Generalized Tricomi Problem for a Quasilinear Mixed Type Equation

In this paper, the Tricomi problem and the generalized Tricomi problem for a quasilinear mixed type equation are studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to these problems is proved. The method developed in this paper can be used to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.     

Global Classical Solutions of the Cauchy Problems for Nonlinear Vorticity Equations and Its Applications

In this paper, the definition of a vortieity equation on a compact Riemannian manifold without boundary is given. Some accurate a priori estimates are obtained with the use of the peculiarity of the equation. Then, the existence and “uniqueness” of global classical solution of the Cauchy problem for the equation is proved. As its application, the existence and “uniqueness” of Cauchy problem for barotropic nondivergent model is obtained. The model is a fundamental one in atmospheric dynamics.     

A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation

In this paper, we consider a Cauchy problem of the time fractional diffusion equation (TFDE) in x ∈ [0, L]. This problem is ubiquitous in science and engineering applications. The illposedness of the Cauchy problem is explained by its solution in frequency domain. Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method. The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hpnorm priori bound assumptions. Finally, numerical examples illustrate the effectiveness of the proposed method.     

A NONLINEAR LAVRENTIEV-BITSADZE MIXED TYPE EQUATION

In this paper the Tricomi problem for a nonlinear mixed type equation is studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to this problem is proved. The method developed in this paper can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.     

On the stability of contact discontinuity for Cauchy problem of compress Navier-Stokes equations with general initial data

In this paper,we study the stability of solutions of the Cauchy problem for 1-D compressible NarvierStokes equations with general initial data.The asymptotic limit of solution is found,under some conditions.The results in this paper imply the case that the limit function of solution as t →∞ is a viscous contact wave in the sense,which approximates the contact discontinuity on any finite-time interval as the heat conduction coefficients toward zero.As a by-product,the decay rates of the solution for the fast diffusion equations are also obtained.The proofs are based on the elementary energy method and the study of asymptotic behavior of the solution to the fast diffusion equation.     

ON THE EQUATION□Φ=|▽Φ|~2 IN FOUR SPACE DIMENSIONS

This paper considers the following Cauchy problem for semilinear wave equations in n space dimensionswhere A is the wave operator, F is quadratic in (?) with (?) = ( ).The minimal value of s is determined such that the above Cauchy problem is locally well-posed in H8. It turns out that for the general equation s must satisfyThis is due to Ponce and Sideris (when n = 3) and Tataru (when n≥5). The purpose of this paper is to supplement with a proof in the case n = 2,4.     

A TWO-SCALE HIGHER-ORDER FINITE ELEMENT DISCRETIZATION FOR SCHROEDINGER EQUATION

In this paper, a two-scale higher-order finite element discretization scheme is proposed and analyzed for a Schroedinger equation on tensor product domains. With the scheme, the solution of the eigenvalue problem on a fine grid can be reduced to an eigenvalue problem on a much coarser grid together with some eigenvalue problems on partially fine grids. It is shown theoretically and numerically that the proposed two-scale higher-order scheme not only significantly reduces the number of degrees of freedom but also produces very accurate approximations.     

The Cauchy problem for non-autonomous nonlinear Schrdinger equations

In this paper we study the Cauchy problem for cubic nonlinear Schrodinger equation with space- and time-dependent coefficients on Rm and Tm. By an approximation argument we prove that for suitable initial values, the Cauchy problem admits unique local solutions. Global existence is discussed in the cases of m = 1,2.     

Local and Parallel Finite Element Algorithms for Eigenvalue Problems

Abstract Some new local and parallel finite element algorithms are proposed and analyzed in this paper foreigenvalue problems.With these algorithms, the solution of an eigenvalue problem on a fine grid is reduced tothe solution of an eigenvalue problem on a relatively coarse grid together with solutions of some linear algebraicsystems on fine grid by using some local and parallel procedure.A theoretical tool for analyzing these algorithmsis some local error estimate that is also obtained in this paper for finite element approximations of eigenvectorson general shape-regular grids.