Superconvergence analysis of the finite element method for nonlinear hyperbolic equations with nonlinear boundary condition

This paper discusses the semidiscrete finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. The superclose property is derived through interpolation instead of the nonlinear H^1 projection and integral identity technique. Meanwhile, the global superconvergence is obtained based on the interpolated postprocessing techniques.

Existence and asymptotic behavior for systems of nonlinear hyperbolic equations

The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.

NONEXISTENCE OF GLOBAL SOLUTIONS OF NONLINEAR HYPERBOLIC EQUATION WITH MATERIAL DAMPING

Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are proved by the energy method, Jensen inequality and the concavity method, respectively. As applications of our main results, three examples are given.

ON NONLINEAR HYPERBOLIC EQUATION IN UNBOUNDED DOMAIN

The following nonlinear hyperbolic equation is discussed in this paper:where A= -? + Iandx∈Rn. The model comes from the transverse deflection equation of an extensible beam. We prove that there exists a unique local solution of the above equation as M depends on x.

Iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations

Presents the iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations, and the application of the computational technique of reproducing kernel space to simplify, the iterative computation and increase the convergence rate and points out that this method is still effective. Even if the initial condition is discrete.

Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations

In this paper,we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)method solving nonlinear hyperbolic equations with smooth solutions.The smoothness-increasing accuracy-conserving(SIAC)filter is a post-processing technique to enhance the accuracy of the discontinuous Galerkin(DG)solutions.This work is the essential step to extend the SIAC filter to the moving mesh for nonlinear problems.By the post-processing theory,the negative norm estimates are vital to get the superconvergence error estimates of the solutions after post-processing in the L2 norm.Although the SIAC filter has been extended to nonuniform mesh,the analysis of fil-tered solutions on the nonuniform mesh is complicated.We prove superconvergence error estimates in the negative norm for the ALE-DG method on moving meshes.The main dif-ficulties of the analysis are the terms in the ALE-DG scheme brought by the grid velocity field,and the time-dependent function space.The mapping from time-dependent cells to reference cells is very crucial in the proof.The numerical results also confirm the theoreti-cal proof.

Blow-up of Solution for a Nonlinear Hyperbolic Equation

This paper gives the suffcient conditions of blow-up of the solution of a nonlinear hyperbolic equation with the initial boundary value conditions in finite time and proves the existence and uniqueness of the local solution of the problem.

IMPROVED ERROR ESTIMATES FOR MIXED FINITE ELEMENT FOR NONLINEAR HYPERBOLIC EQUATIONS: THE CONTINUOUS-TIME CASE

Presents a study which computed the L sup2-error estimates for mixed finite element methods for second order nonlinear hyperbolic equations. Application of computed results to a continuous-time case; Demonstration of the convergence of the values for both the scalar function and the flux; Technique used in the computation.

Nonexistence of the Solution for a Class of Nonlinear Hyperbolic Equation

In this paper,we prove the existence and uniqueness of the local generalized solution of the Cauchy problem for a class of nonlinear hyperbolic equation of higher order are proved.Moreover,we give the sufficient conditions for blow-up of the solution of the problem in finite time will be given.

EXISTENCE AND NONEXISTENCE OF GLOBAL SOLUTIONS FOR NONLINEAR EVOLUTION EQUATION OF FOURTH ORDER

The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation u tt -a 1u xx -a 2u xxt -a 3u xxtt =φ(u x ) x are proved.

FINITE ELEMENT METHOD FOR SOME NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

This paper studies the finite element method for some nonlinear hyperbolic partial differential equations with memory and dampling terms.A Crank\|Nicolson approximation for this kind of equations is presented.By using the elliptic Ritz\|Volterra projection,the analysis of the error estimates for the finite element numerical solutions and the optimal H \+1\|norm error estimate are demonstrated.

On the Propagation of the Analytic Regularity in Strictly Hyperbolic Equations Which Are Lipschitz Continuous With Respect to Time

We consider strictly hyperbolic nonlinear equations which are Lipschitz continuous in the time variable and study the local analytic regularity of the solutions with respect to the space variables.

OSCILLATION OF NEUTRAL NONLINEAR HYPERBOLIC EQUATIONS WITH DOUBLED VARIABLE COEFFICIENTS

In this paper the sufficient conditions are obtained for oscillation of neutral nonlinearhyperbolic equations with doubled variable coefficients. These results are illustrated bysome examples.

A CLASS OF NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR NONLINEAR HYPERBOLIC DIFFERENTIAL EQUATION

The nonlocal singularly perturbed problems for the hyperbolic differential equation are considered. Under suitable conditions, using the fixed point theorem, the asymptotic behavior of solution for the initial boundary value problems is studied.

Convergence Rates to Asymptotic Profile for Solutions of Quasilinear Hyperbolic Equations with Nonlinear Damping

The quasilinear hyperbolic equation with nonlinear damping is considered in this paper,a new asymptotic profile for the solution to the equation is obtained by suitably choosing the initial data of the corresponding parabolic equation,the convergence rates of the new profile are better than that obtained by Nishihara（1997,J.Differential Equations 137,384-395） and H.-J.Zhao（2000,J.Differential Equations 167,467-494）.

HIGH ACCURACY ARITHMETIC AVERAGE TYPE DISCRETIZATION FOR THE SOLUTION OF TWO-SPACE DIMENSIONAL NONLINEAR WAVE EQUATIONS

In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form utt=A(x,y,t)uxx+B(x,y,t)uyy+g(x,y,t,u,ux,uy,ut),0<x,y<1,t>0 subject to appropriate initial and Dirichlet boundary conditions.We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation.The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable.Numerical results are provided to illustrate the usefulness of the proposed method.

BLOW-UP OF THE SOLUTION FOR A KIND OF HIGHER ORDER HYPERBOLIC EVOLUTION SYSTEMS

In this paper, we give some results on the blow-up behaviors of the solution to the mixed problem for some higher nonlinear hyperbolic evolution equation in finite time. By introducing the 'blow-up factor K(u,ut)' we get some new results, which generalize the conclusions of [3] and [4].