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.     

Uniformly convergent numerical method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition

In this article, we consider a class of singularly perturbed differential equations of convection-diffusion type with nonlocal boundary conditions. A uniformly convergent numerical method is constructed via nonstandard finite difference and numerical integration methods to solve the problem. The nonlocal boundary condition is treated using numerical integration techniques. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ϵ -uniformly convergent.     

Nonlinear stability of large k-value shallow spherical shells with a symmetrically distributed line load

In this paper, we treat the nonlinear stability problem of shallow spherical shells with large values ofk(k=12(1–v) · 2f/h,f = shell rise,h = shell thickness) under the action of uniformly distributed line load along a circle concentric with the shell boundary. Load-deflection curves are computed at successive increments of uniformly distributed line loads by using both cubic B-spline approximations and iterative techniques. Our algorithm yields fairly good convergent results for values ofk as large as 400. The limiting case in which shells are loaded along a circle of small radius has been specially investigated and the computed critical loads are compared with those obtained with central point loads by other authors.     

Long Time Estimates in the Mean Field Limit

In this paper, we prove a stability result for measure perturbations of some class of stationary distributions of a Vlasov equation. We use this result to prove that the N particles approximation of these stationary distributions is uniformly valid on a time scale of order N ^1/8, which is much longer than the usual log N scale. We also prove similar results for the approximation of the two-dimensional Euler equation by the vortex blob method.     

THE METHOD OF WEIGHTED DIFFERENCE FOR SINGULAR PERTURBATION PROBLEM

In this paper, we introduce a new difference approximation to the first order derivative u′and give a class of uniformly convergent difference schemes.     

Numerical solution of a singularly perturbed elliptichyperbolic partial differential equation on a nonuniform discretization mesh

In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order convergent, uniformly in the small parameter ε, to the solution of problem (1.1). Numerical results are finally provided.     

The convergent condition and united formula of step reduction method

The step reduction method was first suggested by Prof. Yeh Kai-yuan^[1]. This method has more advantages than other numerical methods. By this method, the analytic expression of solution can he obtained for solving nonuniform elastic mechanics. At the same time, its calculating time is very short and convergent speed very fast. In this paper. the convergent condition and united formula of step reduction method are given by mathematical method. It is proved that the solution of displacement and stress resultants obtained by this method can converge to exact solution uniformly. when the convergent condition is satisfied. By united formula, the analytic solution can be expressed as matrix form, and therefore the former complicated expression can be avoided. Two numerical examples are given at the end of this paper which indicate that by the theory in this paper, a right model can be obtained for step reduction method.Project Supported by Science and Technic Fund of the National Education Committee.The author would like to thank Prof. Yeh Kai-yuan for his directing.     

The necessary and sufficient condition of uniformly convergent difference schemes for the elliptic-parabolic partial differential equation with a small parameter

This paper studies the necessary and sufficient condition of uniformly convergent difference scheme for the elliptic-parabolic partial differential equation with a small parameter.Communicated by Lin Zong-chi.     

Uniformly convergent difference schemes for singular perturbation problem

In this paper, a class of uniformly convergent difference schemes for singular perturbation problem are given.     

On Nonlinear Stability of Isotropic Models in Stellar Dynamics

Consider an aggregation of mass particles in space which attract each other according to Newton's law of attraction. This system can be described by distribution functions satisfying the Vlasov equation and the Poisson laws. We obtain the nonlinear stability of certain stationary states, including those obeying modified Emden's laws. A priori estimates of the energy-Casimir functions around stationary states are established for distribution functions for which the L^5/3L^{5/3}-norms of the density functions are uniformly bounded. A uniform bound on the kinetic energy of the system readily implies that these norms of the density functions are indeed uniformly bounded. In this way we prove nonlinear stability.     

Light diffraction by one ultrasonic wave: Laplace transform method

Light diffraction by sharply bounded ultrasound is studied by solving the Raman-Nath equations. The uniformly convergent series expansions for the amplitudes of the diffracted lightwaves are valid as well in the Raman-Nath as in the Bragg region. Comparison with the NOA method (? > 1) and the earlier found analytical formula (? ? 1) allows clear delimination of the validity of both these approximate solutions.     

NEW THEORY FOR EQUATIONS OF NON-FUCHSIAN TYPE——REPRESENTATION THEOREM OF TREE SERIES SOLUTION (Ⅱ)

Our main result consists in proving the representation theorem. Irregular integral is a new type of analytic function, represented by a compound Taylor-Fourier tree series, in which each coefficient of the Fourier series is a Taylor series, while the Taylor coefficients are tree series in terms of equations parameters, higher order correction terms to each coefficient having tree structure with inexhaustible proliferation.The solution obtained is proved to be convergent absolutely and uniformly in the region defined by coefficient functions of the original equation, provided the structure parameter is less than unity. Direct substitution shows that our tree series solution satisfies the equation explicity generation by generation.As compared with classical theory our method not only furnishes explicit expression of irregular integral, leading to the solution of Poincare problem, but also provides possibility of extending the scope of investigation for analytic theory to equations with various kinds of singularities in a unifying way.Exact explicit analytic expression for irregular integrals can be obtained by means of correspondence principle.It is not difficult to prove the convergence of the tree series solution obtained. Direct substitution shows it satisfies the equation.The tree series is automorphic, which agrees completely with Poincaré’s conjecture.     

Exponential Ordering for Neutral Functional Differential Equations With Non-Autonomous Linear D-Operator

We study neutral functional differential equations with stable linear non-autonomous D-operator. The operator of convolution [^(D)]{\widehat D} transforms BU into BU. We show that, if D is stable, then [^(D)]{\widehat D} is invertible and, besides, [^(D)]{\widehat D} and [^(D)]^-1{\widehat D^{-1}} are uniformly continuous for the compact-open topology on bounded sets. We introduce a new transformed exponential order and, under convenient assumptions, we deduce the 1-covering property of minimal sets. These conclusions are applied to describe the amount of material in a class of compartmental systems extensively studied in the literature.     

The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations

We prove that viscosity solutions in W ^1, of the second order, fully nonlinear, equation F(D ² u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D ² u and from that used by Crandall & Lions [3] and Crandall, Evans & Lions [2] for fully nonlinear first order problems.The research reported here was supported in part by grants from the Alfred P. Sloan Foundation and the National Science Foundation.     

Boundary integral equations for the bending problem of plates on two-parameter foundation

By means of Fourier integral transformation of generalized fimction, the fimdamental solution for the bending problem of plales on two-parameter foundation is derived in this paper, and the fundamental solution is expanded into a uniformly convergent series.On the basis of the above work, two boundary integral eguations which are suitable to arhitrary shapes and arbitrary boundary conditions are established by means of the Rayleigh-Green identity. The content of the paper provides the powerful theories for the application of BEM in this problem.     

The evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic non-conductors

The theory of singular surfaces yields a set of coupled evolution equations for the shock amplitude and the amplitudes of the higher order discontinuities which accompany the shock. To solve these equations, we use perturbation methods with a perturbation parameter characterising the initial shock amplitude. It is shown that for decaying shock waves, if the accompanying second order discontinuity is of order one, the straightforward perturbation procedure yields uniformly valid solutions, but if the accompanying second order discontinuity is of order , the method of multiple scales is needed in order to render the perturbation solutions uniformly valid with respect to the distance of travel. We also construct shock wave solutions from modulated simple wave solutions which are obtained with the aid ofHunter & Keller's Weakly Nonlinear Geometrical Optics method. The two approaches give exactly the same results within their common range of validity. The explicit evolution laws thus obtained enable us to see clearly how weak nonlinear curved shock waves are attenuated because of the effects of geometry and material nonlinearity, and on what length scale these effects are most pronounced. Communicated by C. C. Wang     

Stability of Subsonic Planar Phase Boundaries in a van der Waals Fluid

We are concerned with the structural stability of dynamic phase changes occurring across sharp interfaces in a multidimensional van der Waals fluid. Such phase transitions can be viewed as propagating discontinuities. However, they are usually subsonic, and thus undercompressive. The lacking information lies in an additional jump condition, which may be derived from the viscosity-capillarity criterion. This condition is rather simple in the case of reversible phase transitions, since it reduces to a generalized equal area rule. In a previous work, I proved that reversible planar phase boundaries are weakly linearly stable, in the sense introduced by Majda for shock fronts. This means that they satisfy a generalized Lopatinsky condition but not a uniform one. The aim of this paper is to point out the influence of viscosity on the stability analysis, in order to deal with the more realistic case of dissipative phase transitions. The main difficulty lies in the additional jump condition, which is no longer explicit and depends on the (unknown) internal structure of the interface. We overcome it by using bifurcation arguments on the nondimensional parameter measuring the competition between viscosity and capillarity. We show by perturbation that the positivity of this parameter stabilizes the phase transitions. As a conclusion, we find that dissipative planar phase boundaries are uniformly linearly stable, in the sense of the uniform Lopatinsky condition. Accepted December 14, 1998     

Cublic spline solutions of axisymmetrical nonlinear bending and buckling of circular sandwich plates

IntroductionBruun[1],Huang[2 ]andWang[3]publishedtheirpapersrelatedtolinearlystaticanalysisofcircularsandwichplates .Liuetal.setupnonlinearbendingequationsofacircularsandwichplate[4 ],andsolvedaseriesofnonlinearproblems[5～ 10 ].Sofartheothersneverdiscuss…