Simple Nonlinear Equation Research Articles

Temperature modelling within a thin material sheet involved in conductive‐radiative heat transfer 1

Abstract In this paper we consider a problem about finding of temperature approximation within a thin material sheet involved in conductive‐radiative heat transfer. As result, we found that temperature within the sheet can be approximated in L 2 norm by solution of a simple nonlinear operator equation.

Thermomechanics of Coupled Fields in Passive and Piezoactive Inelastic Bodies Under Harmonic Deformation

ABSTRACT The present article gives a brief overview of studies made at S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine on thermomechanics of coupled fields in nonlinear inelastic passive and piezoactive materials and bodies under monoharmonic deformation. Our attention will be focused on the following subjects: (1) development of the nonlinear models of passive (without piezoeffect) and piezoactive (with piezoeffect) materials for monoharmonic vibrations with accounting for influence of such factors as material nonlinearity, dissipation, and temperature of dissipative heating; (2) development of the thermomechanical models of nonhomogeneous thin-walled inelastic elements under monoharmonic deformation; (3) development of numerical and analytical methods of the solution of coupled problems; (4) investigation of the influence of material nonlinearity, dissipation and temperature of dissipative heating on regularities of vibrations of the bodies with the passive and piezoactive materials, including influence of these factors on the efficiency of vibration control of elements with distributed sensors and actuators. Problem 1 is solved on the basis of concept of complex material characteristics. Accordingly the simplified nonlinear constitutive equations of passive and piezoactive materials are identical to the constitutive equations of linear theory of viscoelasticity, but the complex mechanical and electromechanical characteristics are temperature and field-dependent quantities. Averaged for period the rate of dissipation coincides with averaged electromechanical power. On the basis of these simplified constitutive equations the statement of coupled problems of thermomechanics in the passive and piezoactive inelastic spatial bodies are given. To solve Problem 2, the mechanical hypotheses of Kirchhoff–Love and similar hypotheses for electric field quantities and temperature are introduced. As a result the statement of coupled nonlinear problems of thermomechanics in the passive and piezoactive inelastic thin-walled elements are given. To solve Problem 3 one needs to use some iterative techniques such as method of step-by-step integration in time, method of variable characteristics and accelerating Steffensen-Eitken procedure. At each iteration, the linearized electromechanical and heat conductivity problems are solved by a method of discrete orthogonalization or by a finite element method. By the use of variational methods, the analytical solutions are also obtained. On the basis of the mentioned models, methods, and analytical and numerical solutions obtained, the influence of material nonlinearity, dissipation, and self-heating on the vibrations of nonhomogeneous spatial and thin-walled elements (in particular on the efficiency of active damping of vibrations of the elements) are investigated (Problem 4).

STAT module can function as a biphasic amplitude filter

Signal transducer and actuator of transcription (STATs) are a family of transcription factors activated by various cytokines, growth factors and hormones. They are important mediators of immune responses and growth and differentiation of various cell types. The STAT signalling system represents a defined functional module with a pattern of signalling that is conserved from flies to mammals. In order to probe and gain insights into the signalling properties of the STAT module by computational means, we developed a simple non-linear ordinary differential equations model within the 'Virtual Cell' framework. Our results demonstrate that the STAT module can operate as a 'biphasic amplitude filter' with an ability to amplify input signals within a specific intermediate range. We show that dimerisation of phosphorylated STAT is crucial for signal amplification and the amplitude filtering function. We also demonstrate that maximal amplification at intermediate levels of STAT activation is a moderately robust property of STAT module. We propose that these observations can be extrapolated to the analogous SMAD signalling module.

Nonlinear evolution equations as fast and exact solvers of estimation problems

We develop computationally efficient procedures for solving certain restoration problems in one dimension, including the one-dimensional (1-D) discrete versions of the total variation regularized problem introduced by Sauer and Bouman and the constrained total variation minimization problem introduced by Rudin et al. The procedures are exact and have time complexity O(NlogN) and space complexity O(N), where N is the number of data samples. They are based on a simple nonlinear diffusion equation proposed by Pollak et al. and related to the Perona-Malik equation. A probabilistic interpretation for this diffusion equation in 1-D is provided by showing that it produces optimal solutions to a sequence of estimation problems. We extend our methods to two dimensions, where they no longer have similar optimality properties; however, we experimentally demonstrate their effectiveness for image restoration.

Resonant properties of a nonlinear dissipative layer excited by a vibrating boundary: Q-factor and frequency response

Simplified nonlinear evolution equations describing non-steady-state forced vibrations in an acoustic resonator having one closed end and the other end periodically oscillating are derived. An approach based on a nonlinear functional equation is used. The nonlinear Q-factor and the nonlinear frequency response of the resonator are calculated for steady-state oscillations of both inviscid and dissipative media. The general expression for the mean intensity of the acoustic wave in terms of the characteristic value of a Mathieu function is derived. The process of development of a standing wave is described analytically on the base of exact nonlinear solutions for different laws of periodic motion of the wall. For harmonic excitation the wave profiles are described by Mathieu functions, and their mean energy characteristics by the corresponding eigenvalues. The sawtooth-shaped motion of the boundary leads to a similar process of evolution of the profile, but the solution has a very simple form. Some possibilities to enhance the Q-factor of a nonlinear system by suppression of nonlinear energy losses are discussed.

A Resummed Branching Process Representation for a Class of Nonlinear ODEs

We study some probabilistic representations, based on branching processes, of a simple nonlinear differential equation, i.e. $u'=\lambda u(au^R-1)$. The first approach is basically the same used by Le Jan and Sznitman for 3-d Navier-Stokes equations, which need small initial data to work. In our much simpler setting we are able to make this precise, finding all the cases where their method fails to give the solution. The second approach is based on a resummed representation, which we can prove to give all the solutions of the problem, even those with large initial data.

Solving Integrable Broer-Kaup Equations in (2+1)-Dimensional Spaces via an Improved Variable Separation Approach**The project supported by the State Key Basic Research Program of China under Grant No. 1998030600 and National Natural Science Foundation of China under Grant No. 10072013

Starting from Backlund transformation and using Cole-Hopf transformation, we reduce the integrable Broer-Kaup equations in (2+1)-dimensional spaces to a simple linear evolution equation with two arbitrary functions of and in this paper. And we can obtain some new solutions of the original equations by investigating the simple nonlinear evolution equation, which include the solutions obtained by the variable separation approach.

The soliton-like solutions to the (2+1)-dimensional modified dispersive water-wave system

By a simple transformation, we reduce the (2+1)-dimensional modified dispersive water-wave system to a simple nonlinear partial differential equation. In order to solve this equation by generalized tanh-function method, we only need to solve a simple system of first-order ordinary differential equations and by doing so we can obtain many new soliton-like solutions which include the solutions obtained by using the conventional tanh-function method.

New soliton-like and periodic solution of (2+1)-dimensional higher order Broer–Kaup system

Abstract In this paper, the (2 + 1)-dimensional higher order Broer–Kaup system is reduced to a simple nonlinear partial differential equation by a transformation. Based on the single equation and utilizing new generally projective Riccati equation method, we obtain many new exact solutions for the (2 + 1)-dimensional higher order Broer–Kaup system, including new soliton-like solutions and periodic solutions.

Cornering Stiffness Adaptation for Improved Side Slip Angle Observation

The vehicle body side slip angle (VBSSA) represents a key variable in vehicle dynamics. It allows to assess vehicle stability in critical driving situations. As it cannot directly be measured with standard sensors it ha to be determined using a vehicle model including an observer. This paper describes a method to increase the accuracy of VBSS A-estimation using adaptation of the cornering stiffnesses. The adaptation bases on a simple but efficient nonlinear approximation equation of the lateral wheel force. It is capable to describe the tire behavior up to the stability limit by means of two independent parameters.

A notes on the variable separation approach

Starting from the B-cklund transformation and using Cole-Hopf transformation,we reduce a class of nonlinear evolution equations in (2+1)-dimensional spaces to a simple nonlinear evolution equation with two arbitrary functions of {x,t} and {y,t}.We can obtain some new exact solutions of the original equations by studying the simple nonlinear evolution equation which includes some solutions obtained by the variable separation approach.

A simple method for solving a class of nonlinearly coupled evolution equations in higher dimensional space

In this paper,a class of nonlingearly coupled evolution equations in higher dimensional spaces is reduced to a simple nonlinear evolution equation in lower dimensional spaces by using a simple transformation,and then we can easily obtain the solutions of the original equations by appling existing method to the simple nonlinear evolution equation.

放水路中の越波伝達波について

Transmitted waves due to overtopping and due to waves which pass under the caisson in a discharge flume is sometimes a problem for stable operation of pumping works. The currents make influence on not only the transmitted waves but also the incident waves. The phenomena, therefore, might be quite complicated. To understand the phenomena, the hydraulic model tests were carried out. The transmitted waves due to overtopping in the currents sometime quite larger than the transmitted waves without currents. To simulates the transmitted waves, a new methods is proposed. The methods are based on the Bousinesque equation. The mechanism of the flow under and over the caisson are modelized by a simple pipe flow and non-linear long wave equations. The method is quite capable to represent experimental results.

Solving the (2 + 1)-dimensional higher order Broer–Kaup system via a transformation and tanh-function method

In this paper, the (2 + 1)-dimensional higher order Broer–Kaup system is reduced to a simple nonlinear partial differential equation by a transformation, and utilizing tanh-function method we obtain many new exact solutions for the (2 + 1)-dimensional higher order Broer–Kaup system.

Hillslope‐storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 1. Formulation and characteristic response

Hillslope response to rainfall remains one of the central problems of catchment hydrology. Flow processes in a one‐dimensional sloping aquifer can be described by Boussinesq's hydraulic groundwater theory. Most hillslopes, however, have complex three‐dimensional shapes that are characterized by their plan shape, profile curvature of surface and bedrock, and the soil depth. Field studies and numerical simulation have shown that these attributes are the most significant topographic controls on subsurface flow and saturation along hillslopes. In this paper the Boussinesq equation is reformulated in terms of soil water storage rather than water table height. The continuity and Darcy equations formulated in terms of storage along the hillslope lead to the hillslope‐storage Boussinesq (HSB) equation for subsurface flow. Solutions of the HSB equation account explicitly for plan shape of the hillslope by introducing the hillslope width function and for profile curvature through the bedrock slope angle and the hillslope soil depth function. We investigate the behavior of the HSB model for different hillslope types (uniform, convergent, and divergent) and different slope angles under free drainage conditions after partial initial saturation (drainage scenario) and under constant rainfall recharge conditions (recharge scenario). The HSB equation is solved by means of numerical integration of the partial differential equation. We find that convergent hillslopes drain much more slowly compared to divergent hillslopes. The accumulation of moisture storage near the outlet of convergent hillslopes results in bell‐shaped hydrographs. In contrast, the fast draining divergent hillslopes produce highly peaked hydrographs. In order to investigate the relative importance of the different terms in the HSB equation, several simplified nonlinear and linearized versions are derived, for instance, by recognizing that the width function of a hillslope generally shows smooth transition along the flow direction or by introducing a fitting parameter to account for average storage along the hillslope. The dynamic response of these reduced versions of the HSB equation under free drainage conditions depend strongly on hillslope shape and bedrock slope angle. For flat slopes (of the order of 5%), only the simplified nonlinear HSB equation is able to capture the dynamics of subsurface flow along complex hillslopes. In contrast, for steep slopes (of the order of 30%), we see that all the reduced versions show very similar results compared to the full version. It can be concluded that the complex derivative terms of width with respect to flow distance play a less dominant role with increasing slope angle. Comparison with the hillslope‐storage kinematic wave model of Troch et al. [2002] shows that the diffusive drainage terms of the HSB model become less important for the fast draining divergent hillslopes. These results have important implications for the use of simplified versions of the HSB equation in landscapes and for the development of appropriate analytical solutions for subsurface flow along complex hillslopes.

Explosion en temps fini de la solution d'un problème d'évolution de surface de film mince

In this Note, we are interested in the evolution of a surface of a crystal structure, constituted by an elastic substrate and a thin film. If the crystal is constrained, some morphological instabilities may appear. To study these instabilities, we made use of the model developped in Phys. Rev. B 47 (1993) 9760–9777. There, the map f of the free surface of the film satisfies a parabolic partial differential equation, depending on the elastic displacement of the substrate. For simplicity, the substrate is assumed to be linearly elastic and the structure to be infinite in one direction. Then, under some formal asymptotic assumptions, a formal expansion of the displacement can be determined after some appropriate scalings, allowing to derive a simplified parabolic nonlinear equation as in Lods et al. (Asymptotic Anal. 33 (2003) 67–91). We give here some results about the finite-time blow-up and the existence and uniqueness of the solution in an appropriate space. To validate the theoretical results, we also performed some numerical simulations using a pseudo-spectral method and also compute the initial-profile dependent critical value of the parameter θ involved in the nonlinear equation. To cite this article: M. Boutat et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Rigorous results in selection of steady needle crystals

This paper concerns the existence of solutions to a steady needle crystal growth problem in a one-sided model. We rigorously prove that for small nonzero anisotropy γ, analytic symmetric needle crystal solutions exist in the limit of surface tension ε 2 if only if the stokes constant S for a relatively simple nonlinear differential equation is zero. This Stokes constant S depends on the parameter β=2 9/7 γε −8/7 and earlier numerical calculations by a number of investigators have shown this to be zero for a discrete set of values of β. It is also proven that for γ=0, there can be no symmetric needle crystal solution in the considered space. The methodology consists of two steps. First, the original problem is reduced to a weak half-strip problem for any γ in a compact set of [0,1) by relaxation of the symmetry condition. The weak problem is shown to have a unique solution in the function space considered for any γ∈[0, γ m ] for some γ m >0. When a symmetry is invoked, the weak problem is shown equivalent to the original needle crystal problem. Next, by considering the behavior of the solution in neighborhood of an appropriate complex turning point for γ∈(0, γ m ], we extract an exponentially small term in ε as ε→0 + that generally violates the symmetric condition. We prove that the symmetry condition is satisfied for small ε when the parameter β is constrained appropriately.

Analytic solutions of a nonlinear convective equation in population dynamics

Analytic solutions are presented for a simple nonlinear convective equation of use in population dynamics. In spite of its simplicity the equation predicts rich behavior including a velocity inversion transition. Stability considerations are also presented.

Holographic RG flows and universal structures on the Coulomb branch of supersymmetric large- gauge theory

We report on our results of D3-brane probing a large class of generalised type IIB supergravity solutions presented very recently in the literature. The structure of the solutions is controlled by a single non-linear differential equation. These solutions correspond to renormalisation group flows from pure N=4 supersymmetric gauge theory to an N=2 gauge theory with a massive adjoint scalar. The gauge group is SU(n) with n large. After presenting the general result, we focus on one of the new solutions, solving for the specific coordinates needed to display the explicit metric on the moduli space. We obtain an appropriately holomorphic result for the coupling. We look for the singular locus, and interestingly, the final result again manifests itself in terms of a square root branch cut on the complex plane, as previously found for a set of solutions for which the details are very different. This, together with the existence of the single simple non-linear differential equation, is further evidence in support of an earlier suggestion that there is a very simple model --perhaps a matrix model with relation to the Calogero-Moser integrable system-- underlying this gauge theory physics.

On balanced approximations for time integration of multiple time scale systems

The effect of various numerical approximations used to solve linear and nonlinear problems with multiple time scales is studied in the framework of modified equation analysis (MEA). First, MEA is used to study the effect of linearization and splitting in a simple nonlinear ordinary differential equation (ODE), and in a linear partial differential equation (PDE). Several time discretizations of the ODE and PDE are considered, and the resulting truncation terms are compared analytically and numerically. It is demonstrated quantitatively that both linearization and splitting can result in accuracy degradation when a computational time step larger than any of the competing (fast) time scales is employed. Many of the issues uncovered on the simple problems are shown to persist in more realistic applications. Specifically, several differencing schemes using linearization and/or time splitting are applied to problems in nonequilibrium radiation–diffusion, magnetohydrodynamics, and shallow water flow, and their solutions are compared to those using balanced time integration methods.