Linearized viscoelastic Oldroyd fluid motion in an almost periodic environment

In most of the linear homogenization problems involving convolution terms so far studied, the main tool used to derive the homogenized problem is the Laplace transform. Here we propose a direct approach enabling one to tackle both linear and nonlinear homogenization problems that involve convolution sequences without using Laplace transform. To illustrate this, we investigate in this paper the asymptotic behavior of the solutions of a Stokes–Volterra problem with rapidly oscillating coefficients describing the viscoelastic fluid flow in a fixed domain. Under the almost periodicity assumption on the coefficients of the problem, we prove that the sequence of solutions of our ?‐problem converges in L² to a solution of a rather classical Stokes system. One important fact is that the memory disappears in the limit. To achieve our goal, we use some very recent results about the sigma‐convergence of convolution sequences. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite element approximation of a semilinear elliptic problem with a singular nonlinearity

Given , we consider the following problem: find , such that where or 3, and in . We prove and error bounds for the standard continuous piecewise linear Galerkin finite element approximation with a (weakly) acute triangulation. Our bounds are nearly optimal. In addition, for d = 1 and 2 and we analyze a more practical scheme involving numerical integration on the nonlinear term. We obtain nearly optimal and error bounds for d = 1. For this case we also present some numerical results. Received July 4, 1996 / Revised version received December 18, 1997     

Some results related to complex linear and nonlinear difference equations

In this paper, we investigate the complex oscillation problems of meromorphic solutions to some linear difference equations with meromorphic coefficients, and obtain some results about the relationships between the exponent of convergence of zeros, poles and the order of growth of meromorphic solutions to complex linear difference equations. We also study the existence of solution of certain types of nonlinear differential-difference equations, and partially answer a conjecture concerning the above problem posed by Yang and Laine (C.C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser. A Math. Sci. 86(1) (2010), pp. 10–14).     

Regular and self-similar solutions of nonlinear Schrödinger equations

We study the Cauchy problem for the nonlinear Schrödinger equations with nonlinear term |u|^ou. For some admissible α we show the existence of global solutions and we calculate the regularity of those solutions. Also we give some necessary conditions and some sufficient conditions on initial data for the existence of self-similar solutions.     

Uniqueness via the adjoint problems for systems of conservation laws

We prove a result of uniqueness of the entropy weak solution to the Cauchy problem for a class of nonlinear hyperbolic systems of conservation laws that includes in particular the p-system of isentropic gas dynamics. Our result concerns weak solutions satisfying the, as we call it, Wave Entropy Condition, or WEC for short, introduced in this paper. The main feature of this condition is that it concerns both shock waves and rarefaction waves present in a solution. For the proof of uniqueness, we derive an existence result (respectively a uniqueness result) for the backward (respectively forward) adjoint problem associated with the nonlinear system. Our method also applies to obtain results of existence or uniqueness for some linear hyperbolic systems with discontinuous coefficients. © 1993 John Wiley & Sons, Inc.     

Parallel nonlinear multisplitting methods

Summary Linear multisplitting methods are known as parallel iterative methods for solving a linear systemAx=b. We extend the idea of multisplittings to the problem of solving a nonlinear system of equationsF(x)=0. Our nonlinear multisplittings are based on several nonlinear splittings of the functionF. In a parallel computing environment, each processor would have to calculate the exact solution of an individual nonlinear system belonging to his nonlinear multisplitting and these solutions are combined to yield the next iterate. Although the individual systems are usually much less involved than the original system, the exact solutions will in general not be available. Therefore, we consider important variants where the exact solutions of the individual systems are approximated by some standard method such as Newton's method. Several methods proposed in literature may be regarded as special nonlinear multisplitting methods. As an application of our systematic approach we present a local convergence analysis of the nonlinear multisplitting methods and their variants. One result is that the local convergence of these methods is determined by an induced linear multisplitting of the Jacobian ofF.Dedicated to the memory of Peter Henrici     

Sturm type theorems for general quasi-variational nonlinear problems

Consider the general equation associated to the initial-value problem where is a restoring force and Q represents a nonlinear damping term. Through an analysis of the equation, we give precise estimates of b in terms of the initial data that extend some results derived from Sturm comparison theorems for linear differential equations of second order. In particular, we show some important theorems of non existence of radial solutions of Dirichlet problems in that either significantly improve the former ones, with the m-Laplacian operator, or cover cases never appeared before, with the mean curvature operator. Received July 18, 1996     

On the theory of periodic solution for some nondensely nonautonomous delayed partial differential equations

We first study the Massera problem for the existence of a τ?periodic solution for some nondensely defined partial differential equation, where the autonomous linear part satisfies the Hille‐Yosida condition and the delayed nonlinear part satisfies a locally Lipschitz condition. Second, inspired by an existing study, we prove in the dichotomic case, for τ=1, the existence‐uniqueness and conditional stability of the periodic solution. Moreover, we show the existence of a local stable manifold around such solution. Our theoretical results are finally illustrated by an application.     

Solvability in weighted Hölder spaces for a problem governing the evolution of two compressible fluids

Local (in time) unique solvability of a problem on the motion of two compressible fluids, one of which has finite volume, is obtained in Hölder spaces of functions with a power-like decay at infinity. After passage to Lagrangian coordinates, we arrive at a nonlinear initial boundary value problem with a given closed interface between the liquids. We establish an existence theorem for this problem on the basis of the solvability of a linearized problem by means of the fixed-point theorem. To obtain estimates and to prove the solvability for the linearized problem, we use the Schauder method and an explicit solution of a model linear problem with a plane interface between the liquids. The results are obtained under some restrictions on the fluid density and viscosities, which mean that the fluids are not much different from each other. Bibliography: 8 titles.To Olga Aleksandrovna Ladyzhenskaya on the occasion of her jubilee__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 57–89.     

Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem

Recently, Chen and Tseng extended non-interior continuation/ smooth- ing methods for solving linear/ nonlinear complementarity problems to semidefinite complementarity problems (SDCP). In this paper we propose a non-interior continuation method for solving the monotone SDCP based on the smoothed Fischer—Burmeister function, which is shown to be globally linearly and locally quadratically convergent under suitable assumptions. Our algorithm needs at most to solve a linear system of equations at each iteration. In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Fréchet derivative of the function involved in the SDCP is Lipschitz continuous. For non-interior continuation/ smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence results of the algorithms.     

Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem

Recently, Chen and Tseng extended non-interior continuation/ smooth- ing methods for solving linear/ nonlinear complementarity problems to semidefinite complementarity problems (SDCP). In this paper we propose a non-interior continuation method for solving the monotone SDCP based on the smoothed Fischer—Burmeister function, which is shown to be globally linearly and locally quadratically convergent under suitable assumptions. Our algorithm needs at most to solve a linear system of equations at each iteration. In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Fréchet derivative of the function involved in the SDCP is Lipschitz continuous. For non-interior continuation/ smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence results of the algorithms.     

Separable nonlinear least squares fitting with linear bound constraints and its application in magnetic resonance spectroscopy data quantification

An application in magnetic resonance spectroscopy quantification models a signal as a linear combination of nonlinear functions. It leads to a separable nonlinear least squares fitting problem, with linear bound constraints on some variables. The variable projection (VARPRO) technique can be applied to this problem, but needs to be adapted in several respects. If only the nonlinear variables are subject to constraints, then the Levenberg–Marquardt minimization algorithm that is classically used by the VARPRO method should be replaced with a version that can incorporate those constraints. If some of the linear variables are also constrained, then they cannot be projected out via a closed-form expression as is the case for the classical VARPRO technique. We show how quadratic programming problems can be solved instead, and we provide details on efficient function and approximate Jacobian evaluations for the inequality constrained VARPRO method.     

Existence,uniqueness and stability of solutions for a class of nonlinear integral equations under generalized Lipschitz condition

In this paper, we prove the existence, uniqueness and the stability of solutions for some nonlinear functional-integral equations by using generalized Lipschitz condition. We prove a fixed point theorem to obtain the mentioned aims in Banach space X:= C([a, b],R). As application we study some Volterra integral equations with linear, nonlinear and singular kernel.     

Applications of B‐bounded nonlinear semigroups to particle transport problems

In this paper we study an application of nonlinear B‐bounded semigroups introduced in a previous paper. The application is similar to the particle transport problem which led to B‐bounded linear semigroups. We deal with a nonlinear particle transport problem, which can be solved by using B‐bounded nonlinear semigroups. Copyright © 2010 John Wiley & Sons, Ltd.     

Splitting method for an inverse source problem in parabolic differential equations: Error analysis and applications

In this work, we present a numerical method based on a splitting algorithm to find the solution of an inverse source problem with the integral condition. The source term is reconstructed by using the specified data and by employing the Lie splitting method, we decompose the equation into linear and nonlinear parts. Each subproblem is solved by the Fourier transform and then by combining the solutions of subproblems, the solution of the original problem is computed. Moreover, the framework of strongly continuous semigroup (or C₀-semigroup) is employed in error analysis of operator splitting method for the inverse problem. The convergence of the proposed method is also investigated and proved. Finally, some numerical examples in one, two, and three-dimensional spaces are provided to confirm the efficiency and capability of our work compared with some other well-known methods.     

Analysis of a FitzHugh–Nagumo–Rall model of a neuronal network

Pursuing an investigation started in (Math. Meth. Appl. Sci. 2007; 30 :681–706), we consider a generalization of the FitzHugh–Nagumo model for the propagation of impulses in a network of nerve fibres. To this aim, we consider a whole neuronal network that includes models for axons, somata, dendrites, and synapses (of both inhibitory and excitatory type). We investigate separately the linear part by means of sesquilinear forms, in order to obtain well posedness and some qualitative properties. Once they are obtained, we perturb the linear problem by a nonlinear term and we prove existence of local solutions. Qualitative properties with biological meaning are also investigated. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonlinear Methods of Approximation

Abstract. Our main interest in this paper is nonlinear approximation. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a fixed linear space but are allowed to depend on the function being approximated. While the scope of this paper is mostly theoretical, we should note that this form of approximation appears in many numerical applications such as adaptive PDE solvers, compression of images and signals, statistical classification, and so on. The standard problem in this regard is the problem of m -term approximation where one fixes a basis and looks to approximate a target function by a linear combination of m terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is the starting point for compression algorithms. We are interested in the quantitative aspects of this type of approximation. Namely, we want to understand the properties (usually smoothness) of the function which govern its rate of approximation in some given norm (or metric). We are also interested in stable algorithms for finding good or near best approximations using m terms. Some of our earlier work has introduced and analyzed such algorithms. More recently, there has emerged another more complicated form of nonlinear approximation which we call highly nonlinear approximation. It takes many forms but has the basic ingredient that a basis is replaced by a larger system of functions that is usually redundant. Some types of approximation that fall into this general category are mathematical frames, adaptive pursuit (or greedy algorithms), and adaptive basis selection. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. With this motivation, our recent work and the current activity focuses on nonlinear approximation both in the classical form of m -term approximation (where several important problems remain unsolved) and in the form of highly nonlinear approximation where a theory is only now emerging.     

A Remark on the Existence of Solutions to Nonlinear Equations with Degenerate Mappings

The paper studies nonlinear mappings using methods of p-regularity theory and some concepts and technics of set-valued analysis. The main result addresses to the problem of existence of solutions to nonlinear equations in the degenerate case where a linear part may be singular at the considered initial point.      

Strong-Weak Nonlinear Bilevel Problems: Existence of Solutions in a Sequential Setting

The paper deals with a strong-weak nonlinear bilevel problem which generalizes the well-known weak and strong ones. In general, the study of the existence of solutions to such a problem is a difficult task. So that, for a strong-weak nonlinear bilevel problem, we first give a regularization based on the use of strict ??-solutions of the lower level problem. Then, via this regularization and under sufficient conditions, we show that the problem admits at least one solution. The obtained result is an extension and an improvement of some recent results appeared recently in the literature for both weak nonlinear bilevel programming problems and linear finite dimensional case.     

The effect of a reinforcing ring on the stress-strain state of flexible cylindrical shells

We investigate the effect of a reinforcing ring on the stress-strain state of a cylindrical shell in the geometrically nonlinear problem with a nonaxisymmetric load on the edge. The nonlinear boundary-value problem is reduced to a sequence of linear problems by the quasilinearization method. The linear problems are solved by the discrete orthogonalization method. The results obtained using linear and nonlinear theory are compared.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 92–96, 1985.