SOME DISCRETE NONLINEAR INEQUALITIES AND APPLICATIONS TO DIFFERENCE EQUATIONS

In this article, the authors establish some new nonlinear difference inequalities in two independent variables, which generalize some existing results and can be used as handy tools in the study of qualitative as well as quantitative properties of solutions of certain classes of difference equations.

POINTWISE ESTIMATE OF BOUNDED SOLUTIONS TO SOME DISCRETE INEQUALITIES INVOLVING INFINITE SUMMATION

Explicit bounds on bounded solutions to a new class of Volterra-type linear and nonlinear discrete inequalities involving infinite sums are established. These inequalities can be viewed as discrete analogues of some Volterra-type inequalities having improper integral functionals,which are new to the literature.

Nonlinear Discrete Inequalities of Bihari-type and Applications

Discrete Bihari-type inequalities with n nonlinear terms are discussed, which generalize some known results and may be used in the analysis of certain problems in the theory of difference equations. Examples to illustrate the boundedness of solutions of a difference equation are also given.

Design of observer-based discrete repetitive-control system based on 2D model

A discrete observer-based repetitive control(RC) design method for a linear system with uncertainties was presented based on two-dimensional(2D) system theory. Firstly, a 2D discrete model was established to describe both the control behavior within a repetition period and the learning process taking place between periods. Next, by converting the designing problem of repetitive controller into one of the feedback gains of reconstructed variables, the stable condition was obtained through linear matrix inequality(LMI) and also the gain coefficient of repetitive system. Numerical simulation shows an exceptional feasibility of this proposal with remarkable robustness and tracking speed.

ON SOME DISCRETE INEQUALITIES USEFUL IN THE THEORY OF CERTAINPARTIAL FINITE DIFFERENCE EQUATIONS

The aim of this paper is to establish some new discrete inequalities in two independent variables which can be used as handy tools.in the theory of certain fourth order partial finite difference equations. The analysis used in the proof is elementary and the results established provide new estimates for these types of inequalities.AMS (MOS) Subject Classification (1991 ): Primary 26D15.

Discrete Generalizations of Some N-Independent-Variable Integral Inequalities of Langenhop-Gollwitzer Type

We establish some new n-independent-variable discrete inequalities which are analo- gous to some Langenhop-Gollwitzer type integral inequalities obtained by the present author in J. Math.Anal.Appl.,109(1985),171-181.An application to hyperbolic summary-difference equations in n variables is also sketched.

Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations

This paper proposes a new nonconforming finite difference streamline diffusion method to solve incompressible time-dependent Navier-Stokes equations with a high Reynolds number. The backwards difference in time and the Crouzeix-Raviart （CR） element combined with the P0 element in space are used. The result shows that this scheme has good stabilities and error estimates independent of the viscosity coefficient.

NEW SECOND ORDER NONCONFORMING TRIANGULAR ELEMENT FOR PLANAR ELASTICITY PROBLEMS

In the use of finite element methods to the planar elasticity problems,one diffculty is to overcome locking when elasticity constant λ→∞.In the case of traction boundary condition,another diffculty is to make the discrete Korn＇s second inequality valid.In this paper,a triangular element is presented.We prove that this element is locking-free,the discrete Korn＇s second inequality holds and the convergence order is two.

A NEW NONLINEAR DISCRETE INEQUALITY AND ITS APPLICATION

A new discrete inequality with the power nonlinearity is obtained which unifies and generalizes some known results due to B.G.Pachpatte. A certain initial value problem of a sum-difference equation is also given to convey the usefulness of the inequality obtained.

A New Class of Simple,General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper,a new efficient,and at the same time,very simple and general class of thermodynamically compatiblefinite volume schemes is introduced for the discretization of nonlinear,overdetermined,and thermodynamically compatiblefirst-order hyperbolic systems.By construction,the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm.A very peculiar feature of our approach is that entropy is discretized directly,while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization.The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs,including both,conservative and non-conservative products,as well as potentially stiff algebraic relaxation source terms,provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law,such as the conservation of total energy density.The proposed family offinite volume schemes is based on the seminal work of Abgrall[1],where for thefirst time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented.We apply our new approach to three particular thermodynamically compatible systems:the equations of ideal magnetohydrodynamics(MHD)with thermodynamically compatible generalized Lagrangian multiplier(GLM)divergence cleaning,the unifiedfirst-order hyperbolic model of continuum mechanics proposed by Godunov,Peshkov,and Romenski(GPR model)and thefirst-order hyperbolic model for turbulent shallow waterflows of Gavrilyuk et al.In addition to formal mathematical proofs of the properties of our newfinite volume schemes,we also present a large set of numerical results in order to show their potential,efficiency,and practical applicability.

Some Delay Gronwall Type Inequalities on Time Scales

In this paper, we investigate some delay Cronwall type inequalities on time scales by using Cron- wall＇s inequality. Our results unify and extend some delay integral inequalities and their corresponding discrete analogues. The inequalities given here can be used as handy tools in the qualitative theory of certain classes of delay dynamic equations on time scales.

Nonlinear Discrete Inequality in Two Variables with Delay and Its Application

Delay discrete integral inequalities with n nonlinear terms in two variables are discussed, which generalize some existing results and can be used as powerful tools in the analysis of certain partial difference equations. An application example is also given to show boundedness of solutions of a difference equation.

Linearized Transformed L1 Galerkin FEMs with Unconditional Convergence for Nonlinear Time Fractional Schr¨odinger Equations

A linearized transformed L1 Galerkin finite element method(FEM)is presented for numerically solving the multi-dimensional time fractional Schr¨odinger equations.Unconditionally optimal error estimates of the fully-discrete scheme are proved.Such error estimates are obtained by combining a new discrete fractional Gr¨onwall inequality,the corresponding Sobolev embedding theorems and some inverse inequalities.While the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting approaches.Numerical examples are presented to confirm the theoretical results.

Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

Hyperbolic Conservation Laws on Manifolds.An Error Estimate for Finite Volume Schemes

Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm is of order h1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch＇s theory which was originally developed in the Euclidian setting. We extend the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.

A Compact Difference Scheme on Graded Meshes for the Nonlinear Fractional Integro-differential Equation with Non-smooth Solutions

In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M denote the numbers of grids in temporal and spatial direction,α ∈(0,1)is the fractional order.To recover the full accuracy based on the regularity requirement of the solution,we adopt the L1 method and the trapezoidal product integration(PI)rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral,respectively,further handle the nonlinear term carefully by the Newton linearized method.Based on the discrete fractional Gr¨onwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral,the stability and convergence of the proposed scheme are analyzed by the energy method.Theoretical results are also confirmed by a numerical example.

ESTIMATES ON THE SOLUTIONS OF CERTAIN HIGHER ORDER FINITE DIFFERENCE EQUATIONS

In this paper results on the estimates for solutions of certain higher order finite difference equations are established. The main tool employed in our analysis is based on the applications of the discrete inequality which provides an explicit bound on the unknown function.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A CLASS OF HIGHER ORDER DIFFERENCE EQUATIONS

In this paper,we study the asymptotic behavior of solutions to a class of higher order difference equations.With the aid of the discrete inequality,we obtain some sufficient conditions which ensure that all the solutions to the equation are some high order of infinities,and also that some conditions which guarantee that every oscillatory solution to the equation has the property that the i order L operator of it tends to infinity when its independent variable tends to zero.