高阶拟线性双曲型方程的精确边界能控性

By means of the existence and uniqueness of semi-global C^1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues ,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.     

Consistent Numerical Schemes for Solving Nonlinear Inverse Source Problems with Gradient-Type Algorithms and Newton–Kantorovich Methods

In this paper, algorithms of solving an inverse source problem for systems of production–destruction equations are considered. Numerical schemes that are consistent to satisfy Lagrange’s identity for solving direct and adjoint problems are constructed. With the help of adjoint equations, a sensitivity operator with a discrete analog is constructed. It links perturbations of the measured values with those of the sought-for model parameters. This operator transforms the inverse problem to a quasilinear system of equations and allows applying Newton–Kantorovich methods to it. A numerical comparison of gradient algorithms based on consistent and inconsistent numerical schemes and a Newton–Kantorovich algorithm applied to solving an inverse source problem for a nonlinear Lorenz model is done.     

Lipschitz Continuous Solutions to the Cauchy Problem for Quasi-linear Hyperbolic Systems

Lipschitz continuous solutions to the Cauchy problem for 1-D first order quasilinear hyperbolic systems are considered. Based on the methods of approximation and integral equations,the author gives two...     

The Nonlinear Initial-Boundary Value Problem and the Existence of Multi-Dimension al Shook Wave for Quasilinear Hyperbolic-Parabolic Coupled Systems

For the quasilinear hyperbolie-parabolio coupled system, the nonlinear initial- boundary value problem and the shook wave free boundary problem are considered. By linear iteration, the existence and uniqueness of the local H^m (m\geq [N+1/2]+4) solution are obtained under the assumption that for the fixed boundary problem, the boundary conditions are uniformly Lopatinski well-posed with respect to the hyperbolic and parabolic part, and for the free boundary problem, there exists a linear stable shock front structure. In particular, the local existence of the isothermal shock wave solution for radiative hydrodynamic eqations is proved.     

Initial Boundary Value Problems for Quasilinear Hyperbolio- Parabolio Ooupled Systems in Higher Dimensional Spaces

The initial bounary value problem for quasilinear byperbolie-parabolic coupled systems in higher dimensional spaces, which arises in many mechanical problerns is considered. Under the assumptions that the-hyperbolic part of the coupled system is a quasilinear symmetric hyperbolic system and the parabolic part is a quasilinear parabolic system of second order and suitable assumptions of smoothness and compatibiliy conditions, the existence and uniqueness of local smooth solution is proved in the cases that the boundary of domain is noncharacteristic or uniformly characteristic with respect to the hyperbolic part. As an application, the existence and uniqueness of local smooth solution for the initial boundary problem of the radiation hydrodynamic system, as well as of the viscous compressible hydrodynamic system, with solid wal1 boundary, is obtained.     

Numerical resolution for a compressible two-phase flow model based on the theory of thermodynamically compatible systems

This paper is concerned with the numerical solution of the equations governing two-phase gas-solid mixture in the framework of thermodynamically compatible systems theory. The equations constitute a non-homogeneous system of nonlinear hyperbolic conservation laws. A total variation diminishing (TVD) slope limiter centre (SLIC) numerical scheme, based on the splitting approach, is presented and applied for the solution of the initial-boundary value problem for the equations. The model equations and the numerical methods are systematically assessed through a series of numerical test cases. Strong numerical evidence shows that the model and the methods are accurate, robust and conservative. The model correctly describes the formations of shocks and rarefactions in two-phase gas-solid flow.     

Exact Boundary Controllability and Exact Boundary Observability for a Coupled System of Quasilinear Wave Equations

Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary(null) controllability and the local boundary(weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.     

On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions

Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In this paper, the problem of solving the one-dimensional wave equation subject to given initial and non-local boundary conditions is considered. These non-local conditions arise mainly when the data on the boundary cannot be measured directly. Several finite difference methods with low order have been proposed in other papers for the numerical solution of this one dimensional non-classic boundary value problem. Here, we derive a new family of efficient three-level algorithms with higher order to solve the wave equation and also use a Simpson formula with higher order to approximate the integral conditions. Additionally, the fourth-order formula is also adapted to nonlinear equations, in particular to the well-known nonlinear Klein–Gordon equations which many physical problems can be modeled with. Numerical results are presented and are compared with some existing methods showing the efficiency of the new algorithms.     

Exact Controllability for Nonautonomous First Order Quasilinear Hyperbolic Systems

By means of the theory on the semi-global C1 solution to the mixed initial-boundary value problem (IBVP) for first order quasilinear hyperbolic systems, we establish the exact controllability for general nonautonomous first order quasilinear hyperbolic systems with general nonlinear boundary conditions.     

On a non-linear wave equation and the control of an elastic string from one equilibrium location to another

This paper concerns a non-linear system of wave equations describing the motion in space of an elastic string. We derive the equations, determine the equilibrium solutions and, using the methods of quasilinear hyperbolic systems, prove that the natural initial, boundary value problem has classical solutions existing in neighbourhoods of the “stretched” equilibrium solutions. We then prove that the positions of the endpoints of the string can be controlled in such a way that the string moves from an equilibrium in one location to an equilibrium in another location.     

Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow. __________ Translated from Chinese Annals of Mathematics, Ser. A, 1982, 3(2): 223–232     

Exact boundary observability for nonautonomous first-order quasilinear hyperbolic systems

By means of the theory on the semiglobal C¹ solution to the mixed initial-boundary value problem for first-order quasilinear hyperbolic systems, we establish the local exact boundary observability for general nonautonomous first-order quasilinear hyperbolic systems without zero eigenvalues and reveal the essential difference between nonautonomous hyperbolic systems and autonomous hyperbolic systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Interaction of weak discontinuities and the hodograph method as applied to electric field fractionation of a two-component mixture

The hodograph method is used to construct a solution describing the interaction of weak discontinuities (rarefaction waves) for the problem of mass transfer by an electric field (zonal electrophoresis). Mathematically, the problem is reduced to the study of a system of two first-order quasilinear hyperbolic partial differential equations with data on characteristics (Goursat problem). The solution is constructed analytically in the form of implicit relations. An efficient numerical algorithm is described that reduces the system of quasilinear partial differential equations to ordinary differential equations. For the zonal electrophoresis equations, the Riemann problem with initial discontinuities specified at two different spatial points is completely solved.     

GLOBAL EXISTENCE OF WEAKLY DISCONTINUOUS SOLUTIONS TO A KIND OF MIXED INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.     

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.     

Generalized solutions to partial differential equations of evolution type

This paper deals with a new solution concept for partial differential equations in algebras of generalized functions. Introducing regularized derivatives for generalized functions, we show that the Cauchy problem is wellposed backward and forward in time for every system of linear partial differential equations of evolution type in this sense. We obtain existence and uniqueness of generalized solutions in situations where there is no distributional solution or where even smooth solutions are nonunique. In the case of symmetric hyperbolic systems, the generalized solution has the classical weak solution as macroscopic aspect. Two extensions to nonlinear systems are given: global solutions to quasilinear evolution equations with bounded nonlinearities and local solutions to quasilinear symmetric hyperbolic systems.     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws II

This work is a continuation of our previous work. In the present paper, we study the existence and uniqueness of global piecewise C¹ solutions with shock waves to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping in the presence of a boundary. It is shown that the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping with nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0} admits a unique global piecewise C¹ solution u = u (t, x) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of a self‐similar solution u = U (x /t) of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shock waves but no rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global Exisienoe of Classical Solutions to the Typical Free Boundary Problem for General Quasilinear Hyperbolic Systems and Its Applications

In this paper the authors prove the existence and uniqueness of global classical solutions to the typical free boundary problem for general quasilinear hyperbolic systems. As an application, a unique global discontinuous solution only containing n shocks on t \leq 0 is obtained for a class of generalized Riemann problem for the quasilinear hyperbolic system of n conservation laws.     

在可交换四元数空间中的双曲方程的特征边值问题

讨论了一阶和二阶双曲复方程的特征边值问题.对其中两种线性方程,分别在不同的情况下获得通解和可解条件.而对于拟线性的二阶双曲复方程,证明了解的存在性和唯一性.