scholarly journals Instability, nonexistence, and uniqueness in elasticity with porous dissipation

2006 ◽  
Vol 2006 ◽  
pp. 1-14 ◽  
Author(s):  
M. C. Leseduarte ◽  
R. Quintanilla
Keyword(s):  
Nonlinear Problem ◽  
Exponential Growth ◽  
Linear Problem ◽  
Linear Equations ◽  
Uniqueness Of Solutions ◽  
Time Problem ◽  
Nonexistence Of Solutions

This paper is devoted to the study of the elasticity with porous dissipation. In the context of the nonlinear problem, we prove instability and nonexistence of solutions. In the context of the linear problem, we obtain exponential growth. We also obtain uniqueness of solutions of the backward in time problem of the linear equations.

Cauchy problems involving a Hadamard-type fractional derivative

2018 ◽  
Vol 68 (6) ◽  
pp. 1353-1366
Author(s):  
Rafał Kamocki
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivative ◽  
Unique Solution ◽  
Nonlinear Problem ◽  
Point Theorem ◽  
Linear Problem ◽  
Cauchy Formula ◽  
Cauchy Problems

Abstract In this paper, we investigate some Cauchy problems involving a left-sided Hadamard-type fractional derivative. A theorem on the existence of a unique solution to a nonlinear problem is proved. The main result is obtained using a fixed point theorem due to Banach, as well as the Bielecki norm. A Cauchy formula for the solution of the linear problem is derived.

On the impossibility of localization in linear thermoelasticity

2007 ◽  
Vol 463 (2088) ◽  
pp. 3311-3322 ◽  
Author(s):  
R Quintanilla
Keyword(s):  
Finite Time ◽  
Main Idea ◽  
Type I ◽  
Type Ii ◽  
Exterior Domains ◽  
Uniqueness Of Solutions ◽  
Null Solution ◽  
Time Problem ◽  
Linear Thermoelasticity ◽  
Without Energy Dissipation

In the middle of the 1990s, Green & Naghdi proposed three theories of thermoelasticity that they labelled as types I, II and II. The type II theory, which is also called thermoelasticity without energy dissipation, is conservative and the solutions cannot decay with respect to time. It is well known that, in general, in the linear theories of thermoelasticity of types I and III, the solutions decay with respect to time. In many situations this decay is at least exponential. In this paper we study whether this decay can be fast enough to guarantee the solutions to be zero in a finite time. We investigate the impossibility of the localization in time of the solutions of linear thermoelasticity for the theories of Green & Naghdi. This means that the only solution that vanishes after a finite time is the null solution. The main idea is to show the uniqueness of solutions for the backward in time problem. To be precise, for type III thermoelasticity we will prove the impossibility of localization of solutions in the case of bounded domains, and for the type I thermoelasticity in the case of exterior domains, even when the solutions can be unbounded, whether the spatial variable goes to infinity.

Transient Response in a One-Dimensional Model of Thermoelastic Contact

1996 ◽  
Vol 63 (3) ◽  
pp. 575-581 ◽  
Author(s):  
Z. S. Olesiak ◽  
Yu. A. Pyryev
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Problem ◽  
Transient Response ◽  
Computational Simulation ◽  
Linear Equations ◽  
Numerical Results ◽  
System Of Equations ◽  
Dimensional Model ◽  
One Dimensional ◽  
Method Of Solution

We consider two layers of different materials with the initial gap between them in the field of temperature with imperfect boundary conditions in Barber’s sense. The model we discuss is that of two contacting rods (Barber and Zhang, 1988) which in the case of a single rod was devised and discussed by Dundurs and Comninou (1976, 1979). In this paper we try to make a step further in the investigation of the essentially nonlinear problem. Though we consider a system of the linear equations of thermoelasticity the nonlinearity is induced by the boundary conditions dependent on the solution. We present an algorithm for solving the system of equations based on Laplace’s transform technique. The method of solution can be used also in the dynamical problems with inertial terms taken into account. The numerical results have been obtained by a kind of computational simulation.

Electromagnetic wave propagation in a layer with power nonlinearity

2019 ◽  
Vol 28 (01) ◽  
pp. 1950009 ◽  
Author(s):  
Valeria Kurseeva ◽  
Stanislav Tikhov ◽  
Dmitry Valovik
Keyword(s):  
Electromagnetic Field ◽  
Electromagnetic Wave ◽  
Nonlinear Problem ◽  
Dielectric Waveguide ◽  
Electromagnetic Wave Propagation ◽  
Linear Problem ◽  
Transverse Electric ◽  
Kerr Nonlinearity ◽  
Power Nonlinearity ◽  
Plane Dielectric Waveguide

The paper focuses on a problem that describes propagation of transverse-electric (TE) waves in a plane dielectric waveguide filled with nonlinear medium. The nonlinearity is characterized by the power term [Formula: see text], where [Formula: see text], [Formula: see text] are constants and [Formula: see text] is the electric term of the guided electromagnetic field. The layer is located between two half-spaces filled with linear media having constant permittivities. It is proved that the nonlinear problem has infinitely many propagation constants (PCs), whereas the corresponding linear problem has only a finite number of them. The nonlinearity leads to the occurrence of infinitely many nonperturbative solutions of the nonlinear problem. Results of the paper show that the power nonlinearity (for any [Formula: see text]) and Kerr nonlinearity (for [Formula: see text]) produce qualitatively similar outcomes. In addition, the found results allow one to study very important cases of quintic, septimal, etc. nonlinear permittivities in the focusing regime.

Approximate Solution of Generalized Ginzburg-Landau-Higgs System via Homotopy Perturbation Method

2010 ◽  
Vol 65 (4) ◽  
pp. 301-304 ◽  
Author(s):  
Ju-Hong Lu ◽  
Chun-Long Zheng
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Nonlinear Problem ◽  
Homotopy Perturbation Method ◽  
Linear Equations ◽  
Initial Approximation ◽  
System Of Linear Equations ◽  
Arbitrary Degree ◽  
Ginzburg Landau ◽  
Homotopy Perturbation

Using the homotopy perturbation method, a class of nonlinear generalized Ginzburg-Landau-Higgs systems (GGLH) is considered. Firstly, by introducing a homotopic transformation, the nonlinear problem is changed into a system of linear equations. Secondly, by selecting a suitable initial approximation, the approximate solution with arbitrary degree accuracy to the generalized Ginzburg- Landau-Higgs system is derived. Finally, another type of homotopic transformation to the generalized Ginzburg-Landau-Higgs system reported in previous literature is briefly discussed.

scholarly journals Fundamental Study on Hand Waving Sensors

1990 ◽  
Vol 2 (5) ◽  
pp. 325-334
Author(s):  
Kazue Nishihara ◽  
Keyword(s):  
Differential Equations ◽  
Nonlinear Problem ◽  
Linear Equations ◽  
Angular Acceleration ◽  
Fundamental Study ◽  
Acceleration Signal ◽  
Angular Velocities ◽  
Machine Interface ◽  
Man Machine Interface ◽  
Wave Sensors

In order to develop a dynamic man-machine interface which measures angular motions of multi-link mechanisms, a uniaxial hand wave sensor is experimented with and triaxial hand wave sensors are simulated numerically. It was confirmed that a uniaxial hand wave sensor composed of a pair of uniaxially located accelerometers directly obtains exact angular acceleration by subtracting each acceleration signal. A triaxial hand wave sensor by a six (i.e. three pairs) accelerometer method, however, contains duplex angular velocities influenced by other axes in addition to the exact angular acceleration, so it is necessary to separate those physical values by a software algorithm. Adams-Moulton's method for solving differential equations was best suited to solve this nonlinear problem. A nine accelerometer method obtains linear equations for angular accelerations readily after arithmetic calculations of the nine signals.

Spatial Behavior in Thermoelastodynamics with Double Porosity Structure

2017 ◽  
Vol 09 (07) ◽  
pp. 1750097 ◽  
Author(s):  
Olivia Florea
Keyword(s):  
Upper Bound ◽  
Double Porosity ◽  
Spatial Behavior ◽  
Bounded Domains ◽  
Uniqueness Of Solutions ◽  
Porosity Structure ◽  
Time Problem

In the present study, we consider the theory of thermoelastodynamics with double porosity structure. Two situations are studied: for bounded domains, the impossibility of time localization of solutions is obtained, which is equivalent to the uniqueness of solutions for the backward in time problem. For the second study, in the case of semi-infinite cylinders, a Phragmen–Lindelof alternative, as well as an upper bound for the amplitude term when solutions decay, are obtained.

scholarly journals The Backward in Time Problem of Double Porosity Material with Microtemperature

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 552
Author(s):  
Olivia A. Florea
Keyword(s):  
Double Porosity ◽  
Bounded Domains ◽  
Uniqueness Of Solutions ◽  
Porosity Structure ◽  
Time Problem

In the present study, the theory of thermoelastodynamics is considered in the case of materials with double porosity structure and microtemperature. The novelty of this study consists in the investigation of a backward in time problem associated with double porous thermoelastic materials with microtemperature. In the first part of the paper, in case of the bounded domains the impossibility of time localization of solutions is obtained. This study is equivalent to the uniqueness of solutions for the backward in time problem. In the second part of the paper, a Phragmen-Lindelof alternative in the case of semi-infinite cylinders is obtained.

scholarly journals Unconditional uniqueness results for the nonlinear Schrödinger equation

2019 ◽  
Vol 21 (07) ◽  
pp. 1850058 ◽  
Author(s):  
Sebastian Herr ◽  
Vedran Sohinger
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Linear Equations ◽  
Nonlinear Schrodinger Equation ◽  
Dimensional System ◽  
Uniqueness Of Solutions ◽  
Nonlinear Schrödinger ◽  
Uniqueness Results ◽  
Unconditional Uniqueness

We study the problem of unconditional uniqueness of solutions to the cubic nonlinear Schrödinger equation (NLS). We introduce a new strategy to approach this problem on bounded domains, in particular on rectangular tori. It is a known fact that solutions to the cubic NLS give rise to solutions of the Gross–Pitaevskii (GP) hierarchy, which is an infinite-dimensional system of linear equations. By using the uniqueness analysis of the GP hierarchy, we obtain new unconditional uniqueness results for the cubic NLS on rectangular tori, which cover the full scaling-subcritical regime in high dimensions. In fact, we prove a more general result which is conditional on the domain. In addition, we observe that well-posedness of the cubic NLS in Fourier–Lebesgue spaces implies unconditional uniqueness.

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