Exchange rate uncertainty and trade growth—a comparison of linear and non‐linear (forecasting) models

A huge body of empirical and theoretical literature has emerged on the relationship between foreign exchange (FX) uncertainty and international trade. Empirical findings about the impact of FX uncertainty on trade figures are at best weak and often ambiguous with respect to its direction. Almost all empirical contributions assume and estimate a linear relationship. Possible nonlinearity or state dependence of causal links between FX uncertainty and trade has been mostly ignored yet. In addition, widely used regression models have not been evaluated in terms of ex‐ante forecasting. In this paper we analyse the impact of FX uncertainty on sectoral categories of multilateral exports and imports for 15 industrialized economies. We particularly provide a comparison of linear and non‐linear models with respect to ex‐ante forecasting. In terms of average ranks of absolute forecast errors non‐linear models outperform both, a common linear model and some specification building on the assumption that FX uncertainty and trade growth are uncorrelated. Our results support the view that the relationship of interest might be non‐linear and, moreover, lacks of homogeneity across countries, economic sectors and when contrasting imports vs exports. Copyright © 2005 John Wiley & Sons, Ltd.     

Global non‐existence of solutions of a class of wave equations with non‐linear damping and source terms

In this paper we consider the non‐linear wave equation a,b>0, associated with initial and Dirichlet boundary conditions. We prove, under suitable conditions on α,β,m,p and for negative initial energy, a global non‐existence theorem. This improves a result by Yang (Math. Meth. Appl. Sci. 2002; 25 :825–833), who requires that the initial energy be sufficiently negative and relates the global non‐existence of solutions to the size of Ω. Copyright © 2004 John Wiley & Sons, Ltd.     

A non‐linear and non‐local boundary condition for a diffusion equation in petroleum engineering

This article deals with a boundary value problem for Laplace equation with a non‐linear and non‐local boundary condition. This problem comes from petroleum engineering and is used to obtain an estimation of well productivity. The non‐linear and non‐local boundary condition is written on the well boundary. On the outer reservoir boundaries, we have both Dirichlet and Neumann conditions. In this paper, we prove the existence and uniqueness of a solution to this problem. The existence is proved by Schauder theorem and the uniqueness is obtained under more restricted conditions, when the involved operator is a contraction. Copyright © 2005 John Wiley & Sons, Ltd.     

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

In this paper we will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove blow‐up results or global existence (in time) of small data energy solutions.     

Global existence to a three‐dimensional non‐linear thermoelasticity system arising in shape memory materials

This paper is concerned with the unique global solvability of a three‐dimensional (3‐D) non‐linear thermoelasticity system arising from the study of shape memory materials. The system consists of the coupled evolutionary problems of viscoelasticity with non‐convex elastic energy and non‐linear heat conduction with mechanical dissipation. The present paper extends the previous 2‐D existence result of the authors Reference [1] to 3‐D case. This goal is achieved by means of the Leray–Schauder fixed point theorem using technique based on energy arguments and DeGiorgi method. Copyright © 2004 John Wiley & Sons, Ltd.     

Pointwise stabilization of strings with non‐linear feedback

We consider an initial and boundary value problem for a homogenous string subject to an internal pointwise control. The solution resulting from a non‐linear feedback is studied. We give various explicit decay estimates depending on the control position and the feedback non‐linearity. Copyright © 2004 John Wiley & Sons, Ltd.     

Boundary position feedback control of Kirchhoff's non‐linear strings

This paper is concerned with global stabilization of an undamped non‐linear string in the case where any velocity feedback is not available. The linearized system has an infinite number of poles and zeros on the imaginary axis. In the case where any velocity feedback is not available, a parallel compensator is effective. The stabilizer is constructed for the augmented system which consists of the controlled system and a parallel compensator. It is proved that the string can be stabilized by linear boundary control. Copyright © 2003 John Wiley & Sons, Ltd.     

Control problem for a non‐linear thermoelasticity system

The control problem for a three‐dimensional non‐linear thermoelasticity system is considered. The system may represent, among others, the dynamical model of shape memory materials. As controls we take distributed heat sources and body forces. The goal functional refers to the desired evolution of displacement, strain and temperature. The continuity and differentiability of solutions with respect to controls is studied. The existence of optimal controls is proved and the necessary optimality conditions are formulated. The existence of adjoint state variables is proved under additional regularity of data. Copyright © 2004 John Wiley & Sons, Ltd.     

Global and blow‐up solutions for non‐linear degenerate parabolic systems

In this paper the degenerate parabolic system u_t=u(u_xx+av). vt=v(v_xx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (α₁=α₂) of solution are analysed. For , the blow‐up time, blow‐up rate and blow‐up set of blow‐up solution are estimated and the asymptotic behaviour of solution near the blow‐up time is discussed by using the ‘energy’ method. Copyright © 2003 John Wiley & Sons, Ltd.     

Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems

To further study the Hermitian and non‐Hermitian splitting methods for a non‐Hermitian and positive‐definite matrix, we introduce a so‐called lopsided Hermitian and skew‐Hermitian splitting and then establish a class of lopsided Hermitian/skew‐Hermitian (LHSS) methods to solve the non‐Hermitian and positive‐definite systems of linear equations. These methods include a two‐step LHSS iteration and its inexact version, the inexact Hermitian/skew‐Hermitian (ILHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically prove that the LHSS method converges to the unique solution of the linear system for a loose restriction on the parameter α. Moreover, the contraction factor of the LHSS iteration is derived. The presented numerical examples illustrate the effectiveness of both LHSS and ILHSS iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

Global existence of solutions for non‐small data to non‐linear spherically symmetric thermoviscoelasticity

We consider some initial–boundary value problems for non‐linear equations of thermoviscoelasticity in the three‐dimensional case. Since, we are interested to prove global existence we consider spherically symmetric problem. We examine the Neumann conditions for the temperature and either the Neumann or the Dirichlet boundary conditions for the elasticity equations. Using the energy method, we are able to obtain some energy estimates in appropriate Sobolev spaces enough to prove existence for all time without any restrictions on data. Due to the spherical symmetricity the constants in the above estimates increase with time so the existence for all finite times is proved only. Copyright © 2003 John Wiley & Sons, Ltd.     

Boundary stabilization for a non‐linear beam on elastic bearings

In this paper, we study the equation under non‐linear boundary conditions which model the vibrations of a beam clamped at x=0 and supported by a non‐linear bearing at x=L. By adding only one damping mechanism at x=L, we prove the existence of a global solution and exponential decay of the energy. Copyright © 2001 John Wiley & Sons, Ltd.     

Adaptive stabilization of Kirchhoff's non‐linear strings by boundary displacement feedback

This paper is concerned with adaptive global stabilization of an undamped non‐linear string in the case where any velocity feedback is not available. The linearized system may have an infinite number of poles and zeros on the imaginary axis. In the case where any velocity feedback is not available, a parallel compensator is effective. The adaptive stabilizer is constructed for the augmented system which consists of the controlled system and a parallel compensator. It is proved that the string can be stabilized by adaptive boundary control. Copyright © 2007 John Wiley & Sons, Ltd.     

Exponential stability in one‐dimensional non‐linear thermoelasticity with second sound

In this paper, we consider a one‐dimensional non‐linear system of thermoelasticity with second sound. We establish an exponential decay result for solutions with small ‘enough’ initial data. This work extends the result of Racke (Math. Methods Appl. Sci. 2002; 25 :409–441) to a more general situation. Copyright © 2004 John Wiley & Sons, Ltd.     

The tanh method: a tool for solving certain classes of non‐linear PDEs

The tanh (or hyperbolic tangent) method is a powerful technique to look for travelling waves when dealing with one‐dimensional non‐linear wave and evolution equations. In particular, this method is well suited for those problems where dispersion, convection and reaction–diffusion play an important role. To show the strength of this method we study a coupled set (the so‐called Boussinesq equations) which arises in the theory of non‐linear dispersive water waves. As a result, a solitary wave profile is found which generalizes an earlier result, the famous Korteweg‐de Vries solitary wave solution. Copyright © 2005 John Wiley & Sons, Ltd.     

Unconditional non‐linear stability for a fluid of third grade

The thermal convection in a layer of a third grade fluid is investigated, with viscosity being a general function of temperature. We develop a non‐linear stability analysis and prove that unconditional non‐linear stability criterion is achieved using a natural energy approach. This shows that, in some sense, the equations for a fluid of third grade are preferable to those for a fluid of second grade or a dipolar fluid. Copyright © 2004 John Wiley & Sons, Ltd.     

Local C1 solutions to some non‐linear PDE system

We prove that the quasilinear initial value problem (1) has a unique, local in time, C¹ solution, if the matrices A_i are diagonalizable and commute with each other. Copyright © 2006 John Wiley & Sons, Ltd.     

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation,II

This paper is a continuation of our recent paper 8 . We will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove global existence (in time) of small data energy solutions assuming suitable regularity on the L² scale with additional L¹ regularity for the data. In order to deal with this L² regularity in the non‐linear part, we will develop and employ some tools from Harmonic Analysis.