Derivatives of the energy functional for 2D‐problems with a crack under Signorini and friction conditions

We consider the two‐dimensional elasticity problem for an elastic body with a crack under unilateral constraints imposed at the crack. We assume that both the Signorini condition for non‐penetration of the crack faces and the condition of given friction between them are fulfilled. The problem is non‐linear and can be described by a variational inequality. Varying the shape of the crack by a local coordinate transformation of the domain, the first derivative of the energy functional to the problem with respect to the crack length is obtained, which gives the criterion for the crack growing. The regularity of the solution is discussed and the singular solution is performed. Copyright © 2000 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.     

Local and non‐local boundary quenching

The quenching problem is examined for a one‐dimensional heat equation with a non‐linear boundary condition that is of either local or non‐local type. Sufficient conditions are derived that establish both quenching and non‐quenching behaviour. The growth rate of the solution near quenching is also given for a power‐law non‐linearity. The analysis is conducted in the context of a nonlinear Volterra integral equation that is equivalent to the initial–boundary value problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Blowup of solutions for a class of non‐linear evolution equations with non‐linear damping and source terms

We consider the blowup of solutions of the initial boundary value problem for a class of non‐linear evolution equations with non‐linear damping and source terms. By using the energy compensation method, we prove that when p>max{m, α}, where m, α and p are non‐negative real numbers and m+1, α+1, p+1 are, respectively, the growth orders of the non‐linear strain terms, damping term and source term, under the appropriate conditions, any weak solution of the above‐mentioned problem blows up in finite time. Comparison of the results with the previous ones shows that there exist some clear condition boundaries similar to thresholds among the growth orders of the non‐linear terms, the states of the initial energy and the existence and non‐existence of global weak solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

A linearization‐based approach of homotopy analysis method for non‐linear time‐fractional parabolic PDEs

In this paper, a novel approach, namely, the linearization‐based approach of homotopy analysis method, to analytically treat non‐linear time‐fractional PDEs is proposed. The presented approach suggests a new optimized structure of the homotopy series solution based on a linear approximation of the non‐linear problem. A comparative study between the proposed approach and standard homotopy analysis approach is illustrated by solving two examples involving non‐linear time‐fractional parabolic PDEs. The performed numerical simulations demonstrate that the linearization‐based approach reduces the computational complexity and improves the performance of the homotopy analysis method.     

Asymptotic behaviour of a non‐classical heat conduction problem for a semi‐infinite material

The asymptotic behaviour of a heat conduction problem involving a non‐linear heat source depending on the heat‐flux occurring in the extremum of a semi‐infinite slab is discussed. Conditions are given on the non‐linearity so as to accelerate the convergence of the solution to zero. Copyright © 2000 John Wiley & Sons, Ltd.     

Initial boundary value problem for a class of non‐linear strongly damped wave equations

The paper studies the existence, asymptotic behaviour and stability of global solutions to the initial boundary value problem for a class of strongly damped non‐linear wave equations. By a H00.5ptk‐Galerkin approximation scheme, it proves that the above‐mentioned problem admits a unique classical solution depending continuously on initial data and decaying to zero as t→+∞as long as the non‐linear terms are sufficiently smooth; they, as well as their derivatives or partial derivatives, are of polynomial growth order and the initial energy is properly small. Copyright © 2003 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.     

Mesh‐independent convergence of the modified inexact Newton method for a second order non‐linear problem

In this paper, we consider an inexact Newton method applied to a second order non‐linear problem with higher order non‐linearities. We provide conditions under which the method has a mesh‐independent rate of convergence. To do this, we are required, first, to set up the problem on a scale of Hilbert spaces and second, to devise a special iterative technique which converges in a higher than first order Sobolev norm. We show that the linear (Jacobian) system solved in Newton's method can be replaced with one iterative step provided that the initial non‐linear iterate is accurate enough. The closeness criteria can be taken independent of the mesh size. Finally, the results of numerical experiments are given to support the theory. Published in 2005 by John Wiley & Sons, Ltd.     

Stabilizing and destabilizing effect of small velocity‐dependent forceson non‐conservative systems: new results

A theory of the destabilization paradox in general non‐conservative systems with small dissipative and gyroscopic forces is presented. The problem is investigated by the approach based on the sensitivity analysis of multiple eigenvalues. An explicit asymptotic expression for the critical load as a function of the dissipation and gyroscopic parameters allowing to calculate a jump in the critical load is derived. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Stability of non‐constant steady‐state solutions for non‐isentropic Euler–Maxwell system with a temperature damping term

This work is concerned with the periodic problem for compressible non‐isentropic Euler–Maxwell systems with a temperature damping term arising in plasmas. For this problem, we prove the global in time existence of a smooth solution around a given non‐constant steady state with the help of an induction argument on the order of the mixed time‐space derivatives of solutions in energy estimates. Moreover, we also show the convergence of the solution to this steady state as the time goes to the infinity. This phenomenon on the charge transport shows the essential relation of the systems with the non‐isentropic Euler–Maxwell and the isentropic Euler–Maxwell systems. Copyright © 2015 John Wiley & Sons, Ltd.     

A higher‐order PDE‐based image registration approach

This paper addresses the problem of image registration with higher‐order partial differential equation (PDE) methods. From the study of existing affine‐linear and non‐linear methods, a new framework is proposed that unifies common image registration methods within a generic formulation. Currently image registration strategies are classified into either affine‐linear or non‐linear methods subject to the underlying transformations. The new approach combines both strategies to obtain proper approximations which are invariant under global geometrical distortion (shearing), anisotropic resolution (scale changes), as well as rotation and translation. To achieve this favourable property, a modified gradient flow approach is proposed which uses an operator with a kernel consisting of affine‐linear transformations. An approximation with finite differences leads to a large singular linear system. The pseudo‐inverse solution of this system can be computed efficiently by augmenting the singular system to a regular system. Numerical experiments show the improvements compared to unmodified gradient flow approaches. Copyright © 2005 John Wiley & Sons, Ltd.     

Existence and asymptotic behaviour of solutions for a class of quasi‐linear evolution equations with non‐linear damping and source terms

We consider a class of quasi‐linear evolution equations with non‐linear damping and source terms arising from the models of non‐linear viscoelasticity. By a Galerkin approximation scheme combined with the potential well method we prove that when m<p, where m(?0) and p are, respectively, the growth orders of the non‐linear strain terms and the source term, under appropriate conditions, the initial boundary value problem of the above‐mentioned equations admits global weak solutions and the solutions decay to zero as t→∞. Copyright © 2002 John Wiley & Sons, Ltd.     

One‐level Newton–Krylov–Schwarz algorithm for unsteady non‐linear radiation diffusion problem

In this paper, we present a parallel Newton–Krylov–Schwarz (NKS)‐based non‐linearly implicit algorithm for the numerical solution of the unsteady non‐linear multimaterial radiation diffusion problem in two‐dimensional space. A robust solver technology is required for handling the high non‐linearity and large jumps in material coefficients typically associated with simulations of radiation diffusion phenomena. We show numerically that NKS converges well even with rather large inflow flux boundary conditions. We observe that the approach is non‐linearly scalable, but not linearly scalable in terms of iteration numbers. However, CPU time is more important than the iteration numbers, and our numerical experiments show that the algorithm is CPU‐time‐scalable even without a coarse space given that the mesh is fine enough. This makes the algorithm potentially more attractive than multilevel methods, especially on unstructured grids, where course grids are often not easy to construct. Copyright © 2004 John Wiley & Sons, Ltd.     

On the Odir iterative method for non‐symmetric indefinite linear systems

Several Krylov subspace iterative methods have been proposed for the approximation of the solution of general non‐symmetric linear systems. Odir is such a method. Here we study the restarted version of Odir for non‐symmetric indefinite linear systems and we prove convergence under certain conditions on the matrix of coefficients. These results hold for all the restarted Krylov methods equivalent to Odir. We also introduce a new truncated Odir method which is proved to converge for a large class of non‐symmetric indefinite linear systems. This new method requires one‐half of the storage of the standard Odir. Copyright © 2001 John Wiley & Sons, Ltd.     

Generalized solution of non‐autonomous photon transport problem

The purpose of this paper is to show the existence of a generalized solution of a non‐autonomous transport problem. By means of the theory of equicontinuous evolution system on a sequentially complete locally convex topological vector space, we show that the perturbed abstract non‐autonomous Cauchy problem has a unique solution when the perturbation operator and forcing term function satisfy certain conditions. A consequence of the abstract result is that it can be directly applied to obtain a generalized solution of the non‐autonomous photon transport problem. Copyright © 2009 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.     

Linear thermo‐elasticity in non‐smooth media

An alternative first order formulation of linear thermo‐elasticity is presented, designed to deal with coefficients of low regularity as a model for a large class of media occupying arbitrary non‐empty open sets. By fine‐tuning the concepts of generalized derivatives a general existence, uniqueness and continuous dependence result is obtained in the framework of extrapolation spaces (Sobolev lattices). Copyright © 2005 John Wiley & Sons, Ltd.     

A FEM–DtN formulation for a non‐linear exterior problem in incompressible elasticity

In this paper, we combine the usual finite element method with a Dirichlet‐to‐Neumann (DtN) mapping, derived in terms of an infinite Fourier series, to study the solvability and Galerkin approximations of an exterior transmission problem arising in non‐linear incompressible 2d‐elasticity. We show that the variational formulation can be written in a Stokes‐type mixed form with a linear constraint and a non‐linear main operator. Then, we provide the uniqueness of solution for the continuous and discrete formulations, and derive a Cea‐type estimate for the associated error. In particular, our error analysis considers the practical case in which the DtN mapping is approximated by the corresponding finite Fourier series. Finally, a reliable a posteriori error estimate, well suited for adaptive computations, is also given. Copyright © 2003 John Wiley & Sons, Ltd.