Solutions For Quasilinear Research Articles

ОБ УЧЕТЕ НЕЛИНЕЙНЫХ И СВЯЗАННЫХ ЭФФЕКТОВ ТЕПЛОВОЙ ЗАДАЧИ И ФАЗОВЫХ ПЕРЕХОДОВ ПРИ МОДЕЛИРОВАНИИ ТЕХНОЛОГИИ КОНТАКТНОГО ТЕРМОСИЛОВОГО ПОВЕРХНОСТНОГО УПРОЧНЕНИЯ МЕТАЛЛИЧЕСКИХ СПЛАВОВ

The article considers the problem setting of the coupled thermal-structural problem arising in the process of modelling the surface hardening using electromechanical treatment (EMT). A mathematical model is presented with respect to forming the structure of metallic alloys with EMT based on a joint analysis of the calculated data on the dynamics of temperature fields and continuous cooling transformation diagramby using the Ti6Al2V titanium alloy as an example. Kinetics of the martensitic transition during EMT is described using Koistinen-Marburger equation. An algorithm of solving the thermal problem based on the finite element method in the weak Galerkin form is given. The calculated area is approximated by Zienkiewicz type infinite elements. Based on a series of computational experiments, the influence of the magnitude of the time step on the accuracy of the problem solution was investigated. The significance of the considered coupled and nonlinear effects which are specific for high-speed high-temperature thermal processes are analyzed, e.g. the change of the thermophysical properties of the metal, latent heat of phase transitions, thermal radiation and dependence between the metal physical properties and temperature. Two methods are used to consider the effects aiming to solve the thermo-structural problems: a totally coupled non-linear problem using the direct iteration method of Picard in a combination with the relaxation formulae for the convergence acceleration; the quasilinear solution when the values of the nonlinear terms are calculated on the basis of the temperature distribution obtained in the previous time step. Based on the series of numerical experiments, we analyzed the unsteady temperature fields, as well as the distribution of structural domains after EMT in the titanium pseudo-alpha-alloys.

Phased Array Beam Fields of Nonlinear Rayleigh Surface Waves**Supported by the National Natural Science Foundation of China under Grant Nos 61271356 and 51575541, the National Research Foundation of Korea under Grant Nos 2013-M2A2A9043241 and 2013-R1A2A2A01016042, and the Hunan Provincial Innovation Foundation For Postgraduate under Grant No CX2016B046.

This study concerns calculation of phased array beam fields of the nonlinear Rayleigh surface waves based on the integral solutions for a nonparaxial wave equation. Since the parabolic approximation model for describing the nonlinear Rayleigh waves has certain limitations in modeling the sound beam fields of phased arrays, a more general model equation and integral forms of quasilinear solutions are introduced. Some features of steered and focused beam fields radiated from a linear phased array of the second harmonic Rayleigh wave are presented.

Infinitely many solutions for quasilinear Schrödinger systems with finite and sign-changing potentials

We consider the following quasilinear Schrodinger system in $${\mathbb{R}^N}$$ with $${N\geq3}$$ : P $$\left\{\begin{array}{l}\sum_{i,j=1}^{N}D_j(a_{ij}(u)D_i u)-\frac{1}{2} \sum_{i,j=1}^{N}D_s a_{ij}(u) D_i u D_j u-A(x) u+F_u(u,v)=0 \\ \sum_{i,j=1}^{N}D_j(a_{ij}(v)D_iv)-\frac{1}{2} \sum_{i,j=1}^{N}D_s a_{ij}(v) D_i v D_j v-B(x)v+F_v(u,v)=0,\end{array} \right.$$ where $${D_i=\frac{\partial}{\partial x_i},\ \ D_s a_{ij}(s)=\frac{d}{ds}a_{ij}(s)}$$ , $${F(u,v)}$$ is the coupling term, $${A(x)}$$ and $${B(x)}$$ are finite and sign-changing potential functions. Using an approximation scheme and $${q}$$ -Laplacian regularization, we prove the existence of infinitely many solutions for system $${(P)}$$ .

A more general model equation of nonlinear Rayleigh waves and their quasilinear solutions

A more general two-dimensional wave motion equation with consideration of attenuation and nonlinearity is proposed to describe propagating nonlinear Rayleigh waves of finite amplitude. Based on the quasilinear theory, the numerical solutions for the sound beams of fundamental and second harmonic waves are constructed with Green’s function method. Compared with solutions from the parabolic approximate equation, results from the general equation have more accuracy in both the near distance of the propagation direction and the far distance of the transverse direction, as quasiplane waves are used and non-paraxial Green’s functions are obtained. It is more effective to obtain the nonlinear Rayleigh sound beam distributions accurately with the proposed general equation and solutions. Brief consideration is given to the measurement of nonlinear parameter using nonlinear Rayleigh waves.

Multidimensional quasilinear first-order equations and multivalued solutions of the elliptic and hyperbolic equations

We discuss an extension of the theory of multidimensional second-order equations of the elliptic and hyperbolic types related to multidimensional quasilinear autonomous first-order partial differential equations. Calculating the general integrals of these equations allows constructing exact solutions in the form of implicit functions. We establish a connection with hydrodynamic equations. We calculate the number of free functional parameters of the constructed solutions. We especially construct and analyze implicit solutions of the Laplace and d’Alembert equations in a coordinate space of arbitrary finite dimension. In particular, we construct generalized Penrose–Rindler solutions of the d’Alembert equation in 3+1 dimensions.

The Delta Medial Axis: A fast and robust algorithm for filtered skeleton extraction

In this paper, we present the Delta Medial Axis (DMA), a quasi-linear algorithmic solution addressing several of the main concerns of discrete medial axes (MA) computation. First, its sensitivity to small shape perturbations is counterbalanced by a single parameter (delta), used in a pruning strategy that implicitly takes into account the local topology of the boundary. Second, the discrete nature of images is addressed by approximating the original MA definition in continuous space. Third, to allow real-time performances, a set of algorithmic optimizations is proposed. We compare our approach both qualitatively and quantitatively to recent state of the art solutions, and show that the DMA presents an excellent choice for a wide range of applications. Finally, to allow fast and efficient use of our algorithm, we propose the complete pseudo-code.

Multiplicity of Solutions for Quasilinear $p(x)$-Laplacian Equations in $\mathbb{R}^{N}$

This paper investigates the multiplicity of solutions for quasilinear elliptic equations with $p(x)$-Laplacian in $\mathbb{R}^{N}$ by using the nonsmooth critical point theory. We obtain the existence of critical points for nondifferentiable functionals.

Are Nonlinear Methods Necessary at the Zero Lower Bound?

This paper examines the importance of the zero lower bound (ZLB) constraint on the nominal interest rate by estimating three variants of a small-scale New Keynesian model: (1) a nonlinear model with an occassionally binding ZLB constraint; (2) a constrained linear model, which imposes the constraint in the filter but not the solution; and (3) an unconstrained linear model, which never imposes the constraint. The posterior distributions are similar, but important differences arise in their predictions at the ZLB. The nonlinear model fits the data better at the ZLB and primarily attributes the ZLB to a reduction in household demand due to discount factor shocks. In the linear models, the ZLB is due to large contractionary monetary policy shocks, which is at odds with the Fed?s expansionary policy during the Great Recession. Posterior predictive analysis shows the nonlinear model is partially able to account for the increase in output volatility and the negative skewness in output and inflation that occurred during the ZLB period, whereas the linear models predict almost no changes in those statistics. We also compare the results from our nonlinear model to the quasi-linear solution based on OccBin. The quasi-linear model fits the data better than the linear models, but it still generate too little volatility at the ZLB and predicts that a large policy shock caused the ZLB to bind in 2008Q4.

Are Nonlinear Methods Necessary at the Zero Lower Bound?

This paper examines the importance of the zero lower bound (ZLB) constraint on the nominal interest rate by estimating three variants of a small-scale New Keynesian model: (1) a nonlinear model with an occassionally binding ZLB constraint; (2) a constrained linear model, which imposes the constraint in the filter but not the solution; and (3) an unconstrained linear model, which never imposes the constraint. The posterior distributions are similar, but important differences arise in their predictions at the ZLB. The nonlinear model fits the data better at the ZLB and primarily attributes the ZLB to a reduction in household demand due to discount factor shocks. In the linear models, the ZLB is due to large contractionary monetary policy shocks, which is at odds with the Fed’s expansionary policy during the Great Recession. Posterior predictive analysis shows the nonlinear model is partially able to account for the increase in output volatility and the negative skewness in output and inflation that occurred during the ZLB period, whereas the linear models predict almost no changes in those statistics. We also compare the results from our nonlinear model to the quasi-linear solution based on OccBin. The quasi-linear model fits the data better than the linear models, but it still generate too little volatility at the ZLB and predicts that a large policy shock caused the ZLB to bind in 2008Q4.

Significance of accurate diffraction corrections for the second harmonic wave in determining the acoustic nonlinearity parameter

The accurate measurement of acoustic nonlinearity parameter β for fluids or solids generally requires making corrections for diffraction effects due to finite size geometry of transmitter and receiver. These effects are well known in linear acoustics, while those for second harmonic waves have not been well addressed and therefore not properly considered in previous studies. In this work, we explicitly define the attenuation and diffraction corrections using the multi-Gaussian beam (MGB) equations which were developed from the quasilinear solutions of the KZK equation. The effects of making these corrections are examined through the simulation of β determination in water. Diffraction corrections are found to have more significant effects than attenuation corrections, and the β values of water can be estimated experimentally with less than 5% errors when the exact second harmonic diffraction corrections are used together with the negligible attenuation correction effects on the basis of linear frequency dependence between attenuation coefficients, α2 ≃ 2α1.

Solutions for quasilinear Schrödinger systems with critical exponents

This paper deals with a class of quasilinear Schrodinger system with critical exponents and bounded potentials in \({\mathbb{R}^N (N \geq 3)}\) : $$\left\{\begin{array}{ll}\sum_{i,j=1}^{N}D_j(a_{ij}(u)D_iu)-\frac{1}{2}\sum_{i,j=1}^{N}D_s a_{ij}(u) D_i u D_j u-a(x)u+F_u(u,v)=0\\ \sum_{i,j=1}^{N}D_j(a_{ij}(v)D_iv)-\frac{1}{2}\sum_{i,j=1}^{N}D_s a_{ij}(v) D_i v D_j v-b(x)v+F_v(u,v)=0,\end{array}\right. \quad\quad\quad(P)$$ where \({D_i = \frac{\partial}{\partial x_i},\ \ D_sa_{ij}(s) = \frac{d}{ds}a_{ij}(s), F(u,v)}\) is the coupling term with critical growth, and a(x), b(x) are potential functions. This kind of equations include the so-called Modified Nonlinear Schrodinger Systems. By using a perturbation method, we prove the existence of a ground state positive solution for (P).

Weak Solutions for a Class of (p,q) Laplacian quasilinear Elliptic System with Different Weights

Using variational methods, we study the existence of weak solutions for the degenerate quasilinear

Mechanics Analysis of Cantilever Beam with Uniform Load

According to the mechanics analysis of the flexible cantilever beam in the uniform load distribution, the corresponding mathematical model is established. Applying quasi-linear analysis and perturbation solution in order to convert to complex integral and differential formula into its power series form, then derives the general formula of dimensionless deflection formula. Last we further discussed the deflection formula and analysis the force situation of the beam.

Investigating the wave-fixed and place-fixed origins of the 2f1-f2distortion product otoacoustic emission within a micromechanical cochlear model

The 2f(1)-f(2) distortion product otoacoustic emission (DPOAE) arises within the cochlea due to the nonlinear interaction of two stimulus tones (f(1) and f(2)). It is thought to comprise contributions from a wave-fixed source and a place-fixed source. The generation and transmission of the 2f(1)-f(2) DPOAE is investigated here using quasilinear solutions to an elemental model of the human cochlea with nonlinear micromechanics. The micromechanical parameters and nonlinearity are formulated to match the measured response of the cochlea to single- and two-tone stimulation. The controlled introduction of roughness into the active micromechanics of the model allows the wave- and place-fixed contributions to the DPOAE to be studied separately. It is also possible to manipulate the types of nonlinear suppression that occur within the quasilinear model to investigate the influence of stimulus parameters on DPOAE generation. The model predicts and explains a variety of 2f(1)-f(2) DPOAE phenomena: The dependence of emission amplitude on stimulus parameters, the weakness of experiments designed to quantify cochlear amplifier gain, and the predominant mechanism which gives rise to DPOAE fine structure. In addition, the model is used to investigate the properties of the wave-fixed source and how these properties are influenced by the stimulus parameters.

Torsional oscillation of vortex tangles. Possible applications to oscillations of solid 4He

Torsional oscillation of vessels containing quantum fluids are one of oldest and most popular methods for the study of quantized vortices. One recent and very brilliant example is the discovery of the supersolidity of solid helium. In torsional oscillation experiments, a drop in the period of the oscillations is observed when some low temperature is reached. This effect has been attributed to the appearance of the superfluid component. It depends on many factors and has various explanations. But, assuming (at least hypothetically, at this stage) that the phenomenon of “supersolidity” (dissipationless flow) does occur, we must consider the relaxation of a vortex system (we can call it a vortex tangle, vortex fluid, chaotic set of vortices, etc.). This is necessary because the only way to involve the superfluid component in rotation is through polarized vortices (with nonzero mean polarization along their axis of rotation). Here we consider a vortex tangle relaxation model for the torsional oscillation response of quantum systems, with the aim of using it to study solid 4He. It is shown that the rotation of the superfluid component occurs as a relaxation effect with a relaxation time that depends on the amplitude of the oscillations (as well as on temperature and pressure). This problem has a quasi-linear solution which explains the (amplitude dependent) shift in the period. There is also an imaginary shift of the frequency (also amplitude dependent), which represents an additional dissipation. The theoretical results are compared with recent measurements.

Modeling of Parametric Loudspeakers by Gaussian-Beam Expansion Technique

A quasi-linear analytical solution based on the Gaussian-beam expansion technique is used to model the parametric array. The axial sound pressure levels and the beam pattern of the primary and difference frequency waves of a compact hexagonal array, a spaced hexagonal array and a conventional square array consisting of circular elements are calculated. For comparison, a square piston source with equal radiation area is considered as well. Simulation results show that the beam patterns of the difference frequency sound of four sources are almost the same. The compact hexagonal array is more efficient than the square array because of higher axial sound pressure levels of the difference frequency sound.

Feasibility of simplified integral equation modeling of low-frequency marine CSEM with a resistive target

We have assessed the accuracy of a simplified integral equation (SIE) modeling approach for marine controlled-source electromagnetics (CSEM) with low applied frequencies and a resistive target. The most computationally intensive part of rigorous integral equation (IE) modeling is the computation of the anomalous electric field within the target itself. This leads to a matrix problem with a dense coefficient matrix. It is well known that, in general, the presence of many grid cells creates a computational disadvantage for dense-matrix methods compared to sparse-matrix methods. The SIE approach replaces the dense-matrix part of rigorous IE modeling by sparse-matrix calculations based on an approximation of Maxwell’s equations. The approximation is justified theoretically if a certain dimensionless parameter [Formula: see text] is small. As opposed to approximations of the Born type, the validity of the SIE approach does not rely on the anomalous field to be small in comparison with the background field in the target region. We have calculated [Formula: see text] for a range of parameter values typical for marine CSEM, and compared the SIE approach numerically to the rigorous IE method and to the quasi-linear (QL) and quasi-analytic (QA) approximate solutions. It is found that the SIE approach is very accurate for small [Formula: see text], corresponding to frequencies in the lower range of those typical for marine CSEM for petroleum exploration. In addition, the SIE approach is found to be significantly more accurate than the QL and QA approximations for small [Formula: see text].

High resolution time integration for S N radiation transport

First-order, second-order, and high resolution time discretization schemes are implemented and studied for the discrete ordinates ( S N ) equations. The high resolution method employs a rate of convergence better than first-order, but also suppresses artificial oscillations introduced by second-order schemes in hyperbolic partial differential equations. The high resolution method achieves these properties by nonlinearly adapting the time stencil to use a first-order method in regions where oscillations could be created. We employ a quasi-linear solution scheme to solve the nonlinear equations that arise from the high resolution method. All three methods were compared for accuracy and convergence rates. For non-absorbing problems, both second-order and high resolution converged to the same solution as the first-order with better convergence rates. High resolution is more accurate than first-order and matches or exceeds the second-order method.

Second-harmonic generation of limited diffraction beams

This paper presents an analytical result concerning second-harmonic generation of a general limited diffraction beam in nonlinear media. The result is based on quasilinear solutions of the Khokhlov?Zabolotskaya?Kuznetsov nonlinear wave equation. It is shown that second-harmonic generation by the nonlinear interaction of two arbitrary Bessel beams gives a nearly radially limited diffraction beam. For an arbitrary limited diffraction beam from the linear superposition of basis mode Bessel beams, results also indicate that the second-harmonic beam remains an almost radially or transversely limited diffraction beam.

Quasilinear elliptic inequalities on complete Riemannian manifolds

We prove maximum and comparison principles for weak distributional solutions of quasilinear, possibly singular or degenerate, elliptic differential inequalities in divergence form on complete Riemannian manifolds. A new definition of ellipticity for nonlinear operators on Riemannian manifolds is introduced, covering the standard important examples. As an application, uniqueness results for some related boundary value problems are presented.