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.

ON WELL-CONDITIONED BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF SECOND ORDER DIFFERENCE EQUATIONS

In this paper well-conditioning of boundary value problems for systems of second order difference equa-tions is studied.First,a sufficient condition for the existence of a unique bounded solution (for large enough number of steps) of an associated homogeneous system is given.Finally,a sufficient condition for well-condi-tioning,intrinsically related to the problem data is proposed.

Oscillation of Second Order Delay Difference Equations

In this paper we study the Oscillatory behaviour of the second order delay differenceequation.（1）△（rn△An）+PnAn-k=0,n=n0,n0+1……where{Pn}（?）is a nonnegative Sequenceof real number,（?）is a positive sequence of real number with sum from n=n0 to +∞（1/rn）=+∞,K is a positive integer and △An=An+1-An we prove that each one of following conditions.imples that al solutions of Eq（1）oscillate,where Rn=sum from i=n0 to n（1/ri

Necessary and Sufficient Conditions for Oscillation ofBounded Solutions of Nonlinear Second Order Difference Equations

In this paper, we give necessary and sufficient conditions for oscillation of bounded solutions of nonlinear second order difference equation △(pn△yn)+ qnf(yn-rn) = 0. Obtained results improve theorems in the literature [3,6,7].

A Second Order Difference Scheme with Nonuniform Rectangular Meshes for Nonlinear Parabolic System

In this paper, a difference scheme with nonuniform meshes is proposed for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in both space and time.

OSCILLATION THEOREMS FOR SECOND ORDER NEUTRAL DIFFERENCE EQUATIONS

The aim of this paper is to study the oscillation of second order neutral difference equations.Our results are based on the new comparison theorems,that reduce the problem of the oscillation of the second order equation to that of the first order equation.The comparison principles obtained essentially simplify the examination of the equations.

Experimental investigation of nonlinear process based on cSHG/DFG in PPLN waveguide

Effects of second harmonic generation （SHG） and cascaded second harmonic generation/difference frequency generation（cSHG/DFG） based on the quasi-phase-matching （QPM） condition in periodically poled lithium niobate （PPLN） waveguide were investigated experimentally. SHG conversion efficiency of -13.6dB and QPM bandwidth of 0.45nm were achieved using a 16.1dBm power of fundamental wave at 1550.4nm. Using pulsed all-fiber passive mode locked laser and tunable continuous wave laser, cSHG/DFG effect utilized for optical sampling was observed. Conversion efficiencies were calculated, and 11.88nm-wide QPM bandwidth was achieved through changing the wavelength of input signal. Conversion efficiency of cSHG/DFG effect increased linearly with the total injected power.

ASYMPTOTIC BEHAVIOR AND OSCILLATIONS OF SECOND ORDER DIFFERENCE EQUATIONS WITH DELAY DEPENDING ON THE UNKNOWN FUNCTION

The asymptotic behavior of the nonoscillatory solutions of the difference equations △[r(n)△x(n)]+f(n,x(n),x(r(n,x(n))))=0 is considered. In the case when f is a strongly sublinear (superlinear) function, conditions for oscillations of (1) are also found.

Language Learner Motivation Related Theoretical Framework

Interest in studying individual differences （IDs） is driven by a trend characteristic of educational psychology in general and of second language acquisition （SLA） research in particular. Language learning motivation and self-determination theory （SDT） and an SDT-sub theory framework termed cognitive evaluation theory （CET） are reviewed in this article.

A SECOND ORDER UNCONDITIONALLY CONVERGENT FINITE ELEMENT METHOD FOR THE THERMAL EQUATION WITH JOULE HEATING PROBLEM

In this paper,we study the finite element approximation for nonlinear thermal equation.Because the nonlinearity of the equation,our theoretical analysis is based on the error of temporal and spatial discretization.We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential,and establish optimal L^(2) error estimates for the fully discrete finite element solution without any restriction on the time-step size.The discrete solution is bounded in infinite norm.Finally,several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A Comparison of Semi-Lagrangian and Lagrange-Galerkin hp-FEM Methods in Convection-Diffusion Problems

We perform a comparison in terms of accuracy and CPU time between second order BDF semi-Lagrangian and Lagrange-Galerkin schemes in combination with high order finite element method.The numerical results show that for polynomials of degree 2 semi-Lagrangian schemes are faster than Lagrange-Galerkin schemes for the same number of degrees of freedom,however,for the same level of accuracy both methods are about the same in terms of CPU time.For polynomials of degree larger than 2,Lagrange-Galerkin schemes behave better than semi-Lagrangian schemes in terms of both accuracy and CPU time;specially,for polynomials of degree 8 or larger.Also,we have performed tests on the parallelization of these schemes and the speedup obtained is quasi-optimal even with more than 100 processors.