On the free energy and variational theorem of elastic problem in thermal shock

In some investigations on variational principle for coupled thermoelastic problems, the free energy Φ(e_ij,θ) ,where the state variables are elastic strain e_ij and temperature increment θ, is expressed as Φ(e_ij,θ)=λ/2e_kke_ij=ue_k1e_k1-γe_kkθ-c/2 p θ²/T₀(0.1) This expression is employed only under the condition of |θ|≤T₀(absolute temperature of reference) But the value of temperature increment is great, even greater than T₀ in thermal shock. And the material properties (λ ,μ ,ν ,c , etc.) will not remain constant, they vary with θ. The expression of free energy for this condition.is derived in this paper. Equation (0.1) is its special case.Euler’s equations will be nonlinear while this expression of free energy has been introduced into variational theorem. In order to linearise, the time interval of thermal shock is divided into a number of time elements Δt_k, (Δt_k=t_k-t_k-1,k=1,2…,n), which are so small that the temperature increment θ_k within it is very small, too. Thus, the material properties may be defined by temperature field T_k-1=T(x₁,x₂,x₃,t_k-1) at instant t_k-1 , and the free energy Φ_k expressed by eg. (0.1) may be employed in element Δt_k.Hence the variational theorem will be expressed partly and approximately.     

A singular perturbation problem for periodic boundary partial differential equation

In this paper,we consider a singular perturbation elliptic-parabolic partial differentialequation for periodic boundary value problem,and construct a difference scheme.Using themethod of decomposing the singular term from its solution and combining an asymptoticexpansion of the equation,we prove that the scheme constructed by this paper convergesuniformly to the solution of its original problem with O(τ h~2).     

A UNIFORMLY DIFFERENCE SCHEME OF SINGULAR PERTURBATION PROBLEM FOR A SEMILINEAR ORDINARY DIFFERENTIAL EQUATION WITH MIXED BOUNDARY VALUE CONDITION

ＡＵＮＩＦＯＲＭＬＹＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＯＦＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＰＲＯＢＬＥＭＦＯＲＡＳＥＭＩＬＩＮＥＡＲＯＲＤＩＮＡＲＹＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＷＩＴＨＭＩＸＥＤＢＯＵＮＤＡＲＹＶＡＬＵＥＣＯＮＤＩＴ...     

A UNIFORMLY CONVERGENT DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF A HIGH ORDER ELLIPTIC DIFFERENTIAL EQUATION

ＡＵＮＩＦＯＲＭＬＹＣＯＮＶＥＲＧＥＮＴＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＦＯＲＴＨＥＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＰＲＯＢＬＥＭＯＦＡＨＩＧＨＯＲＤＥＲＥＬＬＩＰＴＩＣＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮ（刘国庆）（苏煜城）ＡＵＮＩＦＯ...     

A uniformly convergent finite difference method for a singularly perturbed initial value problem

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｐｕｒｐｏｓｅｏｆｔｈｉｓｐａｐｅｒｉｓｔｏｄｅｖｅｌｏｐａｃｕｒａｔｅｄｉｆｅｒｅｎｃｅｍｅｔｈｏｄｆｏｒｔｈｅｆｏｌｏｗｉｎｇｉｎｉｔｉａｌｖａｌｕｅｐｒｏｂｌｅｍ,ｗｈｉｃｈｉｓｉｎｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａ...     

The numerical solution of a singularly perturbed problem for quasilinear parabolic differential equation

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L~2 norm is proved and numerical examples are presented.     

Numerical solution of nonlinear ordinary differential equation for a turning point problem

By using the method in[3],several useful estimations of the derivatives of the solutionof the boundary value problem for a nonlinear ordinary differential equation with a turningpoint are obtained.With the help of the technique in[4],the uniform convergence on thesmall parameterεfor a difference scheme is proved.At the end of this paper,a numericalexample is given.The numerical result coincides with theoretical analysis.     

非线性方程的保辛近似算法

保守体系的微分方程可用Hamilton体系的方法描述,其特点是保辛。两个辛矩阵之和不能保辛,两个辛矩阵的乘积仍是辛矩阵。最常用的小参数摄动法用的是加法,因此对辛矩阵不能保辛。从保辛的角度,要用正则变换。本文针对非线性微分方程,运用自变量坐标变换,对原系统进行变换。由此推导出变换后系统的变分原理。引入Hamilton对偶变量,通过数学变换,得到变系数非线性方程。针对该方程,本文提出了保辛摄动算法。通过数值算例,对不同步长下,保辛摄动法、多尺度摄动法、龙格库塔法和精确解的结果做了比较。数值例题表明,对于非线性方程,本文提出的保辛摄动算法有良好的精度。在步长增大的情况下,保辛摄动保持了良好的稳定性。     

A Class of Singularly Perturbed Generalized Boundary Value Problems for Quasi-Linear Elliptic Equation of Higher Order

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅａｕｔｈｏｒｓｔｕｄｉｅｄａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｉｎ [1]- [8].Ｔｈｉｓｐａｐｅｒｉｎｖｏｌｖｅｓｔｈｅｇｅｎｅｒａｌｉｚｅｄｓｏｌｕｔｉｏｎｏｆｔｈｅｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓ.ＮｏｗｗｅｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇＤｉｒｉｃｈｌｅｔｐｒｏｂｌｅｍｆｏｒｔｈｅｑｕａｓｉ＿ｌｉｎｅａｒｅｌｌｉｐｔｉｃｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｏｆｈｉｇｈｅｒｏｒｄｅｒεＬ2ｍ[ｕ] ∑ｎ…     

On Boundary Damping for a Weakly Nonlinear Wave Equation

In this paper an initial-boundary value problem for a weakly nonlinear string(or wave) equation with non-classical boundary conditions is considered. Oneend of the string is assumed to be fixed and the other end of the string isattached to a spring-mass-dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a rather simple model describing oscillationsof flexible structures such as suspension bridges or overhead transmission lines in a windfield. A multiple-timescales perturbation method will be usedto construct formal asymptotic approximations of the solution. It will also beshown that all solutions tend to zero for a sufficiently large value of thedamping parameter. For smaller values of the damping parameter it will be shownhow the string-system eventually will oscillate.     

An exponentially fitted difference scheme for the hyperbolic-hyperbolic singularly perturbed initial-boundary value problem

In this paper we discuss,an initial-boundary value problem of hyperbolic type with firstderivative with respect to x.The asymptotic solution is constructed and its uniform validityis proved under weader compatibility conditions.Then we develop an exponentially fitteddifference scheme and establish discrete energy inequality.Finally,we prove that thesolution of difference problem uniformly converges to the solution of the original problem.     

On a Rayleigh Wave Equation with Boundary Damping

In this paper an initial-boundary value problem for a weakly nonlinear string (or wave) equation with non-classical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a simple model describing oscillations of flexible structures such as overhead transmission lines in a windfield. An asymptotic theory for a class ofinitial-boundary value problems for nonlinear wave equations is presented. Itwill be shown that the problems considered are well-posed for all time t. A multiple time-scales perturbation method incombination with the method of characteristics will be used to construct asymptotic approximations of the solution. It will also be shown that all solutions tend to zero for a sufficiently large value of the damping parameter. For smaller values of the damping parameter it will be shown how the string-system eventually will oscillate. Some numerical results are alsopresented in this paper.     

NUMERICAL SOLUTION OF THE SINGULARLY PERTURBED PROBLEM WITH NONLOCAL BOUNDARY CONDITION

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｉｓｃｏｎｃｅｒｎｅｄｗｉｔｈｔｈｅｎｕｍｅｒｉｃａｌｓｏｌｕｔｉｏｎ ,ｂｙｆｉｎｉｔｅｄｉｆｆｅｒｅｎｃｅｍｅｔｈｏｄ ,ｏｆｔｈｅｆｏｌｌｏｗｉｎｇｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｎｏｎｌｏｃａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ :Ｌｕ≡-εｕ″ +ａ(ｘ)ｕ=ｆ(ｘ)　　 ( 0 <ｘ<ｌ) ,( 1 )Ｌ0 ｕ≡-εｕ′( 0 ) +γｕ( 0 ) =μ0 , ( 2 )Ｌ1ｕ≡ｕ(ｌ) -δｕ(ｄ) =μｌ　　 ( 0 <ｄ<ｌ) ,( 3 )ｗｈｅｒｅεｉｓａｓｍａｌｌｐｏｓｉｔｉｖｅｐａｒａｍｅｔｅｒ,…     

Numerical methods for parabolic equation with a small parameter in time variable

In this paper,we discuss the parabolic equation with a small parameter on thederivative in time variable.We construct difference scheme on the non-uniform meshaccording to Bakhvalov,and prove the one-order uniform convergence of this scheme.Numerical results are presented.     

Singular Perturbation of Boundary Value Problem for Quasilinear Third-Order Ordinary Differential Equations Involving Two Small Parameters

ＩｎｔｒｏｄｕｃｔｉｏｎＡｂｏｕｔｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａｔｉｏｎｏｆｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｓｅｃｏｎｄ＿ｏｒｄｅｒｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｗｏｒｋｓｏｆａｌａｒｇｅｎｕｍｂｅｒｗｅｒｅｃｏｎｓｉｄｅｒｅｄｏｎｌｙｆｏｒｔｈｅｃａｓｅｏｆｉｎｖｏｌｖｉｎｇｏｎｅｓｍａｌｌｐａｒａｍｅｔｅｒ[1- 4].Ｏｎｌｙａｆｅｗｗｏｒｋｓｗｅｒｅｃｏｎｓｉｄｅｒｅｄｆｏｒｔｈｅｃａｓｅｏｆｉｎｖｏｌｖｉｎｇｔｗｏｓｍａｌｌｐａｒａｍｅｔｅｒｓ[5 - 8]…     

Stability of buoyancy-driven convection in an Oldroyd-B fluid-saturated anisotropic porous layer

The nonlinear stability of thermal convection in a layer of an Oldroyd-B fluid-saturated Darcy porous medium with anisotropic permeability and thermal diffusivity is investigated with the perturbation method. A modified Darcy-Oldroyd model is used to describe the flow in a layer of an anisotropic porous medium. The results of the linear instability theory are delineated. The thresholds for the stationary and oscillatory convection boundaries are established, and the crossover boundary between them is demarcated by identifying a codimension-two point in the viscoelastic parameter plane. The stability of the stationary and oscillatory bifurcating solutions is analyzed by deriving the cubic Landau equations. It shows that these solutions always bifurcate supercritically. The heat transfer is estimated in terms of the Nusselt number for the stationary and oscillatory modes. The result shows that, when the ratio of the thermal to mechanical anisotropy parameters increases, the heat transfer decreases.     

The Asymptotic Behavior of Solution for the Singularly Perturbed Initial Boundary Value Problems of the Reaction Diffusion Equations in a Part of Domain

Ｔｈｅａｕｔｈｏｒｓｓｔｕｄｉｅｄａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｉｎ [1 ] -[7] .Ｎｏｗｗｅｒａｉｓｅａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｏｎａｐａｒｔｏｆｄｏｍａｉｎ .Ｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｔｈｅｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ　　ｕｔ-λε(ｘ) ( μ(ｕ)ｕｘ) ｘ Ｋｘ(ｕ) ｆ(ｘ ,ｔ,ｕ) =0 ,(ｔ,ｘ) ∈ ( 0 ,Ｔ)× ( ( 0 ,α) ∪ (…     

Spatial instability of a swirling flow

The instability of a swirling flow of an inviscidand incompressible fluid is studied on the assumption that the wavenumber k=k_r + ik_i of the disturbance is complex while its frequency ω is real. This implies that the disturbance grows with distance along the axis of the swirling flow, but it does not grow with time. The occurrence of such disturbance is called spatial instability, in contrast to the temporal instability, in which k is a real number and ω=ω_r + iω_i is a complex one. The results show that spatial instability analysis is a useful tool for the comprehensive understanding of the instability behaviours of a swirling flow.     

Boundary value problem for a singularly perturbed nonlinear system

I.Intr0ducti0n'Manyresultshavebeeng0tinstudyingthesingu1arperturbation0fboundaryvalueproblemfornonlinearsystemeg#=j(t,g,y',8)(l.l)bythemethodandthetechniqueofdiagonalizationl'~'l.Thepapersarebasedonsucha.conditionthattheeigenvaluesofJacobimatrixlu'offwit…     

Boundary and interior layer behavior for singularly perturbed vector problem

In this paper,we consider the vector nonlinear boundary value problem:εy~v=f(x,y,z,y′,ε),y(0)=A_1,y(1)=B_1εz~v=g(x,y,z,z′,ε),z(0)=A_2,z(1)=B_2whereε>0 is a small parameter,0≤x≤1 ,f and g are continuous functions in R~4,Under appropriate assumptions,by means of the differential inequalities,we demonstratethe existence and estimation,involving boundary and interior layers,of the solutions to theabove problem.