On the generalized exact boundary synchronization for a coupled system of wave equations

In this paper, we deliver a normalized synchronization transformation to study the generalized exact boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. The clear relationship among the generalized exact boundary synchronization, the exact boundary null controllability, and the generalized exactly synchronizable states is precisely obtained. This approach gives further a forthright decomposition for the generalized exact boundary synchronization problem, whereby, we gain directly the determination of generalized exactly synchronizable states.     

Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method

In this paper, we introduce a spectral collocation method based on Lagrange polynomials for spatial derivatives to obtain numerical solutions for some coupled nonlinear evolution equations. The problem is reduced to a system of ordinary differential equations that are solved by the fourth order Runge–Kutta method. Numerical results of coupled Korteweg–de Vries (KdV) equations, coupled modified KdV equations, coupled KdV system and Boussinesq system are obtained. The present results are in good agreement with the exact solutions. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations.     

Finite termination of a Newton-type algorithm for a class of affine variational inequality problems

Many optimization problems can be reformulated as a system of equations. One may use the generalized Newton method or the smoothing Newton method to solve the reformulated equations so that a solution of the original problem can be found. Such methods have been powerful tools to solve many optimization problems in the literature. In this paper, we propose a Newton-type algorithm for solving a class of monotone affine variational inequality problems (AVIPs for short). In the proposed algorithm, the techniques based on both the generalized Newton method and the smoothing Newton method are used. In particular, we show that the algorithm can find an exact solution of the AVIP in a finite number of iterations under an assumption that the solution set of the AVIP is nonempty. Preliminary numerical results are reported.     

Symmetry groups and similarity solutions for the system of equations for a viscous compressible fluid

Here, using Lie group transformations, we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of a compressible fluid in the presence of viscosity and thermal conduction, using the general form of the equation of state. The symmetry groups admitted by the governing system of PDEs are obtained, and the complete Lie algebra of infinitesimal symmetries is established. Indeed, with the use of the entailed similarity solution the problem is transformed to a system of ordinary differential equations(ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.     

The exponential function rational expansion method and exact solutions to nonlinear lattice equations system

We propose an exponential function rational expansion method for solving exact traveling wave solutions to nonlinear differential-difference equations system. By this method, we obtain some exact traveling wave solutions to the relativistic Toda lattice equations system and discuss the significance of these solutions. Finally, we give an open problem.     

Sur l'état de synchronisation exacte d'un système couplé d'équations des ondes

In this Note, we consider the determination of the state of exact synchronization for a coupled system of wave equations. In a special case, the state of exact synchronization can be uniquely determined whatever the boundary controls would be chosen. In the general case, the state of exact synchronization depends on the boundary controls that realize the exact synchronization. However, we can estimate the difference between the state of exact synchronization and the solution to a problem independent of boundary controls. The determination of the state of exact synchronization by groups is also discussed.     

Qualitative Properties of a Certain Kinetic Model of a Binary Gas

We consider a system of kinetic equations with one-dimensional velocity space. The system is a simple mathematical model that describes the evolution of a two-component gas mixture at the molecular level. We study some qualitative properties of its solutions, in particular, the conservation laws and spectrum of the linearized problem. In the spatially homogeneous case we present the widest Lie algebra of admissible operators and construct some exact solutions in closed form. We indicate some methods for constructing numerical schemes conservative with respect to fulfillment of the discrete conservation laws of energy and the concentrations of the components.     

A method of adjoint operators for solving boundary value problems for second-order ordinary differential equations

In this paper, for a linear boundary value problem we propose a method that reduces the differential problem to a discrete (difference) problem. The difference equations, which are an exact analog of the differential equation, are constructed by an adjoint operator method. The adjoint equations are solved by a factorization method.     

New Approach to Linear Exact Model Matching for a Class of Nonlinear Systems

In this paper, a new approach to the linear exact model matching problem for a class of nonlinear systems, using static state feedback, is presented. This approach reduces the problem of determining the state feedback control law to that of solving a system of first-order partial differential equations. Based on these equations, two major issues are resolved: the necessary and sufficient conditions for the problem to have a solution and the general analytical expression for the feedback control law. Furthermore, the proposed approach is extended to solve the same problem via static output feedback.     

Uzawa methods for a class of block three‐by‐three saddle‐point problems

In this work, we consider numerical methods for solving a class of block three‐by‐three saddle‐point problems, which arise from finite element methods for solving time‐dependent Maxwell equations and some other applications. The direct extension of the Uzawa method for solving this block three‐by‐three saddle‐point problem requires the exact solution of a symmetric indefinite system of linear equations at each step. To avoid heavy computations at each step, we propose an inexact Uzawa method, which solves the symmetric indefinite linear system in some inexact way. Under suitable assumptions, we show that the inexact Uzawa method converges to the unique solution of the saddle‐point problem within the approximation level. Two special algorithms are customized for the inexact Uzawa method combining the splitting iteration method and a preconditioning technique, respectively. Numerical experiments are presented, which demonstrated the usefulness of the inexact Uzawa method and the two customized algorithms.     

Factorization method for linear and quasilinear singularly perturbed boundary value problems for ordinary differential equations

For linear singularly perturbed boundary value problems, we come up with a method that reduces solving a differential problem to a discrete (difference) problem. Difference equations, which are an exact analog of differential equations, are constructed by the factorization method. Coefficients of difference equations are calculated by solving Cauchy problems for first-order differential equations. In this case nonlinear Ricatti equations with a small parameter are solved by asymptotic methods, and solving linear equations reduces to computing quadratures. A solution for quasilinear singularly perturbed equations is obtained by means of an implicit relaxation method. A solution to a linearized problem is calculated by analogy with a linear problem at each iterative step. The method is tested against solutions to the known Lagerstrom-Cole problem.     

Approximate Toeplitz Matrix Problem Using Semidefinite Programming

Given a data matrix, we find its nearest symmetric positive-semidefinite Toeplitz matrix. In this paper, we formulate the problem as an optimization problem with a quadratic objective function and semidefinite constraints. In particular, instead of solving the so-called normal equations, our algorithm eliminates the linear feasibility equations from the start to maintain exact primal and dual feasibility during the course of the algorithm. Subsequently, the search direction is found using an inexact Gauss-Newton method rather than a Newton method on a symmetrized system and is computed using a diagonal preconditioned conjugate-gradient-type method. Computational results illustrate the robustness of the algorithm.     

Fundamental mode exact schemes for unsteady problems

The problem of increasing the accuracy of an approximate solution is considered for boundary value problems for parabolic equations. For ordinary differential equations (ODEs), nonstandard finite difference schemes are in common use for this problem. They are based on a modification of standard discretizations of time derivatives and, in some cases, allow to obtain the exact solution of problems. For multidimensional problems, we can consider the problem of increasing the accuracy only for the most important components of the approximate solution. In the present work, new unconditionally stable schemes for parabolic problems are constructed, which are exact for the fundamental mode. Such two‐level schemes are designed via a modification of standard schemes with weights using Padé approximations. Numerical results obtained for a model problem demonstrate advantages of the proposed fundamental mode exact schemes.     

Exact Solutions of Extended Boussinesq Equations

The problem addressed in this paper is the verification of numerical solutions of nonlinear dispersive wave equations such as Boussinesq-like system of equations. A practical verification tool for numerical results is to compare the numerical solution to an exact solution if available. In this work, we derive some exact solitary wave solutions and several invariants of motion for a wide range of Boussinesq-like equations using Maple software. The exact solitary wave solutions can be used to specify initial data for the incident waves in the Boussinesq numerical model and for the verification of the associated computed solution. The invariants of motions can be used as verification tools for the conservation properties of the numerical model.     

Numerical resolution of an exact heat conduction model with a delay term

In this paper we analyze, from the numerical point of view, a dynamic thermoelastic problem. Here, the so-called exact heat conduction model with a delay term is used to obtain the heat evolution. Thus, the thermomechanical problem is written as a coupled system of partial differential equations, and its variational formulation leads to a system written in terms of the velocity and the temperature fields. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A priori error estimates are proved, from which the linear convergence of the algorithm could be derived under suitable additional regularity conditions. Finally, a two-dimensional numerical example is solved to show the accuracy of the approximation and the decay of the discrete energy.     

Quasistationary Approximation in the Rotating Ring Problem

We consider the planar rotation-symmetric motion by inertia of a viscous incompressible fluid in a ring with free boundary. We reduce the corresponding initial-boundary value problem for the Navier–Stokes equations to some problem for a coupled system of one parabolic equation and two ordinary differential equations. We suppose that the coefficient of the derivatives of the sought functions with respect to time (the quasistationary parameter) is small; so the system is singularly perturbed. In this article we construct an asymptotic expansion for a solution to the rotating ring problem in a small quasistationary parameter and obtain a smallness estimate for the difference between the exact and approximate solutions.     

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras II: A nonlinear system arising from the group G 6.1 and its exact integration

In the first paper of this series a correspondence was established between coupled systems of two-dimensional nonlinear wave equations and the six-dimensional simply transitive Lie algebras. In the present paper we make use of this result to construct a Darboux integrable and exactly integrable nonlinear system associated with the six-parameter nilpotent Lie group G _6,1 and we give its exact general solution in terms of four arbitrary functions. The procedure is shown to be an exact linearization of the nonlinear problem.     

A System of Plane Elasticity Canonical Integral Equations and Its Application

In this paper, we obtain a new system of canonical integral equations for the plane elasticity problem over an exterior circular domain, and give its numerical solution. Coupling with the classical finite element method, it can be used for solving general plane elasticity exterior boundary value problems. This system of highly singular equations is also an exact boundary condition on the artificial boundary. It can be approximated by a series of nonsingular integral boundary conditions.     

Controllability for some integrodifferential equations driven by vector measures

In this work, we consider the question of controllability of a class of integrodifferential equations on Hilbert space with measures as controls. We assume that the linear part has a resolvent operator in the sense given by R. Grimmer. We generalize the original work of N. Ahmed on vector measures, and we use it to develop necessary and sufficient conditions for weak and the exact controllability of the integrodifferential equation. Using the latter, we prove that exact controllability of the integrodifferential equation implies exact controllability of a perturbed integrodifferential equation. Controllability problem for the perturbed system is formulated fixed point problem in the space of vector measures. Our results cover impulsive controls as well as regular controls. Copyright © 2016 John Wiley & Sons, Ltd.     

A Chebyshev spectral collocation method for solving Burgers’-type equations

In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary differential equations that are solved by Runge–Kutta method of order four. Numerical results for the nonlinear evolution equations such as 1D Burgers’, KdV–Burgers’, coupled Burgers’, 2D Burgers’ and system of 2D Burgers’ equations are obtained. The numerical results are found to be in good agreement with the exact solutions. Numerical computations for a wide range of values of Reynolds’ number, show that the present method offers better accuracy in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.