A COMPLETE EXPRESSION OF THE ASYMPTOTIC SOLUTION OF DIFFERENTIAL EQUATION WITH A∩-TH ORDER TURNING POINT

A second order linear ordinary differential equation has been studied, and the complete expresson, of the formal uniformly valid asymptotic solutions to the equation near turning point is obtained by using extended Airy function.     

A COMPLETE EXPRESSION OF THE ASYMPTOTIC SOLUTION OF DIFFERENTIAL EQUATION WITH A N－TH ORDER TURNING POINT

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THE EXPONENTIAL ASYMPTOTIC SOLUTION OF DIFFERENTIAL EQUATION

In this paper, the exponential asgmptotic solution (E.A.S.) of differential equation is discussed. Firstly, E.A.S. of the second-order differential equation is studied and the orthogonal conditions of the uniformly valid E.A.S. are found out. Next, E.A.S. in matched asymptotic method is discussed. Finally, some examples are given.     

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR SEMILINEAR PARABOLIC DIFFERENTIAL EQUATION

The numerical solution of a singularly perturbed problem for the semilinear parabolicdifferential equation with parabolic boundary layers is discussed.A nonlinear two-leveldifference scheme is constructed on the special non-uniform grids.The uniform convergenceof this scheme is proved and some numerical examples are given.     

A HIGH ACCURACY DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF THE SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper,combining the idea of difference method and finite element method,weconstruct a difference scheme for a self-adjoint problem in conservation form.Its solutionuniformly converges to that of the original differential equation problem with order h~3.     

THE UNIFORMLY CONVERGENT DIFFERENCE SCHEMES FOR A SINGULAR PERTURBATION PROBLEM OF A SELFADJOINT ORDINARY DIFFERENTIAL EQUATION

In this paper,we construct a class of difference schemes with fitted factors for asingular perturbation problem of a self-adjoint ordinary differential equation.Using adifferent method from[1],by analyzing the truncation errors of schemes,we give thesufficient conditions under which the solution of the difference scheme converges uniformlyto the solution of the differential equation.From this we propose several specific schemesunder weaker conditions,and give much higher order of uniform convergence,and applyingthem to example,obtain the numerical results.     

INSTABILITY OF SOLUTION FOR THE THIRD ORDER LINEAR DIFFEBENTIAL EQUATION WITH VARIED COEFFICIENT

In reference [1] asymptotic stability of dynamic system with slowly changing coefficients for all characteristic roots which have negative real part has been proved by means of Liapunov's second method. In this paper, we give some sufficient conditions of the instability for the third order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part, by means of Liapunov's second method.     

THE BOUNDARY LAYER METHOD FOR THE SOLUTION OF SINGULAR PERTURBATION PROBLEM FOR THE PARABOLIC PARTIAL DIFFERENTIAL EQUATION

In this paper, we discuss the singular perturbation problem of the parabolic partialdifferential equation. As usual, we must reduce the mesh size in the neighbourhood of theboundary layer so that typical feature of the boundary layer will not be lost. Then we needvery large operational quantity when mesh sizes are getting too small.Now we propose the boundary layer scheme, which need not take very fine mesh size inthe neighbourhood of the boundary layer. Numerical examples show that the accuracy canbe satisfied with moderate step size.     

THE ASYMPTOTIC SOLUTION OF A KIND OF STEFAN PROBLEM

In this paper,a kind of Stefan problem subject to general initial condition is studied.The slab considered is divided into three regions.There is a different time scale in each one.By means of PLK method or like multi-scales method,the asympto-tic solution of each one is obtained.Finally we discuss the asymptotic solution and draw some conclusions.     

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...     

ASYMPTOTIC EXPERSSION OF THE SOLUTION OF GENERAL BOUNDARY VALUE PROBLEN FOR HIGHER ORDER ELLIPTIC EQUATION WITH PERTURBATION BOTH IN BOUNDARY AND IN OPERATOR

In this paper basing on (1) and (2),we study thesingular perturbation of general boundary value problemfor higher order elliptic equation with perturbationboth in the boundary and in the operator,so as to es-tablish the asymptotic expression involving two para-meters.Thus,the iterative process of finding theasymptotic solution is derived and the estimation ofthe remainder term is given out,we extend and improvethe previously published papers.     

THE EXISTENCE OF PERIODIC SOLUTION OF THE FOURTH ORDINARY NONLINEAR DIFFERENTIAL EQUATION CAUSED BY FLOW－IN

ＴＨＥＥＸＩＳＴＥＮＣＥＯＦＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＯＦＴＨＥＦＯＵＲＴＨＯＲＤＩＮＡＲＹＮＯＮＬＩＮＥＡＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＣＡＵＳＥＤＢＹＦＬＯＷ－ＩＮＤＵＣＥＤＶＩＢＲＡＴＩＯＮＧｕＱｉｎｇ－ｆａｎｇ（顾清芳）Ｔａ...     

ASYMPTOTIC SOLUTION OF SINGULAR PERTURBATION PROBLEMS FOR THE FOURTH-ORDER ELLIPTIC DIFFERENTIAL EQUATIONS

In this paper we consider the singularly perturbed boundary value problem for thefourth-order elliptic differential equation,establish the energy estimates of the solutionandits derivatives and construct the formal asymptotic solution by Lyuternik-Vishik’s method.Finally,by means of the energy estimates we obtain the bound of the remainder of theasymptotic solution.     

SINGULAR PERTURBATION SOLUTION OF BOUNDARY-VALUE PROBLEM FOR A SECORD-ORDER DIFFERENTIAL-DIFFERENCE EQUATION

In this paper, the method of two-variables expansion is used to construct boundary layer terms of asymptotic solution of the boundary-value problem for a second-order DDE. The n-order formal asymptotic solution is obtained and the error is estimated. Thus the existence of uniformly valid asymptotic solution is proved.     

KANTOROVICH SOLUTION FOR THE PROBLEM OF BENDING OF A LADDER PLATE

ctangular plate in bending, this paper deals with the solutions for the ladder plate with various boundary conditions. The deflection of the plate is expressed in a first-order displacement function ω(x,y)=μ(x,y)υ(y) where the μ(x,y) in x direction is the generalized beam function. By making use of the principle of least potential energy, the variable coefficients differential equations for v(y) may be established. By solvingis, these differential euqations and making use of the boundary conditions, the accurate solutions of v(y) in y direction may be obtained. Then the displacement function w(x,y) is the solution for the problem of the bending of the ladder plate with a better degree of approximation.     

NUMERICAL SOLUTION OF THE SINGULARLY PERTURBED PROBLEM FOR THE HYPERBOLIC EQUATION WITH INITIAL JUMP

In this paper we consider the initial-boundary value problem for a second orderhyperbolic equation with initial jump.The bounds on the derivatives of the exact solutionare given.Then a difference scheme is constructed on a non-uniform grid.Finally,uniformconvergence of the difference solution is proved in the sense of the discrete energy norm.     

TO SOLVE THE SHALLOW SHELL EQUATIONS WITH THE HELP OF THE PLATE’S EQUATION SOLUTION(LOWERING THE ORDER OF PARTIAL DIFFERENTIAL EQUATION FROM 8 TO 2)

If a plate solution is known which has the same boundary and loading conditions with ashallow shell,the solution of that shell can be reduced to a non-homogeneous Helmholtz’sequation in complex domain.Two examples are given to illustrate our method.     

TO SOLVE THE SHALLOW SHELL EQUATIONS WITH THE HELP OF THE PLATE’S EQUATION SOLUTION(LOWERING THE ORDER OF PARTIAL DIFFERENTIAL EQUATION FROM 8 TO 2)

If a plate solution is known which has the same boundary and loading conditions with a shallow shell, the solution of that shell can be reduced to a non-homogeneous Helmholtz’s equation in complex domain. Two examples are given to illustrate our method.     

UNIFORM CONVERGENCE FOR THE DIFFERENCE SCHEME IN THE CONSERVATION FORM OF ORDINARY DIFFERENTIAL EQUATION WITH A SMALL PARAMETER

In this paper,we consider a singular perturbation boundary problem for a self-adjointordinary differential equaiton.We construct a class of difference schemes with fittedfactors,and give the sufficient conditions under which the solution of difference schemeconverges uniformly to the solution of differential equation.From this we propose severalspecific schemes under weaker conditions,and give much higher order of uniformconvergence.     

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

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