, the formula of its scattering coefficient is derived by considering the spectrum of the rough surface with exponential correlation function; the curves of the bistatic scattering coefficient of HH polarization with variation of the scattering angle are obtained by numerical calculation. The influence of the permittivity of layered medium, the mean layer thickness of intermediate medium, the roughness surface parameters and the frequency of the incident wave on the blstatic scattering coefficient is discussed. Numerical results show that the influence of the permittivity of layered medium, the mean layer thickness of intermediate medium, the rms and the correlation length of the rough surface, and the frequency of the incident wave on the bistatic scattering coefficient is very complex.

Coupling of high order multiplication perturbation method and reduction method for variable coefcient singular perturbation problems

Based on the precise integration method （PIM）, a coupling technique of the high order multiplication perturbation method （HOMPM） and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems （TPBVPs） with one boundary layer. First, the inhomogeneous ordinary differential equations （ODEs） are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.

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.

REDUCTIVE PERTURBATION METHOD OF SUPER KdV EQUATIONS

By using reductive perturbation method, super KdV equations are changed into ordinary KdV equations, small amplitude perturbation solutions are obtained.

Application of singular perturbation method in analyzing traffic density waves

Car following model is one of microscopic models for describing traffic flow. Through linear stability analysis, the neutral stability lines and the critical points are obtained for the different types of car following models and two modified models. The singular perturbation method has been used to derive various nonlinear wave equations, such as the Kortewegde-Vries （KdV） equation and the modified Korteweg-de-Vries （mKdV） equation, which could describe different density waves occurring in traffic flows under certain conditions. These density waves are mainly employed to depict the formation of traffic jams in the congested traffic flow. The general soliton solutions are given for the different types of car following models, and the results have been used to the modified models efficiently.

Finding the buckling load of non-uniform columns using the iteration perturbation method

The aim of this study is to calculate the critical load of variable inertia columns. The example studied in this paper can be used as a paradigm for other non-uniform columns. The wavelength of equivalent vibratory system is used to calculate the critical load of the trigonometrically varied inertia column. In doing so, the equilibrium equation of the column is theoretically studied using the perturbation method. Accuracy of the calculated results is evaluated by comparing the solution with numerical results. Effect of improving the initial guess on the solution accuracy is investigated. Effects of varying parameters of the trigonometrically varied inertia and the uniformly tapered columns on their stability behavior are studied. Finally, using the so-called ＂perfectibility＂ parameter, two design goals, i.e., being lightweight and being strong, are studied for the discussed columns.

PERTURBATION METHOD FOR REANALYSIS OF THE MATRIX SINGULAR VALUE DECOMPOSITION

The perturbation method for the reanalysis of the singular value decomposition (SVD) of general real matrices is presented in this paper. This is a simple but efficient reanalysis technique for the SVD, which is of great worth to enhance computational efficiency of the iterative analysis problems that require matrix singular value decomposition repeatedly. The asymptotic estimate formulas for the singular values and the corresponding left and right singular vectors up to second-order perturbation components are derived. At the end of the paper the way to extend the perturbation method to the case of general complex matrices is advanced.

Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems

A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators, in which the perturbation proce- dure is improved for those systems whose exact homoclinic generating solutions cannot be explicitly derived. The generalized hyperbolic functions are employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method. Several strongly nonlinear oscillators with quadratic, cubic, and quartic nonlinearity are studied in detail to illustrate the efficiency and accuracy of the present method.

THE APPLICATION OF MULTI－SCALE PERTURBATION METHOD TO THE STABILITY ANALYSIS OF PLANE COUETTE FLOW

This paper applies the multi-scale perturbation method suggested by Ref [3] toinvestigate the linear stability behavior of distorted plane Couette .flow. Using thismethod, the unstable Tollmien-Schlichting wave in plane Couette flow can be found,but not the most unstable mode. By comparing the results of this paper with those ofRef. [3], the effectiveness of this method is investigated.

Damped Mathieu Equation with a Modulation Property of the Homotopy Perturbation Method

In this article,the main objective is to employ the homotopy perturbation method(HPM)as an alternative to classical perturbation methods for solving nonlinear equations having periodic coefficients.As a simple example,the nonlinear damping Mathieu equation has been investigated.In this investigation,two nonlinear solvability conditions are imposed.One of them was imposed in the first-order homotopy perturbation and used to study the stability behavior at resonance and non-resonance cases.The next level of the perturbation approaches another solvability condition and is applied to obtain the unknowns become clear in the solution for the firstorder solvability condition.The approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science.

He’s Homotopy Perturbation Method and Fractional Complex Transform for Analysis Time Fractional Fornberg-Whitham Equation

In this article,time fractional Fornberg-Whitham equation of He’s fractional derivative is studied.To transform the fractional model into its equivalent differential equation,the fractional complex transform is used and He’s homotopy perturbation method is implemented to get the approximate analytical solutions of the fractional-order problems.The graphs are plotted to analysis the fractional-order mathematical modeling.