Weak Solution of Generalized KdV Equation with High Order Perturbation Terms

By using the theory of compensated compactness,we prove that there exists a sequence {uδε} converges nearly everywhere to the solution of the initial-value problem of generalized KdV equation with high order perturbation terms,namely we prove the existence of the weak solution.

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.

High order multiplication perturbation method for singular perturbation problems

This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.

UNIFORM DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED LINEAR 2ND ORDER HYPERBOLIC PROBLEM WITH ZEROTH ORDER REDUCED EQUATION

In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.

A UNIFORMLY CONVERGENT SECOND ORDER DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.

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.

A POSTERIORI ESTIMATOR OF NONCONFORMING FINITE ELEMENT METHOD FOR FOURTH ORDER ELLIPTIC PERTURBATION PROBLEMS

In this paper, we consider the nonconforming finite element approximations of fourth order elliptic perturbation problems in two dimensions. We present an a posteriori error estimator under certain conditions, and give an h-version adaptive algorithm based on the error estimation. The local behavior of the estimator is analyzed as well. This estimator works for several nonconforming methods, such as the modified Morley method and the modified Zienkiewicz method, and under some assumptions, it is an optimal one. Numerical examples are reported, with a linear stationary Cahn-HiUiard-type equation as a model problem.

General formula for phonon-assisted n-photon absorption in solids

A general formula for phonon-assisted n-photon absorption in solids is obtained by （n ＋ 1）-th order perturbation technique. The complicated calculation process for transition element of n-photon absorption is simply demonstrated by a diagram approach that is proposed in this work. We find that the transition element for the n-photon absorption has a simple form, i.e., it is just the first term of the n-th order fist kind Bessel function.

Prediction of Three Dimensional Crack Path Brittle Fracture in Weldment under Dynamic Load

Three dimensional dynamic stress intensity factors are analyzed for a curved crack with a second order perturbation method. The method is extended to obtain an approximate representation of a three dimensional dynamic stress intensity factors at the tip of a curved crack. Due to three dimensional curved crack growth the dynamic energy release rate can be calculated by using the Irwin＇s formula. A three dimensional curved crack in materials with inhomogeneous fracture toughness are considered. Paths of a brittle three dimensional curved crack propagating along a welded joint are predicted via the present method, where the effects of dynamic applied stresses, residual stresses, and material deterioration due to welding are taken into considerations.

Nonlinear vibration analysis of a rotor supported by magnetic bearings using homotopy perturbation method

In this paper,the effects of nonlinear forces due to the electromagnetic field of bearing and the unbalancing force on nonlinear vibration behavior of a rotor is investigated.The rotor is modeled as a rigid body that is supported by two magnetic bearings with eightpolar structures.The governing dynamics equations of the system that are coupled nonlinear second order ordinary differential equations(ODEs)are derived,and for solving these equations,the homotopy perturbation method(HPM)is used.By applying HPM,the possibility of presenting a harmonic semi-analytical solution,is provided.In fact,with equality the coefficient of auxiliary parameter(p),the system of coupled nonlinear second order and non-homogenous differential equations are obtained so that consists of unbalancing effects.By considering some initial condition for displacement and velocity in the horizontal and vertical directions,free vibration analysis is done and next,the forced vibration analysis under the effect of harmonic forces also is investigated.Likewise,various parameters on the vibration behavior of rotor are studied.Changes in amplitude and response phase per excitation frequency are investigated.Results show that by increasing excitation frequency,the motion amplitude is also increases and by passing the critical speed,it decreases.Also it shows that the magnetic bearing system performance is in stable maintenance of rotor.The parameters affecting on vibration behavior,has been studied and by comparison the results with the other references,which have a good precision up to 2nd order of embedding parameter,it implies the accuracy of this method in current research.

The J Integral of a Slightly Curved Elasticity-plasticity Crack when Enduring Quasi Static Loads

The J Integral of an elastic-plastic curved crack under quasi-static loads has been mainly worked in the paper,and the J Integral has been calculated as a practical application of a second order perturbation method and theorem of surname KA where the effect of quasi-static applied stresses and normal and shear stresses on the boundaries of plasticity area are synthetically taken into consideration.A regular pattern of variations of the J Integral of an elastic-plastic curved crack with the variations of curved crack shape parameters under different quasi-static loads has been mainly carried out.Continuity and unity of a elastic-plastic fracture and linear elastic fracture has been demonstrated from the viewpoint of the mechanical parameter of the curved crack tip J integral.The elastic-plastic slightly curved crack tip J integral has been calculated approximately,an integral loop having been reasonably selected and continuity and unity of the fracture characteristic of centre penetration straight line crack and curved crack have been demonstrated.

Effeet of Hardening on Elastic-plastic Curved Crack Tip Displacement Under Dynamic Load

In this article, elastic-plastic curved crack tip opening displacement maximum in hardened material under dynamic loads has been studied, and curved crack tip opening displacement has been calculated as a practical application of a second order perturbation method and theorem of surname KA, where the effects of dynamic applied stresses and dynamic normal and shear stresses on the boundaries of plastic area are synthetically taken into considerations. Diagrams have been constructed to analyz the transformation relationships between the curved crack tip opening displacement and the work hardening exponents. Curved crack tip opening displacement will decrease with the increasing of hardening exponents in hardened material. The decrease extent of curved crack tip dynamic opening displacement will be more and more severe when hardening exponents increase evenly. Maximum dynamic opening displacement of curved crack tip will decrease when external load decrease with the same hardening exponents