Directory of Open Access Journals (Sweden)
Yusuf Pandir
2013-01-01
Full Text Available We firstly give some new functions called generalized hyperbolic functions. By the using of the generalized hyperbolic functions, new kinds of transformations are defined to discover the exact approximate solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation of the generalized KdV equation and the coupled equal width wave equations (CEWE, we find new exact solutions of two equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions. We think that these solutions are very important to explain some physical phenomena.
Directory of Open Access Journals (Sweden)
A. H. Bhrawy
2014-01-01
Full Text Available One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.
Hyperbolic partial differential equations
Witten, Matthew
1986-01-01
Hyperbolic Partial Differential Equations III is a refereed journal issue that explores the applications, theory, and/or applied methods related to hyperbolic partial differential equations, or problems arising out of hyperbolic partial differential equations, in any area of research. This journal issue is interested in all types of articles in terms of review, mini-monograph, standard study, or short communication. Some studies presented in this journal include discretization of ideal fluid dynamics in the Eulerian representation; a Riemann problem in gas dynamics with bifurcation; periodic M
Oscillations of the Solutions of Nonlinear Delay Hyperbolic Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
LIU An-ping; GUO Yan-feng; YANG Xiang-hui
2004-01-01
In this paper, oscillatory properties of solutions of certain hyperbolic partial differential equations with multi-delays are investigated and a series of sufficient conditions for oscillations of the equations are established. Theresults fully indicate that the oscillations are caused by delays.
Hyperbolic partial differential equations
Lax, Peter D
2006-01-01
The theory of hyperbolic equations is a large subject, and its applications are many: fluid dynamics and aerodynamics, the theory of elasticity, optics, electromagnetic waves, direct and inverse scattering, and the general theory of relativity. This book is an introduction to most facets of the theory and is an ideal text for a second-year graduate course on the subject. The first part deals with the basic theory: the relation of hyperbolicity to the finite propagation of signals, the concept and role of characteristic surfaces and rays, energy, and energy inequalities. The structure of soluti
Nonlinear hyperbolic waves in multidimensions
Prasad, Phoolan
2001-01-01
The propagation of curved, nonlinear wavefronts and shock fronts are very complex phenomena. Since the 1993 publication of his work Propagation of a Curved Shock and Nonlinear Ray Theory, author Phoolan Prasad and his research group have made significant advances in the underlying theory of these phenomena. This volume presents their results and provides a self-contained account and gradual development of mathematical methods for studying successive positions of these fronts.Nonlinear Hyperbolic Waves in Multidimensions includes all introductory material on nonlinear hyperbolic waves and the theory of shock waves. The author derives the ray theory for a nonlinear wavefront, discusses kink phenomena, and develops a new theory for plane and curved shock propagation. He also derives a full set of conservation laws for a front propagating in two space dimensions, and uses these laws to obtain successive positions of a front with kinks. The treatment includes examples of the theory applied to converging wavefronts...
Witten, Matthew
1983-01-01
Hyperbolic Partial Differential Equations, Volume 1: Population, Reactors, Tides and Waves: Theory and Applications covers three general areas of hyperbolic partial differential equation applications. These areas include problems related to the McKendrick/Von Foerster population equations, other hyperbolic form equations, and the numerical solution.This text is composed of 15 chapters and begins with surveys of age specific population interactions, populations models of diffusion, nonlinear age dependent population growth with harvesting, local and global stability for the nonlinear renewal eq
Energy Technology Data Exchange (ETDEWEB)
Wang Qi [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China) and Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080 (China)]. E-mail: wangqi_dlut@yahoo.com.cn; Chen Yong [Nonlinear Science Center, Department of Mathematics, Ningbo University, Ningbo 315211 (China)
2007-01-15
With the aid of symbolic computation, some algorithms are presented for the rational expansion methods, which lead to closed-form solutions of nonlinear partial differential equations (PDEs). The new algorithms are given to find exact rational formal polynomial solutions of PDEs in terms of Jacobi elliptic functions, solutions of the Riccati equation and solutions of the generalized Riccati equation. They can be implemented in symbolic computation system Maple. As applications of the methods, we choose some nonlinear PDEs to illustrate the methods. As a result, we not only can successfully obtain the solutions found by most existing Jacobi elliptic function methods and Tanh-methods, but also find other new and more general solutions at the same time.
Advanced Research Workshop on Nonlinear Hyperbolic Problems
Serre, Denis; Raviart, Pierre-Arnaud
1987-01-01
The field of nonlinear hyperbolic problems has been expanding very fast over the past few years, and has applications - actual and potential - in aerodynamics, multifluid flows, combustion, detonics amongst other. The difficulties that arise in application are of theoretical as well as numerical nature. In fact, the papers in this volume of proceedings deal to a greater extent with theoretical problems emerging in the resolution of nonlinear hyperbolic systems than with numerical methods. The volume provides an excellent up-to-date review of the current research trends in this area.
Monotone method for nonlinear nonlocal hyperbolic problems
Directory of Open Access Journals (Sweden)
Azmy S. Ackleh
2003-02-01
Full Text Available We present recent results concerning the application of the monotone method for studying existence and uniqueness of solutions to general first-order nonlinear nonlocal hyperbolic problems. The limitations of comparison principles for such nonlocal problems are discussed. To overcome these limitations, we introduce new definitions for upper and lower solutions.
Oscillation of solutions to neutral nonlinear impulsive hyperbolic equations with several delays
Directory of Open Access Journals (Sweden)
Jichen Yang
2013-01-01
Full Text Available In this article, we study oscillatory properties of solutions to neutral nonlinear impulsive hyperbolic partial differential equations with several delays. We establish sufficient conditions for oscillation of all solutions.
Nonlinear electrodynamics as a symmetric hyperbolic system
Abalos, Fernando; Goulart, Érico; Reula, Oscar
2015-01-01
Nonlinear theories generalizing Maxwell's electromagnetism and arising from a Lagrangian formalism have dispersion relations in which propagation planes factor into null planes corresponding to two effective metrics which depend on the point-wise values of the electromagnetic field. These effective Lorentzian metrics share the null (generically two) directions of the electromagnetic field. We show that, the theory is symmetric hyperbolic if and only if the cones these metrics give rise to have a non-empty intersection. Namely that there exist families of symmetrizers in the sense of Geroch which are positive definite for all covectors in the interior of the cones intersection. Thus, for these theories, the initial value problem is well-posed. We illustrate the power of this approach with several nonlinear models of physical interest such as Born-Infeld, Gauss-Bonnet and Euler-Heisenberg.
Generalized Hyperbolic Function Solution to a Class of Nonlinear Schrödinger-Type Equations
Directory of Open Access Journals (Sweden)
Zeid I. A. Al-Muhiameed
2012-01-01
Full Text Available With the help of the generalized hyperbolic function, the subsidiary ordinary differential equation method is improved and proposed to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Liu equation are investigated and the exact solutions are derived with the aid of the homogenous balance principle and generalized hyperbolic functions. We study the effect of the generalized hyperbolic function parameters p and q in the obtained solutions by using the computer simulation.
Goloviznin, V. M.; Kanaev, A. A.
2011-05-01
For the CABARET finite difference scheme, a new approach to the construction of convective flows for the one-dimensional nonlinear transport equation is proposed based on the minimum principle of partial local variations. The new approach ensures the monotonicity of solutions for a wide class of problems of a fairly general form including those involving discontinuous and nonconvex functions. Numerical results illustrating the properties of the proposed method are discussed.
HYPERBOLIC-PARABOLIC CHEMOTAXIS SYSTEM WITH NONLINEAR PRODUCT TERMS
Institute of Scientific and Technical Information of China (English)
Chen Hua; Wu Shaohua
2008-01-01
We prove the local existence and uniqueness of week solution of the hyperbolic-parabolic Chemotaxis system with some nonlinear product terms. For one dimensional case, we prove also the global existence and uniqueness of the solution for the problem.
MULTISCALE HOMOGENIZATION OF NONLINEAR HYPERBOLIC EQUATIONS WITH SEVERAL TIME SCALES
Institute of Scientific and Technical Information of China (English)
Jean Louis Woukeng; David Dongo
2011-01-01
We study the multiscale homogenization of a nonlinear hyperbolic equation in a periodic setting. We obtain an accurate homogenization result. We also show that as the nonlinear term depends on the microscopic time variable, the global homogenized problem thus obtained is a system consisting of two hyperbolic equations. It is also shown that in spite of the presence of several time scales, the global homogenized problem is not a reiterated one.
Beyer, Horst Reinhard
2007-01-01
The present volume is self-contained and introduces to the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis. Only some examples require more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Particular stress is on equations of the hyperbolic type since considerably less often treated in the literature. Also, evolution equations from fundamental physics need to be compatible with the theory of special relativity and therefore are of hyperbolic type. Throughout, detailed applications are given to hyperbolic partial differential equations occurring in problems of current theoretical physics, in particular to Hermitian hyperbolic systems. This volume is thus also of interest to readers from theoretical physics.
Hyperbolic function method for solving nonlinear differential-different equations
Institute of Scientific and Technical Information of China (English)
Zhu Jia-Min
2005-01-01
An algorithm is devised to obtained exact travelling wave solutions of differential-different equations by means of hyperbolic function. For illustration, we apply the method to solve the discrete nonlinear (2+1)-dimensional Toda lattice equation and the discretized nonlinear mKdV lattice equation, and successfully constructed some explicit and exact travelling wave solutions.
Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws
Chen, Gui-Qiang; Zhang, Yongqian
2012-01-01
We establish an $L^1$-estimate to validate the weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws with arbitrary initial data of small bounded variation. This implies that the simpler geometric optics expansion function can be employed to study the properties of general entropy solutions to hyperbolic systems of conservation laws. Our analysis involves new techniques which rely on the structure of the approximate equations, besides the properties of the wave-front tracking algorithm and the standard semigroup estimates.
INITIAL BOUNDARY VALUE PROBLEM FOR A DAMPED NONLINEAR HYPERBOLIC EQUATION
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陈国旺
2003-01-01
In the paper, the existence and uniqueness of the generalized global solution and the classical global solution of the initial boundary value problems for the nonlinear hyperbolic equationare proved by Galerkin method and the sufficient conditions of blow-up of solution in finite time are given.
Nonlinear Hyperbolic-Parabolic System Modeling Some Biological Phenomena
Institute of Scientific and Technical Information of China (English)
WU Shaohua; CHEN Hua
2011-01-01
In this paper, we study a nonlinear hyperbolic-parabolic system modeling some biological phenomena. By semigroup theory and Leray-Schauder fixed point argument, the local existence and uniqueness of the weak solutions for this system are proved. For the spatial dimension N = 1, the global existence of the weak solution will be established by the bootstrap argument.
Techniques in Linear and Nonlinear Partial Differential Equations
1991-10-21
nonlinear partial differential equations , elliptic 15. NUMBER OF PAGES hyperbolic and parabolic. Variational methods. Vibration problems. Ordinary Five...NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS FINAL TECHNICAL REPORT PROFESSOR LOUIS NIRENBERG OCTOBER 21, 1991 NT)S CRA&I D FIC ,- U.S. ARMY RESEARCH OFFICE...Analysis and partial differential equations . ed. C. Sadowsky. Marcel Dekker (1990) 567-619. [7] Lin, Fanghua, Asymptotic behavior of area-minimizing
Nonlinear Sigma Models with Compact Hyperbolic Target Spaces
Gubser, Steven; Schoenholz, Samuel S; Stoica, Bogdan; Stokes, James
2015-01-01
We explore the phase structure of nonlinear sigma models with target spaces corresponding to compact quotients of hyperbolic space, focusing on the case of a hyperbolic genus-2 Riemann surface. The continuum theory of these models can be approximated by a lattice spin system which we simulate using Monte Carlo methods. The target space possesses interesting geometric and topological properties which are reflected in novel features of the sigma model. In particular, we observe a topological phase transition at a critical temperature, above which vortices proliferate, reminiscent of the Kosterlitz-Thouless phase transition in the $O(2)$ model. Unlike in the $O(2)$ case, there are many different types of vortices, suggesting a possible analogy to the Hagedorn treatment of statistical mechanics of a proliferating number of hadron species. Below the critical temperature the spins cluster around six special points in the target space known as Weierstrass points. The diversity of compact hyperbolic manifolds suggest...
Institute of Scientific and Technical Information of China (English)
刘安平; 何猛省
2002-01-01
By making use of the integral inequalities and some results of the functional differential equations, oscillatory properties of solutions of certain nonlinear hyperbolic partial differential equations of neutral type with multi-delays were investigated and a series of sufficient conditions for oscillations of the equations were established. The results fully indicate that the oscillations are caused by delay and hence reveal the difference between these equations and those equations without delay.
Some problems on nonlinear hyperbolic equations and applications
Peng, YueJun
2010-01-01
This volume is composed of two parts: Mathematical and Numerical Analysis for Strongly Nonlinear Plasma Models and Exact Controllability and Observability for Quasilinear Hyperbolic Systems and Applications. It presents recent progress and results obtained in the domains related to both subjects without attaching much importance to the details of proofs but rather to difficulties encountered, to open problems and possible ways to be exploited. It will be very useful for promoting further study on some important problems in the future.
AD GALERKIN ANALYSIS FOR NONLINEAR PSEUDO-HYPERBOLIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
Xia Cui
2003-01-01
AD (Alternating direction) Galerkin schemes for d-dimensional nonlinear pseudo-hyperbolic equations are studied. By using patch approximation technique, AD procedure is realized,and calculation work is simplified. By using Galerkin approach, highly computational accuracy is kept. By using various priori estimate techniques for differential equations,difficulty coming from non-linearity is treated, and optimal H1 and L2 convergence properties are demonstrated. Moreover, although all the existed AD Galerkin schemes using patch approximation are limited to have only one order accuracy in time increment, yet the schemes formulated in this paper have second order accuracy in it. This implies an essential advancement in AD Galerkin analysis.
Parabolic Perturbation of a Nonlinear Hyperbolic Problem Arising in Physiology
Colli, P.; Grasselli, M.
We study a transport-diffusion initial value problem where the diffusion codlicient is "small" and the transport coefficient is a time function depending on the solution in a nonlinear and nonlocal way. We show the existence and the uniqueness of a weak solution of this problem. Moreover we discuss its asymptotic behaviour as the diffusion coefficient goes to zero, obtaining a well-posed first-order nonlinear hyperbolic problem. These problems arise from mathematical models of muscle contraction in the framework of the sliding filament theory.
Nonlinear sigma models with compact hyperbolic target spaces
Gubser, Steven; Saleem, Zain H.; Schoenholz, Samuel S.; Stoica, Bogdan; Stokes, James
2016-06-01
We explore the phase structure of nonlinear sigma models with target spaces corresponding to compact quotients of hyperbolic space, focusing on the case of a hyperbolic genus-2 Riemann surface. The continuum theory of these models can be approximated by a lattice spin system which we simulate using Monte Carlo methods. The target space possesses interesting geometric and topological properties which are reflected in novel features of the sigma model. In particular, we observe a topological phase transition at a critical temperature, above which vortices proliferate, reminiscent of the Kosterlitz-Thouless phase transition in the O(2) model [1, 2]. Unlike in the O(2) case, there are many different types of vortices, suggesting a possible analogy to the Hagedorn treatment of statistical mechanics of a proliferating number of hadron species. Below the critical temperature the spins cluster around six special points in the target space known as Weierstrass points. The diversity of compact hyperbolic manifolds suggests that our model is only the simplest example of a broad class of statistical mechanical models whose main features can be understood essentially in geometric terms.
Optimized difference schemes for multidimensional hyperbolic partial differential equations
Directory of Open Access Journals (Sweden)
Adrian Sescu
2009-04-01
Full Text Available In numerical solutions to hyperbolic partial differential equations in multidimensions, in addition to dispersion and dissipation errors, there is a grid-related error (referred to as isotropy error or numerical anisotropy that affects the directional dependence of the wave propagation. Difference schemes are mostly analyzed and optimized in one dimension, wherein the anisotropy correction may not be effective enough. In this work, optimized multidimensional difference schemes with arbitrary order of accuracy are designed to have improved isotropy compared to conventional schemes. The derivation is performed based on Taylor series expansion and Fourier analysis. The schemes are restricted to equally-spaced Cartesian grids, so the generalized curvilinear transformation method and Cartesian grid methods are good candidates.
Finite Volume Evolution Galerkin Methods for Nonlinear Hyperbolic Systems
Lukáčová-Medvid'ová, M.; Saibertová, J.; Warnecke, G.
2002-12-01
We present new truly multidimensional schemes of higher order within the frame- work of finite volume evolution Galerkin (FVEG) methods for systems of nonlinear hyperbolic conservation laws. These methods couple a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of the multidimensional hyperbolic system, such that all of the infinitely many directions of wave propagation are taken into account. Following our previous results for the wave equation system, we derive approximate evolution operators for the linearized Euler equations. The integrals along the Mach cone and along the cell interfaces are evaluated exactly, as well as by means of numerical quadratures. The influence of these numerical quadratures will be discussed. Second-order resolution is obtained using a conservative piecewise bilinear recovery and the midpoint rule approximation for time integration. We prove error estimates for the finite volume evolution Galerkin scheme for linear systems with constant coefficients. Several numerical experiments for the nonlinear. Euler equations, which confirm the accuracy and good multidimensional behavior of the FVEG schemes, are presented as well.
Blow-up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities
Guo, Yanqiu; Rammaha, Mohammad A.; Sakuntasathien, Sawanya
2017-02-01
We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influence of a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as an energy-amplifying supercritical nonlinear source:
A scalar hyperbolic equation with GR-type non-linearity
Khokhlov, A M
2003-01-01
We study a scalar hyperbolic partial differential equation with non-linear terms similar to those of the equations of general relativity. The equation has a number of non-trivial analytical solutions whose existence rely on a delicate balance between linear and non-linear terms. We formulate two classes of second-order accurate central-difference schemes, CFLN and MOL, for numerical integration of this equation. Solutions produced by the schemes converge to exact solutions at any fixed time $t$ when numerical resolution is increased. However, in certain cases integration becomes asymptotically unstable when $t$ is increased and resolution is kept fixed. This behavior is caused by subtle changes in the balance between linear and non-linear terms when the equation is discretized. Changes in the balance occur without violating second-order accuracy of discretization. We thus demonstrate that a second-order accuracy, althoug necessary for convergence at finite $t$, does not guarantee a correct asymptotic behavior...
Nonlinear sigma models with compact hyperbolic target spaces
Energy Technology Data Exchange (ETDEWEB)
Gubser, Steven [Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (United States); Saleem, Zain H. [Department of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA 19104 (United States); National Center for Physics, Quaid-e-Azam University Campus,Islamabad 4400 (Pakistan); Schoenholz, Samuel S. [Department of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA 19104 (United States); Stoica, Bogdan [Walter Burke Institute for Theoretical Physics, California Institute of Technology,452-48, Pasadena, CA 91125 (United States); Stokes, James [Department of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA 19104 (United States)
2016-06-23
We explore the phase structure of nonlinear sigma models with target spaces corresponding to compact quotients of hyperbolic space, focusing on the case of a hyperbolic genus-2 Riemann surface. The continuum theory of these models can be approximated by a lattice spin system which we simulate using Monte Carlo methods. The target space possesses interesting geometric and topological properties which are reflected in novel features of the sigma model. In particular, we observe a topological phase transition at a critical temperature, above which vortices proliferate, reminiscent of the Kosterlitz-Thouless phase transition in the O(2) model V.L. Berezinskii, Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group II. Quantum systems, Sov. Phys. JETP 34 (1972) 610. J.M. Kosterlitz and D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6 (1973) 1181 [http://inspirehep.net/search?p=find+J+%22J.Phys.,C6,1181%22]. . Unlike in the O(2) case, there are many different types of vortices, suggesting a possible analogy to the Hagedorn treatment of statistical mechanics of a proliferating number of hadron species. Below the critical temperature the spins cluster around six special points in the target space known as Weierstrass points. The diversity of compact hyperbolic manifolds suggests that our model is only the simplest example of a broad class of statistical mechanical models whose main features can be understood essentially in geometric terms.
Institute of Scientific and Technical Information of China (English)
Yang Jun; Wang Chunyan; Li Jing; Meng Zhijuan
2006-01-01
This paper is concerned with the oscillations of neutral hyperbolic partial differential equations with delays. Necessary and sufficient conditions are obtained for the oscillations of all solutions of the equations, and these results are illustrated by some examples.
OSCILLATION OF IMPULSIVE HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION WITH DELAY
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
In this paper, oscillation properties of the solutions of impulsive hyperbolic equation with delay are investigated via the method of differential inequalities. Sufficient conditions for oscillations of the solutions are established.
Application of Variational Iteration Method to Fractional Hyperbolic Partial Differential Equations
Directory of Open Access Journals (Sweden)
Fadime Dal
2009-01-01
Full Text Available The solution of the fractional hyperbolic partial differential equation is obtained by means of the variational iteration method. Our numerical results are compared with those obtained by the modified Gauss elimination method. Our results reveal that the technique introduced here is very effective, convenient, and quite accurate to one-dimensional fractional hyperbolic partial differential equations. Application of variational iteration technique to this problem has shown the rapid convergence of the sequence constructed by this method to the exact solution.
The Full—Discrete Mixed Finite Element Methods for Nonlinear Hyperbolic Equations
Institute of Scientific and Technical Information of China (English)
YanpingCHEN; YunqingHUANG
1998-01-01
This article treats mixed finite element methods for second order nonlinear hyperbolic equations.A fully discrete scheme is presented and improved L2-error estimates are established.The convergence of both the function value andthe flux is demonstrated.
The approximate weak inertial manifolds of a class of nonlinear hyperbolic dynamical systems
Institute of Scientific and Technical Information of China (English)
赵怡
1996-01-01
Some concepts about approximate and semi-approximate weak inertial manifolds are introduced and the existence of global attractor and semi-approximate weak inertial manifolds is obtained for a class of nonlinear hyperbolic dynamical systems by means of some topologically homeomorphic mappings and techniques. Using these results, the existence of approximate weak inertial manifolds is also presented for a kind of nonlinear hyperbolic system arising from relativistic quantum mechanics. The regularization problem is proposed finally.
Mohammed, Mogtaba; Sango, Mamadou
2016-07-01
This paper deals with the homogenization of a linear hyperbolic stochastic partial differential equation (SPDE) with highly oscillating periodic coefficients. We use Tartar’s method of oscillating test functions and deep probabilistic compactness results due to Prokhorov and Skorokhod. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized linear hyperbolic SPDE with constant coefficients. We also prove the convergence of the associated energies.
A HIGH ORDER ADAPTIVE FINITE ELEMENT METHOD FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS
Institute of Scientific and Technical Information of China (English)
Zhengfu Xu; Jinchao Xu; Chi-Wang Shu
2011-01-01
In this note,we apply the h-adaptive streamline diffusion finite element method with a small mesh-dependent artificial viscosity to solve nonlinear hyperbolic partial differential equations,with the objective of achieving high order accuracy and mesh efficiency.We compute the numerical solution to a steady state Burgers equation and the solution to a converging-diverging nozzle problem.The computational results verify that,by suitably choosing the artificial viscosity coefficient and applying the adaptive strategy based on a posterior error estimate by Johnson et al.,an order of N-3/2 accuracy can be obtained when continuous piecewise linear elements are used,where N is the number of elements.
Jiwari, Ram
2015-08-01
In this article, the author proposed two differential quadrature methods to find the approximate solution of one and two dimensional hyperbolic partial differential equations with Dirichlet and Neumann's boundary conditions. The methods are based on Lagrange interpolation and modified cubic B-splines respectively. The proposed methods reduced the hyperbolic problem into a system of second order ordinary differential equations in time variable. Then, the obtained system is changed into a system of first order ordinary differential equations and finally, SSP-RK3 scheme is used to solve the obtained system. The well known hyperbolic equations such as telegraph, Klein-Gordon, sine-Gordon, Dissipative non-linear wave, and Vander Pol type non-linear wave equations are solved to check the accuracy and efficiency of the proposed methods. The numerical results are shown in L∞ , RMS andL2 errors form.
Institute of Scientific and Technical Information of China (English)
Peng QU; Cunming LIU
2012-01-01
For a kind of partially dissipative quasilinear hyperbolic systems without Shizuta-Kawashima condition,in which all the characteristics,except a weakly linearly degenerate one,are involved in the dissipation,the global existence of H2 classical solution to the Cauchy problem with small initial data is obtained.
Discreet Solutions for Matrix Systems in Partial Derivatives Hyperbolic and Singular
Salazar, Manuel J
2011-01-01
In this paper we study the construction of a discrete solution for a hyperbolic system of partial differentials of the strongly coupled type. In its construction, the discrete separation of matricial variable method was followed. Two separate equations in differences were obtained: a singular matricial and the other one a Sturm Liouville vectorial problem, which by the superposition principle yield a stable discrete solution.
$C_{0}$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain
Jacob, Birgit; Morris, Kirsten; Zwart, Hans
2015-01-01
Hyperbolic partial differential equations on a one-dimensional spatial domain are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of nonhomogeneous transmission lines. The main result of this paper is a simple test for $C_{0}$-semigrou
Extended Trial Equation Method for Nonlinear Partial Differential Equations
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
Shock structure simulation using hyperbolic moment models in partially-conservative form
Koellermeier, Julian; Torrilhon, Manuel
2016-11-01
The Boltzmann equation is often used to model rarefied gas flow in the transition or kinetic regime for moderate to large Knudsen numbers. However, standard moment methods like Grad's approach lack hyperbolicity of the equations. We point out the failure of Grad's method and overcome the deficiencies with the help of the new hyperbolic moment models called QBME and HME, derived by an operator projection framework. The new model equations are in partially-conservative form meaning that a subset of the equations cannot be written in conservative form due to some changes in these equations. This leads to additional numerical difficulties. The influence of the partially-conservative terms on the solution is analyzed and we present a numerical scheme for the solution of the partially-conservative PDE systems, namely the PRICE-C scheme by Canestrelli. Furthermore, a shock structure test case is used to compare the accuracy of the different hyperbolic moment models to a discrete velocity reference solution. The results show that the new hyperbolic models achieve higher accuracy than the standard Grad model despite the fact that the model equations cannot be fully written in conservative form.
Banks, H. T.; Kunisch, K.
1982-01-01
Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.
A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
BAI Cheng-Lin
2003-01-01
For the Noyes-Fields equations, two-dimensional hyperbolic equations of conversation laws, and theBurgers-KdV equation, a class of traveling wave solutions has been obtained by constructing appropriate functiontransformations. The main idea of solving the equations is that nonlinear partial differential equations are changed intosolving algebraic equations. This method has a wide-rangingpracticability.
A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
BAICheng-Lin
2003-01-01
For the Noyes-Fields equations, two-dimenslonal hyperbolic equations of conversation laww and the Burgers-KdV equation, a class of travellng wave solutions has been obtained by constructhag appropriate function transformations. The main idea of solving the equations is that nonlinear partial differential equations are changed into solving algebraic equations. This method has a wide-ranging practicability.
Pettersson, Mass Per; Nordström, Jan
2015-01-01
This monograph presents computational techniques and numerical analysis to study conservation laws under uncertainty using the stochastic Galerkin formulation. With the continual growth of computer power, these methods are becoming increasingly popular as an alternative to more classical sampling-based techniques. The approach described in the text takes advantage of stochastic Galerkin projections applied to the original conservation laws to produce a large system of modified partial differential equations, the solutions to which directly provide a full statistical characterization of the effect of uncertainties. Polynomial Chaos Methods of Hyperbolic Partial Differential Equations focuses on the analysis of stochastic Galerkin systems obtained for linear and non-linear convection-diffusion equations and for a systems of conservation laws; a detailed well-posedness and accuracy analysis is presented to enable the design of robust and stable numerical methods. The exposition is restricted to one spatial dime...
Interaction of Tangent Conormal Waves for Higher-Order Nonlinear Strictly Hyperbolic Equations
Institute of Scientific and Technical Information of China (English)
尹会成; 仇庆久
1994-01-01
In this paper we deal with the interaction of three conormal waves for a class of third-order nonlinear strictly hyperbolic equations, in which two conormal waves are tangent. By the same argument, we may also discuss the similar problem for equation system of compressible fluid flow and obtain similar conclusions.
The Superconvergence of Mixed Finite Element Methods for Nonlinear Hyperbolic Equations
Institute of Scientific and Technical Information of China (English)
YanpingCHEN; YunqingHUANG
1998-01-01
Imprioved L2-error estimates are computed for mixed finte element methods for second order nonlinear hyperbolic equations.Superconvergence results,L∞ in time and discrete L2 in space,are derived for both the solution and gradients on the rectangular domain.Results are given for the continuous-time case.
Fast-Slow Partially Hyperbolic Systems Versus Freidlin-Wentzell Random Systems
de Simoi, Jacopo; Liverani, Carlangelo; Poquet, Christophe; Volk, Denis
2016-09-01
We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Freidlin-Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation.
Generalized solutions of nonlinear partial differential equations
Rosinger, EE
1987-01-01
During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concernin
Biala, T A; Jator, S N
2015-01-01
In this article, the boundary value method is applied to solve three dimensional elliptic and hyperbolic partial differential equations. The partial derivatives with respect to two of the spatial variables (y, z) are discretized using finite difference approximations to obtain a large system of ordinary differential equations (ODEs) in the third spatial variable (x). Using interpolation and collocation techniques, a continuous scheme is developed and used to obtain discrete methods which are applied via the Block unification approach to obtain approximations to the resulting large system of ODEs. Several test problems are investigated to elucidate the solution process.
Global attractors for damped abstract nonlinear hyperbolic systems
Pinter, Gabriella Agnes
1997-12-01
This dissertation is concerned with the long time dynamics of a class of damped abstract hyperbolic systems that arise in the study of certain smart material structures, namely elastomers. The term smart material refers to a material capable of both sensing and responding actively to outside excitation. These properties make smart materials a prime canditate for actuation and sensing in next generation control systems. However, modeling and numerically simulating their behavior poses several difficulties. In this work we consider a model for elastomers developed by H. T. Banks, N. J. Lybeck, B. C. Munoz, L. C. Yanyo, formulate this model as an abstract evolution system, and study the long time behavior of its solutions. We remark that the question of existence and uniqueness of solutions for this class of systems is a challenging problem and was only recently solved by H. T. Banks, D. S. Gilliam and V. I. Shubov. Concerning the long time dynamics of the problem, we first prove that the system generates a weak dynamical system, and possesses a weak global attractor. Our main result is the existence of a "strong" dynamical system which has a compact global attractor. With the help of a Lyapunov function we are able to characterize the structure of this attractor. We also give a theorem that guarantees the stability of the global attractor with respect to varying parameters in the system. Our last result concerns the uniform differentiability of the dynamical system.
Generation of Photon-Plasmon Quantum States in Nonlinear Hyperbolic Metamaterials
Poddubny, Alexander N.; Iorsh, Ivan V.; Sukhorukov, Andrey A.
2016-09-01
We develop a general theoretical framework of integrated paired photon-plasmon generation through spontaneous wave mixing in nonlinear plasmonic and metamaterial nanostructures, rigorously accounting for material dispersion and losses in quantum regime through the electromagnetic Green function. We identify photon-plasmon correlations in layered metal-dielectric structures with 70% internal heralding quantum efficiency, and reveal novel mechanism of broadband generation enhancement due to topological transition in hyperbolic metamaterials.
Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws
Xu, Jiang; Kawashima, Shuichi
2014-02-01
The basic existence theory of Kato and Majda enables us to obtain local-in-time classical solutions to generally quasilinear hyperbolic systems in the framework of Sobolev spaces (in x) with higher regularity. However, it remains a challenging open problem whether classical solutions still preserve well-posedness in the case of critical regularity. This paper is concerned with partially dissipative hyperbolic system of balance laws. Under the entropy dissipative assumption, we establish the local well-posedness and blow-up criterion of classical solutions in the framework of Besov spaces with critical regularity with the aid of the standard iteration argument and Friedrichs' regularization method. Then we explore the theory of function spaces and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner spaces (mixed space-time Besov spaces). This fact allows us to capture the dissipation rates generated from the partial dissipative source term and further obtain the global well-posedness and stability by assuming at all times the Shizuta-Kawashima algebraic condition. As a direct application, the corresponding well-posedness and stability of classical solutions to the compressible Euler equations with damping are also obtained.
Maj, Omar
2008-01-01
This is the second part of a work aimed to study complex-phase oscillatory solutions of nonlinear symmetric hyperbolic systems. We consider, in particular, the case of one space dimension. That is a remarkable case, since one can always satisfy the \\emph{naive} coherence condition on the complex phases, which is required in the construction of the approximate solution. Formally the theory applies also in several space dimensions, but the \\emph{naive} coherence condition appears to be too restrictive; the identification of the optimal coherence condition is still an open problem.
Institute of Scientific and Technical Information of China (English)
Yan-ping Chen; Yun-qing Huang
2001-01-01
Improved L2-error estimates are computed for mixed finite element methods for second order nonlinear hyperbolic equations. Results are given for the continuous-time case. The convergence of the values for both the scalar function and the flux is demonstrated. The technique used here covers the lowest-order Raviart-Thomas spaces, as well as the higherorder spaces. A second paper will present the analysis of a fully discrete scheme (Numer.Math. J. Chinese Univ. vol.9, no.2, 2000, 181-192).
Otway, Thomas H
2015-01-01
This text is a concise introduction to the partial differential equations which change from elliptic to hyperbolic type across a smooth hypersurface of their domain. These are becoming increasingly important in diverse sub-fields of both applied mathematics and engineering, for example: • The heating of fusion plasmas by electromagnetic waves • The behaviour of light near a caustic • Extremal surfaces in the space of special relativity • The formation of rapids; transonic and multiphase fluid flow • The dynamics of certain models for elastic structures • The shape of industrial surfaces such as windshields and airfoils • Pathologies of traffic flow • Harmonic fields in extended projective space They also arise in models for the early universe, for cosmic acceleration, and for possible violation of causality in the interiors of certain compact stars. Within the past 25 years, they have become central to the isometric embedding of Riemannian manifolds and the prescription of Gauss curvatur...
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Moawad, S. M., E-mail: smmoawad@hotmail.com [Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef (Egypt)
2015-02-15
In this paper, we present a solution method for constructing exact analytic solutions to magnetohydrodynamics (MHD) equations. The method is constructed via all the trigonometric and hyperbolic functions. The method is applied to MHD equilibria with mass flow. Applications to a solar system concerned with the properties of coronal mass ejections that affect the heliosphere are presented. Some examples of the constructed solutions which describe magnetic structures of solar eruptions are investigated. Moreover, the constructed method can be applied to a variety classes of elliptic partial differential equations which arise in plasma physics.
Numerical methods for nonlinear partial differential equations
Bartels, Sören
2015-01-01
The description of many interesting phenomena in science and engineering leads to infinite-dimensional minimization or evolution problems that define nonlinear partial differential equations. While the development and analysis of numerical methods for linear partial differential equations is nearly complete, only few results are available in the case of nonlinear equations. This monograph devises numerical methods for nonlinear model problems arising in the mathematical description of phase transitions, large bending problems, image processing, and inelastic material behavior. For each of these problems the underlying mathematical model is discussed, the essential analytical properties are explained, and the proposed numerical method is rigorously analyzed. The practicality of the algorithms is illustrated by means of short implementations.
A universal asymptotic regime in the hyperbolic nonlinear Schr\\"odinger equation
Ablowitz, Mark J; Rumanov, Igor
2016-01-01
The appearance of a fundamental long-time asymptotic regime in the two space one time dimensional hyperbolic nonlinear Schr\\"odinger (HNLS) equation is discussed. Based on analytical and extensive numerical simulations an approximate self-similar solution is found for a wide range of initial conditions -- essentially for initial lumps of small to moderate energy. Even relatively large initial amplitudes, which imply strong nonlinear effects, eventually lead to local structures resembling those of the self-similar solution, with appropriate small modifications. These modifications are important in order to properly capture the behavior of the phase of the solution. This solution has aspects that suggest it is a universal attractor emanating from wide ranges of initial data.
Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions
Jameson Graber, P.; Shomberg, Joseph L.
2016-04-01
We establish the well-posedness of a strongly damped semilinear wave equation equipped with nonlinear hyperbolic dynamic boundary conditions. Results are carried out with the presence of a parameter distinguishing whether the underlying operator is analytic, α >0 , or only of Gevrey class, α =0 . We establish the existence of a global attractor for each α \\in ≤ft[0,1\\right], and we show that the family of global attractors is upper-semicontinuous as α \\to 0. Furthermore, for each α \\in ≤ft[0,1\\right] , we show the existence of a weak exponential attractor. A weak exponential attractor is a finite dimensional compact set in the weak topology of the phase space. This result ensures the corresponding global attractor also possesses finite fractal dimension in the weak topology; moreover, the dimension is independent of the perturbation parameter α. In both settings, attractors are found under minimal assumptions on the nonlinear terms.
Decay of solutions of a nonlinear hyperbolic system in noncylindrical domain
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Tania Nunes Rabello
1994-01-01
Full Text Available In this paper we study the existence of solutions of the following nonlinear hyperbolic svstem|u″+A(tu+b(xG(u=f in Qu=0 on Σu(0=uο u1(0=u1where Q is a noncylindrical domain of ℝn+1 with lateral boundary Σ, u−(u1,u2 a vector defined on Q, {A(t, 0≤t≤+∞} is a family of operators in ℒ(Hο1(Ω,H−1(Ω, where A(tu=(A(tu1,A(tu2 and G:ℝ2→ℝ2 a continuous function such that x.G(x≥0, for x∈ℝ2.
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Bapurao C. Dhage
2006-03-01
Full Text Available In this paper, we prove an existence theorem for hyperbolic differential equations in Banach algebras under Lipschitz and Caratheodory conditions. The existence of extremal solutions is also proved under certain monotonicity conditions.
Darboux problem for implicit impulsive partial hyperbolic fractional order differential equations
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Said Abbas
2011-11-01
Full Text Available In this article we investigate the existence and uniqueness of solutions for the initial value problems, for a class of hyperbolic impulsive fractional order differential equations by using some fixed point theorems.
Chen, G.; Zheng, Q.; Coleman, M.; Weerakoon, S.
1983-01-01
This paper briefly reviews convergent finite difference schemes for hyperbolic initial boundary value problems and their applications to boundary control systems of hyperbolic type which arise in the modelling of vibrations. These difference schemes are combined with the primal and the dual approaches to compute the optimal control in the unconstrained case, as well as the case when the control is subject to inequality constraints. Some of the preliminary numerical results are also presented.
New Exact Solutions for New Model Nonlinear Partial Differential Equation
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A. Maher
2013-01-01
Full Text Available In this paper we propose a new form of Padé-II equation, namely, a combined Padé-II and modified Padé-II equation. The mapping method is a promising method to solve nonlinear evaluation equations. Therefore, we apply it, to solve the combined Padé-II and modified Padé-II equation. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions, and elliptic functions.
EXISTENCE AND UNIQUENESS OF THE ENTROPY SOLUTION TO A NONLINEAR HYPERBOLIC EQUATION
Institute of Scientific and Technical Information of China (English)
R.EYMARD; T.Gallouёt; R.Herbin
1995-01-01
This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: ut+div(vf(u)） = 0 in IRN×（0, T）, with initial data u(-, 0) = u0(-) in IRN, where u0 ∈ L∞（IRN) is a given function, v is a divergence-free bounded fnnction of class C1 from IRN × [0, T] to IRN, and f is a 5motion of class C1 from IR to IR. It also gives a result of convergence of a numerical scheme for the discretization of this equation. The authors first show the existence of a “process” solution (which generalizes the concept of entropy weak solutions, and can be obtained by passing to the limit of solutions ofthe numerical scheme). The uniqueness of this entropy process solution is then proven; it isalso proven that the entropy process solution is in fact an entropy weak solution. Hence the existence and uniqueness of the entropy weak solution are proven.
On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations
Glatt-Holtz, Nathan; Mattingly, Jonathan C.; Richards, Geordie
2017-02-01
We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.
On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations
Glatt-Holtz, Nathan; Mattingly, Jonathan C.; Richards, Geordie
2016-08-01
We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.
Bifurcation and stability for a nonlinear parabolic partial differential equation
Chafee, N.
1973-01-01
Theorems are developed to support bifurcation and stability of nonlinear parabolic partial differential equations in the solution of the asymptotic behavior of functions with certain specified properties.
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Ji Juan-Juan
2017-01-01
Full Text Available A table lookup method for solving nonlinear fractional partial differential equations (fPDEs is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.
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S. Vaidyanathan
2013-09-01
Full Text Available This research work describes the modelling of two novel 3-D chaotic systems, the first with a hyperbolic sinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (A and the second with a hyperbolic cosinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (B. In this work, a detailed qualitative analysis of the novel chaotic systems (A and (B has been presented, and the Lyapunov exponents and Kaplan-Yorke dimension of these chaotic systems have been obtained. It is found that the maximal Lyapunov exponent (MLE for the novel chaotic systems (A and (B has a large value, viz. for the system (A and for the system (B. Thus, both the novel chaotic systems (A and (B display strong chaotic behaviour. This research work also discusses the problem of finding adaptive controllers for the global chaos synchronization of identical chaotic systems (A, identical chaotic systems (B and nonidentical chaotic systems (A and (B with unknown system parameters. The adaptive controllers for achieving global chaos synchronization of the novel chaotic systems (A and (B have been derived using adaptive control theory and Lyapunov stability theory. MATLAB simulations have been shown to illustrate the novel chaotic systems (A and (B, and also the adaptive synchronization results derived for the novel chaotic systems (A and (B.
Exact solutions for some nonlinear systems of partial differential equations
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Darwish, A.A. [Department of Mathematics, Faculty of Science, Helwan University (Egypt)], E-mail: profdarwish@yahoo.com; Ramady, A. [Department of Mathematics, Faculty of Science, Beni-Suef University (Egypt)], E-mail: aramady@yahoo.com
2009-04-30
A direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear systems of partial differential equations (PDEs) is used and implemented in a computer algebraic system. New solutions for some nonlinear partial differential equations (NLPDEs) are obtained. Graphs of the solutions are displayed.
Exact solutions for nonlinear partial fractional differential equations
Institute of Scientific and Technical Information of China (English)
Khaled A.Gepreel; Saleh Omran
2012-01-01
In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved (G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.
Chou, Shih-Wei; Lin, Ying-Chieh
2017-08-01
In this paper, we investigate the Cauchy problem for a nonlinear hyperbolic system of balance laws with sources ax g and at h. To get the approximate solutions of our problem, we consider a version of generalized Riemann problem that concentrates the variation of a on a thin T-shaped region of each grid. A new version of Glimm scheme is introduced to construct the approximate solutions and its stability is proved by considering two types of conditions on a. Finally, we verify the consistency of the scheme and the entropy inequality to establish the global existence of entropy solutions.
Jagtap, Ameya Dilip
2015-01-01
A novel explicit and implicit Kinetic Streamlined-Upwind Petrov Galerkin (KSUPG) scheme is presented for hyperbolic equations such as Burgers equation and compressible Euler equations. The proposed scheme performs better than the original SUPG stabilized method in multi-dimensions. To demonstrate the numerical accuracy of the scheme, various numerical experiments have been carried out for 1D and 2D Burgers equation as well as for 1D and 2D Euler equations using Q4 and T3 elements. Furthermore, spectral stability analysis is done for the explicit 2D formulation. Finally, a comparison is made between explicit and implicit versions of the KSUPG scheme.
Analytic solutions of a class of nonlinear partial differential equations
Institute of Scientific and Technical Information of China (English)
ZHANG Hong-qing; DING Qi
2008-01-01
An approach is presented for computing the adjoint operator vector of a class of nonlinear (that is,partial-nonlinear) operator matrices by using the properties of conjugate operators to generalize a previous method proposed by the author.A unified theory is then given to solve a class of nonlinear (partial-nonlinear and including all linear)and non-homogeneous differential equations with a mathematical mechanization method.In other words,a transformation is constructed by homogenization and triangulation,which reduces the original system to a simpler diagonal system.Applications are given to solve some elasticity equations.
Singh, Paramjeet
2010-01-01
Explicit numerical methods based on Lax-Friedrichs and Leap-Frog finite difference approximations are constructed to find the numerical solution of the first-order hyperbolic partial differential equation with point-wise delay or advance, i.e., shift in space. The differential equation involving point-wise delay and advance models the distribution of the time intervals between successive neuronal firings. We construct higher order numerical approximations and discuss their consistency, stability and convergence. The numerical approximations constructed in this paper are consistent, stable under CFL condition, and convergent. We also extend our methods to the higher space dimensions. Some test examples are included to illustrate our approach. These examples verify the theoretical estimates and shows the effect of point-wise delay on the solution.
Global classical solutions for partially dissipative hyperbolic system of balance laws
Xu, Jiang
2012-01-01
This work is concerned with ($N$-component) hyperbolic system of balance laws in arbitrary space dimensions. Under entropy dissipative assumption and the Shizuta-Kawashima algebraic condition, a general theory on the well-posedness of classical solutions in the framework of Chemin-Lerner's spaces with critical regularity is established. To do this, we first explore the functional space theory and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner's spaces. Then this fact allows to prove the local well-posedness for general data and global well-posedness for small data by using the Fourier frequency-localization argument. Finally, we apply the new existence theory to a specific fluid model-the compressible Euler equations with damping, and obtain the corresponding results in critical spaces.
Modified Homotopy Analysis Method for Nonlinear Fractional Partial Differential Equations
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D. Ziane
2017-05-01
Full Text Available In this paper, a combined form of natural transform with homotopy analysis method is proposed to solve nonlinear fractional partial differential equations. This method is called the fractional homotopy analysis natural transform method (FHANTM. The FHANTM can easily be applied to many problems and is capable of reducing the size of computational work. The fractional derivative is described in the Caputo sense. The results show that the FHANTM is an appropriate method for solving nonlinear fractional partial differentia equation.
Approximate solution of a nonlinear partial differential equation
Vajta, M.
2007-01-01
Nonlinear partial differential equations (PDE) are notorious to solve. In only a limited number of cases can we find an analytic solution. In most cases, we can only apply some numerical scheme to simulate the process described by a nonlinear PDE. Therefore, approximate solutions are important for t
Auxiliary equation method for solving nonlinear partial differential equations
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Sirendaoreji,; Jiong, Sun
2003-03-31
By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.
Logarithmic singularities of solutions to nonlinear partial differential equations
Tahara, Hidetoshi
2007-01-01
We construct a family of singular solutions to some nonlinear partial differential equations which have resonances in the sense of a paper due to T. Kobayashi. The leading term of a solution in our family contains a logarithm, possibly multiplied by a monomial. As an application, we study nonlinear wave equations with quadratic nonlinearities. The proof is by the reduction to a Fuchsian equation with singular coefficients.
A three-point backward finite-difference method has been derived for a system of mixed hyperbolic¯¯parabolic (convection¯¯diffusion) partial differential equations (mixed PDEs). The method resorts to the three-point backward differenci...
A three-point backward finite-difference method has been derived for a system of mixed hyperbolic¯¯parabolic (convection¯¯diffusion) partial differential equations (mixed PDEs). The method resorts to the three-point backward differenci...
Entropy and convexity for nonlinear partial differential equations.
Ball, John M; Chen, Gui-Qiang G
2013-12-28
Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue.
Partially Linearizable Class of Nonlinear System with Uncertainty
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Joo, Sung Jun [Samsung Electronics Coporation (Korea, Republic of); Seo, Jin H. [Seoul National University (Korea, Republic of)
1998-03-01
In this paper the problem of robust stabilizing control for nonlinear SISO systems in the presence of uncertainties is studied and we give some geometric conditions for this problem. We also show that if and only if the systems satisfy the proposed conditions it can be transformed into a partially linearized system with unknown parameter using the nominal transformation and nominal feedback linearizing controller. In this paper, we call the above considered class of nonlinear system as partially linearizable system. We design the robust controller which stabilizes the partially linearizable system. (author). 14 refs.
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Jan Hesthaven
2012-02-06
Final report for DOE Contract DE-FG02-98ER25346 entitled Parallel High Order Accuracy Methods Applied to Non-Linear Hyperbolic Equations and to Problems in Materials Sciences. Principal Investigator Jan S. Hesthaven Division of Applied Mathematics Brown University, Box F Providence, RI 02912 Jan.Hesthaven@Brown.edu February 6, 2012 Note: This grant was originally awarded to Professor David Gottlieb and the majority of the work envisioned reflects his original ideas. However, when Prof Gottlieb passed away in December 2008, Professor Hesthaven took over as PI to ensure proper mentoring of students and postdoctoral researchers already involved in the project. This unusual circumstance has naturally impacted the project and its timeline. However, as the report reflects, the planned work has been accomplished and some activities beyond the original scope have been pursued with success. Project overview and main results The effort in this project focuses on the development of high order accurate computational methods for the solution of hyperbolic equations with application to problems with strong shocks. While the methods are general, emphasis is on applications to gas dynamics with strong shocks.
Scalable nonlinear iterative methods for partial differential equations
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Cai, X-C
2000-10-29
We conducted a six-month investigation of the design, analysis, and software implementation of a class of singularity-insensitive, scalable, parallel nonlinear iterative methods for the numerical solution of nonlinear partial differential equations. The solutions of nonlinear PDEs are often nonsmooth and have local singularities, such as sharp fronts. Traditional nonlinear iterative methods, such as Newton-like methods, are capable of reducing the global smooth nonlinearities at a nearly quadratic convergence rate but may become very slow once the local singularities appear somewhere in the computational domain. Even with global strategies such as line search or trust region the methods often stagnate at local minima of {parallel}F{parallel}, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u* of F(u) = 0, we solve, instead, an equivalent nonlinearly preconditioned system G(F(u*)) = 0 whose nonlinearities are more balanced. In this project, we proposed and studied a nonlinear additive Schwarz based parallel nonlinear preconditioner and showed numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, when a traditional inexact Newton method fails.
Residual models for nonlinear partial differential equations
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Garry Pantelis
2005-11-01
Full Text Available Residual terms that appear in nonlinear PDEs that are constructed to generate filtered representations of the variables of the fully resolved system are examined by way of a consistency condition. It is shown that certain commonly used empirical gradient models for the residuals fail the test of consistency and therefore cannot be validated as approximations in any reliable sense. An alternate method is presented for computing the residuals. These residual models are independent of free or artificial parameters and there direct link with the functional form of the system of PDEs which describe the fully resolved system are established.
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Vaidyanathan Sundarapandian
2014-09-01
Full Text Available In this research work, a six-term 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities has been proposed, and its qualitative properties have been detailed. The Lyapunov exponents of the novel jerk system are obtained as L1 = 0.07765,L2 = 0, and L3 = −0.87912. The Kaplan-Yorke dimension of the novel jerk system is obtained as DKY = 2.08833. Next, an adaptive backstepping controller is designed to stabilize the novel jerk chaotic system with two unknown parameters. Moreover, an adaptive backstepping controller is designed to achieve complete chaos synchronization of the identical novel jerk chaotic systems with two unknown parameters. Finally, an electronic circuit realization of the novel jerk chaotic system using Spice is presented in detail to confirm the feasibility of the theoretical model
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V. Ramachandra Prasad
2016-01-01
Full Text Available This article presents the nonlinear free convection boundary layer flow and heat transfer of an incompressible Tangent Hyperbolic non-Newtonian fluid from a vertical porous plate with velocity slip and thermal jump effects. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. The influence of a number of emerging non-dimensional parameters, namely the Weissenberg number (We, the power law index (n, Velocity slip (Sf, Thermal jump (ST, Prandtl number (Pr and dimensionless tangential coordinate ( on velocity and temperature evolution in the boundary layer regime are examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated. Validation with earlier Newtonian studies is presented and excellent correlation achieved. It is found that velocity, skin friction and heat transfer rate (Nusselt number is increased with increasing Weissenberg number (We, whereas the temperature is decreased. Increasing power law index (n enhances velocity and heat transfer rate but decreases temperature and skin friction. An increase in Thermal jump (ST is observed to decrease velocity, temperature, local skin friction and Nusselt number. Increasing Velocity slip (Sf is observed to increase velocity and heat transfer rate but decreases temperature and local skin friction. An increasing Prandtl number, (Pr, is found to decrease both velocity and temperature. The study is relevant to chemical materials processing applications.
Rhebergen, S.; Bokhove, O.; Vegt, van der J.J.W.
2008-01-01
We present space- and space-time discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the weak formulation is that if the system of nonconservative partial diffe
Rhebergen, S.; Bokhove, O.; Vegt, van der J.J.W.
2007-01-01
We present space- and space-time discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the formulation is that if the system of nonconservative partial differenti
Fuhry, Martin; Krivodonova, Lilia
2016-01-01
We present a novel implementation of the modal discontinuous Galerkin (DG) method for hyperbolic conservation laws in two dimensions on graphics processing units (GPUs) using NVIDIA's Compute Unified Device Architecture (CUDA). Both flexible and highly accurate, DG methods accommodate parallel architectures well as their discontinuous nature produces element-local approximations. High performance scientific computing suits GPUs well, as these powerful, massively parallel, cost-effective devices have recently included support for double-precision floating point numbers. Computed examples for Euler equations over unstructured triangle meshes demonstrate the effectiveness of our implementation on an NVIDIA GTX 580 device. Profiling of our method reveals performance comparable to an existing nodal DG-GPU implementation for linear problems.
Zingan, Valentin Nikolaevich
This work develops a discontinuous Galerkin finite element discretization of non- linear hyperbolic conservation equations with efficient and robust high order stabilization built on an entropy-based artificial viscosity approximation. The solutions of equations are represented by elementwise polynomials of an arbitrary degree p > 0 which are continuous within each element but discontinuous on the boundaries. The discretization of equations in time is done by means of high order explicit Runge-Kutta methods identified with respective Butcher tableaux. To stabilize a numerical solution in the vicinity of shock waves and simultaneously preserve the smooth parts from smearing, we add some reasonable amount of artificial viscosity in accordance with the physical principle of entropy production in the interior of shock waves. The viscosity coefficient is proportional to the local size of the residual of an entropy equation and is bounded from above by the first-order artificial viscosity defined by a local wave speed. Since the residual of an entropy equation is supposed to be vanishingly small in smooth regions (of the order of the Local Truncation Error) and arbitrarily large in shocks, the entropy viscosity is almost zero everywhere except the shocks, where it reaches the first-order upper bound. One- and two-dimensional benchmark test cases are presented for nonlinear hyperbolic scalar conservation laws and the system of compressible Euler equations. These tests demonstrate the satisfactory stability properties of the method and optimal convergence rates as well. All numerical solutions to the test problems agree well with the reference solutions found in the literature. We conclude that the new method developed in the present work is a valuable alternative to currently existing techniques of viscous stabilization.
REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS
Institute of Scientific and Technical Information of China (English)
YeCaier; PanZuliang
2003-01-01
Nonlinear partial differetial equation(NLPDE)is converted into ordinary differential equation(ODE)via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.
Exact periodic wave solutions for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
El-Wakil, S.A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt); Elgarayhi, A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)]. E-mail: elgarayhi@yahoo.com; Elhanbaly, A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)
2006-08-15
The periodic wave solutions for some nonlinear partial differential equations, including generalized Klein-Gordon equation, Kadomtsev-Petviashvili (KP) equation and Boussinesq equations, are obtained by using the solutions of Jacobi elliptic equation. Under limit conditions, exact solitary wave solutions, shock wave solutions and triangular periodic wave solutions have been recovered.
Exact solutions for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Peng, Yan-Ze
2003-08-11
Exact solutions to some nonlinear partial differential equations, including (2+1)-dimensional breaking soliton equation, sine-Gordon equation and double sine-Gordon equation, are studied by means of the mapping method proposed by the author recently. Many new results are presented. A simple review of the method is finally given.
Advances in nonlinear partial differential equations and stochastics
Kawashima, S
1998-01-01
In the past two decades, there has been great progress in the theory of nonlinear partial differential equations. This book describes the progress, focusing on interesting topics in gas dynamics, fluid dynamics, elastodynamics etc. It contains ten articles, each of which discusses a very recent result obtained by the author. Some of these articles review related results.
Large Time Asymptotics for Solutions of Nonlinear Partial Differential Equations
Sachdev, PL
2010-01-01
A large number of physical phenomena are modeled by nonlinear partial differential equations, subject to appropriate initial/boundary conditions. This title presents the constructive mathematical techniques. It deals with the asymptotic methods which include self-similarity, balancing argument, and matched asymptotic expansions
Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations.
Baranwal, Vipul K; Pandey, Ram K; Singh, Om P
2014-01-01
We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.
Brahim Tellab; Kamel Haouam
2016-01-01
In this paper, we investigate the existence and uniqueness of solutions for second order nonlinear fractional differential equation with integral boundary conditions. Our result is an application of the Banach contraction principle and the Krasnoselskii fixed point theorem.
Dynamics of kinks in one- and two-dimensional hyperbolic models with quasidiscrete nonlinearities.
Rotstein, H G; Mitkov, I; Zhabotinsky, A M; Epstein, I R
2001-06-01
We study the evolution of fronts in the Klein-Gordon equation when the nonlinear term is inhomogeneous. Extending previous works on homogeneous nonlinear terms, we describe the derivation of an equation governing the front motion, which is strongly nonlinear, and, for the two-dimensional case, generalizes the damped Born-Infeld equation. We study the motion of one- and two-dimensional fronts finding a much richer dynamics than in the homogeneous system case, leading, in most cases, to the stabilization of one phase inside the other. For a one-dimensional front, the function describing the inhomogeneity of the nonlinear term acts as a "potential function" for the motion of the front, i.e., a front initially placed between two of its local maxima asymptotically approaches the intervening minimum. Two-dimensional fronts, with radial symmetry and without dissipation can either shrink to a point in finite time, grow unboundedly, or their radius can oscillate, depending on the initial conditions. When dissipation effects are present, the oscillations either decay spirally or not depending on the value of the damping dissipation parameter. For fronts with a more general shape, we present numerical simulations showing the same behavior.
Painlevé analysis for nonlinear partial differential equations
Musette, M
1998-01-01
The Painlevé analysis introduced by Weiss, Tabor and Carnevale (WTC) in 1983 for nonlinear partial differential equations (PDE's) is an extension of the method initiated by Painlevé and Gambier at the beginning of this century for the classification of algebraic nonlinear differential equations (ODE's) without movable critical points. In these lectures we explain the WTC method in its invariant version introduced by Conte in 1989 and its application to solitonic equations in order to find algorithmically their associated so-called ``integrable'' equations but they are generically no more valid for equations modelising physical phenomema. Belonging to this second class, some equations called ``partially integrable'' sometimes keep remnants of integrability. In that case, the singularity analysis may also be useful for building closed form analytic solutions, which necessarily % Conte agree with the singularity structure of the equations. We display the privileged role played by the Riccati equation and syste...
Nonlinear partial differential equations for scientists and engineers
Debnath, Lokenath
1997-01-01
"An exceptionally complete overview. There are numerous examples and the emphasis is on applications to almost all areas of science and engineering. There is truly something for everyone here. This reviewer feels that it is a very hard act to follow, and recommends it strongly. [This book] is a jewel." ---Applied Mechanics Review (Review of First Edition) This expanded and revised second edition is a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied applications. Building upon the successful material of the first book, this edition contains updated modern examples and applications from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation. Methods and properties of solutions are presented, along with their physical significance, making the book more useful for a diverse readership. Topics and key features: * Thorough coverage of derivation and methods of soluti...
Nonclassical Symmetries for Nonlinear Partial Differential Equations via Compatibility
Institute of Scientific and Technical Information of China (English)
Mostafa F. El-Sabbagh; Ahmad T. Ali
2011-01-01
The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The （2＋1）-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.
Kernel Partial Least Squares for Nonlinear Regression and Discrimination
Rosipal, Roman; Clancy, Daniel (Technical Monitor)
2002-01-01
This paper summarizes recent results on applying the method of partial least squares (PLS) in a reproducing kernel Hilbert space (RKHS). A previously proposed kernel PLS regression model was proven to be competitive with other regularized regression methods in RKHS. The family of nonlinear kernel-based PLS models is extended by considering the kernel PLS method for discrimination. Theoretical and experimental results on a two-class discrimination problem indicate usefulness of the method.
Symposium on Nonlinear Semigroups, Partial Differential Equations and Attractors
Zachary, Woodford
1987-01-01
The original idea of the organizers of the Washington Symposium was to span a fairly narrow range of topics on some recent techniques developed for the investigation of nonlinear partial differential equations and discuss these in a forum of experts. It soon became clear, however, that the dynamical systems approach interfaced significantly with many important branches of applied mathematics. As a consequence, the scope of this resulting proceedings volume is an enlarged one with coverage of a wider range of research topics.
Boundary layers for self-similar viscous approximations of nonlinear hyperbolic systems
Christoforou, Cleopatra
2011-01-01
We provide a precise description of the set of residual boundary conditions generated by the self-similar viscous approximation introduced by Dafermos et al. We then apply our results, valid for both conservative and non conservative systems, to the analysis of the boundary Riemann problem and we show that, under appropriate assumptions, the limits of the self-similar and the classical vanishing viscosity approximation coincide. We require neither genuinely nonlinearity nor linear degeneracy of the characteristic fields.
The Mathematical Analysis for Peristaltic Flow of Hyperbolic Tangent Fluid in a Curved Channel
Institute of Scientific and Technical Information of China (English)
S.Nadeem; E.N.Maraj
2013-01-01
In the present paper,we have investigated the peristaltic flow of hyperbolic tangent fluid in a curved channel.The governing equations of hyperbolic tangent fluid model for curved channel are derived including the effects of curvature.The highly nonlinear partial differential equations are simplified by using the wave frame transformation,long wave length and low Reynolds number assumptions.The reduced nonlinear partial differential equation is solved analytically with the help of homotopy perturbation method (HPM).The physical features of pertinent parameters have been discussed by plotting the graphs of pressure rise and stream functions.
Nonlinear evolution operators and semigroups applications to partial differential equations
Pavel, Nicolae H
1987-01-01
This research monograph deals with nonlinear evolution operators and semigroups generated by dissipative (accretive), possibly multivalued operators, as well as with the application of this theory to partial differential equations. It shows that a large class of PDE's can be studied via the semigroup approach. This theory is not available otherwise in the self-contained form provided by these Notes and moreover a considerable part of the results, proofs and methods are not to be found in other books. The exponential formula of Crandall and Liggett, some simple estimates due to Kobayashi and others, the characterization of compact semigroups due to Brézis, the proof of a fundamental property due to Ursescu and the author and some applications to PDE are of particular interest. Assuming only basic knowledge of functional analysis, the book will be of interest to researchers and graduate students in nonlinear analysis and PDE, and to mathematical physicists.
Solutions in the large for some nonlinear hyperbolic conservation laws of gas dynamics
Temple, J. B.
1980-03-01
The constraints under which a gas at a certain state will evolve can be given by three partial differential equations which express the conservation of mass, momentum, and energy. A particular energy function was discovered for which there is a global weak solution for bounded measurable data having finite total variation. This energy function models an ideal gas, and is given by the formula e = - lambda eta V + (S/R). The following general existence theorem is also obtained: let e sub epsilon (v,S) be any smooth one parameter family of energy functions such that at epsilon = 0 the energy is given by e (v,S) = - lambda eta V + (S/R). It is proven that there exists a constant C independent of epsilon, such that, if the total variation of the inertial data C, then there exists a global weak solution to the equations. An existence theorem for polytropic gases was also obtained.
On a Nonlinear Partial Integro-Differential Equation
Abergel, Frederic
2009-01-01
Consistently fitting vanilla option surfaces is an important issue when it comes to modelling in finance. Local volatility models introduced by Dupire in 1994 are widely used to price and manage the risks of structured products. However, the inconsistencies observed between the dynamics of the smile in those models and in real markets motivate researches for stochastic volatility modelling. Combining both those ideas to form Local and Stochastic Volatility models is of interest for practitioners. In this paper, we study the calibration of the vanillas in those models. This problem can be written as a nonlinear and nonlocal partial differential equation, for which we prove short-time existence of solutions.
Exact travelling wave solutions of nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish) Suez Canal University, AL-Arish 45111 (Egypt)]. E-mail: asoliman_99@yahoo.com; Abdou, M.A. [Theoretical Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-04-15
An extended Fan-sub equation method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. The key idea of this method is to take full advantage of the general elliptic equation, involving five parameters, which has more new solutions and whose degeneracies can lead to special sub equation involving three parameters. As an illustration of the extended Fan method, more new solutions are obtained for three models namely, generalized KdV, Drinfeld-Sokolov system and RLW equation.
Approximation on computing partial sum of nonlinear differential eigenvalue problems
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
In computing the electronic structure and energy band in a system of multi-particles, quite a large number of problems are referred to the acquisition of obtaining the partial sum of densities and energies using the “first principle”. In the conventional method, the so-called self-consistency approach is limited to a small scale because of high computing complexity. In this paper, the problem of computing the partial sum for a class of nonlinear differential eigenvalue equations is changed into the constrained functional minimization. By space decomposition and perturbation method, a secondary approximating formula for the minimal is provided. It is shown that this formula is more precise and its quantity of computation can be reduced significantly
Cantor-Type Sets in Hyperbolic Numbers
Balankin, A. S.; Bory-Reyes, J.; Luna-Elizarrarás, M. E.; Shapiro, M.
2016-12-01
The construction of the ternary Cantor set is generalized into the context of hyperbolic numbers. The partial order structure of hyperbolic numbers is revealed and the notion of hyperbolic interval is defined. This allows us to define a general framework of the fractal geometry on the hyperbolic plane. Three types of the hyperbolic analogues of the real Cantor set are identified. The complementary nature of the real Cantor dust and the real Sierpinski carpet on the hyperbolic plane are outlined. The relevance of these findings in the context of modern physics are briefly discussed.
Stability and boundary stabilization of 1-D hyperbolic systems
Bastin, Georges
2016-01-01
This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices. The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary...
DIFFERENCE METHODS FOR A NON-LINEAR ELLIPTIC SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS,
DIFFERENCE EQUATIONS, ITERATIONS), (*ITERATIONS, DIFFERENCE EQUATIONS), (* PARTIAL DIFFERENTIAL EQUATIONS , BOUNDARY VALUE PROBLEMS), EQUATIONS, FUNCTIONS(MATHEMATICS), SEQUENCES(MATHEMATICS), NONLINEAR DIFFERENTIAL EQUATIONS
Vassiliev, V. A.
2016-10-01
We enumerate the local Petrovskii lacunas (that is, the domains of local regularity of the principal fundamental solutions of strictly hyperbolic PDEs with constant coefficients in {R}^N) close to parabolic singular points of their wavefronts (that is, at the points of types P_8^1, P_8^2, +/- X_9, X_9^1, X_9^2, J10^1 and J10^3). These points form the next most difficult family of classes in the natural classification of singular points after the so-called simple singularities A_k, D_k, E_6, E_7 and E_8, which have been investigated previously. Also we present a computer program which counts the topologically distinct morsifications of critical points of smooth functions, and hence also the local components of the complement of a generic wavefront at its singular points. Bibliography: 22 titles.
Mathematical Methods in Wave Propagation: Part 2--Non-Linear Wave Front Analysis
Jeffrey, Alan
1971-01-01
The paper presents applications and methods of analysis for non-linear hyperbolic partial differential equations. The paper is concluded by an account of wave front analysis as applied to the piston problem of gas dynamics. (JG)
一类具阻尼的非线性双曲方程解的blow-up%The blow-up of solutions of a class of nonlinear damped hyperbolic equation
Institute of Scientific and Technical Information of China (English)
呼青英; 陆军
2003-01-01
The blow-up property of a nonlinear damped hyperbolic equation,which describes the motion of the neo-Hookean elastomer rod,is proven.%本文讨论了一类描述新胡克弹性杆运动的具阻尼的非线性双曲方程解的blow up性质.
Energy Technology Data Exchange (ETDEWEB)
Zhao Xiqiang [Department of Mathematics, Ocean University of China, Qingdao Shandong 266071 (China)] e-mail: zhaodss@yahoo.com.cn; Wang Limin [Shandong University of Technology, Zibo Shandong 255049 (China); Sun Weijun [Shandong University of Technology, Zibo Shandong 255049 (China)
2006-04-01
In this letter, a new method, called the repeated homogeneous balance method, is proposed for seeking the traveling wave solutions of nonlinear partial differential equations. The Burgers-KdV equation is chosen to illustrate our method. It has been confirmed that more traveling wave solutions of nonlinear partial differential equations can be effectively obtained by using the repeated homogeneous balance method.
Recent topics in non-linear partial differential equations 4
Mimura, M
1989-01-01
This fourth volume concerns the theory and applications of nonlinear PDEs in mathematical physics, reaction-diffusion theory, biomathematics, and in other applied sciences. Twelve papers present recent work in analysis, computational analysis of nonlinear PDEs and their applications.
Sahadevan, R.; Prakash, P.
2017-01-01
We show how invariant subspace method can be extended to time fractional coupled nonlinear partial differential equations and construct their exact solutions. Effectiveness of the method has been illustrated through time fractional Hunter-Saxton equation, time fractional coupled nonlinear diffusion system, time fractional coupled Boussinesq equation and time fractional Whitman-Broer-Kaup system. Also we explain how maximal dimension of the time fractional coupled nonlinear partial differential equations can be estimated.
Discontinuous Galerkin Method for Hyperbolic Conservation Laws
Mousikou, Ioanna
2016-11-11
Hyperbolic conservation laws form a special class of partial differential equations. They describe phenomena that involve conserved quantities and their solutions show discontinuities which reflect the formation of shock waves. We consider one-dimensional systems of hyperbolic conservation laws and produce approximations using finite difference, finite volume and finite element methods. Due to stability issues of classical finite element methods for hyperbolic conservation laws, we study the discontinuous Galerkin method, which was recently introduced. The method involves completely discontinuous basis functions across each element and it can be considered as a combination of finite volume and finite element methods. We illustrate the implementation of discontinuous Galerkin method using Legendre polynomials, in case of scalar equations and in case of quasi-linear systems, and we review important theoretical results about stability and convergence of the method. The applications of finite volume and discontinuous Galerkin methods to linear and non-linear scalar equations, as well as to the system of elastodynamics, are exhibited.
ON THE PARTIAL EQUIASYMPTOTIC STABILITY OF NONLINEAR TIME-VARYING DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
JianJigui; JiangMinghui; ShenYanjun
2005-01-01
In this paper, the problem of partial equiasymptotic stability for nonlinear time-varying differential equations are analyzed. A sufficient condition of partial stability and a set of sufficient conditions of partial equiasymptotic stability are given. Some of these conditions allow the derivative of Lyapunov function to be positive. Finally, several numerical examples are also given to illustrate the main results.
Directory of Open Access Journals (Sweden)
Elsayed M.E. Zayed
2016-02-01
Full Text Available In this article, the modified extended tanh-function method is employed to solve fractional partial differential equations in the sense of the modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into nonlinear ordinary differential equations of integer orders. For illustrating the validity of this method, we apply it to four nonlinear equations namely, the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space–time fractional nonlinear Whitham–Broer–Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations.
An effective analytic approach for solving nonlinear fractional partial differential equations
Ma, Junchi; Zhang, Xiaolong; Liang, Songxin
2016-08-01
Nonlinear fractional differential equations are widely used for modelling problems in applied mathematics. A new analytic approach with two parameters c1 and c2 is first proposed for solving nonlinear fractional partial differential equations. These parameters are used to improve the accuracy of the resulting series approximations. It turns out that much more accurate series approximations are obtained by choosing proper values of c1 and c2. To demonstrate the applicability and effectiveness of the new method, two typical fractional partial differential equations, the nonlinear gas dynamics equation and the nonlinear KdV-Burgers equation, are solved.
The numerical dynamic for highly nonlinear partial differential equations
Lafon, A.; Yee, H. C.
1992-01-01
Problems associated with the numerical computation of highly nonlinear equations in computational fluid dynamics are set forth and analyzed in terms of the potential ranges of spurious behaviors. A reaction-convection equation with a nonlinear source term is employed to evaluate the effects related to spatial and temporal discretizations. The discretization of the source term is described according to several methods, and the various techniques are shown to have a significant effect on the stability of the spurious solutions. Traditional linearized stability analyses cannot provide the level of confidence required for accurate fluid dynamics computations, and the incorporation of nonlinear analysis is proposed. Nonlinear analysis based on nonlinear dynamical systems complements the conventional linear approach and is valuable in the analysis of hypersonic aerodynamics and combustion phenomena.
Nonlinear Characteristics of an Intense Laser Pulse Propagating in Partially Stripped Plasmas
Institute of Scientific and Technical Information of China (English)
HU Qiang-Lin; LIU Shi-Bing; CHEN Tao; JIANG Yi-Jian
2005-01-01
The nonlinear optic characteristics of an intense laser pulse propagating in partially stripped plasmas are investigated analytically. The phase and group velocity of the laser pulse propagation as well as the three general expressions governing the nonlinear optic behavior, based on the photon number conservation, are obtained by considering the partially stripped plasma as a nonlinear optic medium. The numerical result shows that the presence of the bound electrons in partially stripped plasma can significantly change the propagating property of the intense laser pulse.
Oscillation criteria for a class of nonlinear partial differential equations
Directory of Open Access Journals (Sweden)
Robert Marik
2002-03-01
Full Text Available This paper presents sufficient conditions on the function $c(x$ to ensure that every solution of partial differential equation $$ sum_{i=1}^{n}{partial over partial x_i} Phi_{p}({partial u over partial x_i}+B(x,u=0, quad Phi_p(u:=|u|^{p-1}mathop{ m sgn} u. quad p>1 $$ is weakly oscillatory, i.e. has zero outside of every ball in $mathbb{R}^n$. The main tool is modified Riccati technique developed for Schrodinger operator by Noussair and Swanson [11].
Height in Splittings of Hyperbolic Groups
Indian Academy of Sciences (India)
Mahan Mitra
2004-02-01
Suppose is a hyperbolic subgroup of a hyperbolic group . Assume there exists > 0 such that the intersection of essentially distinct conjugates of is always finite. Further assume splits over with hyperbolic vertex and edge groups and the two inclusions of are quasi-isometric embeddings. Then is quasiconvex in . This answers a question of Swarup and provides a partial converse to the main theorem of [23].
H-OSCILLATION OF NEUTRAL VECTOR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS%中立型向量双曲偏微分方程的H-振动性
Institute of Scientific and Technical Information of China (English)
罗李平; 俞元洪
2011-01-01
In this article, we investigate the oscillations of boundary value problems for a class of neutral vector hyperbolic partial differential equations. By employing the concept of H-oscillation introduced by Domslak and the method of reducing dimension with scalar product, we reduce the multi-dimensional oscillation problems to the nonexistence problems of positive solutions of one-dimensional functional differential inequalities, some new sufficient conditions for the H-oscillation of all solutions of the boundary value problems are obtained under Dirichlet boundary condition, where H is a unit vector of Rm.%本文研究了一类中立型向量双曲偏微分方程边值问题的振动性.利用Domslak引进的H-振动的概念及内积降维的方法,将多维振动问题化为一维泛函微分不等式正解的不存在性问题,获得了该类边值问题在Dirichlet边界条件下所有解H-振动的若干新的充分条件,其中H是Rm中的单位向量.
Forced hyperbolic mean curvature flow
Mao, Jing
2012-01-01
In this paper, we investigate two hyperbolic flows obtained by adding forcing terms in direction of the position vector to the hyperbolic mean curvature flows in \\cite{klw,hdl}. For the first hyperbolic flow, as in \\cite{klw}, by using support function, we reduce it to a hyperbolic Monge-Amp$\\grave{\\rm{e}}$re equation successfully, leading to the short-time existence of the flow by the standard theory of hyperbolic partial differential equation. If the initial velocity is non-negative and the coefficient function of the forcing term is non-positive, we also show that there exists a class of initial velocities such that the solution of the flow exists only on a finite time interval $[0,T_{max})$, and the solution converges to a point or shocks and other propagating discontinuities are generated when $t\\rightarrow{T_{max}}$. These generalize the corresponding results in \\cite{klw}. For the second hyperbolic flow, as in \\cite{hdl}, we can prove the system of partial differential equations related to the flow is ...
Hitting spheres on hyperbolic spaces
Cammarota, Valentina
2011-01-01
For a hyperbolic Brownian motion on the Poincar\\'e half-plane $\\mathbb{H}^2$, starting from a point of hyperbolic coordinates $z=(\\eta, \\alpha)$ inside a hyperbolic disc $U$ of radius $\\bar{\\eta}$, we obtain the probability of hitting the boundary $\\partial U$ at the point $(\\bar \\eta,\\bar \\alpha)$. For $\\bar{\\eta} \\to \\infty$ we derive the asymptotic Cauchy hitting distribution on $\\partial \\mathbb{H}^2$ and for small values of $\\eta$ and $\\bar \\eta$ we obtain the classical Euclidean Poisson kernel. The exit probabilities $\\mathbb{P}_z\\{T_{\\eta_1}
Poddubny, Alexander; Iorsh, Ivan; Belov, Pavel; Kivshar, Yuri
2013-12-01
Electromagnetic metamaterials, artificial media created by subwavelength structuring, are useful for engineering electromagnetic space and controlling light propagation. Such materials exhibit many unusual properties that are rarely or never observed in nature. They can be employed to realize useful functionalities in emerging metadevices based on light. Here, we review hyperbolic metamaterials -- one of the most unusual classes of electromagnetic metamaterials. They display hyperbolic (or indefinite) dispersion, which originates from one of the principal components of their electric or magnetic effective tensor having the opposite sign to the other two principal components. Such anisotropic structured materials exhibit distinctive properties, including strong enhancement of spontaneous emission, diverging density of states, negative refraction and enhanced superlensing effects.
A higher order lattice BGK model for simulating some nonlinear partial differential equations
Institute of Scientific and Technical Information of China (English)
LAI HuiLin; MA ChangFeng
2009-01-01
In this paper, we consider a one-dimensional nonlinear partial differential equation that has the form ut + αuux+βunux-γuxx+δuxxx= F(U). A higher order lattice Bhatnager-Gross-Krook (BGK) model with an amending-function is proposed. With the Chapman-Enskog expansion, different kinds of nonlinear partial differential equations are recovered correctly from the continuous Boltzmann equation. The numerical results show that this method is very effective.
A higher order lattice BGK model for simulating some nonlinear partial differential equations
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
In this paper, we consider a one-dimensional nonlinear partial differential equation that has the form ut + αuux + βunux - γuxx + δuxxx = F(u). A higher order lattice Bhatnager-Gross-Krook (BGK) model with an amending-function is proposed. With the Chapman-Enskog expansion, different kinds of nonlinear partial differential equations are recovered correctly from the continuous Boltzmann equation. The numerical results show that this method is very effective.
Analytic continuation of solutions of some nonlinear convolution partial differential equations
Directory of Open Access Journals (Sweden)
Hidetoshi Tahara
2015-01-01
Full Text Available The paper considers a problem of analytic continuation of solutions of some nonlinear convolution partial differential equations which naturally appear in the summability theory of formal solutions of nonlinear partial differential equations. Under a suitable assumption it is proved that any local holomorphic solution has an analytic extension to a certain sector and its extension has exponential growth when the variable goes to infinity in the sector.
A unified lattice Boltzmann model for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Chai Zhenhua [State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074 (China); Shi Baochang [Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074 (China)], E-mail: sbchust@126.com; Zheng Lin [Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074 (China)
2008-05-15
In this paper, a unified and novel lattice Boltzmann model is proposed for solving nonlinear partial differential equation that has the form DU{sub t} + {alpha}UU{sub x} + {beta}U{sup n}U{sub x} - {gamma}U{sub xx} + {delta} U{sub xxx} = F(x,t). Numerical results agree well with the analytical solutions and results derived by existing literature, which indicates the present model is satisfactory and efficient on solving nonlinear partial differential equations.
Applications of algebraic method to exactly solve some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Darwish, A.A. [Department of Mathematics, Faculty of Science, Helwan University (Egypt)]. E-mail: profdarwish@yahoo.com; Ramady, A. [Department of Mathematics, Faculty of Science, Beni-Suef University (Egypt)]. E-mail: aramady@yahoo.com
2007-08-15
A direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear evolution equations is used and implemented in a computer algebraic system. New solutions for some nonlinear partial differential equations (NLPDE's) are obtained. Graphs of the solutions are displayed.
Institute of Scientific and Technical Information of China (English)
杜宁
2001-01-01
Mixed finite element method is used to treat a kind of second-order nonlinear hyperbolic equations with absorbing boundary conditions. explicit-intime procedures are formulated and analyzed. Optimal L2-in-space error estimates are derived.
New Exact Travelling Wave and Periodic Solutions of Discrete Nonlinear Schr(o)dinger Equation
Institute of Scientific and Technical Information of China (English)
YANG Qin; DAI Chao-Qing; ZHANG Jie-Fang
2005-01-01
Some new exact travelling wave and period solutions of discrete nonlinear Schrodinger equation are found by using a hyperbolic tangent function approach, which was usually presented to find exact travelling wave solutions of certain nonlinear partial differential models. Now we can further extend the new algorithm to other nonlinear differentialdifferent models.
Solving Nonlinear Partial Differential Equations with Maple and Mathematica
Shingareva, Inna K
2011-01-01
The emphasis of the book is given in how to construct different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple an
Building systems from simple hyperbolic ones
Zwart, H.; Le Gorrec, Y.; Maschke, B.
2016-01-01
In this article we introduce a technique that derives from the existence and uniqueness of solutions to a simple hyperbolic partial differential equation (p.d.e.) the existence and uniqueness of solutions to hyperbolic and parabolic p.d.e.’s. Among others, we show that starting with an impedance pas
Building systems from simple hyperbolic ones
Zwart, Heiko J.; Le Gorrec, Y.; Maschke, B.
In this article we introduce a technique that derives from the existence and uniqueness of solutions to a simple hyperbolic partial differential equation (p.d.e.) the existence and uniqueness of solutions to hyperbolic and parabolic p.d.e.’s. Among others, we show that starting with an impedance
Nonlinear partial differential equations: Integrability, geometry and related topics
Krasil'shchik, Joseph; Rubtsov, Volodya
2017-03-01
Geometry and Differential Equations became inextricably entwined during the last one hundred fifty years after S. Lie and F. Klein's fundamental insights. The two subjects go hand in hand and they mutually enrich each other, especially after the "Soliton Revolution" and the glorious streak of Symplectic and Poisson Geometry methods in the context of Integrability and Solvability problems for Non-linear Differential Equations.
Non-Linearity in Wide Dynamic Range CMOS Image Sensors Utilizing a Partial Charge Transfer Technique
Directory of Open Access Journals (Sweden)
Izhal Abdul Halin
2009-11-01
Full Text Available The partial charge transfer technique can expand the dynamic range of a CMOS image sensor by synthesizing two types of signal, namely the long and short accumulation time signals. However the short accumulation time signal obtained from partial transfer operation suffers of non-linearity with respect to the incident light. In this paper, an analysis of the non-linearity in partial charge transfer technique has been carried, and the relationship between dynamic range and the non-linearity is studied. The results show that the non-linearity is caused by two factors, namely the current diffusion, which has an exponential relation with the potential barrier, and the initial condition of photodiodes in which it shows that the error in the high illumination region increases as the ratio of the long to the short accumulation time raises. Moreover, the increment of the saturation level of photodiodes also increases the error in the high illumination region.
Non-Linearity in Wide Dynamic Range CMOS Image Sensors Utilizing a Partial Charge Transfer Technique
Shafie, Suhaidi; Kawahito, Shoji; Halin, Izhal Abdul; Hasan, Wan Zuha Wan
2009-01-01
The partial charge transfer technique can expand the dynamic range of a CMOS image sensor by synthesizing two types of signal, namely the long and short accumulation time signals. However the short accumulation time signal obtained from partial transfer operation suffers of non-linearity with respect to the incident light. In this paper, an analysis of the non-linearity in partial charge transfer technique has been carried, and the relationship between dynamic range and the non-linearity is studied. The results show that the non-linearity is caused by two factors, namely the current diffusion, which has an exponential relation with the potential barrier, and the initial condition of photodiodes in which it shows that the error in the high illumination region increases as the ratio of the long to the short accumulation time raises. Moreover, the increment of the saturation level of photodiodes also increases the error in the high illumination region. PMID:22303133
Institute of Scientific and Technical Information of China (English)
Ronghua Huan; Lincong Chen; Weiliang Jin; Weiqiu Zhu
2009-01-01
An optimal vibration control strategy for partially observable nonlinear quasi Hamil-tonian systems with actuator saturation is proposed. First, a controlled partially observable non-linear system is converted into a completely observable linear control system of finite dimension based on the theorem due to Charalambous and Elliott. Then the partially averaged Ito stochas-tic differential equations and dynamical programming equation associated with the completely observable linear system are derived by using the stochastic averaging method and stochastic dynamical programming principle, respectively. The optimal control law is obtained from solving the final dynamical programming equation. The results show that the proposed control strategy has high control effectiveness and control efficiency.
Modified extended tanh-function method for solving nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
El-Wakil, S.A. [Department of Physics, Faculty of Science, Theoretical Research Group, Mansoura University, 35516 Mansoura (Egypt); Abdou, M.A. [Department of Physics, Faculty of Science, Theoretical Research Group, Mansoura University, 35516 Mansoura (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-03-15
Based on computerized symbolic computation, modified extended tanh-method for constructing multiple travelling wave solutions of nonlinear evolution equations is presented and implemented in a computer algebraic system. Applying this method, with the aid of Maple, we consider some nonlinear evolution equations in mathematical physics such as the nonlinear partial differential equation, nonlinear Fisher-type equation, ZK-BBM equation, generalized Burgers-Fisher equation and Drinfeld-Sokolov system. As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extended tanh-function method and other more sophisticated methods.
Investigation of the nonlinear dynamics of a partially cracked plate
Energy Technology Data Exchange (ETDEWEB)
Israr, A [School of Engineering and Physical Sciences, Heriot Watt University - Dubai Campus, Block 2, Dubai International Academic City, P O Box 294345, Dubai (United Arab Emirates); Atepor, L, E-mail: a.israr@hw.ac.u, E-mail: katepor@yahoo.co [Department of Mechanical Engineering, James Watt South Building, University of Glasgow, Glasgow, G12 8QQ Scotland (United Kingdom)
2009-08-01
In this paper the nonlinear vibration of an aircraft panel structure modelled as an isotropic cracked plate and subjected to transverse harmonic excitation is considered for studying the dynamic response, both analytically and experimentally. A crack is arbitrarily located at the centre of the plate, consisting of a continuous line. This mathematical model is in the form of Duffing equation with a cubic nonlinear term. The perturbation method of multiple scales is used to solve the algebraic equation, and then investigated with the results of the direct integration within Mathematica{sup TM} and finite element analysis in ABAQUS for the first mode only. In addition, experimental measurements are also carried out to verify the dependence of the cracked plate's fundamental mode shape and resonance frequency on the vibration displacement amplitude. An extermely close agreement between these results is observed.
Directory of Open Access Journals (Sweden)
M. R.Odekunle
2014-08-01
Full Text Available Tau method which is an economized polynomial technique for solving ordinary and partial differential equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution of linearizedPartial differential Equation without first rewriting them in terms of other known functions as commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique. The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial differential equations.
Calculus of variations and nonlinear partial differential equations
Marcellini, Paolo
2008-01-01
This volume provides the texts of lectures given by L. Ambrosio, L. Caffarelli, M. Crandall, L.C. Evans, N. Fusco at the Summer course held in Cetraro (Italy) in 2005. These are introductory reports on current research by world leaders in the fields of calculus of variations and partial differential equations. The topics discussed are transport equations for nonsmooth vector fields, homogenization, viscosity methods for the infinite Laplacian, weak KAM theory and geometrical aspects of symmetrization. A historical overview of all CIME courses on the calculus of variations and partial differential equations is contributed by Elvira Mascolo.
Zhong, Xian-Qiong; Zhang, Xiao-Xia; Du, Xian-Tong; Liu, Yong; Cheng, Ke
2015-10-01
The approximate analytical frequency chirps and the critical distances for cross-phase modulation induced optical wave breaking (OWB) of the initial hyperbolic-secant optical pulses propagating in optical fibers with quintic nonlinearity (QN) are presented. The pulse evolutions in terms of the frequency chirps, shapes and spectra are numerically calculated in the normal dispersion regime. The results reveal that, depending on different QN parameters, the traditional OWB or soliton or soliton pulse trains may occur. The approximate analytical critical distances are found to be in good agreement with the numerical ones only for the traditional OWB whereas the approximate analytical frequency chirps accords well with the numerical ones at the initial evolution stages of the pulses. Supported by the Postdoctoral Fund of China under Grant No. 2011M501402, the Key Project of Chinese Ministry of Education under Grant No. 210186, the Major Project of Natural Science Supported by the Educational Department of Sichuan Province under Grant No. 13ZA0081, the Key Project of National Natural Science Foundation of China under Grant No 61435010, and the National Natural Science Foundation of China under Grant No. 61275039
Differential geometry techniques for sets of nonlinear partial differential equations
Estabrook, Frank B.
1990-01-01
An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.
Nonlinear Programming Approach to Optimal Scaling of Partially Ordered Categories
Nishisato, Shizuhiko; Arri, P. S.
1975-01-01
A modified technique of separable programming was used to maximize the squared correlation ratio of weighted responses to partially ordered categories. The technique employs a polygonal approximation to each single-variable function by choosing mesh points around the initial approximation supplied by Nishisato's method. Numerical examples were…
Bijl, Hester; Meister, Andreas; Sonar, Thomas
2013-01-01
In January 2012 an Oberwolfach workshop took place on the topic of recent developments in the numerics of partial differential equations. Focus was laid on methods of high order and on applications in Computational Fluid Dynamics. The book covers most of the talks presented at this workshop.
Anomalous diffraction in hyperbolic materials
Alberucci, Alessandro; Boardman, Allan D; Assanto, Gaetano
2016-01-01
We demonstrate that light is subject to anomalous (i.e., negative) diffraction when propagating in the presence of hyperbolic dispersion. We show that light propagation in hyperbolic media resembles the dynamics of a quantum particle of negative mass moving in a two-dimensional potential. The negative effective mass implies time reversal if the medium is homogeneous. Such property paves the way to diffraction compensation, spatial analogue of dispersion compensating fibers in the temporal domain. At variance with materials exhibiting standard elliptic dispersion, in inhomogeneous hyperbolic materials light waves are pulled towards regions with a lower refractive index. In the presence of a Kerr-like optical response, bright (dark) solitons are supported by a negative (positive) nonlinearity.
Anomalous diffraction in hyperbolic materials
Alberucci, Alessandro; Jisha, Chandroth P.; Boardman, Allan D.; Assanto, Gaetano
2016-09-01
We demonstrate that light is subject to anomalous (i.e., negative) diffraction when propagating in the presence of hyperbolic dispersion. We show that light propagation in hyperbolic media resembles the dynamics of a quantum particle of negative mass moving in a two-dimensional potential. The negative effective mass implies time reversal if the medium is homogeneous. Such property paves the way to diffraction compensation, i.e., spatial analog of dispersion compensating fibers in the temporal domain. At variance with materials exhibiting standard elliptic dispersion, in inhomogeneous hyperbolic materials light waves are pulled towards regions with a lower refractive index. In the presence of a Kerr-like optical response, bright (dark) solitons are supported by a negative (positive) nonlinearity.
Partial differential equations
Evans, Lawrence C
2010-01-01
This text gives a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDEs) with particular emphasis on nonlinear equations. The exposition is divided into three parts: representation formulas for solutions; theory for linear partial differential equations; and theory for nonlinear partial differential equations. Included are complete treatments of the method of characteristics; energy methods within Sobolev spaces; regularity for second-order elliptic, parabolic, and hyperbolic equations; maximum principles; the multidimensional calculus of variations; viscosity solutions of Hamilton-Jacobi equations; shock waves and entropy criteria for conservation laws; and, much more.The author summarizes the relevant mathematics required to understand current research in PDEs, especially nonlinear PDEs. While he has reworked and simplified much of the classical theory (particularly the method of characteristics), he primarily emphasizes the modern interplay between funct...
Front tracking for hyperbolic conservation laws
Holden, Helge
2002-01-01
Hyperbolic conservation laws are central in the theory of nonlinear partial differential equations and in science and technology. The reader is given a self-contained presentation using front tracking, which is also a numerical method. The multidimensional scalar case and the case of systems on the line are treated in detail. A chapter on finite differences is included. "It is already one of the few best digests on this topic. The present book is an excellent compromise between theory and practice. Students will appreciate the lively and accurate style." D. Serre, MathSciNet "I have read the book with great pleasure, and I can recommend it to experts as well as students. It can also be used for reliable and very exciting basis for a one-semester graduate course." S. Noelle, Book review, German Math. Soc. "Making it an ideal first book for the theory of nonlinear partial differential equations...an excellent reference for a graduate course on nonlinear conservation laws." M. Laforest, Comp. Phys. Comm.
PARTIAL STABILIZATION OF A CLASS OF CONTINUOUS NONLINEAR CONTROL SYSTEMS WITH SEPARATED VARIABLES
Institute of Scientific and Technical Information of China (English)
Jigui JIAN; Xiaoxin LIAO
2005-01-01
In this paper, the partial stabilization problem for a class of nonlinear continuous control systems with separated variables is investigated. Several stabilizing controllers are constructed based on the partial stability theory of Lyapunov and the property of M-matrix, and some of these stabilizing controllers are only related to partial state variables. The controllers constructed here are shown to guarantee partial asymptotic stability of the closed-loop systems and these sufficient conditions may give some instructions to actual engineering application. A example is also given to illustrate the design method.
Directory of Open Access Journals (Sweden)
Jun Shuai
2013-11-01
Full Text Available A new approach using optimization technique for constructing low-dimensional dynamical systems of nonlinear partial differential equations (PDEs is presented. After the spatial basis functions of the nonlinear PDEs are chosen, spatial basis functions expansions combined with weighted residual methods are used for time/space separation and truncation to obtain a high-dimensional dynamical system. Secondly, modes of lower-dimensional dynamical systems are obtained by linear combination from the modes of the high-dimensional dynamical systems (ordinary differential equations of nonlinear PDEs. An error function for matrix of the linear combination coefficients is derived, and a simple algorithm to determine the optimal combination matrix is also introduced. A numerical example shows that the optimal dynamical system can use much smaller number of modes to capture the dynamics of nonlinear partial differential equations.
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2016-07-01
Full Text Available In this paper, we improve the extended trial equation method to construct the exact solutions for nonlinear coupled system of partial differential equations in mathematical physics. We use the extended trial equation method to find some different types of exact solutions such as the Jacobi elliptic function solutions, soliton solutions, trigonometric function solutions and rational, exact solutions to the nonlinear coupled Schrodinger Boussinesq equations when the balance number is a positive integer. The performance of this method is reliable, effective and powerful for solving more complicated nonlinear partial differential equations in mathematical physics. The balance number of this method is not constant as we have in other methods. This method allows us to construct many new types of exact solutions. By using the Maple software package we show that all obtained solutions satisfy the original partial differential equations.
The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions
Directory of Open Access Journals (Sweden)
Shoukry Ibrahim Atia El-Ganaini
2013-01-01
Full Text Available The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1-dimensional hyperbolic nonlinear Schrodinger (HNLS equation, the generalized nonlinear Schrodinger (GNLS equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner.
Hyperbolic Chaos A Physicist’s View
Kuznetsov, Sergey P
2012-01-01
"Hyperbolic Chaos: A Physicist’s View” presents recent progress on uniformly hyperbolic attractors in dynamical systems from a physical rather than mathematical perspective (e.g. the Plykin attractor, the Smale – Williams solenoid). The structurally stable attractors manifest strong stochastic properties, but are insensitive to variation of functions and parameters in the dynamical systems. Based on these characteristics of hyperbolic chaos, this monograph shows how to find hyperbolic chaotic attractors in physical systems and how to design a physical systems that possess hyperbolic chaos. This book is designed as a reference work for university professors and researchers in the fields of physics, mechanics, and engineering. Dr. Sergey P. Kuznetsov is a professor at the Department of Nonlinear Processes, Saratov State University, Russia.
Passivation and control of partially known SISO nonlinear systems via dynamic neural networks
Reyes-Reyes J.; yu W.; Poznyak A. S.
2000-01-01
In this paper, an adaptive technique is suggested to provide the passivity property for a class of partially known SISO nonlinear systems. A simple Dynamic Neural Network (DNN), containing only two neurons and without any hidden-layers, is used to identify the unknown nonlinear system. By means of a Lyapunov-like analysis the new learning law for this DNN, guarantying both successful identification and passivation effects, is derived. Based on this adaptive DNN model, an adaptive feedback con...
A Maple Package for the Painlevé Test of Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
徐桂琼; 李志斌
2003-01-01
A Maple package, named PLtest, is presented to study whether or not nonlinear partial differential equations the standard WTC algorithm and the Kruskal simplification algorithm. Therefore, we not only study whether the given PDEs pass the test or not, but also obtain its truncated expansion form related to some integrability properties. Several well-known nonlinear models with physical interests illustrate the effectiveness of this package.
Analysis of boundary layer flow over a porous nonlinearly stretching sheet with partial slip at
Directory of Open Access Journals (Sweden)
Swati Mukhopadhyay
2013-12-01
Full Text Available The boundary layer flow of a viscous incompressible fluid toward a porous nonlinearly stretching sheet is considered in this analysis. Velocity slip is considered instead of no-slip condition at the boundary. Similarity transformations are used to convert the partial differential equation corresponding to the momentum equation into nonlinear ordinary differential equation. Numerical solution of this equation is obtained by shooting method. It is found that the horizontal velocity decreases with increasing slip parameter.
On invariant analysis of some time fractional nonlinear systems of partial differential equations. I
Singla, Komal; Gupta, R. K.
2016-10-01
An investigation of Lie point symmetries for systems of time fractional partial differential equations including Ito system, coupled Burgers equations, coupled Korteweg de Vries equations, Hirota-Satsuma coupled KdV equations, and coupled nonlinear Hirota equations has been done. Using the obtained symmetries, each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving Erdélyi-Kober fractional differential operator depending on a parameter α.
Clawpack: building an open source ecosystem for solving hyperbolic PDEs
Directory of Open Access Journals (Sweden)
Kyle T. Mandli
2016-08-01
Full Text Available Clawpack is a software package designed to solve nonlinear hyperbolic partial differential equations using high-resolution finite volume methods based on Riemann solvers and limiters. The package includes a number of variants aimed at different applications and user communities. Clawpack has been actively developed as an open source project for over 20 years. The latest major release, Clawpack 5, introduces a number of new features and changes to the code base and a new development model based on GitHub and Git submodules. This article provides a summary of the most significant changes, the rationale behind some of these changes, and a description of our current development model.
Clawpack: building an open source ecosystem for solving hyperbolic PDEs
Mandli, Kyle T.
2016-08-08
Clawpack is a software package designed to solve nonlinear hyperbolic partial differential equations using high-resolution finite volume methods based on Riemann solvers and limiters. The package includes a number of variants aimed at different applications and user communities. Clawpack has been actively developed as an open source project for over 20 years. The latest major release, Clawpack 5, introduces a number of new features and changes to the code base and a new development model based on GitHub and Git submodules. This article provides a summary of the most significant changes, the rationale behind some of these changes, and a description of our current development model.
Soliton solution for nonlinear partial differential equations by cosine-function method
Energy Technology Data Exchange (ETDEWEB)
Ali, A.H.A. [Mathematics Department, Faculty of Science, Menoufia University, Shebein El-Koom (Egypt); Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish), Suez Canal University, AL-Arish 45111 (Egypt)], E-mail: asoliman_99@yahoo.com; Raslan, K.R. [Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo (Egypt)
2007-08-20
In this Letter, we established a traveling wave solution by using Cosine-function algorithm for nonlinear partial differential equations. The method is used to obtain the exact solutions for five different types of nonlinear partial differential equations such as, general equal width wave equation (GEWE), general regularized long wave equation (GRLW), general Korteweg-de Vries equation (GKdV), general improved Korteweg-de Vries equation (GIKdV), and Coupled equal width wave equations (CEWE), which are the important soliton equations.
Nonlinear grid error effects on numerical solution of partial differential equations
Dey, S. K.
1980-01-01
Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.
Passivation and control of partially known SISO nonlinear systems via dynamic neural networks
Directory of Open Access Journals (Sweden)
Reyes-Reyes J.
2000-01-01
Full Text Available In this paper, an adaptive technique is suggested to provide the passivity property for a class of partially known SISO nonlinear systems. A simple Dynamic Neural Network (DNN, containing only two neurons and without any hidden-layers, is used to identify the unknown nonlinear system. By means of a Lyapunov-like analysis the new learning law for this DNN, guarantying both successful identification and passivation effects, is derived. Based on this adaptive DNN model, an adaptive feedback controller, serving for wide class of nonlinear systems with an a priori incomplete model description, is designed. Two typical examples illustrate the effectiveness of the suggested approach.
Variational iteration method for solving non-linear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Hemeda, A.A. [Department of Mathematics, Faculty of Science, University of Tanta, Tanta (Egypt)], E-mail: aahemeda@yahoo.com
2009-02-15
In this paper, we shall use the variational iteration method to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV-MKdV equation and Camassa-Holm equation. The variational iteration method is superior than the other non-linear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.
Front tracking for hyperbolic conservation laws
Holden, Helge
2015-01-01
This is the second edition of a well-received book providing the fundamentals of the theory hyperbolic conservation laws. Several chapters have been rewritten, new material has been added, in particular, a chapter on space dependent flux functions, and the detailed solution of the Riemann problem for the Euler equations. Hyperbolic conservation laws are central in the theory of nonlinear partial differential equations and in science and technology. The reader is given a self-contained presentation using front tracking, which is also a numerical method. The multidimensional scalar case and the case of systems on the line are treated in detail. A chapter on finite differences is included. From the reviews of the first edition: "It is already one of the few best digests on this topic. The present book is an excellent compromise between theory and practice. Students will appreciate the lively and accurate style." D. Serre, MathSciNet "I have read the book with great pleasure, and I can recommend it to experts ...
Modeling Solution of Nonlinear Dispersive Partial Differential Equations using the Marker Method
Energy Technology Data Exchange (ETDEWEB)
Jerome L.V. Lewandowski
2005-01-25
A new method for the solution of nonlinear dispersive partial differential equations is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details.
Lobachevsky geometry and modern nonlinear problems
Popov, Andrey
2014-01-01
This monograph presents the basic concepts of hyperbolic Lobachevsky geometry and their possible applications to modern nonlinear applied problems in mathematics and physics, summarizing the findings of roughly the last hundred years. The central sections cover the classical building blocks of hyperbolic Lobachevsky geometry, pseudo spherical surfaces theory, net geometrical investigative techniques of nonlinear differential equations in partial derivatives, and their applications to the analysis of the physical models. As the sine-Gordon equation appears to have profound “geometrical roots” and numerous applications to modern nonlinear problems, it is treated as a universal “object” of investigation, connecting many of the problems discussed. The aim of this book is to form a general geometrical view on the different problems of modern mathematics, physics and natural science in general in the context of non-Euclidean hyperbolic geometry.
Nonlinear partial least squares with Hellinger distance for nonlinear process monitoring
Harrou, Fouzi
2017-02-16
This paper proposes an efficient data-based anomaly detection method that can be used for monitoring nonlinear processes. The proposed method merges advantages of nonlinear projection to latent structures (NLPLS) modeling and those of Hellinger distance (HD) metric to identify abnormal changes in highly correlated multivariate data. Specifically, the HD is used to quantify the dissimilarity between current NLPLS-based residual and reference probability distributions. The performances of the developed anomaly detection using NLPLS-based HD technique is illustrated using simulated plug flow reactor data.
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained.
Institute of Scientific and Technical Information of China (English)
WANG Peng-Zhou; ZHANG Shun-Li
2008-01-01
We present basic theory of variable separation for (1 + 1)-dimensional nonlinear evolution equations with mixed partial derivatives. As an application, we classify equations uxt = A(u, ux)uxxx + B(u, ux) that admits derivative-dependent functional separable solutions (DDFSSs) and illustrate how to construct those DDFSSs with some examples.
Nonlinear perturbations of systems of partial differential equations with constant coefficients
Directory of Open Access Journals (Sweden)
Carmen J. Vanegas
2000-01-01
Full Text Available In this article, we show the existence of solutions to boundary-value problems, consisting of nonlinear systems of partial differential equations with constant coefficients. For this purpose, we use the right inverse of an associated operator and a fix point argument. As illustrations, we apply this method to Helmholtz equations and to second order systems of elliptic equations.
Special Conditional Similarity Reduction Solutions for Two Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
We present a method of special conditional similarity reduction solutions for nonlinear partial differential equations. As concrete examples of its application, we apply this method to the (2+1)-dimensional modified Broer-Kaup equations and the variable coefficient KdV-mKdV equation, which have extensive physics backgrounds, and obtain abundant exact solutions derived from some reduction equations.
Energy Technology Data Exchange (ETDEWEB)
Zhang Huiqun [College of Mathematical Science, Qingdao University, Qingdao, Shandong 266071 (China)], E-mail: hellozhq@yahoo.com.cn
2009-02-15
By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger-KdV equations and the Hirota-Maccari equations. New exact complex solutions are obtained.
Picone's identity for a system of first-order nonlinear partial differential equations
Directory of Open Access Journals (Sweden)
Jaroslav Jaros
2013-06-01
Full Text Available We established a Picone identity for systems of nonlinear partial differential equations of first-order. With the help of this formula, we obtain qualitative results such as an integral inequality of Wirtinger type and the existence of zeros for the first components of solutions in a given bounded domain.
Application of homotopy-perturbation to non-linear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Cveticanin, L. [Faculty of Technical Sciences, 21000 Novi Sad, Trg D. Obradovica 6 (Serbia)], E-mail: cveticanin@uns.ns.ac.yu
2009-04-15
In this paper He's homotopy perturbation method has been adopted for solving non-linear partial differential equations. An approximate solution of the differential equation which describes the longitudinal vibration of a beam is obtained. The solution is compared with that found using the variational iteration method introduced by He. The difference between the two solutions is negligible.
Energy Technology Data Exchange (ETDEWEB)
Zhou Yubin; Wang Mingliang; Miao Tiande
2004-03-15
The periodic wave solutions for a class of nonlinear partial differential equations, including the Davey-Stewartson equations and the generalized Zakharov equations, are obtained by using the F-expansion method, which can be regarded as an overall generalization of the Jacobi elliptic function expansion method recently proposed. In the limit cases the solitary wave solutions of the equations are also obtained.
Directory of Open Access Journals (Sweden)
Yusuf Pandir
2012-01-01
Full Text Available We obtain the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method. In discussion, we propose a more general trial equation method for nonlinear partial differential equations with generalized evolution.
Directory of Open Access Journals (Sweden)
Abaker. A. Hassaballa.
2015-10-01
Full Text Available - In recent years, many more of the numerical methods were used to solve a wide range of mathematical, physical, and engineering problems linear and nonlinear. This paper applies the homotopy perturbation method (HPM to find exact solution of partial differential equation with the Dirichlet and Neumann boundary conditions.
Directory of Open Access Journals (Sweden)
Qi Song
2013-01-01
Full Text Available This paper proposes a partial refactorization for faster nonlinear analysis based on sparse matrix solution, which is nowadays the default solution choice in finite element analysis and can solve finite element models up to millions degrees of freedom. Among various fill-in’s reducing strategies for sparse matrix solution, the graph partition is in general the best in terms of resultant fill-ins and floating-point operations and furthermore produces a particular graph of sparse matrix that prevents local change of entries from wide spreading in factorization. Based on this feature, an explicit partial triangular refactorization with local change is efficiently constructed with limited additional storage requirement in row-sparse storage scheme. The partial refactorization of the changed stiffness matrix inherits a big percentage of the original factor and is carried out only on partial factor entries. The proposed method provides a new possibility for faster nonlinear analysis and is mainly suitable for material nonlinear problems and optimization problems. Compared to full factorization, it can significantly reduce the factorization time and can make nonlinear analysis more efficient.
Fokas, A. S.; De Lillo, S.
2014-03-01
So-called inverse scattering provides a powerful method for analyzing the initial value problem for a large class of nonlinear evolution partial differential equations which are called integrable. In the late 1990s, the first author, motivated by inverse scattering, introduced a new method for analyzing boundary value problems. This method provides a unified treatment for linear, linearizable and integrable nonlinear partial differential equations. Here, this method, which is often referred to as the unified transform, is illustrated for the following concrete cases: the heat equation on the half-line; the nonlinear Schrödinger equation on the half-line; Burger's equation on the half-line; and Burger's equation on a moving boundary.
Stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
The stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems is investigated. First, the stochastic optimal control problem of a partially observable nonlinear quasi-integrable Hamiltonian system is converted into that of a completely observable linear system based on a theorem due to Charalambous and Elliot. Then, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. The response of the controlled quasi Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation and the Riccati equation for the estimated error of system states. As an example to illustrate the procedure and effectiveness of the proposed method, the stochastic optimal control problem of a partially observable two-degree-of-freedom quasi-integrable Hamiltonian system is worked out in detail.
Nonlinear Spline Kernel-based Partial Least Squares Regression Method and Its Application
Institute of Scientific and Technical Information of China (English)
JIA Jin-ming; WEN Xiang-jun
2008-01-01
Inspired by the traditional Wold's nonlinear PLS algorithm comprises of NIPALS approach and a spline inner function model,a novel nonlinear partial least squares algorithm based on spline kernel(named SK-PLS)is proposed for nonlinear modeling in the presence of multicollinearity.Based on the iuner-product kernel spanned by the spline basis functions with infinite numher of nodes,this method firstly maps the input data into a high dimensional feature space,and then calculates a linear PLS model with reformed NIPALS procedure in the feature space and gives a unified framework of traditional PLS"kernel"algorithms in consequence.The linear PLS in the feature space corresponds to a nonlinear PLS in the original input (primal)space.The good approximating property of spline kernel function enhances the generalization ability of the novel model,and two numerical experiments are given to illustrate the feasibility of the proposed method.
Bove, Antonio; Murthy, MK Venkatesha
2009-01-01
This collection of original articles and surveys addresses the recent advances in linear and nonlinear aspects of the theory of partial differential equations. The key topics include operators as "sums of squares" of real and complex vector fields, nonlinear evolution equations, local solvability, and hyperbolic questions.
Infinitesimal Lyapunov functions and singular-hyperbolicity
Araujo, Vitor
2012-01-01
We present an extension of the notion of infinitesimal Lyapunov function to singular flows on three-dimensional manifolds, and show how this technique provides a characterization of partially hyperbolic structures for invariant sets for such flows, and also of singular-hyperbolicity. In the absence of singularities, we can also rephrase uniform hyperbolicity with the language of infinitesimal Lyapunov functions. These conditions are expressed using the vector field X and its space derivative DX together with an infinitesimal Lyapunov function only and are reduced to checking that a certain symmetric operator is positive definite on the trapping region: we show how to express partial hyperbolicity using only the interplay between the infinitesimal generator X of the flow X_t, its derivative DX and the infinitesimal Lyapunov function.
Institute of Scientific and Technical Information of China (English)
2008-01-01
Using functional derivative technique in quantum field theory, the algebraic dy-namics approach for solution of ordinary differential evolution equations was gen-eralized to treat partial differential evolution equations. The partial differential evo-lution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynam-ics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new nu-merical algorithm—algebraic dynamics algorithm was proposed for partial differ-ential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experi-ments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.
Institute of Scientific and Technical Information of China (English)
WANG Shundin; ZHANG Hua
2008-01-01
Using functional derivative technique In quantum field theory,the algebraic dy-namics approach for solution of ordinary differential evolution equations was gen-eralized to treat partial differential evolution equations.The partial differential evo-lution equations were lifted to the corresponding functional partial differential equations in functional space by Introducing the time translation operator.The functional partial differential evolution equations were solved by algebraic dynam-ics.The algebraic dynamics solutions are analytical In Taylor series In terms of both initial functions and time.Based on the exact analytical solutions,a new nu-merical algorithm-algebraic dynamics algorithm was proposed for partial differ-ential evolution equations.The difficulty of and the way out for the algorithm were discussed.The application of the approach to and computer numerical experi-ments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.
Hyperbolic differential operators and related problems
Ancona, Vincenzo
2003-01-01
Presenting research from more than 30 international authorities, this reference provides a complete arsenal of tools and theorems to analyze systems of hyperbolic partial differential equations. The authors investigate a wide variety of problems in areas such as thermodynamics, electromagnetics, fluid dynamics, differential geometry, and topology. Renewing thought in the field of mathematical physics, Hyperbolic Differential Operators defines the notion of pseudosymmetry for matrix symbols of order zero as well as the notion of time function. Surpassing previously published material on the top
He-Laplace Method for Linear and Nonlinear Partial Differential Equations
Directory of Open Access Journals (Sweden)
Hradyesh Kumar Mishra
2012-01-01
Full Text Available A new treatment for homotopy perturbation method is introduced. The new treatment is called He-Laplace method which is the coupling of the Laplace transform and the homotopy perturbation method using He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The method is implemented on linear and nonlinear partial differential equations. It is found that the proposed scheme provides the solution without any discretization or restrictive assumptions and avoids the round-off errors.
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.
Exact solutions of some nonlinear partial differential equations using functional variable method
Indian Academy of Sciences (India)
A Nazarzadeh; M Eslami; M Mirzazadeh
2013-08-01
The functional variable method is a powerful solution method for obtaining exact solutions of some nonlinear partial differential equations. In this paper, the functional variable method is used to establish exact solutions of the generalized forms of Klein–Gordon equation, the (2 + 1)-dimensional Camassa–Holm Kadomtsev–Petviashvili equation and the higher-order nonlinear Schrödinger equation. By using this useful method, we found some exact solutions of the above-mentioned equations. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions. It is shown that the proposed method is effective and general.
Grothaus, Martin
2012-01-01
In this paper a length-conserving numerical scheme for a nonlinear fourth order system of partial differential algebraic equations arising in technical textile industry is studied. Applying a semidiscretization in time, the resulting sequence of nonlinear elliptic systems with algebraic constraint is reformulated as constrained optimization problems in a Hilbert space setting that admit a solution at each time level. Stability and convergence of the scheme are proved. The numerical realization is performed by projected gradient methods on finite element spaces which determine the computational effort and approximation quality of the algorithm. Simulation results are presented and discussed in view of the application of an elastic inextensible fiber motion.
Energy Technology Data Exchange (ETDEWEB)
Ravi Kanth, A.S.V. [Applied Mathematics Division, School of Science and Humanities, V.I.T. University, Vellore-632 014, Tamil Nadu (India)], E-mail: asvravikanth@yahoo.com; Aruna, K. [Applied Mathematics Division, School of Science and Humanities, V.I.T. University, Vellore-632 014, Tamil Nadu (India)
2008-11-17
In this Letter, we propose a reliable algorithm to develop exact and approximate solutions for the linear and non-linear systems of partial differential equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.
Simple equation method for nonlinear partial differential equations and its applications
Directory of Open Access Journals (Sweden)
Taher A. Nofal
2016-04-01
Full Text Available In this article, we focus on the exact solution of the some nonlinear partial differential equations (NLPDEs such as, Kodomtsev–Petviashvili (KP equation, the (2 + 1-dimensional breaking soliton equation and the modified generalized Vakhnenko equation by using the simple equation method. In the simple equation method the trial condition is the Bernoulli equation or the Riccati equation. It has been shown that the method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems.
Kannan, Rohit; Tangirala, Arun K.
2014-06-01
Identification of directional influences in multivariate systems is of prime importance in several applications of engineering and sciences such as plant topology reconstruction, fault detection and diagnosis, and neurosciences. A spectrum of related directionality measures, ranging from linear measures such as partial directed coherence (PDC) to nonlinear measures such as transfer entropy, have emerged over the past two decades. The PDC-based technique is simple and effective, but being a linear directionality measure has limited applicability. On the other hand, transfer entropy, despite being a robust nonlinear measure, is computationally intensive and practically implementable only for bivariate processes. The objective of this work is to develop a nonlinear directionality measure, termed as KPDC, that possesses the simplicity of PDC but is still applicable to nonlinear processes. The technique is founded on a nonlinear measure called correntropy, a recently proposed generalized correlation measure. The proposed method is equivalent to constructing PDC in a kernel space where the PDC is estimated using a vector autoregressive model built on correntropy. A consistent estimator of the KPDC is developed and important theoretical results are established. A permutation scheme combined with the sequential Bonferroni procedure is proposed for testing hypothesis on absence of causality. It is demonstrated through several case studies that the proposed methodology effectively detects Granger causality in nonlinear processes.
Directory of Open Access Journals (Sweden)
Jialin Wang
2013-01-01
Full Text Available This paper is concerned with partial regularity to nonlinear subelliptic systems with Dini continuous coefficients under quadratic controllable growth conditions in the Heisenberg group ℍn. Based on a generalization of the technique of -harmonic approximation introduced by Duzaar and Steffen, partial regularity to the sub-elliptic system is established in the Heisenberg group. Our result is optimal in the sense that in the case of Hölder continuous coefficients we establish the optimal Hölder exponent for the horizontal gradients of the weak solution on its regular set.
Covariant Hyperbolization of Force-free Electrodynamics
Carrasco, Federico
2016-01-01
Force-Free Flectrodynamics (FFE) is a non-linear system of equations modeling the evolution of the electromagnetic field, in the presence of a magnetically dominated relativistic plasma. This configuration arises on several astrophysical scenarios, which represent exciting laboratories to understand physics in extreme regimes. We show that this system, when restricted to the correct constraint submanifold, is symmetric hyperbolic. In numerical applications is not feasible to keep the system in that submanifold, and so, it is necessary to analyze its structure first in the tangent space of that submanifold and then in a whole neighborhood of it. As already shown by Pfeiffer, a direct (or naive) formulation of this system (in the whole tangent space) results in a weakly hyperbolic system of evolution equations for which well-possednes for the initial value formulation does not follows. Using the generalized symmetric hyperbolic formalism due to Geroch, we introduce here a covariant hyperbolization for the FFE s...
On stochastic optimal control of partially observable nonlinear quasi Hamiltonian systems
Institute of Scientific and Technical Information of China (English)
朱位秋; 应祖光
2004-01-01
A stochastic optimal control strategy for partially observable nonlinear quasi Hamiltonian systems is proposed.The optimal control forces consist of two parts. The first part is determined by the conditions under which the stochastic optimal control problem of a partially observable nonlinear system is converted into that of a completely observable linear system. The second part is determined by solving the dynamical programming equation derived by applying the stochastic averaging method and stochastic dynamical programming principle to the completely observable linear control system. The response of the optimally controlled quasi Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation associated with the optimally controlled completely observable linear system and solving the Riccati equation for the estimated error of system states. An example is given to illustrate the procedure and effectiveness of the proposed control strategy.
Institute of Scientific and Technical Information of China (English)
朱位秋; 应祖光
2004-01-01
A stochastic optimal control strategy for partially observable nonlinear quasi Hamiltonian systems is proposed. The optimal control forces consist of two parts. The first part is determined by the conditions under which the stochastic optimal control problem of a partially observable nonlinear system is converted into that of a completely observable linear system. The second part is determined by solving the dynamical programming equation derived by applying the stochastic averaging method and stochastic dynamical programming principle to the completely observable linear control system. The response of the optimally controlled quasi Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation associated with the optimally controlled completely observable linear system and solving the Riccati equation for the estimated error of system states. An example is given to illustrate the procedure and effectiveness of the proposed control strategy.
Sharma, Dinkar; Singh, Prince; Chauhan, Shubha
2016-01-01
In this paper, a combined form of the Laplace transform method with the homotopy perturbation method (HPTM) is applied to solve nonlinear systems of partial differential equations viz. the system of third order KdV Equations and the systems of coupled Burgers' equations in one- and two- dimensions. The nonlinear terms can be easily handled by the use of He's polynomials. The results shows that the HPTM is very efficient, simple and avoids the round-off errors. Four test examples are considered to illustrate the present scheme. Further the results are compared with Homotopy perturbation method (HPM) which shows that this method is a suitable method for solving systems of partial differential equations.
An ansatz for solving nonlinear partial differential equations in mathematical physics.
Akbar, M Ali; Ali, Norhashidah Hj Mohd
2016-01-01
In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.
Directory of Open Access Journals (Sweden)
Fukang Yin
2013-01-01
Full Text Available This paper develops a modified variational iteration method coupled with the Legendre wavelets, which can be used for the efficient numerical solution of nonlinear partial differential equations (PDEs. The approximate solutions of PDEs are calculated in the form of a series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The main advantage of the new method is that it can avoid solving the nonlinear algebraic system and symbolic computation. Furthermore, the developed vector-matrix form makes it computationally efficient. The results show that the proposed method is very effective and easy to implement.
Directory of Open Access Journals (Sweden)
Veyis Turut
2013-01-01
Full Text Available Two tecHniques were implemented, the Adomian decomposition method (ADM and multivariate Padé approximation (MPA, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM, then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.
Study of coupled nonlinear partial differential equations for finding exact analytical solutions.
Khan, Kamruzzaman; Akbar, M Ali; Koppelaar, H
2015-07-01
Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G'/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.
Energy Technology Data Exchange (ETDEWEB)
Zhang, Sheng [Department of Mathematics, Bohai University, Jinzhou 121000 (China)]. E-mail: zhshaeng@yahoo.com.cn; Xia, Tiecheng [Department of Mathematics, Bohai University, Jinzhou 121000 (China); Department of Mathematics, Shanghai University, Shanghai 200444 (China)
2007-04-09
In this Letter, a generalized new auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the combined KdV-mKdV equation and the (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equations to illustrate the validity and advantages of the method. As a result, many new and more general exact solutions are obtained.
Homotopy perturbation method for nonlinear partial differential equations of fractional order
Energy Technology Data Exchange (ETDEWEB)
Momani, Shaher [Department of Mathematics and Physics, Qatar University (Qatar)]. E-mail: shahermm@yahoo.com; Odibat, Zaid [Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa' Applied University, Salt (Jordan)]. E-mail: odibat@bau.edu.jo
2007-06-11
The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative. The fractional derivative is described in the Caputo sense. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient and easy to implement.
A new mapping method and its applications to nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Zeng Xin [Department of Mathematics, Zhengzhou University, Zhengzhou 450052 (China)], E-mail: zeng79723@163.com; Yong Xuelin [Department of Mathematics and Physics, North China Electric Power University, Beijing 102206 (China)
2008-10-27
In this Letter, a new mapping method is proposed for constructing more exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2+1)-dimensional Konopelchenko-Dubrovsky equation and the (2+1)-dimensional KdV equations to illustrate the validity and advantages of the method. As a result, many new and more general exact solutions are obtained.
An ansatz for solving nonlinear partial differential equations in mathematical physics
Akbar, M. Ali; Ali, Norhashidah Hj Mohd
2016-01-01
In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin–Bona–Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general sol...
Hyperbolicity in Median Graphs
Indian Academy of Sciences (India)
José M Sigarreta
2013-11-01
If is a geodesic metric space and $x_1,x_2,x_3\\in X$, a geodesic triangle $T=\\{x_1,x_2,x_3\\}$ is the union of the three geodesics $[x_1 x_2],[x_2 x_3]$ and $[x_3 x_1]$ in . The space is -hyperbolic (in the Gromov sense) if any side of is contained in a -neighborhood of the union of the two other sides, for every geodesic triangle in . If is hyperbolic, we denote by () the sharp hyperbolicity constant of , i.e.,$(X)=\\inf\\{≥ 0: X \\quad\\text{is}\\quad -\\text{hyperbolic}\\}$. In this paper we study the hyperbolicity of median graphs and we also obtain some results about general hyperbolic graphs. In particular, we prove that a median graph is hyperbolic if and only if its bigons are thin.
Constructive role of sensors nonlinearities in the acquisition of partially polarized speckle images
Energy Technology Data Exchange (ETDEWEB)
Delahaies, Agnes; Rousseau, David; Chapeau-Blondeau, Francois [Laboratoire d' Ingenierie des Systemes Automatises (LISA), Universite d' Angers, 62 avenue Notre Dame du Lac, 49000 Angers (France); Gindre, Denis, E-mail: david.rousseau@univ-angers.f [Laboratoire des Proprietes Optiques des Materiaux et Applications (POMA), Universite d' Angers, 2 boulevard Lavoisier, 49000 Angers (France)
2010-02-01
We study the impact of the level of the speckle noise on data acquisition in a partially polarized coherent imaging system with the presence of a nonlinearity in the imaging sensor characteristic. In perfectly linear acquisition conditions, due to the essentially multiplicative action of the speckle, the image contrast is unchanged as the speckle noise level increases, and so it has no impact on the quality of the acquired images. On the contrary, in nonlinear conditions the acquisition is affected by the speckle noise level. However, this effect of the speckle is not always detrimental. We show that, in definite nonlinear conditions, there is usually an optimal level of the speckle noise that leads to a maximum quality of the acquired images. We theoretically analyze such nonlinear regimes with partially polarized speckled images. We specifically exhibit the existence of an optimal speckle noise level in the interesting case of images realized only by a depolarization contrast. Illustrations are given with a simple 1-bit hard limiter and binary images. Then, we propose and discuss as perspectives an experimental optical setup to confront theory and experiment.
Hyperbolic Metamaterials with Bragg Polaritons
Sedov, Evgeny S.; Iorsh, I. V.; Arakelian, S. M.; Alodjants, A. P.; Kavokin, Alexey
2015-06-01
We propose a novel mechanism for designing quantum hyperbolic metamaterials with the use of semiconductor Bragg mirrors containing periodically arranged quantum wells. The hyperbolic dispersion of exciton-polariton modes is realized near the top of the first allowed photonic miniband in such a structure which leads to the formation of exciton-polariton X waves. Exciton-light coupling provides a resonant nonlinearity which leads to nontrivial topologic solutions. We predict the formation of low amplitude spatially localized oscillatory structures: oscillons described by kink shaped solutions of the effective Ginzburg-Landau-Higgs equation. The oscillons have direct analogies in gravitational theory. We discuss implementation of exciton-polariton Higgs fields for the Schrödinger cat state generation.
Second-order hyperbolic Fuchsian systems. General theory
Beyer, Florian; LeFloch, Philippe G.
2010-01-01
We introduce a class of singular partial differential equations, the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First of all, we analyze a class of equations in which hyperbolicity is not assumed and we construct asymptotic solutions of arbitrary order. Second, for the proposed class of second-order hyperbolic Fuchsian systems, we establish the existence of solutions with prescribed asymptotic beh...
Diagnosing nonlinearities in the local and remote responses to partial Amazon deforestation
Badger, Andrew M.; Dirmeyer, Paul A.
2016-08-01
Using a set of fully coupled climate model simulations, the response to partial deforestation over the Amazon due to agricultural expansion has been analyzed. Three variations of 50% deforestation (all of western half, all of eastern half, and half of each grid box) were compared with total deforestation to determine the degree and character of nonlinearity of the climate response to partial deforestation. A metric is developed to quantify the degree and distribution of nonlinearity in the response, applicable to any variable. The metric also quantifies whether the response is saturating or accelerating, meaning significantly either more or less than 50% of the simulated response to total deforestation is attained at 50% deforestation. The spatial structure of the atmospheric response to Amazon deforestation reveals large areas across the tropics that exhibit a significant nonlinear component, particularly for temperature and geopotential height. Over the domain between 45°S and 45°N across all longitudes, 50% deforestation generally provides less than half of the total response to deforestation over oceans, indicating the marine portion of climate system is somewhat resilient to progressive deforestation. However, over continents there are both accelerating and saturating responses to 50% Amazon deforestation, and the response is different depending on whether the eastern or western half of Amazonia is deforested or half of the forest is removed uniformly across the region.
Fast Simulation of Nonlinear Circuits with an Algorithm of Partial Matrix Elimination (PME)
Institute of Scientific and Technical Information of China (English)
CAI Xingjian; MAO Junfa; YANG Xiaoping; LI Zhengfan
2001-01-01
In this paper a theorem is proven thatthe contribution of any linear circuit elements to theadmittance matrix [Y] in the modified nodal admit-tance (MNA) equation of circuit analysis is invariantwith time.Based on this theorem a fast time-domainalgorithm of partial matrix elimination (PME) is de-veloped for simulation of high-speed nonlinear circuitswith interconnects.In PME the matrix [Y] is ar-ranged to consist of two submatrices.The lower sub-matrix that contains entries from only linear circuit el-ements is partially triangulated by Gauss eliminationfor the first-time-step transient analysis.At the fol-lowing time steps of transient analysis,Gauss elimina-tion is needed only for the upper submatrix containingcontributions of nonlinear circuit devices.Efficiencyanalysis of PME is given in detail,showing that it canaccelerate circuit simulation by a factor of at least 2.5for linear or lightly nonlinear circuits without employ-ing any numerical approximation.Moreover,PME isuseful for circuits debugging.
Existence for a class of discrete hyperbolic problems
Directory of Open Access Journals (Sweden)
Luca Rodica
2006-01-01
Full Text Available We investigate the existence and uniqueness of solutions to a class of discrete hyperbolic systems with some nonlinear extreme conditions and initial data, in a real Hilbert space.
Magnetic hyperbolic optical metamaterials.
Kruk, Sergey S; Wong, Zi Jing; Pshenay-Severin, Ekaterina; O'Brien, Kevin; Neshev, Dragomir N; Kivshar, Yuri S; Zhang, Xiang
2016-04-13
Strongly anisotropic media where the principal components of electric permittivity or magnetic permeability tensors have opposite signs are termed as hyperbolic media. Such media support propagating electromagnetic waves with extremely large wave vectors exhibiting unique optical properties. However, in all artificial and natural optical materials studied to date, the hyperbolic dispersion originates solely from the electric response. This restricts material functionality to one polarization of light and inhibits free-space impedance matching. Such restrictions can be overcome in media having components of opposite signs for both electric and magnetic tensors. Here we present the experimental demonstration of the magnetic hyperbolic dispersion in three-dimensional metamaterials. We measure metamaterial isofrequency contours and reveal the topological phase transition between the elliptic and hyperbolic dispersion. In the hyperbolic regime, we demonstrate the strong enhancement of thermal emission, which becomes directional, coherent and polarized. Our findings show the possibilities for realizing efficient impedance-matched hyperbolic media for unpolarized light.
Filimonov, M. Yu.
2016-12-01
An analytical method for representation of solutions of nonlinear partial differential equations in the form of special series with recurrently computed coefficients is presented. The coefficients recurrent obtaining from linear differential equations is achieved by specificity of the considered equations. It turns out that due to the functional arbitrariness which possibly is contained in special series, one can prove global convergence of the constructed series to solution of considered nonlinear partial differential equations.
Exactly integrable hyperbolic equations of Liouville type
Energy Technology Data Exchange (ETDEWEB)
Zhiber, A V [Institute of Mechanics, Ufa Centre of the Russian Academy of Sciences, Ufa (Russian Federation); Sokolov, Vladimir V [Centre for Non-linear Studies Landau Institute for Theoretical Physics, Moscow (Russian Federation)
2001-02-28
This is a survey of the authors' results concerning non-linear hyperbolic equations of Liouville type. The definition is based on the condition that the chain of Laplace invariants of the linearized equation be two-way finite. New results include a procedure for finding the general solution and a solution of the classification problem for Liouville type equations.
Diffusion and dispersion of numerical schemes for Hyperbolic problems
Petit, H.A.H
2001-01-01
In the following text an overview is given of numerical schemes which can be used to solve hyperbolic partial differential equations. The overview is far from extensive and the analysis of the schemes is limited to the application on the simplest hyperbolic equation conceivable, namely the so called
Lu, Bin
2012-06-01
In this Letter, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Bäcklund transformation of fractional Riccati equation are employed for constructing the exact solutions of nonlinear fractional partial differential equations. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations.
Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions.
Fokas, A S
2006-05-19
The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.
Directory of Open Access Journals (Sweden)
S. S. Motsa
2014-01-01
Full Text Available This paper presents a new application of the homotopy analysis method (HAM for solving evolution equations described in terms of nonlinear partial differential equations (PDEs. The new approach, termed bivariate spectral homotopy analysis method (BISHAM, is based on the use of bivariate Lagrange interpolation in the so-called rule of solution expression of the HAM algorithm. The applicability of the new approach has been demonstrated by application on several examples of nonlinear evolution PDEs, namely, Fisher’s, Burgers-Fisher’s, Burger-Huxley’s, and Fitzhugh-Nagumo’s equations. Comparison with known exact results from literature has been used to confirm accuracy and effectiveness of the proposed method.
Directory of Open Access Journals (Sweden)
Yang Zhang
2013-01-01
Full Text Available We introduce a continuum modeling method to approximate a class of large wireless networks by nonlinear partial differential equations (PDEs. This method is based on the convergence of a sequence of underlying Markov chains of the network indexed by N, the number of nodes in the network. As N goes to infinity, the sequence converges to a continuum limit, which is the solution of a certain nonlinear PDE. We first describe PDE models for networks with uniformly located nodes and then generalize to networks with nonuniformly located, and possibly mobile, nodes. Based on the PDE models, we develop a method to control the transmissions in nonuniform networks so that the continuum limit is invariant under perturbations in node locations. This enables the networks to maintain stable global characteristics in the presence of varying node locations.
Partial and total actuator faults accommodation for input-affine nonlinear process plants.
Mihankhah, Amin; Salmasi, Farzad R; Salahshoor, Karim
2013-05-01
In this paper, a new fault-tolerant control system is proposed for input-affine nonlinear plants based on Model Reference Adaptive System (MRAS) structure. The proposed method has the capability to accommodate both partial and total actuator failures along with bounded external disturbances. In this methodology, the conventional MRAS control law is modified by augmenting two compensating terms. One of these terms is added to eliminate the nonlinear dynamic, while the other is reinforced to compensate the distractive effects of the total actuator faults and external disturbances. In addition, no Fault Detection and Diagnosis (FDD) unit is needed in the proposed method. Moreover, the control structure has good robustness capability against the parameter variation. The performance of this scheme is evaluated using a CSTR system and the results were satisfactory.
Intrinsic Nonlinearities and Layout Impacts of 100 V Integrated Power MOSFETs in Partial SOI Process
DEFF Research Database (Denmark)
Fan, Lin; Knott, Arnold; Jørgensen, Ivan Harald Holger
Parasitic capacitances of power semiconductors are a part of the key design parameters of state-of-the-art very high frequency (VHF) power supplies. In this poster, four 100 V integrated power MOSFETs with different layout structures are designed, implemented, and analyzed in a 0.18 ȝm partial...... Silicon-on-Insulator (SOI) process with a die area 2.31 mm2. A small-signal model of power MOSFETs is proposed to systematically analyze the nonlinear parasitic capacitances in different transistor states: off-state, sub-threshold region, and on-state in the linear region. 3D plots are used to summarize...
Institute of Scientific and Technical Information of China (English)
FAN En-Gui
2001-01-01
Two new applications of homogeneous balance (HB) method are presented.It is shown that HB methodcan be extended to search for the Backlund transformations and similarity reductions of nonlinear partial differentialequations.The close relations among the HB method,Weiss-Tabor-Carnevale method and Clarkson-Kruskal directreduction method are also found.KdV-MKdV equation is considered as an illustrative example,and its one kind of Backlund transformation,three kinds of similarity reductions and several kinds of travelling wave solutions are obtained by using extended HB method.
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
In this paper, an extended method is proposed for constructing new forms ofexact travelling wave solutions to nonlinear partial differential equations by making a more general transformation. For illustration, we apply the method to the asymmetric Nizhnik-Novikov-Vesselov equation and the coupled Drinfel'd-Sokolov-Wilson equation and successfully cover the previously known travelling wave solutions found by Chen's method [Y. Chen, et al. Chaos, Solitons and Fractals 22 (2004) 675; Y. Chen, et al. Int. J. Mod. Phys. C 4 (2004) 595].
Calatroni, Luca
2013-08-01
We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H -1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation.
Directory of Open Access Journals (Sweden)
Abdon Atangana
2014-01-01
Full Text Available A novel approach is proposed to deal with a class of nonlinear partial equations including integer and noninteger order derivative. This class of equations cannot be handled with any other commonly used analytical technique. The proposed method is based on the multi-Laplace transform. We solved as an example some complicated equations. Three illustrative examples are presented to confirm the applicability of the proposed method. We have presented in detail the stability, the convergence and the uniqueness analysis of some examples.
Huang, Qing; Wang, Li-Zhen; Zuo, Su-Li
2016-02-01
In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie's modified Riemann-Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada-Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed. Supported by the National Natural Science Foundation of China under Grant Nos. 11101332, 11201371, 11371293 and the Natural Science Foundation of Shaanxi Province under Grant No. 2015JM1037
Directory of Open Access Journals (Sweden)
Shoukry Ibrahim Atia El-Ganaini
2013-01-01
Full Text Available The first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including classical Drinfel'd-Sokolov-Wilson system (DSWE, (2 + 1-dimensional Davey-Stewartson system, and generalized Hirota-Satsuma coupled KdV system. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner. This method can also be applied to nonintegrable equations as well as integrable ones.
Numerical approximation on computing partial sum of nonlinear Schroedinger eigenvalue problems
Institute of Scientific and Technical Information of China (English)
JiachangSUN; DingshengWANG; 等
2001-01-01
In computing electronic structure and energy band in the system of multiparticles,quite a large number of problems are to obtain the partial sum of the densities and energies by using “First principle”。In the ordinary method,the so-called self-consistency approach,the procedure is limited to a small scale because of its high computing complexity.In this paper,the problem of computing the partial sum for a class of nonlinear Schroedinger eigenvalue equations is changed into the constrained functional minimization.By space decompostion and Rayleigh-Schroedinger method,one approximating formula for the minimal is provided.The numerical experiments show that this formula is more precise and its quantity of computation is smaller.
Institute of Scientific and Technical Information of China (English)
LI Hua-Mei
2003-01-01
In this paper, we extend the mapping deformation method proposed by Lou. It is used to find new exacttravelling wave solutions of nonlinear partial differential equation or coupled nonlinear partial differential equations(PDEs). Based on the idea of the homogeneous balance method, we construct the general mapping relation betweenthe solutions of the PDEs and those of the cubic nonlinear Klein-Gordon (NKG) equation. By using this relation andthe abundant solutions of the cubic NKG equation, many explicit and exact travelling wave solutions of three systemsof coupled PDEs, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic functionsolutions, and rational solutions, are obtained.
Bifurcation of hyperbolic planforms
Chossat, Pascal; Faugeras, Olivier
2010-01-01
Motivated by a model for the perception of textures by the visual cortex in primates, we analyse the bifurcation of periodic patterns for nonlinear equations describing the state of a system defined on the space of structure tensors, when these equations are further invariant with respect to the isometries of this space. We show that the problem reduces to a bifurcation problem in the hyperbolic plane D (Poincar\\'e disc). We make use of the concept of periodic lattice in D to further reduce the problem to one on a compact Riemann surface D/T, where T is a cocompact, torsion-free Fuchsian group. The knowledge of the symmetry group of this surface allows to carry out the machinery of equivariant bifurcation theory. Solutions which generically bifurcate are called "H-planforms", by analogy with the "planforms" introduced for pattern formation in Euclidean space. This concept is applied to the case of an octagonal periodic pattern, where we are able to classify all possible H-planforms satisfying the hypotheses o...
Hyperbolic metamaterials based on Bragg polariton structures
Sedov, E. S.; Charukhchyan, M. V.; Arakelyan, S. M.; Alodzhants, A. P.; Lee, R.-K.; Kavokin, A. V.
2016-07-01
A new hyperbolic metamaterial based on a modified semiconductor Bragg mirror structure with embedded periodically arranged quantum wells is proposed. It is shown that exciton polaritons in this material feature hyperbolic dispersion in the vicinity of the second photonic band gap. Exciton-photon interaction brings about resonant nonlinearity leading to the emergence of nontrivial topological polaritonic states. The formation of spatially localized breather-type structures (oscillons) representing kink-shaped solutions of the effective Ginzburg-Landau-Higgs equation slightly oscillating along one spatial direction is predicted.
Generalised hyperbolicity in conical space-times
Vickers, J A
2000-01-01
Solutions of the wave equation in a space-time containing a thin cosmic string are examined in the context of non-linear generalised functions. Existence and uniqueness of solutions to the wave equation in the Colombeau algebra G is established for a conical space-time and this solution is shown to be associated to a distributional solution. A concept of generalised hyperbolicity, based on test fields, can be defined for such singular space-times and it is shown that a conical space-time is G-hyperbolic.
Hyperbolic neighborhoods as organizers of finite-time exponential stretching
Balasuriya, Sanjeeva; Ouellette, Nicholas
2016-11-01
Hyperbolic points and their unsteady generalization, hyperbolic trajectories, drive the exponential stretching that is the hallmark of nonlinear and chaotic flow. Typical experimental and observational velocity data is unsteady and available only over a finite time interval, and in such situations hyperbolic trajectories will move around in the flow, and may lose their hyperbolicity at times. Here we introduce a way to determine their region of influence, which we term a hyperbolic neighborhood, which marks fluid elements whose dynamics are instantaneously dominated by the hyperbolic trajectory. We establish, using both theoretical arguments and numerical verification from model and experimental data, that the hyperbolic neighborhoods profoundly impact Lagrangian stretching experienced by fluid elements. In particular, we show that fluid elements traversing a flow experience exponential boosts in stretching while within these time-varying regions, that greater residence time within hyperbolic neighborhoods is directly correlated to larger Finite-Time Lyapunov Exponent (FTLE) values, and that FTLE diagnostics are reliable only when the hyperbolic neighborhoods have a geometrical structure which is regular in a specific sense. Future Fellowship Grant FT130100484 from the Australian Research Council (SB), and a Terman Faculty Fellowship from Stanford University (NO).
Non-linear partially massless symmetry in an SO(1,5) continuation of conformal gravity
Apolo, Luis
2016-01-01
We construct a non-linear theory of interacting spin-2 fields that is invariant under the partially massless (PM) symmetry to all orders. This theory is based on the SO(1,5) group, in analogy with the SO(2,4) formulation of conformal gravity, but has a quadratic spectrum free of ghost instabilities. The action contains a vector field associated to a local SO(2) symmetry which is manifest in the vielbein formulation of the theory. We show that, in a perturbative expansion, the SO(2) symmetry transmutes into the PM transformations of a massive spin-2 field. In this context, the vector field is crucial to circumvent earlier obstructions to an order-by-order construction of PM symmetry. Although the non-linear theory lacks enough first class constraints to remove all helicity-0 modes from the spectrum, the PM transformations survive to all orders. The absence of ghosts and strong coupling effects at the non-linear level are not addressed here.
Jarlebring, Elias; Michiels, Wim
2012-01-01
The partial Schur factorization can be used to represent several eigenpairs of a matrix in a numerically robust way. Different adaptions of the Arnoldi method are often used to compute partial Schur factorizations. We propose here a technique to compute a partial Schur factorization of a nonlinear eigenvalue problem (NEP). The technique is inspired by the algorithm in [8], now called the infinite Arnoldi method. The infinite Arnoldi method is a method designed for NEPs, and can be interpreted as Arnoldi's method applied to a linear infinite-dimensional operator, whose reciprocal eigenvalues are the solutions to the NEP. As a first result we show that the invariant pairs of the operator are equivalent to invariant pairs of the NEP. We characterize the structure of the invariant pairs of the operator and show how one can carry out a modification of the infinite Arnoldi method by respecting the structure. This also allows us to naturally add the feature known as locking. We nest this algorithm with an outer iter...
Fourth order difference methods for hyperbolic IBVP's
Gustafsson, Bertil; Olsson, Pelle
1994-01-01
Fourth order difference approximations of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second order viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth order methods with a standard second order one and with a third order TVD-method. The results show that the fourth order methods are the only ones that give good results for all the considered test problems.
Optimal boundary control and boundary stabilization of hyperbolic systems
Gugat, Martin
2015-01-01
This brief considers recent results on optimal control and stabilization of systems governed by hyperbolic partial differential equations, specifically those in which the control action takes place at the boundary. The wave equation is used as a typical example of a linear system, through which the author explores initial boundary value problems, concepts of exact controllability, optimal exact control, and boundary stabilization. Nonlinear systems are also covered, with the Korteweg-de Vries and Burgers Equations serving as standard examples. To keep the presentation as accessible as possible, the author uses the case of a system with a state that is defined on a finite space interval, so that there are only two boundary points where the system can be controlled. Graduate and post-graduate students as well as researchers in the field will find this to be an accessible introduction to problems of optimal control and stabilization.
Weakly asymptotically hyperbolic manifolds
Allen, Paul T; Lee, John M; Allen, Iva Stavrov
2015-01-01
We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to $-1$ and are $C^0$, but are not necessarily $C^1$, conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves as an obstruction to "higher order decay" of the Riemann curvature operator. Finally, we establish Fredholm results for geometric elliptic operators, extending the work of Rafe Mazzeo and John M. Lee to this setting. As an application, we show that any weakly asymptotically hyperbolic metric is conformally related to a weakly asymptotically hyperbolic metric of constant negative curvature.
3D early embryogenesis image filtering by nonlinear partial differential equations.
Krivá, Z; Mikula, K; Peyriéras, N; Rizzi, B; Sarti, A; Stasová, O
2010-08-01
We present nonlinear diffusion equations, numerical schemes to solve them and their application for filtering 3D images obtained from laser scanning microscopy (LSM) of living zebrafish embryos, with a goal to identify the optimal filtering method and its parameters. In the large scale applications dealing with analysis of 3D+time embryogenesis images, an important objective is a correct detection of the number and position of cell nuclei yielding the spatio-temporal cell lineage tree of embryogenesis. The filtering is the first and necessary step of the image analysis chain and must lead to correct results, removing the noise, sharpening the nuclei edges and correcting the acquisition errors related to spuriously connected subregions. In this paper we study such properties for the regularized Perona-Malik model and for the generalized mean curvature flow equations in the level-set formulation. A comparison with other nonlinear diffusion filters, like tensor anisotropic diffusion and Beltrami flow, is also included. All numerical schemes are based on the same discretization principles, i.e. finite volume method in space and semi-implicit scheme in time, for solving nonlinear partial differential equations. These numerical schemes are unconditionally stable, fast and naturally parallelizable. The filtering results are evaluated and compared first using the Mean Hausdorff distance between a gold standard and different isosurfaces of original and filtered data. Then, the number of isosurface connected components in a region of interest (ROI) detected in original and after the filtering is compared with the corresponding correct number of nuclei in the gold standard. Such analysis proves the robustness and reliability of the edge preserving nonlinear diffusion filtering for this type of data and lead to finding the optimal filtering parameters for the studied models and numerical schemes. Further comparisons consist in ability of splitting the very close objects which
Directory of Open Access Journals (Sweden)
Jagdev Singh
2014-01-01
Full Text Available The main aim of this work is to present a user friendly numerical algorithm based on homotopy perturbation Sumudu transform method for nonlinear fractional partial differential arising in spatial diffusion of biological populations in animals. The movements are made generally either by mature animals driven out by invaders or by young animals just reaching maturity moving out of their parental territory to establish breeding territory of their own. The homotopy perturbation Sumudu transform method is a combined form of the Sumudu transform method and homotopy perturbation method. The obtained results are compared with Sumudu decomposition method. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. These results reveal that the proposed method is computationally very attractive.
Magnetic hyperbolic optical metamaterials
Kruk, Sergey S; Pshenay-Severin, Ekaterina; O'Brien, Kevin; Neshev, Dragomir N; Kivshar, Yuri S; Zhang, Xiang
2015-01-01
Strongly anisotropic media where the principal components of the electric permittivity and/or magnetic permeability tensor have opposite signs are termed as hyperbolic media. Such media support propagating electromagnetic waves with extremely large wavevectors, and therefore they exhibit unique optical properties. However in all artificial and natural optical structures studied to date the hyperbolic dispersion originates solely from their electric response. This restricts functionality of these materials for only one polarization of light and inhibits impedance matching with free space. Such restrictions can be overcome in media having components of opposite signs in both electric and magnetic tensors. Here we present the first experimental demonstration of the magnetic hyperbolic dispersion in three-dimensional metamaterials. We measure experimentally metamaterial's dispersion and trace the topological transition between the elliptic and hyperbolic regimes. We experimentally demonstrate that due to the uniq...
Hyperbolic conservation laws in continuum physics
Dafermos, Constantine M
2016-01-01
This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conser...
Khan, Zulfiqar Hasan; Gu, Irene Yu-Hua
2013-12-01
This paper proposes a novel Bayesian online learning and tracking scheme for video objects on Grassmann manifolds. Although manifold visual object tracking is promising, large and fast nonplanar (or out-of-plane) pose changes and long-term partial occlusions of deformable objects in video remain a challenge that limits the tracking performance. The proposed method tackles these problems with the main novelties on: 1) online estimation of object appearances on Grassmann manifolds; 2) optimal criterion-based occlusion handling for online updating of object appearances; 3) a nonlinear dynamic model for both the appearance basis matrix and its velocity; and 4) Bayesian formulations, separately for the tracking process and the online learning process, that are realized by employing two particle filters: one is on the manifold for generating appearance particles and another on the linear space for generating affine box particles. Tracking and online updating are performed in an alternating fashion to mitigate the tracking drift. Experiments using the proposed tracker on videos captured by a single dynamic/static camera have shown robust tracking performance, particularly for scenarios when target objects contain significant nonplanar pose changes and long-term partial occlusions. Comparisons with eight existing state-of-the-art/most relevant manifold/nonmanifold trackers with evaluations have provided further support to the proposed scheme.
Partial synchronization in networks of non-linearly coupled oscillators: The Deserter Hubs Model
Energy Technology Data Exchange (ETDEWEB)
Freitas, Celso, E-mail: cbnfreitas@gmail.com; Macau, Elbert, E-mail: elbert.macau@inpe.br [Associate Laboratory for Computing and Applied Mathematics - LAC, Brazilian National Institute for Space Research - INPE (Brazil); Pikovsky, Arkady, E-mail: pikovsky@uni-potsdam.de [Department of Physics and Astronomy, University of Potsdam, Germany and Department of Control Theory, Nizhni Novgorod State University, Gagarin Av. 23, 606950, Nizhni Novgorod (Russian Federation)
2015-04-15
We study the Deserter Hubs Model: a Kuramoto-like model of coupled identical phase oscillators on a network, where attractive and repulsive couplings are balanced dynamically due to nonlinearity of interactions. Under weak force, an oscillator tends to follow the phase of its neighbors, but if an oscillator is compelled to follow its peers by a sufficient large number of cohesive neighbors, then it actually starts to act in the opposite manner, i.e., in anti-phase with the majority. Analytic results yield that if the repulsion parameter is small enough in comparison with the degree of the maximum hub, then the full synchronization state is locally stable. Numerical experiments are performed to explore the model beyond this threshold, where the overall cohesion is lost. We report in detail partially synchronous dynamical regimes, like stationary phase-locking, multistability, periodic and chaotic states. Via statistical analysis of different network organizations like tree, scale-free, and random ones, we found a measure allowing one to predict relative abundance of partially synchronous stationary states in comparison to time-dependent ones.
Directory of Open Access Journals (Sweden)
Brajesh Kumar Singh
2016-01-01
Full Text Available This paper deals with an analytical solution of an initial value system of time dependent linear and nonlinear partial differential equations by implementing reduced differential transform (RDT method. The effectiveness and the convergence of RDT method are tested by means of five test problems, which indicates the validity and great potential of the reduced differential transform method for solving system of partial differential equations.
Institute of Scientific and Technical Information of China (English)
张帆; 刘志红
2014-01-01
非线性双曲Schrödinger函数在计算机编码和通用处理器设计中具有广阔用途，双曲微分方程求解作为单浮点Cache加速器设计的核心算法。传统方法采用稀疏矩阵向量乘方法进行单浮点Cache加速器设计，当微分矩阵的阶数较大时，不能直接进行随机搜索，对具有单浮点数据格式的算法加速性能不好。提出一种基于非线性双曲Schrödinger函数的单浮点Cache加速器优化设计方法，采用非线性双曲Schrödinger函数进行矩阵重排序方法，并进行非线性编码，引入稀疏矩阵向量乘单浮点Cache数据结构。采用三级流水线结构设计单浮点Cache加速器，基于Xilinx Virtex-5平台进行数据并行处理性能测试，得出该算法单浮点Cache并行运算加速比比传统方法大1.37~2.60倍，且有优越的数据吞吐性能，稳定性好，在数据库执行算法的并行处理和外部存储器的带宽利用率提高等方面有很好的应用价值。%Nonlinear hyperbolic Schrödinger function has a wide use in the computer code and general processor design, hy-perbolic differential equations is the a core algorithm for single floating-point Cache accelerator design. Traditional method using sparse matrix vector multiplication method for single floating-point Cache accelerator design, when the differential matrix is large, it cannot directly carry out random search, acceleration performance is not good. A kind of single floating-point Cache accelerator optimization method sis designed is proposed base on nonlinear hyperbolic Schrödinger function, the matrix reordering method is used, and nonlinear coding is taken, the sparse matrix vector multiplication single floating-point Cache data structure is introduced. Three stage pipeline structures are used to design a single floating-point Cache accelerator, data parallel processing performance test is taken based on Xilinx Virtex-5, and it shows that the parallel com
Boundary Layer to a System of Viscous Hyperbolic Conservation Laws
Institute of Scientific and Technical Information of China (English)
2008-01-01
In this paper, we investigate the large-time behavior of solutions to the initial-boundary value problem for nxn hyperbolic system of conservation laws with artificial viscosity in the half line (0, ∞). We first show that a boundary layer exists if the corresponding hyperbolic part contains at least one characteristic field with negative propagation speed. We further show that such boundary layer is nonlinearly stable under small initial perturbation. The proofs are given by an elementary energy method.
Finite volume evolution Galerkin (FVEG) methods for hyperbolic systems
Lukácová-Medvid'ová, Maria; Morton, K.W.; Warnecke, Gerald
2003-01-01
The subject of the paper is the derivation and analysis of new multidimensional, high-resolution, finite volume evolution Galerkin (FVEG) schemes for systems of nonlinear hyperbolic conservation laws. Our approach couples a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of the multidimensional hyperbolic system, such that all of the infinitely many directions of wave propagation are taken into account. In particular, we p...
Gromov Hyperbolicity of Riemann Surfaces
Institute of Scientific and Technical Information of China (English)
José M. RODR(I)GUEZ; Eva TOUR(I)S
2007-01-01
We study the hyperbolicity in the Gromov sense of Riemann surfaces. We deduce the hyperbolicity of a surface from the hyperbolicity of its "building block components". We also prove the equivalence between the hyperbolicity of a Riemann surface and the hyperbolicity of some graph associated with it. These results clarify how the decomposition of a Riemann surface into Y-pieces and funnels affects the hyperbolicity of the surface. The results simplify the topology of the surface and allow us to obtain global results from local information.
Institute of Scientific and Technical Information of China (English)
胡业民; 胡希伟
2001-01-01
Numerical analyses for the nonlinear evolutions of stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS) processes are given. Various effects of the second- and third-order nonlinear susceptibilities on the SRS and SBS processes are studied. The nonlinear evolutions of SRS and SBS processes are atfected more efficiently than their linear growth rates by the nonlinear susceptibility.
Computing the Gromov hyperbolicity constant of a discrete metric space
Ismail, Anas
2012-07-01
Although it was invented by Mikhail Gromov, in 1987, to describe some family of groups[1], the notion of Gromov hyperbolicity has many applications and interpretations in different fields. It has applications in Biology, Networking, Graph Theory, and many other areas of research. The Gromov hyperbolicity constant of several families of graphs and geometric spaces has been determined. However, so far, the only known algorithm for calculating the Gromov hyperbolicity constant of a discrete metric space is the brute force algorithm with running time O (n4) using the four- point condition. In this thesis, we first introduce an approximation algorithm which calculates a O (log n)-approximation of the hyperbolicity constant , based on a layering approach, in time O (n2), where n is the number of points in the metric space. We also calculate the fixed base point hyperbolicity constant r for a fixed point r using a (max; min)matrix multiplication algorithm by Duan in time O (n2:688) [2]. We use this result to present a 2-approximation algorithm for calculating the hyperbolicity constant in time O (n2:688). We also provide an exact algorithm to compute the hyperbolicity constant in time O (n3:688) for a discrete metric space. We then present some partial results we obtained for designing some approximation algorithms to compute the hyperbolicity constant.
Hyperbolic conservation laws and numerical methods
Leveque, Randall J.
1990-01-01
The mathematical structure of hyperbolic systems and the scalar equation case of conservation laws are discussed. Linear, nonlinear systems and the Riemann problem for the Euler equations are also studied. The numerical methods for conservation laws are presented in a nonstandard manner which leads to large time steps generalizations and computations on irregular grids. The solution of conservation laws with stiff source terms is examined.
Katzav, E; Nechaev, S; Vasilyev, O
2007-06-01
We report some observations concerning the statistics of longest increasing subsequences (LIS). We argue that the expectation of LIS, its variance, and apparently the full distribution function appears in statistical analysis of some simple nonlinear stochastic partial differential equation in the limit of very low noise intensity.
Institute of Scientific and Technical Information of China (English)
WU YUE-XIANG; HUO YAN-MEI; WU YA-KUN
2012-01-01
The main purpose of this paper is to examine the existence of coupled solutions and coupled minimal-maximal solutions for a kind of nonlinear operator equations in partial ordered linear topology spaces by employing the semi-order method.Some new existence results are obtained.
Spectral methods for time dependent partial differential equations
Gottlieb, D.; Turkel, E.
1983-01-01
The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.
Institute of Scientific and Technical Information of China (English)
伦淑娴; 张化光
2005-01-01
This paper develops delay-independent fuzzy hyperbolic guaranteed cost control for nonlinear continuous-time systems with parameter uncertainties. Fuzzy hyperbolic model (FHM) can be used to establish the model for certain unknown complex system. The main advantage of using FHM over Takagi-Sugeno (T-S) fuzzy model is that no premise structure identification is needed and no completeness design of premise variables space is needed. In addition, an FHM is not only a kind of valid global description but also a kind of nonlinear model in nature. A nonlinear quadratic cost function is developed as a performance measurement of the closed-loop fuzzy system based on FHM.Based on delay-independent Lyapunov functional approach, some sufficient conditions for the existence of such a fuzzy hyperbolic guaranteed cost controller via state feedback are provided. These conditions are given in terms of the feasibility of linear matrix inequalities (LMIs). A simulation example is provided to illustrate the design procedure of the proposed method.
Hyperbolic Metamaterials with Complex Geometry
DEFF Research Database (Denmark)
Lavrinenko, Andrei; Andryieuski, Andrei; Zhukovsky, Sergei
2016-01-01
We investigate new geometries of hyperbolic metamaterialssuch as highly corrugated structures, nanoparticle monolayer assemblies, super-structured or vertically arranged multilayersand nanopillars. All structures retain basic propertiesof hyperbolic metamaterials, but have functionality improved...
Thermal hyperbolic metamaterials.
Guo, Yu; Jacob, Zubin
2013-06-17
We explore the near-field radiative thermal energy transfer properties of hyperbolic metamaterials. The presence of unique electromagnetic states in a broad bandwidth leads to super-planckian thermal energy transfer between metamaterials separated by a nano-gap. We consider practical phonon-polaritonic metamaterials for thermal engineering in the mid-infrared range and show that the effect exists in spite of the losses, absorption and finite unit cell size. For thermophotovoltaic energy conversion applications requiring energy transfer in the near-infrared range we introduce high temperature hyperbolic metamaterials based on plasmonic materials with a high melting point. Our work paves the way for practical high temperature radiative thermal energy transfer applications of hyperbolic metamaterials.
Luminescent hyperbolic metasurfaces
Smalley, J. S. T.; Vallini, F.; Montoya, S. A.; Ferrari, L.; Shahin, S.; Riley, C. T.; Kanté, B.; Fullerton, E. E.; Liu, Z.; Fainman, Y.
2017-01-01
When engineered on scales much smaller than the operating wavelength, metal-semiconductor nanostructures exhibit properties unobtainable in nature. Namely, a uniaxial optical metamaterial described by a hyperbolic dispersion relation can simultaneously behave as a reflective metal and an absorptive or emissive semiconductor for electromagnetic waves with orthogonal linear polarization states. Using an unconventional multilayer architecture, we demonstrate luminescent hyperbolic metasurfaces, wherein distributed semiconducting quantum wells display extreme absorption and emission polarization anisotropy. Through normally incident micro-photoluminescence measurements, we observe absorption anisotropies greater than a factor of 10 and degree-of-linear polarization of emission >0.9. We observe the modification of emission spectra and, by incorporating wavelength-scale gratings, show a controlled reduction of polarization anisotropy. We verify hyperbolic dispersion with numerical simulations that model the metasurface as a composite nanoscale structure and according to the effective medium approximation. Finally, we experimentally demonstrate >350% emission intensity enhancement relative to the bare semiconducting quantum wells.
Hyperbolic Resonances of Metasurface Cavities
Keene, David
2015-01-01
We propose a new class of optical resonator structures featuring one or two metasurface reflectors or metacavities and predict that such resonators support novel hyperbolic resonances. As an example of such resonances we introduce hyperbolic Tamm plasmons (HTPs) and hyperbolic Fabry-Perot resonances (HFPs). The hyperbolic optical modes feature low-loss incident power re-distribution over TM and TE polarization output channels, clover-leaf anisotropic dispersion, and other unique properties which are tunable and are useful for multiple applications.
Stability and Convergence of Relaxation Schemes to Hyperbolic Balance Laws via a Wave Operator
Miroshnikov, Alexey; Trivisa, Konstantina
2014-01-01
This article deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of D...
Invariant Surfaces under Hyperbolic Translations in Hyperbolic Space
Directory of Open Access Journals (Sweden)
Mahmut Mak
2014-01-01
Full Text Available We consider hyperbolic rotation (G0, hyperbolic translation (G1, and horocyclic rotation (G2 groups in H3, which is called Minkowski model of hyperbolic space. Then, we investigate extrinsic differential geometry of invariant surfaces under subgroups of G0 in H3. Also, we give explicit parametrization of these invariant surfaces with respect to constant hyperbolic curvature of profile curves. Finally, we obtain some corollaries for flat and minimal invariant surfaces which are associated with de Sitter and hyperbolic shape operator in H3.
Global Hyperbolicity and Completeness
Choquet-Bruhat, Y; Choquet-Bruhat, Yvonne; Cotsakis, Spiros
2002-01-01
We prove global hyperbolicity of spacetimes under generic regularity conditions on the metric. We then show that these spacetimes are timelike and null geodesically complete if the gradient of the lapse and the extrinsic curvature $K$ are integrable. This last condition is required only for the tracefree part of $K$ if the universe is expanding.
Sources of hyperbolic geometry
Stillwell, John
1996-01-01
This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincaré that brought hyperbolic geometry into the mainstream of mathematics. A recognition of Beltrami comparable to that given the pioneering works of Bolyai and Lobachevsky seems long overdue-not only because Beltrami rescued hyperbolic geometry from oblivion by proving it to be logically consistent, but because he gave it a concrete meaning (a model) that made hyperbolic geometry part of ordinary mathematics. The models subsequently discovered by Klein and Poincaré brought hyperbolic geometry even further down to earth and paved the way for the current explosion of activity in low-dimensional geometry and topology. By placing the works of these three mathematicians side by side and providing commentaries, this book gives the student, historian, or professional geometer a bird's-eye view of one of the great episodes in mathematics. The unified setting and historical context reveal the insights of Beltrami, Klein, and Po...
Proof mining in ${\\mathbb R}$-trees and hyperbolic spaces
Leustean, Laurentiu
2008-01-01
This paper is part of the general project of proof mining, developed by Kohlenbach. By "proof mining" we mean the logical analysis of mathematical proofs with the aim of extracting new numerically relevant information hidden in the proofs. We present logical metatheorems for classes of spaces from functional analysis and hyperbolic geometry, like Gromov hyperbolic spaces, ${\\mathbb R}$-trees and uniformly convex hyperbolic spaces. Our theorems are adaptations to these structures of previous metatheorems of Gerhardy and Kohlenbach, and they guarantee a-priori, under very general logical conditions, the existence of uniform bounds. We give also an application in nonlinear functional analysis, more specifically in metric fixed-point theory. Thus, we show that the uniform bound on the rate of asymptotic regularity for the Krasnoselski-Mann iterations of nonexpansive mappings in uniformly convex hyperbolic spaces obtained in a previous paper is an instance of one of our metatheorems.
Hyperbolic groupoids: definitions and duality
Nekrashevych, Volodymyr
2011-01-01
We define a notion of a hyperbolic groupoid (pseudogroup) generalizing actions of Gromov hyperbolic groups on their boundaries. We show that the boundary of a Gromov hyperbolic groupoid has a natural local product structure and that actions of hyperbolic groupoids on their boundaries can be described axiomatically as generalized Smale spaces (which we call Smale quasi-flows). The original groupoid is equivalent to the projection of the corresponding Smale quasi-flow onto the stable direction of the local product structure. The projection onto the unstable direction is called the dual of the groupoid. Examples of pairs of mutually dual hyperbolic groupoids and associated Smale quasi-flows are described.
ON THE DIFFUSION PHENOMENON OF QUASILINEAR HYPERBOLIC WAVES
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping utt+ut-div(a(△u)△u)=01and show that, at least when n≤3, they tend, as t→+∞, to those of the nonlinear parabolic equation ut-div(a(△u)△u)=01in the sense that the norm ‖u(.,t)-v(.,t)‖L∞(Rn)of the difference u-v decays faster than that of either u or v. This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves, first observed by Hsiao, L. and Liu Taiping (see [1,2]).
Almaraz, Pablo; Green, Andy J; Aguilera, Eduardo; Rendón, Miguel A; Bustamante, Javier
2012-09-01
1. Understanding the impact of environmental variability on migrating species requires the estimation of sequential abiotic effects in different geographic areas across the life cycle. For instance, waterfowl (ducks, geese and swans) usually breed widely dispersed throughout their breeding range and gather in large numbers in their wintering headquarters, but there is a lack of knowledge on the effects of the sequential environmental conditions experienced by migrating birds on the long-term community dynamics at their wintering sites. 2. Here, we analyse multidecadal time-series data of 10 waterfowl species wintering in the Guadalquivir Marshes (SW Spain), the single most important wintering site for waterfowl breeding in Europe. We use a multivariate state-space approach to estimate the effects of biotic interactions, local environmental forcing during winter and large-scale climate during breeding and migration on wintering multispecies abundance fluctuations, while accounting for partial observability (observation error and missing data) in both population and environmental data. 3. The joint effect of local weather and large-scale climate explained 31·6% of variance at the community level, while the variability explained by interspecific interactions was negligible (observations through data augmentation increased the estimated magnitude of environmental forcing by an average 30·1% and reduced the impact of stochasticity by 39·3% when accounting for observation error. Interestingly however, the impact of environmental forcing on community dynamics was underestimated by an average 15·3% and environmental stochasticity overestimated by 14·1% when ignoring both observation error and data augmentation. 5. These results provide a salient example of sequential multiscale environmental forcing in a major migratory bird community, which suggests a demographic link between the breeding and wintering grounds operating through nonlinear environmental effects
Energy Technology Data Exchange (ETDEWEB)
Lu, Bin, E-mail: lubinhb@163.com [School of Mathematical Sciences, Anhui University, Hefei 230601 (China)
2012-06-04
In this Letter, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the Bäcklund transformation of fractional Riccati equation are employed for constructing the exact solutions of nonlinear fractional partial differential equations. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations. -- Highlights: ► Backlund transformation of fractional Riccati equation is presented. ► A new method for solving nonlinear fractional differential equations is proposed. ► Three important fractional differential equations are solved successfully. ► Some new exact solutions of the fractional differential equations are obtained.
Zia, Haider
2017-06-01
This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized derivative non-linear Schrödinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies to is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material.
Directory of Open Access Journals (Sweden)
Elsayed Mohamed Elsayed ZAYED
2014-07-01
Full Text Available In this article, many new exact solutions of the (2+1-dimensional nonlinear Boussinesq-Kadomtsev-Petviashvili equation and the (1+1-dimensional nonlinear heat conduction equation are constructed using the Riccati equation mapping method. By means of this method, many new exact solutions are successfully obtained. This method can be applied to many other nonlinear evolution equations in mathematical physics.doi:10.14456/WJST.2014.14
Kramer, Sean; Bollt, Erik M
2013-09-01
Given multiple images that describe chaotic reaction-diffusion dynamics, parameters of a partial differential equation (PDE) model are estimated using autosynchronization, where parameters are controlled by synchronization of the model to the observed data. A two-component system of predator-prey reaction-diffusion PDEs is used with spatially dependent parameters to benchmark the methods described. Applications to modeling the ecological habitat of marine plankton blooms by nonlinear data assimilation through remote sensing are discussed.
Institute of Scientific and Technical Information of China (English)
LIU Chun-Ping; LING Zhi
2005-01-01
By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m → 1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.
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Hasibun Naher
2012-01-01
Full Text Available We construct new analytical solutions of the (3+1-dimensional modified KdV-Zakharov-Kuznetsev equation by the Exp-function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp-function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher-dimensional nonlinear partial differential equation.
Chouhan, Romita; Baraskar, Priyanka; Agrawal, Arpana; Gupta, Mukul; Sen, Pranay K.; Sen, Pratima
2017-07-01
We report annealing induced sign reversal of dispersive optical nonlinearity in ion beam sputtered NiO thin films deposited at 30% and 70% oxygen partial pressures. In the Ultraviolet-visible spectra of the samples, the transmission peak corresponding to d-d transitions is observed near 2 eV. A shift in this peak towards higher energy was observed when the same films were annealed at 523 K. The near resonant photoinduced transitions produced giant nonlinear optical susceptibilities of both third- and fifth- orders when the annealed film was irradiated by a continuous wave 632.8 nm He-Ne laser. The role of the thermo-optic effect has been examined critically. Experimental studies further reveal that the oxygen partial pressure influences the growth direction of the grains in the thin films. The well known Z-scan experimental procedure has been followed for measurements of optical nonlinearities in all the NiO films. The nonlinear refractive indices of both the as-deposited and annealed NiO thin films are defined in terms of the thermo-optic coefficients (d/nd T ) T =T0 and (d/2nd T2 ) T =T0 .
Double-wave solutions to quasilinear hyperbolic systems of first-order PDEs
Curró, C.; Manganaro, N.
2017-10-01
A reduction procedure for determining double-wave exact solutions to first-order hyperbolic systems of PDEs is proposed. The basic idea is to reduce the integration of the governing hyperbolic set of N partial differential equations to that of a 2 × 2 reduced hyperbolic model along with a further differential constraint. Therefore, the method of differential constraints is used in order to solve the auxiliary 2 × 2 system. An example of interest to viscoelasticity is presented.
Institute of Scientific and Technical Information of China (English)
姜久红; 王军; 王志伟
2011-01-01
建立二自由度非线性产品包装系统模型，得到冲击动力学方程并数值求解，研究了双曲正切包装系统关键部件的矩形脉冲响应特性。运用数值求解得到关键部件破损边界曲面，并讨论了名义频率比、阻尼、脉冲激励幅值和系统参数对关键部件破损边界的影响规律，结果表明，频率比、阻尼、脉冲激励幅值和系统参数对关键部件破损边界影响显著，研究结论为产品包装设计提供科学依据。%The shock characteristic of the hyperbolic tangent nonlinear packaging system with critical component were investigated under the action of rectangular acceleration pulse. The dynamical model of the system was developed. And the numerical results of the dynamical equations were got. The damage boundary surface of critical component was obtained based on the results. And the effect of the pulse duration, the frequency ratio, the dmaping ratio, the pulse peak acceleration in additional to the defined system parameter on the DBS of critical component was discussed. It's shown that all of their effects are noticeable. The results lead to some insights into the design of cushioning packaging.
Institute of Scientific and Technical Information of China (English)
李永献; 杨晓侠
2016-01-01
研究了非协调类Carey元对非线性伪双曲方程的Galerkin逼近.利用该元在能量模意义下非协调误差比插值误差高一阶的特殊性质,线性三角形元的高精度分析结果,平均值技巧和插值后处理技术,在抛弃传统的Ritz投影的情形下,得到了半离散格式能量模意义下的超逼近性质和整体超收敛结果.同时,针对方程中系数为线性的情形建立一个具有二阶精度的全离散逼近格式,导出了相应的超逼近和超收敛结果.%The Galerkin finite element mothed for nonlinear pseudo-hyperbolic equations with nonconforming quasi-Carey element is studied.Based on the special property of the element(i.e.the consistency error is one order higher than its interpolation error in the energy norm) and high accuracy analysis result of the linear triangular element,the superclose property and global superconvergence result in energy norm with order O(h2) for semi-discrete scheme are obtained employing the mean-value technique and interpolated postprocessing approach.At the same time,a second order fully-discrete scheme is established for quasi-linear case.The corresponding superclose property and superconvergence result of order O(h2 +r2) are deduced.Here,h and r are parameters of subdivision in space and time step respectively.
Nonpoint Symmetry and Reduction of Nonlinear Evolution and Wave Type Equations
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Ivan Tsyfra
2015-01-01
Full Text Available We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ansätze reducing nonlinear evolution equations to system of ordinary differential equations. The ansätze are constructed by using operators of nonpoint classical and conditional symmetry. Then we find solution to nonlinear heat equation which cannot be obtained in the framework of the classical Lie approach. By using operators of Lie-Bäcklund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too.
Hyperbolicity of projective hypersurfaces
Diverio, Simone
2016-01-01
This book presents recent advances on Kobayashi hyperbolicity in complex geometry, especially in connection with projective hypersurfaces. This is a very active field, not least because of the fascinating relations with complex algebraic and arithmetic geometry. Foundational works of Serge Lang and Paul A. Vojta, among others, resulted in precise conjectures regarding the interplay of these research fields (e.g. existence of Zariski dense entire curves should correspond to the (potential) density of rational points). Perhaps one of the conjectures which generated most activity in Kobayashi hyperbolicity theory is the one formed by Kobayashi himself in 1970 which predicts that a very general projective hypersurface of degree large enough does not contain any (non-constant) entire curves. Since the seminal work of Green and Griffiths in 1979, later refined by J.-P. Demailly, J. Noguchi, Y.-T. Siu and others, it became clear that a possible general strategy to attack this problem was to look at particular algebr...
Asymptotically hyperbolic connections
Fine, Joel; Krasnov, Kirill; Scarinci, Carlos
2015-01-01
General Relativity in 4 dimensions can be equivalently described as a dynamical theory of SO(3)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analog of the Fefferman-Graham expansion in the language of connections. As in the metric setup, one can solve the arising "evolution" equations order by order in the expansion in powers of the radial coordinate. The solution in the connection setting is arguably simpler, and very straightforward algebraic manipulations allow one to see how the obstruction appears at third order in the expansion. Another interesting feature of the connection formulation is that the "counter terms" required in the computation of the renormalised volume all combine into the Chern-Simons functional of the restriction of the connection to the boundary. As the Chern-Simons invariant is only defined modulo large gauge transformations, the requirement that the path integral over asymptotically hyperbolic connections is well-d...
Pilyugin, Sergei Yu
2017-01-01
Focusing on the theory of shadowing of approximate trajectories (pseudotrajectories) of dynamical systems, this book surveys recent progress in establishing relations between shadowing and such basic notions from the classical theory of structural stability as hyperbolicity and transversality. Special attention is given to the study of "quantitative" shadowing properties, such as Lipschitz shadowing (it is shown that this property is equivalent to structural stability both for diffeomorphisms and smooth flows), and to the passage to robust shadowing (which is also equivalent to structural stability in the case of diffeomorphisms, while the situation becomes more complicated in the case of flows). Relations between the shadowing property of diffeomorphisms on their chain transitive sets and the hyperbolicity of such sets are also described. The book will allow young researchers in the field of dynamical systems to gain a better understanding of new ideas in the global qualitative theory. It will also be of int...
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Jose Ernie C. Lope
2013-12-01
Full Text Available In their 2012 work, Lope, Roque, and Tahara considered singular nonlinear partial differential equations of the form tut = F(t; x; u; ux, where the function F is assumed to be continuous in t and holomorphic in the other variables. They have shown that under some growth conditions on the coefficients of the partial Taylor expansion of F as t 0, the equation has a unique solution u(t; x with the same growth order as that of F(t; x; 0; 0. Koike considered systems of partial differential equations using the Banach fixed point theorem and the iterative method of Nishida and Nirenberg. In this paper, we prove the result obtained by Lope and others using the method of Koike, thereby avoiding the repetitive step of differentiating a recursive equation with respect to x as was done by the aforementioned authors.
Asymptotically hyperbolic connections
Fine, Joel; Herfray, Yannick; Krasnov, Kirill; Scarinci, Carlos
2016-09-01
General relativity in four-dimensions can be equivalently described as a dynamical theory of {SO}(3)˜ {SU}(2)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analogue of the Fefferman-Graham expansion in the language of connections. As in the metric setup, one can solve the arising ‘evolution’ equations order by order in the expansion in powers of the radial coordinate. The solution in the connection setting is arguably simpler, and very straightforward algebraic manipulations allow one to see how the unconstrained by Einstein equations ‘stress-energy tensor’ appears at third order in the expansion. Another interesting feature of the connection formulation is that the ‘counter terms’ required in the computation of the renormalised volume all combine into the Chern-Simons functional of the restriction of the connection to the boundary. As the Chern-Simons invariant is only defined modulo large gauge transformations, the requirement that the path integral over asymptotically hyperbolic connections is well-defined requires the cosmological constant to be quantised. Finally, in the connection setting one can deform the 4D Einstein condition in an interesting way, and we show that asymptotically hyperbolic connection expansion is universal and valid for any of the deformed theories.
Directory of Open Access Journals (Sweden)
Shaheed N. Huseen
2013-01-01
Full Text Available A modified q-homotopy analysis method (mq-HAM was proposed for solving nth-order nonlinear differential equations. This method improves the convergence of the series solution in the nHAM which was proposed in (see Hassan and El-Tawil 2011, 2012. The proposed method provides an approximate solution by rewriting the nth-order nonlinear differential equation in the form of n first-order differential equations. The solution of these n differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.
Variational principles for some nonlinear partial differential equations with variable coefficients
Energy Technology Data Exchange (ETDEWEB)
He Jihuan E-mail: jhhe@dhu.edu.cn
2004-03-01
Variational principles for generalized Korteweg-de Vries equation and nonlinear Schroedinger's equation are obtained by the semi-inverse method. The most interesting features of the proposed method are its extreme simplicity and concise forms of variational functionals for a wide range of nonlinear problems. Comparison with the results obtained by the Noether's theorem is made, revealing the present theorem is a straightforward and attracting mathematical tool.
LOCAL EXACT BOUNDARY CONTROLLABILITY FOR A CLASS OFQUASILINEAR HYPERBOLIC SYSTEMS
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
For a class of mixed initial-boundary value problem for general quasilinear hyperbolic systems, this paper establishes the local exact boundary controllability with boundary controls only acting on one end. As an application, the authors show the local exact boundary controllability for a kind of nonlinear vibrating string problem.
Directory of Open Access Journals (Sweden)
T. D. Frank
2016-12-01
Full Text Available In physics, several attempts have been made to apply the concepts and tools of physics to the life sciences. In this context, a thermostatistic framework for active Nambu systems is proposed. The so-called free energy Fokker–Planck equation approach is used to describe stochastic aspects of active Nambu systems. Different thermostatistic settings are considered that are characterized by appropriately-defined entropy measures, such as the Boltzmann–Gibbs–Shannon entropy and the Tsallis entropy. In general, the free energy Fokker–Planck equations associated with these generalized entropy measures correspond to nonlinear partial differential equations. Irrespective of the entropy-related nonlinearities occurring in these nonlinear partial differential equations, it is shown that semi-analytical solutions for the stationary probability densities of the active Nambu systems can be obtained provided that the pumping mechanisms of the active systems assume the so-called canonical-dissipative form and depend explicitly only on Nambu invariants. Applications are presented both for purely-dissipative and for active systems illustrating that the proposed framework includes as a special case stochastic equilibrium systems.
Heat Conduction: Hyperbolic Self-similar Shock-waves in Solid Medium
Barna, IF; Kersner, R.
2016-01-01
Analytic solutions for cylindrical thermal waves in solid medium are given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations. The Fourier-Cattaneo phenomenological law is generalized where the relaxation time and heat propagation coefficient have a general power law temperature dependence. From such laws one cannot form a second order parabolic or telegraph-type equation.We consider the original non-linear hyperbolic system itself w...
Heat conduction: hyperbolic self-similar shock-waves in solids
Barna, Imre Ferenc; Kersner, Robert
2012-01-01
Analytic solutions for cylindrical thermal waves in solid medium is given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations. The Fourier-Cattaneo phenomenological law is generalized where the relaxation time and heat propagation coefficient have a general power law temperature dependence. From such laws one cannot form a second order parabolic or telegraph-type equation. We consider the original non-linear hyperbolic system itself with the self...
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Hui Cao
2014-01-01
Full Text Available Quantitative analysis for the flue gas of natural gas-fired generator is significant for energy conservation and emission reduction. The traditional partial least squares method may not deal with the nonlinear problems effectively. In the paper, a nonlinear partial least squares method with extended input based on radial basis function neural network (RBFNN is used for components prediction of flue gas. For the proposed method, the original independent input matrix is the input of RBFNN and the outputs of hidden layer nodes of RBFNN are the extension term of the original independent input matrix. Then, the partial least squares regression is performed on the extended input matrix and the output matrix to establish the components prediction model of flue gas. A near-infrared spectral dataset of flue gas of natural gas combustion is used for estimating the effectiveness of the proposed method compared with PLS. The experiments results show that the root-mean-square errors of prediction values of the proposed method for methane, carbon monoxide, and carbon dioxide are, respectively, reduced by 4.74%, 21.76%, and 5.32% compared to those of PLS. Hence, the proposed method has higher predictive capabilities and better robustness.
Cao, Hui; Yan, Xingyu; Li, Yaojiang; Wang, Yanxia; Zhou, Yan; Yang, Sanchun
2014-01-01
Quantitative analysis for the flue gas of natural gas-fired generator is significant for energy conservation and emission reduction. The traditional partial least squares method may not deal with the nonlinear problems effectively. In the paper, a nonlinear partial least squares method with extended input based on radial basis function neural network (RBFNN) is used for components prediction of flue gas. For the proposed method, the original independent input matrix is the input of RBFNN and the outputs of hidden layer nodes of RBFNN are the extension term of the original independent input matrix. Then, the partial least squares regression is performed on the extended input matrix and the output matrix to establish the components prediction model of flue gas. A near-infrared spectral dataset of flue gas of natural gas combustion is used for estimating the effectiveness of the proposed method compared with PLS. The experiments results show that the root-mean-square errors of prediction values of the proposed method for methane, carbon monoxide, and carbon dioxide are, respectively, reduced by 4.74%, 21.76%, and 5.32% compared to those of PLS. Hence, the proposed method has higher predictive capabilities and better robustness.
On hyperbolic Bessel processes and beyond
Wisniewolski, Maciej
2011-01-01
We investigate distributions of hyperbolic Bessel processes. We find links between the hyperbolic cosinus of the hyperbolic Bessel processes and the functionals of geometric Brownian motion. We present an explicit formula of Laplace transform of hyperbolic cosinus of hyperbolic Bessel processes and some interesting different probabilistic representations of this Laplace transform. We express the one-dimensional distribution of hyperbolic Bessel process in terms of other, known and independent processes. We present some applications including a new proof of Bougerol's identity and it's generalization. We characterize the distribution of the process being hyperbolic sinus of hyperbolic Bessel processes.
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Taha Aziz
2012-01-01
Full Text Available The unsteady unidirectional flow of an incompressible fourth grade fluid bounded by a suddenly moved rigid plate is studied. The governing nonlinear higher order partial differential equation for this flow in a semiinfinite domain is modelled. Translational symmetries in variables and are employed to construct two different classes of closed-form travelling wave solutions of the model equation. A conditional symmetry solution of the model equation is also obtained. The physical behavior and the properties of various interesting flow parameters on the structure of the velocity are presented and discussed. In particular, the significance of the rheological effects are mentioned.
Barbillon, Grégory; Biehs, Svend-Age; Ben-Abdallah, Philippe
2016-01-01
A thermal antenna is an electromagnetic source which emits in its surrounding, a spatially coherent field in the infrared frequency range. Usually, its emission pattern changes with the wavelength so that the heat flux it radiates is weakly directive. Here, we show that a class of hyperbolic materials, possesses a Brewster angle which is weakly dependent on the wavelength, so that they can radiate like a true thermal antenna with a highly directional heat flux. The realization of these sources could open a new avenue in the field of thermal management in far-field regime.
Evolutes of Hyperbolic Plane Curves
Institute of Scientific and Technical Information of China (English)
Shyuichi IZUMIYA; Dong He PEI; Takashi SANO; Erika TORII
2004-01-01
We define the notion of evolutes of curves in a hyperbolic plane and establish the relationships between singularities of these subjects and geometric invariants of curves under the action of the Lorentz group. We also describe how we can draw the picture of an evolute of a hyperbolic plane curve in the Poincar(e) disk.
Hyperbolic semi-adequate links
Futer, David; Kalfagianni, Efstratia; Purcell, Jessica S.
2013-01-01
We provide a diagrammatic criterion for semi-adequate links to be hyperbolic. We also give a conjectural description of the satellite structures of semi-adequate links. One application of our result is that the closures of sufficiently complicated positive braids are hyperbolic links.
MULTIDIMENSIONAL RELAXATION APPROXIMATIONS FOR HYPERBOLIC SYSTEMS OF CONSERVATION LAWS
Institute of Scientific and Technical Information of China (English)
Mohammed Sea(l)d
2007-01-01
We construct and implement a non-oscillatory relaxation scheme for multidimensional hyperbolic systems of conservation laws. The method transforms the nonlinear hyperbolic system to a semilinear model with a relaxation source term and linear characteristics which can be solved numerically without using either Riemann solver or linear iterations.To discretize the relaxation system we consider a high-resolution reconstruction in space and a TVD Runge-Kutta time integration. Detailed formulation of the scheme is given for problems in three space dimensions and numerical experiments are implemented in both scalar and system cases to show the effectiveness of the method.
DECAY OF POSITIVE WAVES OF HYPERBOLIC BALANCE LAWS
Institute of Scientific and Technical Information of China (English)
Cleopatra Christoforou; Konstantina Trivisa
2012-01-01
Historically,decay rates have been used to provide quantitative and qualitative information on the solutions to hyperbolic conservation laws.Quantitative results include the establishment of convergence rates for approximating procedures and numerical schemes.Qualitative results include the establishment of results on uniqueness and regularity as well as the ability to visualize the waves and their evolution in time.This work presents two decay estimates on the positive waves for systems of hyperbolic and genuinely nonlinear balance laws satisfying a dissipative mechanism.The result is obtained by employing the continuity of Glimm-type functionals and the method of generalized characteristics [7,17,24].
Viscosity solutions of fully nonlinear second-order elliptic partial differential equations
Ishii, H.; Lions, P. L.
We investigate comparison and existence results for viscosity solutions of fully nonlinear, second-order, elliptic, possibly degenerate equations. These results complement those recently obtained by R. Jensen and H. Ishii. We consider various boundary conditions like for instance Dirichlet and Neumann conditions. We also apply these methods and results to quasilinear Monge-Ampère equations. Finally, we also address regularity questions.
Symbolic dynamics and hyperbolic groups
Coornaert, Michel
1993-01-01
Gromov's theory of hyperbolic groups have had a big impact in combinatorial group theory and has deep connections with many branches of mathematics suchdifferential geometry, representation theory, ergodic theory and dynamical systems. This book is an elaboration on some ideas of Gromov on hyperbolic spaces and hyperbolic groups in relation with symbolic dynamics. Particular attention is paid to the dynamical system defined by the action of a hyperbolic group on its boundary. The boundary is most oftenchaotic both as a topological space and as a dynamical system, and a description of this boundary and the action is given in terms of subshifts of finite type. The book is self-contained and includes two introductory chapters, one on Gromov's hyperbolic geometry and the other one on symbolic dynamics. It is intended for students and researchers in geometry and in dynamical systems, and can be used asthe basis for a graduate course on these subjects.
Emergent Hyperbolic Network Geometry
Bianconi, Ginestra; Rahmede, Christoph
2017-02-01
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a discretization of spacetime. Here, by extending our knowledge of growing complex networks to growing simplicial complexes we investigate the nature of the emergent geometry of complex networks and explore whether this geometry is hyperbolic. Specifically we show that an hyperbolic network geometry emerges spontaneously from models of growing simplicial complexes that are purely combinatorial. The statistical and geometrical properties of the growing simplicial complexes strongly depend on their dimensionality and display the major universal properties of real complex networks (scale-free degree distribution, small-world and communities) at the same time. Interestingly, when the network dynamics includes an heterogeneous fitness of the faces, the growing simplicial complex can undergo phase transitions that are reflected by relevant changes in the network geometry.
Mittal, R. C.; Jain, R. K.
2012-12-01
In this paper, a numerical method is proposed to approximate the solution of the nonlinear parabolic partial differential equation with Neumann's boundary conditions. The method is based on collocation of cubic B-splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply cubic B-splines for spatial variable and its derivatives, which produce a system of first order ordinary differential equations. We solve this system by using SSP-RK3 scheme. The numerical approximate solutions to the nonlinear parabolic partial differential equations have been computed without transforming the equation and without using the linearization. Four illustrative examples are included to demonstrate the validity and applicability of the technique. In numerical test problems, the performance of this method is shown by computing L∞andL2error norms for different time levels. Results shown by this method are found to be in good agreement with the known exact solutions.
Cortes, Adriano Mauricio
2014-01-01
In this paper we present PetIGA, a high-performance implementation of Isogeometric Analysis built on top of PETSc. We show its use in solving nonlinear and time-dependent problems, such as phase-field models, by taking advantage of the high-continuity of the basis functions granted by the isogeometric framework. In this work, we focus on the Cahn-Hilliard equation and the phase-field crystal equation.
3-D zebrafish embryo image filtering by nonlinear partial differential equations.
Rizzi, Barbara; Campana, Matteo; Zanella, Cecilia; Melani, Camilo; Cunderlik, Robert; Krivá, Zuzana; Bourgine, Paul; Mikula, Karol; Peyriéras, Nadine; Sarti, Alessandro
2007-01-01
We discuss application of nonlinear PDE based methods to filtering of 3-D confocal images of embryogenesis. We focus on the mean curvature driven and the regularized Perona-Malik equations, where standard as well as newly suggested edge detectors are used. After presenting the related mathematical models, the practical results are given and discussed by visual inspection and quantitatively using the mean Hausdorff distance.
Superdiffusions and positive solutions of non-linear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Dynkin, E B [Cornell University, New York (United States)
2004-02-28
By using super-Brownian motion, all positive solutions of the non-linear differential equation {delta}u=u{sup {alpha}} with 1<{alpha}{<=}2 in a bounded smooth domain E are characterized by their (fine) traces on the boundary. This solves a problem posed by the author a few years ago. The special case {alpha}=2 was treated by B. Mselati in 2002.
An Efficient Implementation of Partial Condensing for Nonlinear Model Predictive Control
DEFF Research Database (Denmark)
Frison, Gianluca; Kouzoupis, Dimitris; Jørgensen, John Bagterp
2016-01-01
Partial (or block) condensing is a recently proposed technique to reformulate a Model Predictive Control (MPC) problem into a form more suitable for structure-exploiting Quadratic Programming (QP) solvers. It trades off horizon length for input vector size, and this degree of freedom can be emplo......Partial (or block) condensing is a recently proposed technique to reformulate a Model Predictive Control (MPC) problem into a form more suitable for structure-exploiting Quadratic Programming (QP) solvers. It trades off horizon length for input vector size, and this degree of freedom can...
Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems
Adjerid, Slimane; Weinhart, Thomas
2009-01-01
In this manuscript we present an error analysis for the discontinuous Galerkin discretization error of multi-dimensional first-order linear symmetric hyperbolic systems of partial differential equations. We perform a local error analysis by writing the local error as a series and showing that its le
Nonlinear Parasitic Capacitance Modelling of High Voltage Power MOSFETs in Partial SOI Process
DEFF Research Database (Denmark)
Fan, Lin; Knott, Arnold; Jørgensen, Ivan Harald Holger
2016-01-01
: off-state, sub-threshold region, and on-state in the linear region. A high voltage power MOSFET is designed in a partial Silicon on Insulator (SOI) process, with the bulk as a separate terminal. 3D plots and contour plots of the capacitances versus bias voltages for the transistor summarize...
Optimization of Nonlinear Figure-of-Merits of Integrated Power MOSFETs in Partial SOI Process
DEFF Research Database (Denmark)
Fan, Lin; Jørgensen, Ivan Harald Holger; Knott, Arnold
2016-01-01
different operating conditions. A systematic analysis of the optimization of these FOMs has not been previously established. The optimization methods are verified on a 100 V power MOSFET implemented in a 0.18 µm partial SOI process. Its FOMs are lowered by 1.3-18.3 times and improved by 22...
Fourth-Order Difference Methods for Hyperbolic IBVPs
Gustafsson, Bertil; Olsson, Pelle
1995-03-01
In this paper we consider fourth-order difference approximations of initial-boundary value problems for hyperbolic partial differential equations. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics; the second one is used for modeling shocks and rarefaction waves. The time discretization is done with a third-order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second-order viscosity. In case of the non-linear Burgers' equation we use a flux splitting technique that results in an energy estimate for certain difference approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth-order methods with a standard second-order one and with a third-order TVD method. The results show that the fourth-order methods are the only ones that give good results for all the considered test problems.
ANTI-PERIODIC SOLUTIONS FOR FIRST AND SECOND ORDER NONLINEAR EVOLUTION EQUATIONS IN BANACH SPACES
Institute of Scientific and Technical Information of China (English)
WEI Wei; XIANG Xiaoling
2004-01-01
In this paper, a new existence theorem of anti-periodic solutions for a class ofstrongly nonlinear evolution equations in Banach spaces is presentedThe equations con-tain nonlinear monotone operators and a nonmonotone perturbationMoreover, throughan appropriate transformation, the existence of anti-periodic solutions for a class of second-order nonlinear evolution equations is verifiedOur abstract results are illustrated by anexample from quasi-linear partial differential equations with time anti-periodic conditionsand an example from quasi-linear anti-periodic hyperbolic differential equations.
Manton, Nicholas
2014-01-01
We construct a number of explicit examples of hyperbolic monopoles, with various charges and often with some platonic symmetry. The fields are obtained from instanton data in four-dimensional Euclidean space that are invariant under a circle action, and the monopole charge is equal to the instanton charge. A key ingredient is the identification of a new set of constraints on ADHM instanton data that are sufficient to ensure the circle invariance. Algebraic formulae for the Higgs field magnitude are given and from these we compute and illustrate the energy density of the monopoles. For particular monopoles, the explicit formulae provide a proof that the number of zeros of the Higgs field is greater than the monopole charge. We also present some one-parameter families of monopoles analogous to known scattering events for Euclidean monopoles within the geodesic approximation.
Radiative flow of a tangent hyperbolic fluid with convective conditions and chemical reaction
Hayat, Tasawar; Qayyum, Sajid; Ahmad, Bashir; Waqas, Muhammad
2016-12-01
The objective of present paper is to examine the thermal radiation effects in the two-dimensional mixed convection flow of a tangent hyperbolic fluid near a stagnation point. The analysis is performed in the presence of heat generation/absorption and chemical reaction. Convective boundary conditions for heat and mass transfer are employed. The resulting partial differential equations are reduced into nonlinear ordinary differential equations using appropriate transformations. Series solutions of momentum, energy and concentration equations are computed. The characteristics of various physical parameters on the distributions of velocity, temperature and concentration are analyzed graphically. Numerical values of skin friction coefficient, local Nusselt and Sherwood numbers are computed and examined. It is observed that larger values of thermal and concentration Biot numbers enhance the temperature and concentration distributions.
Observer-Based Bilinear Control of First-Order Hyperbolic PDEs: Application to the Solar Collector
Mechhoud, Sarra
2015-12-18
In this paper, we investigate the problem of bilinear control of a solar collector plant using the available boundary and solar irradiance measurements. The solar collector is described by a first-order 1D hyperbolic partial differential equation where the pump volumetric flow rate acts as the plant control input. By combining a boundary state observer and an internal energy-based control law, a nonlinear observer based feedback controller is proposed. With a feed-forward control term, the effect of the solar radiation is cancelled. Using the Lyapunov approach we prove that the proposed control guarantees the global exponential stability of both the plant and the tracking error. Simulation results are provided to illustrate the performance of the proposed method.
Hyperbolically Shaped Centrifugal Compressor
Institute of Scientific and Technical Information of China (English)
Romuald Puzyrewski; Pawel Flaszy(n)ski
2003-01-01
Starting from the classical centrifugal compressor, cone shaped in meridional cross section, two modifications are considered on the basis of results from 2D and 3D flow models. The first modification is the change of the meridional cross section to hyperbolically shaped channel. The second modification, proposed on the basis of 2D axisymmetric solution, concerns the shape of blading. On the strength of this solution the blades are formed as 3D shaped blades, coinciding with the recent tendency in 3D designs. Two aims were considered for the change of meridional compressor shape. The first was to remove the separation zone which appears as the flow tums from axial to radial direction. The second aim is to uniformize the flow at exit of impeller. These two goals were considered within the frame of 2D axisymmetric model. Replacing the cone shaped compressor by a hyperbolically shaped one, the separation at the corner was removed. The disc and shroud shape of the compressor was chosen in the way which satisfies the condition of most uniform flow at the compressor exit. The uniformity of exit flow from the rotor can be considered as the factor which influences the performance of the diffuser following the rotor. In the 2D model a family of stream surfaces of S1 type is given in order to find S2 surfaces which may be identified with the midblade surfaces of compressor blading. A computation of 3D type has been performed in order to establish the relations between 2D and 3D models in the calculation of flow parameters. In the presented example the 2D model appears as the inverse model which leads to 3D shape of blading whereas the 3D model has been used for the direct solution. In the presented example the confrontation of two models, 2D and 3D, leads to a better understanding of the application of these models to the design procedure.
Jameson, A.
1976-01-01
A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.
Nonlinear Second-Order Partial Differential Equation-Based Image Smoothing Technique
Directory of Open Access Journals (Sweden)
Tudor Barbu
2016-09-01
Full Text Available A second-order nonlinear parabolic PDE-based restoration model is provided in this article. The proposed anisotropic diffusion-based denoising approach is based on some robust versions of the edge-stopping function and of the conductance parameter. Two stable and consistent approximation schemes are then developed for this differential model. Our PDE-based filtering technique achieves an efficient noise removal while preserving the edges and other image features. It outperforms both the conventional filters and also many PDE-based denoising approaches, as it results from the successful experiments and method comparison applied.
Salahuddin, T.; Khan, Imad; Malik, M. Y.; Khan, Mair; Hussain, Arif; Awais, Muhammad
2017-05-01
The present work examines the internal resistance between fluid particles of tangent hyperbolic fluid flow due to a non-linear stretching sheet with heat generation. Using similarity transformations, the governing system of partial differential equations is transformed into a coupled non-linear ordinary differential system with variable coefficients. Unlike the current analytical works on the flow problems in the literature, the main concern here is to numerically work out and find the solution by using Runge-Kutta-Fehlberg coefficients improved by Cash and Karp (Naseer et al., Alexandria Eng. J. 53, 747 (2014)). To determine the relevant physical features of numerous mechanisms acting on the deliberated problem, it is sufficient to have the velocity profile and temperature field and also the drag force and heat transfer rate all as given in the current paper.
Arqub, Omar Abu; El-Ajou, Ahmad; Momani, Shaher
2015-07-01
Building fractional mathematical models for specific phenomena and developing numerical or analytical solutions for these fractional mathematical models are crucial issues in mathematics, physics, and engineering. In this work, a new analytical technique for constructing and predicting solitary pattern solutions of time-fractional dispersive partial differential equations is proposed based on the generalized Taylor series formula and residual error function. The new approach provides solutions in the form of a rapidly convergent series with easily computable components using symbolic computation software. For method evaluation and validation, the proposed technique was applied to three different models and compared with some of the well-known methods. The resultant simulations clearly demonstrate the superiority and potentiality of the proposed technique in terms of the quality performance and accuracy of substructure preservation in the construct, as well as the prediction of solitary pattern solutions for time-fractional dispersive partial differential equations.
Energy Technology Data Exchange (ETDEWEB)
Zenchuk, A I, E-mail: zenchuk@itp.ac.r [Institute of Problems of Chemical Physics, RAS Acad. Semenov av., 1 Chernogolovka, Moscow region 142432 (Russian Federation)
2010-06-18
We develop a new integration technique allowing one to construct a rich manifold of particular solutions to multidimensional generalizations of classical C- and S-integrable partial differential equations (PDEs). Generalizations of (1+1)-dimensional C-integrable and (2+1)-dimensional S-integrable N-wave equations are derived among examples. Examples of multidimensional second-order PDEs are represented as well.
Exact travelling wave solutions for a class of nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Xie Fuding E-mail: xiefd@sohu.com; Gao Xiaoshan
2004-03-01
In this paper, the tanh-method is improved by means of a proper rational transformation based upon a coupled projective Riccati equations. The ansatz can be applied to find more and new exact solutions of the partial differential equations with the aid of symbolic computation system, Maple. We choose an example, which includes phi{sup 4} equation, Klein-Gordon equation, Duffing equation, Landau-Ginburg-Higgs equation and Sine-Gordon equation, to illustrate the method.
LARGE TIME BEHAVIOR OF SOLUTIONS TO NONLINEAR VISCOELASTIC MODEL WITH FADING MEMORY
Institute of Scientific and Technical Information of China (English)
Yanni Zeng
2012-01-01
We study the Cauchy problem of a one-dimensional nonlinear viscoelastic model with fading memory. By introducing appropriate new variables we convert the integro-partial differential equations into a hyperbolic system of balance laws.When it is a perturbation of a constant state,the solution is shown time asymptotically approaching to predetermined diffusion waves.Pointwise estimates on the convergence details are obtained.
Analytical Approximation Method for the Center Manifold in the Nonlinear Output Regulation Problem
Suzuki, Hidetoshi; Sakamoto, Noboru; Celikovský, Sergej
In nonlinear output regulation problems, it is necessary to solve the so-called regulator equations consisting of a partial differential equation and an algebraic equation. It is known that, for the hyperbolic zero dynamics case, solving the regulator equations is equivalent to calculating a center manifold for zero dynamics of the system. The present paper proposes a successive approximation method for obtaining center manifolds and shows its effectiveness by applying it for an inverted pendulum example.
Trivalent expanders and hyperbolic surfaces
Ivrissimtzis, Ioannis; Vdovina, Alina
2012-01-01
We introduce a family of trivalent expanders which tessellate compact hyperbolic surfaces with large isometry groups. We compare this family with Platonic graphs and modifications of them and prove topological and spectral properties of these families.
Output regulation problem for a class of regular hyperbolic systems
Xu, Xiaodong; Dubljevic, Stevan
2016-01-01
This paper investigates the output regulation problem for a class of regular first-order hyperbolic partial differential equation (PDE) systems. A state feedback and an error feedback regulator are considered to force the output of the hyperbolic PDE plant to track a periodic reference trajectory generated by a neutrally stable exosystem. A new explanation is given to extend the results in the literature to solve the regulation problem associated with the first-order hyperbolic PDE systems. Moreover, in order to provide the closed-loop stability condition for the solvability of the regulator problems, the design of stabilising feedback gain and its dual problem design of stabilising output injection gain are considered in this paper. This paper develops an easy method to obtain an adjustable stabilising feedback gain and stabilising output injection gain with the aid of the operator Riccati equation.
Second-order hyperbolic Fuchsian systems. I. General theory
Beyer, Florian
2010-01-01
We introduce a class of singular partial differential equations, the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First of all, we analyze a class of equations in which hyperbolicity is not assumed and we construct asymptotic solutions of arbitrary order. Second, for the proposed class of second-order hyperbolic Fuchsian systems, we establish the existence of solutions with prescribed asymptotic behavior on the singularity. Our proof is based on a new scheme which is also suitable to design numerical approximations. Furthermore, as shown in a follow-up paper, the second-order Fuchsian framework is appropriate to handle Einstein's field equations for Gowdy symmetric spacetimes and allows us to recover (and slightly generalize) earlier results by Rendall and collaborators, while providing a direct approach leading to accurate numerical solutions. The proposed framework is also robust enough to encompass matter models ...
Directory of Open Access Journals (Sweden)
Jacqueline Fleckinger
2001-12-01
Full Text Available We study the asymptotic behavior of positive solutions $u$ of $$ -Delta_p u(x = V(x u(x^{p-1}, quad p>1; x in Omega,$$ and related partial differential inequalities, as well as conditions for existence of such solutions. Here, $Omega$ contains the exterior of a ball in $mathbb{R}^N$ $1
Hyperbolic Methods for Einstein's Equations
Directory of Open Access Journals (Sweden)
Reula Oscar
1998-01-01
Full Text Available I review evolutionary aspects of general relativity, in particular those related to the hyperbolic character of the field equations and to the applications or consequences that this property entails. I look at several approaches to obtaining symmetric hyperbolic systems of equations out of Einstein's equations by either removing some gauge freedoms from them, or by considering certain linear combinations of a subset of them.
Hyperbolically Discounted Temporal Difference Learning
Alexander, William H.; Brown, Joshua W.
2010-01-01
Hyperbolic discounting of future outcomes is widely observed to underlie choice behavior in animals. Additionally, recent studies (Kobayashi & Schultz, 2008) have reported that hyperbolic discounting is observed even in neural systems underlying choice. However, the most prevalent models of temporal discounting, such as temporal difference learning, assume that future outcomes are discounted exponentially. Exponential discounting has been preferred largely because it is able to be expressed r...
Beck, Christian; E, Weinan; Jentzen, Arnulf
2017-01-01
High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, ...
Nonoscillatory Central Schemes for Hyperbolic Systems of Conservation Laws in Three-Space Dimensions
Directory of Open Access Journals (Sweden)
Andrew N. Guarendi
2013-01-01
Full Text Available We extend a family of high-resolution, semidiscrete central schemes for hyperbolic systems of conservation laws to three-space dimensions. Details of the schemes, their implementation, and properties are presented together with results from several prototypical applications of hyperbolic conservation laws including a nonlinear scalar equation, the Euler equations of gas dynamics, and the ideal magnetohydrodynamic equations. Parallel scaling analysis and grid-independent results including contours and isosurfaces of density and velocity and magnetic field vectors are shown in this study, confirming the ability of these types of solvers to approximate the solutions of hyperbolic equations efficiently and accurately.
Owolabi, Kolade M.
2017-03-01
In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x ∈ [0, L], x = x(x , y , z) and t ∈ [0, T] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space-fractional derivatives of order α, for 0 < α < 2. Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method, and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 < α < 1) and super-diffusive (1 < α < 2) scenarios. It is observed that computer simulations of SFORDE give enough evidence that pattern formation in fractional medium at certain parameter value is practically the same as in the standard reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed.
7th International Conference on Hyperbolic Problems Theory, Numerics, Applications
Jeltsch, Rolf
1999-01-01
These proceedings contain, in two volumes, approximately one hundred papers presented at the conference on hyperbolic problems, which has focused to a large extent on the laws of nonlinear hyperbolic conservation. Two-fifths of the papers are devoted to mathematical aspects such as global existence, uniqueness, asymptotic behavior such as large time stability, stability and instabilities of waves and structures, various limits of the solution, the Riemann problem and so on. Roughly the same number of articles are devoted to numerical analysis, for example stability and convergence of numerical schemes, as well as schemes with special desired properties such as shock capturing, interface fitting and high-order approximations to multidimensional systems. The results in these contributions, both theoretical and numerical, encompass a wide range of applications such as nonlinear waves in solids, various computational fluid dynamics from small-scale combustion to relativistic astrophysical problems, multiphase phe...
Complex geometric optics for symmetric hyperbolic systems I: linear theory
Maj, Omar
2008-01-01
We obtain an asymptotic solution for $\\ep \\to 0$ of the Cauchy problem for linear first-order symmetric hyperbolic systems with oscillatory initial values written in the eikonal form of geometric optics with frequency $1/\\ep$, but with complex phases. For the most common linear wave propagation models, this kind on Cauchy problems are well-known in the applied literature and their asymptotic theory, referred to as complex geometric optics, is attracting interest for applications. In this work, which is the first of a series of papers dedicated to complex geometric optics for nonlinear symmetric hyperbolic systems, we develop a rigorous linear theory and set the basis for the subsequent nonlinear analysis.
Partial Differential Equations
1988-01-01
The volume contains a selection of papers presented at the 7th Symposium on differential geometry and differential equations (DD7) held at the Nankai Institute of Mathematics, Tianjin, China, in 1986. Most of the contributions are original research papers on topics including elliptic equations, hyperbolic equations, evolution equations, non-linear equations from differential geometry and mechanics, micro-local analysis.
非线性偏微分方程的约化和精确解%REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS
Institute of Scientific and Technical Information of China (English)
叶彩儿; 潘祖梁
2003-01-01
Nonlinear partial differetial equation(NLPDE) is converted into ordinary differentialequation (ODE) via a new ansatzUsing undetermined function method ,the ODE obtained aboveis replaced by a set of algebraic equations which are solved out with the aid of MathematicaTheexact solutions and solitary solutions of NLPDE are obtained.
Advanced fabrication of hyperbolic metamaterials
Shkondin, Evgeniy; Sukham, Johneph; Panah, Mohammad E. Aryaee; Takayama, Osamu; Malureanu, Radu; Jensen, Flemming; Lavrinenko, Andrei V.
2017-09-01
Hyperbolic metamaterials can provide unprecedented properties in accommodation of high-k (high wave vector) waves and enhancement of the optical density of states. To reach such performance the metamaterials have to be fabricated with as small imperfections as possible. Here we report on our advances in two approaches in fabrication of optical metamaterials. We deposit ultrathin ultrasmooth gold layers with the assistance of organic material (APTMS) adhesion layer. The technology supports the stacking of such layers in a multiperiod construction with alumina spacers between gold films, which is expected to exhibit hyperbolic properties in the visible range. As the second approach we apply the atomic layer deposition technique to arrange vertical alignment of layers or pillars of heavily doped ZnO or TiN, which enables us to produce hyperbolic metamaterials for the near- and mid-infrared ranges.
The spectrum of hyperbolic surfaces
Bergeron, Nicolas
2016-01-01
This text is an introduction to the spectral theory of the Laplacian on compact or finite area hyperbolic surfaces. For some of these surfaces, called “arithmetic hyperbolic surfaces”, the eigenfunctions are of arithmetic nature, and one may use analytic tools as well as powerful methods in number theory to study them. After an introduction to the hyperbolic geometry of surfaces, with a special emphasis on those of arithmetic type, and then an introduction to spectral analytic methods on the Laplace operator on these surfaces, the author develops the analogy between geometry (closed geodesics) and arithmetic (prime numbers) in proving the Selberg trace formula. Along with important number theoretic applications, the author exhibits applications of these tools to the spectral statistics of the Laplacian and the quantum unique ergodicity property. The latter refers to the arithmetic quantum unique ergodicity theorem, recently proved by Elon Lindenstrauss. The fruit of several graduate level courses at Orsay...
Gualdesi, Lavinio
2017-04-01
Mooring lines in the Ocean might be seen as a pretty simple seamanlike activity. Connecting valuable scientific instrumentation to it transforms this simple activity into a sophisticated engineering support which needs to be accurately designed, developed, deployed, monitored and hopefully recovered with its precious load of scientific data. This work is an historical travel along the efforts carried out by scientists all over the world to successfully predict mooring line behaviour through both mathematical simulation and experimental verifications. It is at first glance unexpected how many factors one must observe to get closer and closer to a real ocean situation. Most models have dual applications for mooring lines and towed bodies lines equations. Numerous references are provided starting from the oldest one due to Isaac Newton. In his "Philosophiae Naturalis Principia Matematica" (1687) the English scientist, while discussing about the law of motion for bodies in resistant medium, is envisaging a hyperbolic fitting to the phenomenon including asymptotic behaviour in non-resistant media. A non-exhaustive set of mathematical simulations of the mooring lines trajectory prediction is listed hereunder to document how the subject has been under scientific focus over almost a century. Pode (1951) Prior personal computers diffusion a tabular form of calculus of cable geometry was used by generations of engineers keeping in mind the following limitations and approximations: tangential drag coefficients were assumed to be negligible. A steady current flow was assumed as in the towed configuration. Cchabra (1982) Finite Element Method that assumes an arbitrary deflection angle for the top first section and calculates equilibrium equations down to the sea floor iterating up to a compliant solution. Gualdesi (1987) ANAMOOR. A Fortran Program based on iterative methods above including experimental data from intensive mooring campaign. Database of experimental drag
Exact Solutions for Einstein's Hyperbolic Geometric Flow
Institute of Scientific and Technical Information of China (English)
HE Chun-Lei
2008-01-01
In this paper we investigate the Einstein's hyperbolic geometric flow and obtain some interesting exact solutions for this kind of flow. Many interesting properties of these exact solutions have also been analyzed and we believe that these properties of Einstein's hyperbolic geometric flow are very helpful to understanding the Einstein equations and the hyperbolic geometric flow.
DETERMINISTIC HOMOGENIZATION OF QUASILINEAR DAMPED HYPERBOLIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
Gabriel Nguetseng; Hubert Nnang; Nils Svanstedt
2011-01-01
Deterministic homogenization is studied for quasilinear monotone hyperbolic problems with a linear damping term.It is shown by the sigma-convergence method that the sequence of solutions to a class of multi-scale highly oscillatory hyperbolic problems converges to the solution to a homogenized quasilinear hyperbolic problem.
Quasi-Hyperbolicity and Delay Semigroups
Directory of Open Access Journals (Sweden)
Shard Rastogi
2016-01-01
Full Text Available We study quasi-hyperbolicity of the delay semigroup associated with the equation u′(t=Bu(t+Φut, where ut is the history function and (B,D(B is the generator of a quasi-hyperbolic semigroup. We give conditions under which the associated solution semigroup of this equation generates a quasi-hyperbolic semigroup.
Second-harmonic generation from hyperbolic plasmonic nanorod metamaterial slab
Marino, Giuseppe; Krasavin, Alexey V; Ginzburg, Pavel; Olivier, Nicolas; Wurtz, Gregory A; Zayats, Anatoly V
2015-01-01
Hyperbolic plasmonic metamaterials provide numerous opportunities for designing unusual linear and nonlinear optical properties. We show that the modal overlap of fundamental and second-harmonic light in an anisotropic plasmonic metamaterial slab results in the broadband enhancement of radiated second-harmonic intensity by up to 2 and 11 orders of magnitudes for TM- and TE-polarized fundamental light, respectively, compared to a smooth Au film under TM-polarised illumination. The results open up possibilities to design tuneable frequency-doubling metamaterial with the goal to overcome limitations associated with classical phase matching conditions in thick nonlinear crystals.
Modeling and analysis of linear hyperbolic systems of balance laws
Bartecki, Krzysztof
2016-01-01
This monograph focuses on the mathematical modeling of distributed parameter systems in which mass/energy transport or wave propagation phenomena occur and which are described by partial differential equations of hyperbolic type. The case of linear (or linearized) 2 x 2 hyperbolic systems of balance laws is considered, i.e., systems described by two coupled linear partial differential equations with two variables representing physical quantities, depending on both time and one-dimensional spatial variable. Based on practical examples of a double-pipe heat exchanger and a transportation pipeline, two typical configurations of boundary input signals are analyzed: collocated, wherein both signals affect the system at the same spatial point, and anti-collocated, in which the input signals are applied to the two different end points of the system. The results of this book emerge from the practical experience of the author gained during his studies conducted in the experimental installation of a heat exchange cente...
Cao, Hui; Li, Yao-Jiang; Zhou, Yan; Wang, Yan-Xia
2014-11-01
To deal with nonlinear characteristics of spectra data for the thermal power plant flue, a nonlinear partial least square (PLS) analysis method with internal model based on neural network is adopted in the paper. The latent variables of the independent variables and the dependent variables are extracted by PLS regression firstly, and then they are used as the inputs and outputs of neural network respectively to build the nonlinear internal model by train process. For spectra data of flue gases of the thermal power plant, PLS, the nonlinear PLS with the internal model of back propagation neural network (BP-NPLS), the non-linear PLS with the internal model of radial basis function neural network (RBF-NPLS) and the nonlinear PLS with the internal model of adaptive fuzzy inference system (ANFIS-NPLS) are compared. The root mean square error of prediction (RMSEP) of sulfur dioxide of BP-NPLS, RBF-NPLS and ANFIS-NPLS are reduced by 16.96%, 16.60% and 19.55% than that of PLS, respectively. The RMSEP of nitric oxide of BP-NPLS, RBF-NPLS and ANFIS-NPLS are reduced by 8.60%, 8.47% and 10.09% than that of PLS, respectively. The RMSEP of nitrogen dioxide of BP-NPLS, RBF-NPLS and ANFIS-NPLS are reduced by 2.11%, 3.91% and 3.97% than that of PLS, respectively. Experimental results show that the nonlinear PLS is more suitable for the quantitative analysis of glue gas than PLS. Moreover, by using neural network function which can realize high approximation of nonlinear characteristics, the nonlinear partial least squares method with internal model mentioned in this paper have well predictive capabilities and robustness, and could deal with the limitations of nonlinear partial least squares method with other internal model such as polynomial and spline functions themselves under a certain extent. ANFIS-NPLS has the best performance with the internal model of adaptive fuzzy inference system having ability to learn more and reduce the residuals effectively. Hence, ANFIS-NPLS is an
Variable Lebesgue spaces and hyperbolic systems
2014-01-01
This book targets graduate students and researchers who want to learn about Lebesgue spaces and solutions to hyperbolic equations. It is divided into two parts. Part 1 provides an introduction to the theory of variable Lebesgue spaces: Banach function spaces like the classical Lebesgue spaces but with the constant exponent replaced by an exponent function. These spaces arise naturally from the study of partial differential equations and variational integrals with non-standard growth conditions. They have applications to electrorheological fluids in physics and to image reconstruction. After an introduction that sketches history and motivation, the authors develop the function space properties of variable Lebesgue spaces; proofs are modeled on the classical theory. Subsequently, the Hardy-Littlewood maximal operator is discussed. In the last chapter, other operators from harmonic analysis are considered, such as convolution operators and singular integrals. The text is mostly self-contained, with only some mor...
Visible-frequency hyperbolic metasurface
High, Alexander A.; Devlin, Robert C.; Dibos, Alan; Polking, Mark; Wild, Dominik S.; Perczel, Janos; de Leon, Nathalie P.; Lukin, Mikhail D.; Park, Hongkun
2015-06-01
Metamaterials are artificial optical media composed of sub-wavelength metallic and dielectric building blocks that feature optical phenomena not present in naturally occurring materials. Although they can serve as the basis for unique optical devices that mould the flow of light in unconventional ways, three-dimensional metamaterials suffer from extreme propagation losses. Two-dimensional metamaterials (metasurfaces) such as hyperbolic metasurfaces for propagating surface plasmon polaritons have the potential to alleviate this problem. Because the surface plasmon polaritons are guided at a metal-dielectric interface (rather than passing through metallic components), these hyperbolic metasurfaces have been predicted to suffer much lower propagation loss while still exhibiting optical phenomena akin to those in three-dimensional metamaterials. Moreover, because of their planar nature, these devices enable the construction of integrated metamaterial circuits as well as easy coupling with other optoelectronic elements. Here we report the experimental realization of a visible-frequency hyperbolic metasurface using single-crystal silver nanostructures defined by lithographic and etching techniques. The resulting devices display the characteristic properties of metamaterials, such as negative refraction and diffraction-free propagation, with device performance greatly exceeding those of previous demonstrations. Moreover, hyperbolic metasurfaces exhibit strong, dispersion-dependent spin-orbit coupling, enabling polarization- and wavelength-dependent routeing of surface plasmon polaritons and two-dimensional chiral optical components. These results open the door to realizing integrated optical meta-circuits, with wide-ranging applications in areas from imaging and sensing to quantum optics and quantum information science.
Hyperbolic Formulation of General Relativity
Abrahams, A M; Choquet-Bruhat, Y; York, J W; Abrahams, Andrew; Anderson, Arlen; Choquet-Bruhat, Yvonne; York, James W.
1998-01-01
Two geometrical well-posed hyperbolic formulations of general relativity are described. One admits any time-slicing which preserves a generalized harmonic condition. The other admits arbitrary time-slicings. Both systems have only the physical characteristic speeds of zero and the speed of light.
Hyperbolic metamaterials: fundamentals and applications.
Shekhar, Prashant; Atkinson, Jonathan; Jacob, Zubin
2014-01-01
Metamaterials are nano-engineered media with designed properties beyond those available in nature with applications in all aspects of materials science. In particular, metamaterials have shown promise for next generation optical materials with electromagnetic responses that cannot be obtained from conventional media. We review the fundamental properties of metamaterials with hyperbolic dispersion and present the various applications where such media offer potential for transformative impact. These artificial materials support unique bulk electromagnetic states which can tailor light-matter interaction at the nanoscale. We present a unified view of practical approaches to achieve hyperbolic dispersion using thin film and nanowire structures. We also review current research in the field of hyperbolic metamaterials such as sub-wavelength imaging and broadband photonic density of states engineering. The review introduces the concepts central to the theory of hyperbolic media as well as nanofabrication and characterization details essential to experimentalists. Finally, we outline the challenges in the area and offer a set of directions for future work.
Nonlocal response of hyperbolic metasurfaces.
Correas-Serrano, D; Gomez-Diaz, J S; Tymchenko, M; Alù, A
2015-11-16
We analyze and model the nonlocal response of ultrathin hyperbolic metasurfaces (HMTSs) by applying an effective medium approach. We show that the intrinsic spatial dispersion in the materials employed to realize the metasurfaces imposes a wavenumber cutoff on the hyperbolic isofrequency contour, inversely proportional to the Fermi velocity, and we compare it with the cutoff arising from the structure granularity. In the particular case of HTMSs implemented by an array of graphene nanostrips, we find that graphene nonlocality can become the dominant mechanism that closes the hyperbolic contour - imposing a wavenumber cutoff at around 300k(0) - in realistic configurations with periodicity Lnonlocal response is mainly relevant in hyperbolic metasurfaces and metamaterials with periodicity below a few nm, being very weak in practical scenarios. In addition, we investigate how spatial dispersion affects the spontaneous emission rate of emitters located close to HMTSs. Our results establish an upper bound set by nonlocality to the maximum field confinement and light-matter interactions achievable in practical HMTSs, and may find application in the practical development of hyperlenses, sensors and on-chip networks.
Global Existence for a Parabolic-hyperbolic Free Boundary Problem Modelling Tumor Growth
Institute of Scientific and Technical Information of China (English)
Shang-bin Cui; Xue-mei Wei
2005-01-01
In this paper we study a free boundary problem modelling tumor growth, proposed by A. Friedman in 2004. This free boundary problem involves a nonlinear second-order parabolic equation describing the diffusion of nutrient in the tumor, and three nonlinear first-order hyperbolic equations describing the evolution of proliferative cells, quiescent cells and dead cells, respectively. By applying Lp theory of parabolic equations, the characteristic theory of hyperbolic equations, and the Banach fixed point theorem, we prove that this problem has a unique global classical solution.
Heat conduction: hyperbolic self-similar shock-waves in solids
Barna, Imre Ferenc
2012-01-01
Analytic solutions for cylindrical thermal waves in solid medium is given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations. The Fourier-Cattaneo phenomenological law is generalized where the relaxation time and heat propagation coefficient have a general power law temperature dependence. From such laws one cannot form a second order parabolic or telegraph-type equation. We consider the original non-linear hyperbolic system itself with the self-similar Ansatz for the temperature distribution and for the heat flux. As results continuous and shock-wave solutions are presented. For physical establishment numerous materials with various temperature dependent heat conduction coefficients are mentioned.
Hyperbolic Conservation Laws and Related Analysis with Applications
Holden, Helge; Karlsen, Kenneth
2014-01-01
This book presents thirteen papers, representing the most significant advances and current trends in nonlinear hyperbolic conservation laws and related analysis with applications. Topics covered include a survey on multidimensional systems of conservation laws as well as novel results on liquid crystals, conservation laws with discontinuous flux functions, and applications to sedimentation. Also included are articles on recent advances in the Euler equations and the Navier-Stokes-Fourier-Poisson system, in addition to new results on collective phenomena described by the Cucker-Smale model. The Workshop on Hyperbolic Conservation Laws and Related Analysis with Applications at the International Centre for Mathematical Sciences (Edinburgh, UK) held in Edinburgh, September 2011, produced this fine collection of original research and survey articles. Many leading mathematicians attended the event and submitted their contributions for this volume. It is addressed to researchers and graduate students inter...
Institute of Scientific and Technical Information of China (English)
何雪梅; 吕百达
2012-01-01
一些实验表明,实际大气会偏离理想Kolmogorov模型.本文基于广义Huygens-Fresnel原理和Toselli等提出的非Kolmogorov湍流模型,推导出部分相干双曲正弦-Gauss（HSG）涡旋光束通过非Kolmogorov大气湍流的解析传输公式,并用以对两束部分相干HSG涡旋光束相干叠加和非相干叠加形成的合成相干涡旋在非Kolmogorov大气湍流中的动态演化进行了研究.结果表明,合成光束平均光强的演化过程与非Kolmogorov湍流的广义指数α,源平面上叠加涡旋光束拓扑电荷的符号,以及叠加方式有关.合成相干涡旋在非Kolmogorov大气湍流中传输时会出现移动、产生和湮灭.广义指数α,拓扑电荷符号,以及叠加方式都会影响其演化行为.最后,将本文所得结果与相关文献做了比较.%Some experiments show that the practical atmosphere deviates from ideal Kolmogorov model. In this paper, based on the extended Huygens-Fresnel principle and the non-Kolmogorov turbulence model proposed by Toselli et al., the analytical expression for the propagation of partially coherent hyperbolic-sine-Gaussian vortex beams through non-Kolmogorov atmospheric turbulence is derived and used to study the dynamic evolutions of composite coherence vortices formed by coherent and incoherent superpositions of two partially coherent hyperbolic-sine-Gaussian vortex beams in non-Kolmogorov atmospheric turbulence. It is shown that the evolution process of the average intensity of the superimposed beam depends on the general exponent c~ of the non-Kolmogorov turbulence, the sign of the topological charge of the superimposed vortex beam in the source plane, and superposition scheme. The motion, the creation and the annihilation of composite coherence vortices may take place upon propagation through non-Kolmogorov turbulence, and the general exponent a, sign of the topological charge and superposition scheme affect the evolution behavior. Finally, the results are
Directory of Open Access Journals (Sweden)
ömer gözükızıl
2015-12-01
Full Text Available In this paper, the G'/G expansion method with the aid of computer algebraic system Maple is used for constructing exact travelling wave solutions and new kinds of solutions for the modified dispersive water wave equations, the Abrahams-Tsuneto reaction diffusion system, and for a class of reaction diffusion models. The method is straightforward and concise, and it be also applied to other nonlinear partial differential equations.
Caplan, R M
2011-01-01
An easy to implement modulus-squared Dirichlet (MSD) boundary condition is formulated for numerical simulations of time-dependent complex partial differential equations in multidimensional settings. The MSD boundary condition approximates a constant modulus-square value of the solution at the boundaries. Application of the MSD boundary condition to the nonlinear Schr\\"odinger equation is shown, and numerical simulations are performed to demonstrate its usefulness and advantages over other simple boundary conditions.
Hyperbolic Lagrangian coherent structures align with contours of path-averaged scalars
Farazmand, Mohammad
2015-01-01
We prove that, in area-preserving two-dimensional flows, hyperbolic Lagrangian Coherent Structures (LCS) align with the contours of path-averaged scalars, i.e. the time average of scalar fields along the trajectories of the dynamical system. The alignment is a consequence of the fact that the length of a repelling (respectively, attracting) LCS shrinks rapidly under advection in forward (respectively, backward) time. As a result, the points along a hyperbolic LCS sample similar values of the scalar field, leading to almost uniform distribution of the path-averaged scalar along the LCS. Our results illuminate the relation between the variational theory of hyperbolic LCSs and a significant subset of mixing diagnostics which are obtained as path-averaged scalars. We illustrate the theoretical results on a direct numerical simulation of two-dimensional Navier--Stokes equations. Furthermore, our results provide partial explanation for a recent observation that hyperbolic LCSs separate dynamically distinct regions ...
Gromov Hyperbolicity in Cartesian Product Graphs
Indian Academy of Sciences (India)
Junior Michel; José M Rodríguez; José M Sigarreta; María Villeta
2010-11-01
If is a geodesic metric space and $x_1,x_2,x_3\\in X$, a geodesic triangle $T=\\{x_1,x_2,x_3\\}$ is the union of the three geodesics $[x_1x_2], [x_2x_3]$ and $[x_3x_1]$ in . The space is -hyperbolic (in the Gromov sense) if any side of is contained in a -neighborhood of the union of the two other sides, for every geodesic triangle in . If is hyperbolic, we denote by () the sharp hyperbolicity constant of , i.e. $(X)=\\inf\\{≥ 0:X\\, \\text{is}-\\text{hyperbolic}\\}$. In this paper we characterize the product graphs 1 × 2 which are hyperbolic, in terms of 1 and 2: the product graph 1 × 2 is hyperbolic if and only if 1 is hyperbolic and 2 is bounded or 2 is hyperbolic and 1 is bounded. We also prove some sharp relations between the hyperbolicity constant of $1 × 2,(1),(2) and the diameters of 1 and 2 (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the precise value of the hyperbolicity constant for many product graphs.
Green function for hyperbolic media
Potemkin, Andrey S; Belov, Pavel A; Kivshar, Yuri S
2012-01-01
We revisit the problem of the electromagnetic Green function for homogeneous hyperbolic media, where longitudinal and transverse components of the dielectric permittivity tensor have different signs. We analyze the dipole emission patterns for both dipole orientations with respect to the symmetry axis and for different signs of dielectric constants, and show that the emission pattern is highly anisotropic and has a characteristic cross-like shape: the waves are propagating within a certain cone and are evanescent outside this cone. We demonstrate the coexistence of the cone-like pattern due to emission of the extraordinary TM-polarized waves and elliptical pattern due to emission of ordinary TE-polarized waves. We find a singular complex term in the Green function, proportional to the $\\delta-$function and governing the photonic density of states and Purcell effect in hyperbolic media.
Closure constraints for hyperbolic tetrahedra
Charles, Christoph
2015-01-01
We investigate the generalization of loop gravity's twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant SU(2) gauge theory. Its classical states are graphs provided with algebraic data. In particular closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in flat space R^3. One then glues them allowing for both curvature and torsion. It was recently conjectured that q-deforming the gauge group SU(2) would allow to account for a non-vanishing cosmological constant Lambda, and in particular that deforming the loop gravity phase space with real parameter q>0 would lead to a generalization of twisted geometries to a hyperbolic curvature. Following this insight, we look for generalization of the closure constraints to the hyperbolic case. In particular, we introduce two new closure constraints for hyperbolic tetrahe...
Partial differential equations mathematical techniques for engineers
Epstein, Marcelo
2017-01-01
This monograph presents a graduate-level treatment of partial differential equations (PDEs) for engineers. The book begins with a review of the geometrical interpretation of systems of ODEs, the appearance of PDEs in engineering is motivated by the general form of balance laws in continuum physics. Four chapters are devoted to a detailed treatment of the single first-order PDE, including shock waves and genuinely non-linear models, with applications to traffic design and gas dynamics. The rest of the book deals with second-order equations. In the treatment of hyperbolic equations, geometric arguments are used whenever possible and the analogy with discrete vibrating systems is emphasized. The diffusion and potential equations afford the opportunity of dealing with questions of uniqueness and continuous dependence on the data, the Fourier integral, generalized functions (distributions), Duhamel's principle, Green's functions and Dirichlet and Neumann problems. The target audience primarily comprises graduate s...
Institute of Scientific and Technical Information of China (English)
Zha Qi-Lao; Sirendaoreji
2006-01-01
Some new exact solitary wave solutions of the Hybrid lattice and discrete mKdV lattice are obtained by using a hyperbolic function approach.This approach can also be applied to other nonlinear differential-difference equations.
Bounds on Gromov Hyperbolicity Constant in Graphs
Indian Academy of Sciences (India)
José M Rodríguez; José M Sigarreta
2012-02-01
If is a geodesic metric space and 1,2,3 $\\in$ , a geodesic triangle ={1,2,3} is the union of the three geodesics [1,2], [2,3] and [31] in . The space is -hyperbolic (in the Gromov sense) if any side of is contained in a -neighborhood of the union of two other sides, for every geodesic triangle in . If is hyperbolic, we denote by () the sharp hyperbolicity constant of , i.e. ()=$inf{$≥ 0$ : is -hyperbolic}. In this paper we relate the hyperbolicity constant of a graph with some known parameters of the graph, as its independence number, its maximum and minimum degree and its domination number. Furthermore, we compute explicitly the hyperbolicity constant of some class of product graphs.
8th International Conference on Hyperbolic Problems : Theory, Numerics, Applications
Warnecke, Gerald
2001-01-01
The Eighth International Conference on Hyperbolic Problems - Theory, Nu merics, Applications, was held in Magdeburg, Germany, from February 27 to March 3, 2000. It was attended by over 220 participants from many European countries as well as Brazil, Canada, China, Georgia, India, Israel, Japan, Taiwan, und the USA. There were 12 plenary lectures, 22 further invited talks, and around 150 con tributed talks in parallel sessions as well as posters. The speakers in the parallel sessions were invited to provide a poster in order to enhance the dissemination of information. Hyperbolic partial differential equations describe phenomena of material or wave transport in physics, biology and engineering, especially in the field of fluid mechanics. Despite considerable progress, the mathematical theory is still strug gling with fundamental open problems concerning systems of such equations in multiple space dimensions. For various applications the development of accurate and efficient numerical schemes for computat...
Explicit Traveling Wave Solutions to Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
Linghai ZHANG
2011-01-01
First of all,some technical tools are developed. Then the author studies explicit traveling wave solutions to nonlinear dispersive wave equations,nonlinear dissipative dispersive wave equations,nonlinear convection equations,nonlinear reaction diffusion equations and nonlinear hyperbolic equations,respectively.
Lyapunov type characterization of hyperbolic behavior
Barreira, Luis; Dragičević, Davor; Valls, Claudia
2017-09-01
We give a complete characterization of the uniform hyperbolicity and nonuniform hyperbolicity of a cocycle with values in the space of bounded linear operators acting on a Hilbert space in terms of the existence of appropriate quadratic forms. Our work unifies and extends many results in the literature by considering the general case of not necessarily invertible cocycles acting on a Hilbert space over an arbitrary invertible dynamics. As a nontrivial application of, we study the persistence of hyperbolicity under small linear perturbations.
Institute of Scientific and Technical Information of China (English)
Xu Yumei
2005-01-01
In this paper we consider the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Without restriction on characteristics with constant multiplicity (＞ 1), a blow-up result is obtained for the C1 solution to the Cauchy problem under the assumptions where there is a simple genuinely nonlinear characteristic and the initial data possess certain weaker decaying properties.
Institute of Scientific and Technical Information of China (English)
王利彬
2003-01-01
In this paper we consider the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Without restriction on characteristics with constant multiplicity(＞ 1), under the assumptions that there is a genuinely nonlinear simple characteristic and the initial data possess certain decaying properties, the blow-up result is obtained for the C,1 solution to the Cauchy problem.
Analytic smoothing effect for the cubic hyperbolic Schrodinger equation in two space dimensions
Directory of Open Access Journals (Sweden)
Gaku Hoshino
2016-01-01
Full Text Available We study the Cauchy problem for the cubic hyperbolic Schrodinger equation in two space dimensions. We prove existence of analytic global solutions for sufficiently small and exponential decaying data. The method of proof depends on the generalized Leibniz rule for the generator of pseudo-conformal transform acting on pseudo-conformally invariant nonlinearity.
Cauchy Problem for Quasilinear Hyperbolic Systems with Higher Order Dissipative Terms
Institute of Scientific and Technical Information of China (English)
Wei-guo Zhang
2003-01-01
In this paper, the author studies the global existence, singularities and life span of smooth solutions of the Cauchy problem for a class of quasilinear hyperbolic systems with higher order dissipative terms and gives their applications to nonlinear wave equations with higher order dissipative terms.
Laplace-Type Semi-Invariants for a System of Two Linear Hyperbolic Equations by Complex Methods
Directory of Open Access Journals (Sweden)
F. M. Mahomed
2011-01-01
Full Text Available In 1773 Laplace obtained two fundamental semi-invariants, called Laplace invariants, for scalar linear hyperbolic partial differential equations (PDEs in two independent variables. He utilized this in his integration theory for such equations. Recently, Tsaousi and Sophocleous studied semi-invariants for systems of two linear hyperbolic PDEs in two independent variables. Separately, by splitting a complex scalar ordinary differential equation (ODE into its real and imaginary parts PDEs for two functions of two variables were obtained and their symmetry structure studied. In this work we revisit semi-invariants under equivalence transformations of the dependent variables for systems of two linear hyperbolic PDEs in two independent variables when such systems correspond to scalar complex linear hyperbolic equations in two independent variables, using the above-mentioned splitting procedure. The semi-invariants under linear changes of the dependent variables deduced for this class of hyperbolic linear systems correspond to the complex semi-invariants of the complex scalar linear (1+1 hyperbolic equation. We show that the adjoint factorization corresponds precisely to the complex splitting. We also study the reductions and the inverse problem when such systems of two linear hyperbolic PDEs arise from a linear complex hyperbolic PDE. Examples are given to show the application of this approach.
On hyperbolicity violations in cosmological models with vector fields
Golovnev, Alexey
2014-01-01
Cosmological models with vector fields received much attention in recent years. Unfortunately, most of them are plagued with severe instabilities or other problems. In particular, it was noted by G. Esposito-Farese, C. Pitrou and J.-Ph. Uzan in arXiv:0912.0481 that the models with a non-linear function of the Maxwellian kinetic term do always imply violations of hyperbolicity somewhere in the phase space. In this work we make this statement more precise in several respects and show that those violations may not be present around spatially homogeneous configurations of the vector field.
Elmetennani, Shahrazed
2016-08-09
In this paper, the problem of estimating the distributed profile of the temperature along the tube of a concentrated distributed solar collector from boundary measurements is addressed. A nonlinear observer is proposed based on a nonlinear integral transformation. The objective is to force the estimation error to follow some stable transport dynamics. Convergence conditions are derived in order to determine the observer gain ensuring the stabilization of the estimation error in a finite time. Numerical simulations are given to show the effectiveness of the proposed algorithm under different working conditions. (C) 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Paths of algebraic hyperbolic curves
Institute of Scientific and Technical Information of China (English)
Ya-juan LI; Li-zheng LU; Guo-zhao WANG
2008-01-01
Cubic algebraic hyperbolic (AH) Bezier curves and AH spline curves are defined with a positive parameter α in the space spanned by {1, t, sinht, cosht}. Modifying the value of α yields a family of AH Bezier or spline curves with the family parameter α. For a fixed point on the original curve, it will move on a defined curve called "path of AH curve" (AH Bezier and AH spline curves) when α changes. We describe the geometric effects of the paths and give a method to specify a curve passing through a given point.
Plasmonic waveguides cladded by hyperbolic metamaterials
DEFF Research Database (Denmark)
Ishii, Satoshi; Shalaginov, Mikhail Y.; Babicheva, Viktoriia E.
2014-01-01
Strongly anisotropic media with hyperbolic dispersion can be used for claddings of plasmonic waveguides (PWs). In order to analyze the fundamental properties of such waveguides, we analytically study 1D waveguides arranged from a hyperbolic metamaterial (HMM) in a HMM-Insulator-HMM (HIH) structure...
IMPLEMENTATIONS AND PRACTICAL APPLICATIONS OF HYPERBOLIC METAMATERIALS
Directory of Open Access Journals (Sweden)
A. V. Shchelokova
2014-03-01
Full Text Available The paper presents a review on hyperbolic metamaterials which are normally described by the permittivity and permeability tensors with the components of the opposite sign. Therefore, the hyperbolic metamaterials have the hyperbolic isofrequency surfaces in the wave vector space. It leads to a number of unusual properties, such as the negative refraction, the diverging density of photonic states, ultra-high rate of spontaneous emission and increasing of subwavelength fields. The presence of the unique properties mentioned above makes the concept of hyperbolic metamaterials promising for research in modern science and explains the attempts of research groups around the world to realize structures with hyperbolic isofrequency curve suitable for applications in different frequency ranges. Hyperbolic metamaterials realized as layered metal-dielectric structures, arrays of nanowires, graphene layers, as well as artificial transmission lines are considered in the paper. Possible practical applications of hyperbolic metamaterials are described including hyperlens able to increase the nanoscale objects; wire mediums applied for spectroscopy to improve the resolution and increasing the distance to the object being scanned. Hyperbolic metamaterials are noted to be extremely promising for applications in nanophotonics, including single-photon generation, sensing and photovoltaics.
A high-order accurate embedded boundary method for first order hyperbolic equations
Mattsson, Ken; Almquist, Martin
2017-04-01
A stable and high-order accurate embedded boundary method for first order hyperbolic equations is derived. Where the grid-boundaries and the physical boundaries do not coincide, high order interpolation is used. The boundary stencils are based on a summation-by-parts framework, and the boundary conditions are imposed by the SAT penalty method, which guarantees linear stability for one-dimensional problems. Second-, fourth-, and sixth-order finite difference schemes are considered. The resulting schemes are fully explicit. Accuracy and numerical stability of the proposed schemes are demonstrated for both linear and nonlinear hyperbolic systems in one and two spatial dimensions.
Optic axis-driven new horizons for hyperbolic metamaterials
Directory of Open Access Journals (Sweden)
Boardman Allan D.
2015-01-01
Full Text Available The broad assertion here is that the current hyperbolic metamaterial world is only partially served by investigations that incorporate only some limited version of anisotropy. Even modest deviations of the optic axis from the main propagation axis lead to new phase shifts, which not only compete with those created by absorption but end up dominating them. Some progress has been attempted in the literature by introducing the terms “asymmetric hyperbolic media”, but it appears that this kind of asymmetry only involves an optic axis at an angle to the interface of a uniaxial crystal. From a device point of view, many new prospects should appear and the outcomes of the investigations presented here yield a new general theory. It is emphasised that the orientation of the optic axis is a significant determinant in the resulting optical properties. Whereas for conventional anisotropic waveguides homogeneous propagating waves occur over a limited range of angular dispositions of the optic axis it is shown that for a hyperbolic guide a critical angular setting exists, above which the guided waves are always homogeneous. This has significant implications for metawaveguide designs. The resulting structures are more tolerant to optic axis misalignment.
Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces
Daudé, Thierry; Kamran, Niky; Nicoleau, Francois
2015-12-01
In this paper, we study an inverse scattering problem on Liouville surfaces having two asymptotically hyperbolic ends. The main property of Liouville surfaces consists of the complete separability of the Hamilton-Jacobi equations for the geodesic flow. An important related consequence is the fact that the stationary wave equation can be separated into a system of radial and angular ODEs. The full scattering matrix at fixed energy associated to a scalar wave equation on asymptotically hyperbolic Liouville surfaces can be thus simplified by considering its restrictions onto the generalized harmonics corresponding to the angular separated ODE. The resulting partial scattering matrices consists in a countable set of 2 × 2 matrices whose coefficients are the so called transmission and reflection coefficients. It is shown that the reflection coefficients are nothing but generalized Weyl-Titchmarsh (WT) functions for the radial ODE in which the generalized angular momentum is seen as the spectral parameter. Using the complex angular momentum method and recent results on 1D inverse problem from generalized WT functions, we show that the knowledge of the reflection operators at a fixed non-zero energy is enough to determine uniquely the metric of the asymptotically hyperbolic Liouville surface under consideration.
The art and science of hyperbolic tessellations.
Van Dusen, B; Taylor, R P
2013-04-01
The visual impact of hyperbolic tessellations has captured artists' imaginations ever since M.C. Escher generated his Circle Limit series in the 1950s. The scaling properties generated by hyperbolic geometry are different to the fractal scaling properties found in nature's scenery. Consequently, prevalent interpretations of Escher's art emphasize the lack of connection with nature's patterns. However, a recent collaboration between the two authors proposed that Escher's motivation for using hyperbolic geometry was as a method to deliberately distort nature's rules. Inspired by this hypothesis, this year's cover artist, Ben Van Dusen, embeds natural fractals such as trees, clouds and lightning into a hyperbolic scaling grid. The resulting interplay of visual structure at multiple size scales suggests that hybridizations of fractal and hyperbolic geometries provide a rich compositional tool for artists.
Hyperbolic monopoles, JNR data and spectral curves
Bolognesi, Stefano; Sutcliffe, Paul
2014-01-01
A large class of explicit hyperbolic monopole solutions can be obtained from JNR instanton data, if the curvature of hyperbolic space is suitably tuned. Here we provide explicit formulae for both the monopole spectral curve and its rational map in terms of JNR data. Examples with platonic symmetry are presented, together with some one-parameter families with cyclic and dihedral symmetries. These families include hyperbolic analogues of geodesics that describe symmetric monopole scatterings in Euclidean space and we illustrate the results with energy density isosurfaces. There is a metric on the moduli space of hyperbolic monopoles, defined using the abelian connection on the boundary of hyperbolic space, and we provide a simple integral formula for this metric on the space of JNR data.
Dumbser, Michael; Peshkov, Ilya; Romenski, Evgeniy; Zanotti, Olindo
2016-06-01
This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics recently proposed by Peshkov and Romenski [110], further denoted as HPR model. In that framework, the viscous stresses are computed from the so-called distortion tensor A, which is one of the primary state variables in the proposed first order system. A very important key feature of the HPR model is its ability to describe at the same time the behavior of inviscid and viscous compressible Newtonian and non-Newtonian fluids with heat conduction, as well as the behavior of elastic and visco-plastic solids. Actually, the model treats viscous and inviscid fluids as generalized visco-plastic solids. This is achieved via a stiff source term that accounts for strain relaxation in the evolution equations of A. Also heat conduction is included via a first order hyperbolic system for the thermal impulse, from which the heat flux is computed. The governing PDE system is hyperbolic and fully consistent with the first and the second principle of thermodynamics. It is also fundamentally different from first order Maxwell-Cattaneo-type relaxation models based on extended irreversible thermodynamics. The HPR model represents therefore a novel and unified description of continuum mechanics, which applies at the same time to fluid mechanics and solid mechanics. In this paper, the direct connection between the HPR model and the classical hyperbolic-parabolic Navier-Stokes-Fourier theory is established for the first time via a formal asymptotic analysis in the stiff relaxation limit. From a numerical point of view, the governing partial differential equations are very challenging, since they form a large nonlinear hyperbolic PDE system that includes stiff source terms and non-conservative products. We apply the successful family of one-step ADER-WENO finite volume (FV) and ADER discontinuous Galerkin (DG) finite element schemes to the HPR model in the stiff
Asymptotically exact Discontinuous Galerkin error estimates for linear symmetric hyperbolic systems
Adjerid, S.; Weinhart, T.
2014-01-01
We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetric hyperbolic systems of partial differential equations with smooth solutions. We perform a local error analysis by writing the local error as a series and showing that its lead
Quasi-linear symmetric hyperbolic Fuchsian systems in several space dimensions
Ames, Ellery; Isenberg, James; LeFloch, Philippe G
2012-01-01
We establish well-posedness results for the singular initial value problem associated with a class of quasi-linear, symmetric hyperbolic, partial differential equations of Fuchsian type in several space dimensions. This is an extension of earlier work by the authors for the same problem in one space dimension.
Observability estimate and state observation problems for stochastic hyperbolic equations
2013-01-01
In this paper, we derive a boundary and an internal observability inequality for stochastic hyperbolic equations with nonsmooth lower order terms. The required inequalities are obtained by global Carleman estimate for stochastic hyperbolic equations. By these inequalities, we study a state observation problem for stochastic hyperbolic equations. As a consequence, we also establish a unique continuation property for stochastic hyperbolic equations.
Institute of Scientific and Technical Information of China (English)
Ahmed Awad; Wang Haoping
2016-01-01
The acceleration autopilot design for skid-to-turn (STT) missile faces a great challenge owing to coupling effect among planes, variation of missile velocity and its parameters, inexistence of a complete state vector, and nonlinear aerodynamics. Moreover, the autopilot should be designed for the entire flight envelope where fast variations exist. In this paper, a design of inte-grated roll-pitch-yaw autopilot based on global fast terminal sliding mode control (GFTSMC) with a partial state nonlinear observer (PSNLO) for STT nonlinear time-varying missile model, is employed to address these issues. GFTSMC with a novel sliding surface is proposed to nullify the integral error and the singularity problem without application of the sign function. The pro-posed autopilot consisting of two-loop structure, controls STT maneuver and stabilizes the rolling with a PSNLO in order to estimate the immeasurable states as an output while its inputs are missile measurable states and control signals. The missile model considers the velocity variation, gravity effect and parameters’ variation. Furthermore, the environmental conditions’ dynamics are mod-eled. PSNLO stability and the closed loop system stability are studied. Finally, numerical simula-tion is established to evaluate the proposed autopilot performance and to compare it with existing approaches in the literature.
Directory of Open Access Journals (Sweden)
Awad Ahmed
2016-10-01
Full Text Available The acceleration autopilot design for skid-to-turn (STT missile faces a great challenge owing to coupling effect among planes, variation of missile velocity and its parameters, inexistence of a complete state vector, and nonlinear aerodynamics. Moreover, the autopilot should be designed for the entire flight envelope where fast variations exist. In this paper, a design of integrated roll-pitch-yaw autopilot based on global fast terminal sliding mode control (GFTSMC with a partial state nonlinear observer (PSNLO for STT nonlinear time-varying missile model, is employed to address these issues. GFTSMC with a novel sliding surface is proposed to nullify the integral error and the singularity problem without application of the sign function. The proposed autopilot consisting of two-loop structure, controls STT maneuver and stabilizes the rolling with a PSNLO in order to estimate the immeasurable states as an output while its inputs are missile measurable states and control signals. The missile model considers the velocity variation, gravity effect and parameters’ variation. Furthermore, the environmental conditions’ dynamics are modeled. PSNLO stability and the closed loop system stability are studied. Finally, numerical simulation is established to evaluate the proposed autopilot performance and to compare it with existing approaches in the literature.
On Asymptotic Completeness of Scattering in the Nonlinear Lamb System, II
Komech, A I
2012-01-01
We establish the asymptotic completeness in the nonlinear Lamb system for hyperbolic stationary states. For the proof we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ODE) converging to a hyperbolic stationary point using the Inverse Function Theorem in a Banach space. We give the counterexamples showing nonexistence of such trajectories for nonhyperbolic stationary points.
基于核PLS方法的非线性过程在线监控%Online nonlinear process monitoring using kernel partial least squares
Institute of Scientific and Technical Information of China (English)
胡益; 王丽; 马贺贺; 侍洪波
2011-01-01
针对过程监控数据的非线性特点,提出了一种基于核偏最小二乘(KPLS)的监控方法.KPLS方法是将原始输入数据通过核函数映射到高维特征空间,然后在高维特征空间再进行偏最小二乘(PLS)运算.与线性PIS相比,KPLS方法能充分利用样本空间信息,建立起输入输出变量之间的非线性关系.与其他非线性PLS方法不同,KPLS方法只需要进行线性运算,从而避免非线性优化问题.在对过程进行监控时,首先采用KPLS方法建立模型,得到得分向量,然后计算出T2和SPE统计量及其相应的控制限.Tennessee Eastman (TE)模型上的仿真研究结果表明,所提方法比线性PLS方法具有更好的过程监控性能.%To handle the nonlinear problem for process monitoring, a new technique based on kernel partial least squares (KPLS) is developed. KPLS is an improved partial least squares (PLS) method, and its main idea is to first map the input space into a high-dimensional feature space via a nonlinear kernel function and then to use the standard PLS in that feature space. Compared to linear PLS, KPLS can make full use of the sample space information, and effectively capture the nonlinear relationship between input variables and output variables. Different from other nonlinear PLS, KPLS requires only linear algebra and does not involve any nonlinear optimization. For process data, firstly KPLS was used to derive regression model and got the score vectors, and then two statistics, T2 and SPE, and corresponding control limits were calculated. A case study of the Tennessee-Eastman (TE) process illustrated that the proposed approach showed superior process monitoring performance compared to linear PLS.
Nogueiro, Pedro; Bento, Rita; Silva, Luís Simões da
2006-01-01
The performance of a structural system can be evaluated resorting to non-linear static analysis, also commonly referred to as Pushover Analysis, because of the nature of application of lateral loads while defining the capacity of the structure. This analysis involves the estimation of the structural strength and deformation demands and the comparison with the available capacities at desired performance levels. This paper aims at evaluating the seismic response of three steel struc...
Bollt, Erik
2012-01-01
Given multiple images that describe chaotic reaction-diffusion dynamics, parameters of a PDE model are estimated using autosynchronization, where parameters are controlled by synchronization of the model to the observed data. A two-component system of predator-prey reaction-diffusion PDEs is used with spatially dependent parameters to benchmark the methods described. Applications to modelling the ecological habitat of marine plankton blooms by nonlinear data assimilation through remote sensing is discussed.
Fan, Quan-Yong; Yang, Guang-Hong
2016-01-01
This paper is concerned with the problem of integral sliding-mode control for a class of nonlinear systems with input disturbances and unknown nonlinear terms through the adaptive actor-critic (AC) control method. The main objective is to design a sliding-mode control methodology based on the adaptive dynamic programming (ADP) method, so that the closed-loop system with time-varying disturbances is stable and the nearly optimal performance of the sliding-mode dynamics can be guaranteed. In the first step, a neural network (NN)-based observer and a disturbance observer are designed to approximate the unknown nonlinear terms and estimate the input disturbances, respectively. Based on the NN approximations and disturbance estimations, the discontinuous part of the sliding-mode control is constructed to eliminate the effect of the disturbances and attain the expected equivalent sliding-mode dynamics. Then, the ADP method with AC structure is presented to learn the optimal control for the sliding-mode dynamics online. Reconstructed tuning laws are developed to guarantee the stability of the sliding-mode dynamics and the convergence of the weights of critic and actor NNs. Finally, the simulation results are presented to illustrate the effectiveness of the proposed method.
Hyperbolic Divergence Cleaning for SPH
Tricco, Terrence S
2012-01-01
We present SPH formulations of Dedner et al's hyperbolic/parabolic divergence cleaning scheme for magnetic and velocity fields. Our implementation preserves the conservation properties of SPH which is important for stability. This is achieved by deriving an energy term for the Psi field, and imposing energy conservation on the cleaning subsystem of equations. This necessitates use of conjugate operators for divB and gradPsi in the numerical equations. For both the magnetic and velocity fields, the average divergence error in the system is reduced by an order of magnitude with our cleaning algorithm. Divergence errors in SPMHD are maintained to < 1%, even for realistic 3D applications with a corresponding gain in numerical stability. Density errors for an oscillating elliptic water drop using weakly compressible SPH are reduced by a factor of two.
A Gyrovector Space Approach to Hyperbolic Geometry
Ungar, Abraham
2009-01-01
The mere mention of hyperbolic geometry is enough to strike fear in the heart of the undergraduate mathematics and physics student. Some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. The mission of this book is to open that door by making the hyperbolic geometry of Bolyai and Lobachevsky, as well as the special relativity theory of Einstein that it regulates, accessible to a wider audience in terms of novel analogies that the modern and unknown share with the classical and familiar. T
Hyperbolic Structures and the Stickiness Effect
Institute of Scientific and Technical Information of China (English)
周济林; 周礼勇; 孙义燧
2002-01-01
The stickiness effect of invariant tori in the phase space is widely studied, and extended to the slow-down of orbital diffusion due to some other invariant sets, such as Cantori, island-chains and hyperbolic periodic orbits.We report on two models in which hyperbolic periodic orbits show the stickiness effect. We discuss the generalized stickiness effects caused by different invariant sets. We believe that the main cause of the generalized stickiness effects is the hyperbolic structures in the phase space of the dynamical systems.
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
Dahmani, F; Osin, D
2017-01-01
The authors introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the latter one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, Out(F_n), and the Cremona group. Other examples can be found among groups acting geometrically on CAT(0) spaces, fundamental groups of graphs of groups, etc. The authors obtain a number of general results about rotating families and hyperbolically embedded subgroups; although their technique applies to a wide class of groups, it is capable of producing new results even for well-studied particular classes. For instance, the authors solve two open problems about mapping class groups, and obtain some results which are n...
Feng, Jie; Wang, Zhe; Li, Lizhi; Li, Zheng; Ni, Weidou
2013-03-01
A nonlinearized multivariate dominant factor-based partial least-squares (PLS) model was applied to coal elemental concentration measurement. For C concentration determination in bituminous coal, the intensities of multiple characteristic lines of the main elements in coal were applied to construct a comprehensive dominant factor that would provide main concentration results. A secondary PLS thereafter applied would further correct the model results by using the entire spectral information. In the dominant factor extraction, nonlinear transformation of line intensities (based on physical mechanisms) was embedded in the linear PLS to describe nonlinear self-absorption and inter-element interference more effectively and accurately. According to the empirical expression of self-absorption and Taylor expansion, nonlinear transformations of atomic and ionic line intensities of C were utilized to model self-absorption. Then, the line intensities of other elements, O and N, were taken into account for inter-element interference, considering the possible recombination of C with O and N particles. The specialty of coal analysis by using laser-induced breakdown spectroscopy (LIBS) was also discussed and considered in the multivariate dominant factor construction. The proposed model achieved a much better prediction performance than conventional PLS. Compared with our previous, already improved dominant factor-based PLS model, the present PLS model obtained the same calibration quality while decreasing the root mean square error of prediction (RMSEP) from 4.47 to 3.77%. Furthermore, with the leave-one-out cross-validation and L-curve methods, which avoid the overfitting issue in determining the number of principal components instead of minimum RMSEP criteria, the present PLS model also showed better performance for different splits of calibration and prediction samples, proving the robustness of the present PLS model.
Penrose type inequalities for asymptotically hyperbolic graphs
Dahl, Mattias; Sakovich, Anna
2013-01-01
In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space $\\bH^n$. The graphs are considered as subsets of $\\bH^{n+1}$ and carry the induced metric. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence we estimate the mass by an integral over an inner boundary. In case the inner boundary satisfies a convexity condition this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam's article concerning the asymptotically Euclidean case.
Rothe's method to semilinear hyperbolic integrodifferential equations
Directory of Open Access Journals (Sweden)
D. Bahaguna
1990-01-01
Full Text Available In this paper we consider an application of Rothe's method to abstract semi-linear hyperbolic integrodifferential equations in Hilbert spaces. With the aid of Rothe's method we establish the existence of a unique strong solution.
Classical and quantum resonances for hyperbolic surfaces
Guillarmou, Colin; Hilgert, Joachim; Weich, Tobias
2016-01-01
For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.
Shiryaeva, E V
2014-01-01
In paper [S.I. Senashov, A. Yakhno. 2012. SIGMA. Vol.8. 071] the variant of the hodograph method based on the conservation laws for two hyperbolic quasilinear equations of the first order is described. Using these results we propose a method which allows to reduce the Cauchy problem for the two quasilinear PDE's to the Cauchy problem for ODE's. The proposed method is actually some similar method of characteristics for a system of two hyperbolic quasilinear equations. The method can be used effectively in all cases, when the linear hyperbolic equation in partial derivatives of the second order with variable coefficients, resulting from the application of the hodograph method, has an explicit expression for the Riemann-Green function. One of the method's features is the possibility to construct a multi-valued solutions. In this paper we present examples of method application for solving the classical shallow water equations.
Institute of Scientific and Technical Information of China (English)
洪翔; 卢立新
2012-01-01
建立了二自由度的双曲正切包装系统模型，并应用四阶龙格一库塔法对得到的冲击动力学方程进行数值求解，研究了其冲击响应特性。采用后峰锯齿脉冲作为激励，得到了关键部件的三维；中击谱，进一步讨论了频率比、脉冲激励幅值、包装材料阻尼和脉冲周期对关键部件冲击谱的影响规律。结果表明，频率比、脉冲激励幅值、包装材料阻尼和脉冲周期均对关键部件冲击响应峰值有显著影响。%Two degree-of-freedom nonlinear packaging system model was established. The shock dynamic equations were obtained and their numerical solutions were solved by the forth order Runge-Kutta integration method. The shock response characteristics of hyperbolic tangent packaging system with critical component were in- vestigated. Final peak saw tooth shock pulse was chosen as excitation in the computer simulation and three-di- mensional shock spectra of critical components were obtained. The influences of frequency ratio, amplitude of pulse excitation, package material damp, and shock period on shock response of critical components were discussed. The results showed that frequency ratio, amplitude of pulse excitation, package material damp, and shock period all have significant influence on the peak value of shock response of critical components.
Institute of Scientific and Technical Information of China (English)
洪翔; 卢立新; 王军
2011-01-01
Two-degree-of-freedom nonlinear packaging system model was established. The shock dynamic equations were obtained and their numerical solutions were solved. The shock response characteristics of hyperbolic tangent packaging system with critical component were investigated. The rectangular acceleration pulse was chosen in the experiment due to its highest severity among all kinds of pulse. The influences of frequency ratio, amplitude of pulse excitation, damp, and system parameters on shock response of critical components were studied. The results showed that frequency ratio, amplitude of pulse excitation, damp, and system parameters all have significant influence on the peak value of shock response of critical components. The purpose was to provide reference on cushion packaging design.%建立了二自由度非线性包装系统模型，得到了冲击动力学方程并对其进行了数值求解，在此基础上，研究了带有关键部件双曲正切包装系统的冲击响应特性。矩形脉冲为各种脉冲激励中最严酷的，因此选矩形脉冲为研究对象，得到了关键部件的三维冲击谱图。从谱图中进一步讨论了频率比、脉冲激励幅值、阻尼和系统参数对关键部件冲击谱的影响规律，结果表明频率比、脉冲激励幅值、阻尼和系统参数对关键部件冲击响应峰值均有显著影响。为产品缓冲包装设计提供一定的科学依据。
Model Reduction for Complex Hyperbolic Networks
Himpe, Christian; Ohlberger, Mario
2013-01-01
We recently introduced the joint gramian for combined state and parameter reduction [C. Himpe and M. Ohlberger. Cross-Gramian Based Combined State and Parameter Reduction for Large-Scale Control Systems. arXiv:1302.0634, 2013], which is applied in this work to reduce a parametrized linear time-varying control system modeling a hyperbolic network. The reduction encompasses the dimension of nodes and parameters of the underlying control system. Networks with a hyperbolic structure have many app...
Absorbing Boundary Conditions for Hyperbolic Systems
Institute of Scientific and Technical Information of China (English)
Matthias Ehrhardt
2010-01-01
This paper deals with absorbing boundary conditions for hyperbolic systems in one and two space dimensions. We prove the strict well-posedness of the resulting initial boundary value problem in 1D. Afterwards we establish the GKS-stability of the corresponding Lax-Wendroff-type finite difference scheme. Hereby, we have to extend the classical proofs, since the (discretized) absorbing boundary conditions do not fit the standard form of boundary conditions for hyperbolic systems.
Hyperbolicity of semigroups and Fourier multipliers
Latushkin, Yuri; Shvidkoy, Roman
2001-01-01
We present a characterization of hyperbolicity for strongly continuous semigroups on Banach spaces in terms of Fourier multiplier properties of the resolvent of the generator. Hyperbolicity with respect to classical solutions is also considered. Our approach unifies and simplifies the M. Kaashoek-- S. Verduyn Lunel theory and multiplier-type results previously obtained by S. Clark, M. Hieber, S. Montgomery-Smith, F. R\\"{a}biger, T. Randolph, and L. Weis.
Toward optical sensing with hyperbolic metamaterials
Mackay, Tom G.
2015-06-01
A possible means of optical sensing, based on a porous hyperbolic material that is infiltrated by a fluid containing an analyte to be sensed, was theoretically investigated. The sensing mechanism relies on the observation that extraordinary plane waves propagate in the infiltrated hyperbolic material only in directions enclosed by a cone aligned with the optic axis of the infiltrated hyperbolic material. The angle this cone subtends to the plane perpendicular to the optic axis is θc. The sensitivity of θc to changes in the refractive index of the infiltrating fluid, namely nb, was explored; also considered were the permittivity parameters and porosity of the hyperbolic material, as well as the shape and size of its pores. Sensitivity was gauged by the derivative dθc/dnb. In parametric numerical studies, values of dθc/dnb in excess of 500 deg per refractive index unit were computed, depending upon the constitutive parameters of the porous hyperbolic material and infiltrating fluid and the nature of the porosity. In particular, it was observed that exceeding large values of dθc/dnb could be attained as the negative-valued eigenvalue of the infiltrated hyperbolic material approached zero.
Hyperbolic metamaterials: Novel physics and applications
Smolyaninov, Igor I.; Smolyaninova, Vera N.
2017-10-01
Hyperbolic metamaterials were originally introduced to overcome the diffraction limit of optical imaging. Soon thereafter it was realized that hyperbolic metamaterials demonstrate a number of novel phenomena resulting from the broadband singular behavior of their density of photonic states. These novel phenomena and applications include super resolution imaging, new stealth technologies, enhanced quantum-electrodynamic effects, thermal hyperconductivity, superconductivity, and interesting gravitation theory analogues. Here we briefly review typical material systems, which exhibit hyperbolic behavior and outline important novel applications of hyperbolic metamaterials. In particular, we will describe recent imaging experiments with plasmonic metamaterials and novel VCSEL geometries, in which the Bragg mirrors may be engineered in such a way that they exhibit hyperbolic metamaterial properties in the long wavelength infrared range, so that they may be used to efficiently remove excess heat from the laser cavity. We will also discuss potential applications of three-dimensional self-assembled photonic hypercrystals, which are based on cobalt ferrofluids in external magnetic field. This system bypasses 3D nanofabrication issues, which typically limit metamaterial applications. Photonic hypercrystals combine the most interesting features of hyperbolic metamaterials and photonic crystals.
Hyperbolic phonon polaritons in hexagonal boron nitride
Dai, Siyuan
2015-03-01
Uniaxial materials whose axial and tangential permittivities have opposite signs are referred to as indefinite or hyperbolic media. While hyperbolic responses are normally achieved with metamaterials, hexagonal boron nitride (hBN) naturally possesses this property due to the anisotropic phonons in the mid-infrared. Using scattering-type scanning near-field optical microscopy, we studied polaritonic phenomena in hBN. We performed infrared nano-imaging of highly confined and low-loss hyperbolic phonon polaritons in hBN. The polariton wavelength was shown to be governed by the hBN thickness according to a linear law persisting down to few atomic layers [Science, 343, 1125-1129 (2014)]. Additionally, we carried out the modification of hyperbolic response in heterostructures comprised of a mononlayer graphene deposited on hBN. Electrostatic gating of the top graphene layer allows for the modification of wavelength and intensity of hyperbolic phonon polaritons in bulk hBN. The physics of the modification originates from the plasmon-phonon coupling in the hyperbolic medium. Furthermore, we demonstrated the ``hyperlens'' for subdiffractional imaging and focusing using a slab of hBN.
Nonlinear dynamics of hydrostatic internal gravity waves
Energy Technology Data Exchange (ETDEWEB)
Stechmann, Samuel N.; Majda, Andrew J. [New York University, Courant Institute of Mathematical Sciences, NY (United States); Khouider, Boualem [University of Victoria, Department of Mathematics and Statistics, Victoria, BC (Canada)
2008-11-15
Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden-Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are
Indian Academy of Sciences (India)
M Lakshmanan; T Kanna
2001-11-01
Coupled nonlinear Schrödinger equations (CNLS) very often represent wave propagation in optical media such as multicore ﬁbers, photorefractive materials and so on. We consider speciﬁcally the pulse propagation in integrable CNLS equations (generalized Manakov systems). We point out that these systems possess novel exact soliton type pulses which are shape changing under collision leading to an intensity redistribution. The shape changes correspond to linear fractional transformations allowing for the possibility of construction of logic gates and Turing equivalent all optical computers in homogeneous bulk media as shown by Steiglitz recently. Special cases of such solitons correspond to the recently much discussed partially coherent stationary solitons (PCS). In this paper, we review critically the recent developments regarding the above properties with particular reference to 2-CNLS.
Institute of Scientific and Technical Information of China (English)
罗李平
2007-01-01
In this paper, oscillatory properties for solutions of the systems of certain quasilinear impulsive delay hyperbolic equations with nonlinear diffusion coefficient are investigated. A sufficient criterion for oscillations of such systems is obtained.
HYPERBOLIC MEAN CURVATURE FLOW:EVOLUTION OF PLANE CURVES
Institute of Scientific and Technical Information of China (English)
Kong Dexing; Liu Kefeng; Wang Zenggui
2009-01-01
In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves.More precisely, we consider a family of closed curves F: S1×[0,T)→R2 which satisfies the following evolution equationε2F/εt2(u,t)=k(u,t)-▽p(u,t),▽(u,t)∈ S1X[0,T) with the initial data F(u,0)=Fo(u) and εF/εt(u,0)=F(u)No,Where k is the mean curvature and N is the unit inner normal vector of the plane curve F(u,t)f(u) and No are the initial velocity and the unit inner normal vector of the initial convex closed curve Fo,respectively,and ▽p is given by ▽p≧(ε2F/εsε和/εF/ε和)T,in which T stands for the unit tangent vector.The above problem is an initial value problem for a system of partial differential equations for F,it can be completely reduced to an initial value problem for a single partial differential equation for its support function.The latter equation is a hyperbolic Monge-Ampere equation.Based on this,we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at the a finite time interval[0,Tmax)and when t goes to Tmax,either the solution converges to a point or shocks and other propagating discontinuities are generated.Furthermore,we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support function adn the curvature of the curve.In the end,we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-timeR1,1.
Kim, Inkang
2012-01-01
In this note, we study deformations of a non-uniform real hyperbolic lattice in quaternionic hyperbolic spaces. Specially we show that the representations of the fundamental group of the figure eight knot complement into PU(2,1) cannot be deformed in $PSp(2,1)$ out of PU(2,1) up to conjugacy.
Shih, Tsan-Hsing; Liu, nan-Suey
2010-01-01
A brief introduction of the temporal filter based partially resolved numerical simulation/very large eddy simulation approach (PRNS/VLES) and its distinct features are presented. A nonlinear dynamic subscale model and its advantages over the linear subscale eddy viscosity model are described. In addition, a guideline for conducting a PRNS/VLES simulation is provided. Results are presented for three turbulent internal flows. The first one is the turbulent pipe flow at low and high Reynolds numbers to illustrate the basic features of PRNS/VLES; the second one is the swirling turbulent flow in a LM6000 single injector to further demonstrate the differences in the calculated flow fields resulting from the nonlinear model versus the pure eddy viscosity model; the third one is a more complex turbulent flow generated in a single-element lean direct injection (LDI) combustor, the calculated result has demonstrated that the current PRNS/VLES approach is capable of capturing the dynamically important, unsteady turbulent structures while using a relatively coarse grid.
Ramoelo, A.; Skidmore, A. K.; Cho, M. A.; Mathieu, R.; Heitkönig, I. M. A.; Dudeni-Tlhone, N.; Schlerf, M.; Prins, H. H. T.
2013-08-01
Grass nitrogen (N) and phosphorus (P) concentrations are direct indicators of rangeland quality and provide imperative information for sound management of wildlife and livestock. It is challenging to estimate grass N and P concentrations using remote sensing in the savanna ecosystems. These areas are diverse and heterogeneous in soil and plant moisture, soil nutrients, grazing pressures, and human activities. The objective of the study is to test the performance of non-linear partial least squares regression (PLSR) for predicting grass N and P concentrations through integrating in situ hyperspectral remote sensing and environmental variables (climatic, edaphic and topographic). Data were collected along a land use gradient in the greater Kruger National Park region. The data consisted of: (i) in situ-measured hyperspectral spectra, (ii) environmental variables and measured grass N and P concentrations. The hyperspectral variables included published starch, N and protein spectral absorption features, red edge position, narrow-band indices such as simple ratio (SR) and normalized difference vegetation index (NDVI). The results of the non-linear PLSR were compared to those of conventional linear PLSR. Using non-linear PLSR, integrating in situ hyperspectral and environmental variables yielded the highest grass N and P estimation accuracy (R2 = 0.81, root mean square error (RMSE) = 0.08, and R2 = 0.80, RMSE = 0.03, respectively) as compared to using remote sensing variables only, and conventional PLSR. The study demonstrates the importance of an integrated modeling approach for estimating grass quality which is a crucial effort towards effective management and planning of protected and communal savanna ecosystems.
On Differential Equations Describing 3-Dimensional Hyperbolic Spaces
Institute of Scientific and Technical Information of China (English)
WU Jun-Yi; DING Qing; Keti Tenenblat
2006-01-01
In this paper, we introduce the notion of a (2+1)-dimensional differential equation describing threedimensional hyperbolic spaces (3-h.s.). The (2+1)-dimensional coupled nonlinear Schrodinger equation and its sister equation, the (2+1)-dimensional coupled derivative nonlinear Schrodinger equation, are shown to describe 3-h.s. The (2+1)-dimensional generalized HF model: St = (1/2i [S, Sy] + 2iσS)x, σx = 1-4itr(SSxSy), in which S ∈ GLC(2)/GLC(1)×GLC(1),provides another example of (2+1)-dimensional differential equations describing 3-h.s. As a direct consequence, the geometric construction of an infinite number of conservation laws of such equations is illustrated. Furthermore we display a new infinite number of conservation laws of the (2+1)-dimensional nonlinear Schrodinger equation and the (2+1)odimensional derivative nonlinear Schrodinger equation by a geometric way.
Elementary quadratic chaotic flows with a single non-hyperbolic equilibrium
Energy Technology Data Exchange (ETDEWEB)
Wei, Zhouchao, E-mail: weizhouchao@163.com [School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074 (China); College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124 (China); Mathematical Institute, University of Oxford, Oxford (United Kingdom); Sprott, J.C., E-mail: sprott@physics.wisc.edu [Department of Physics, University of Wisconsin, Madison, WI 53706 (United States); Chen, Huai [Faculty of Earth Sciences, China University of Geosciences, Wuhan, 430074 (China)
2015-10-02
Highlights: • A kind of jerk equations are proposed. • Chaos can occur with all types of a non-hyperbolic equilibrium. • The mechanism of generating chaos is discussed. • Feigenbaum's constant explains the identified chaotic flows. - Abstract: This paper describes a class of third-order explicit autonomous differential equations, called jerk equations, with quadratic nonlinearities that can generate a catalog of nine elementary dissipative chaotic flows with the unusual feature of having a single non-hyperbolic equilibrium. They represent an interesting sub-class of dynamical systems that can exhibit many major features of regular and chaotic motion. The proposed systems are investigated through numerical simulations and theoretical analysis. For these jerk dynamical systems, a certain amount of nonlinearity is sufficient to produce chaos through a sequence of period-doubling bifurcations.
Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation uxt=F(x,t,u,ux
Directory of Open Access Journals (Sweden)
L. Byszewski
1990-01-01
Full Text Available The aim of the paper is to give two theorems about existence and uniqueness of continuous solutions for hyperbolic nonlinear differential problems with nonlocal conditions in bounded and unbounded domains. The results obtained in this paper can be applied in the theory of elasticity with better effect than analogous known results with classical initial conditions.
Institute of Scientific and Technical Information of China (English)
LIHua-Mei; LINJi; XUYou-Sheng
2005-01-01
In this paper, we extend the hyperbolic function approach for constructing the exact solutions of nonlinear differential-difference equation (NDDE) in a unified way. Applying the extended approach and with the aid of Maple,we have studied the discrete complex Ginzburg-Landau equation (dCGLE). As a result, we find a set of exact solutions which include bright and dark soliton solutions.
Anticontrol of chaos for discrete-time fuzzy hyperbolic model with uncertain parameters
Institute of Scientific and Technical Information of China (English)
Zhao Yan; Zhang Hua-Guang; Zheng Cheng-De
2008-01-01
This paper proposes a new method to chaotify the discrete-time fuzzy hyperbolic model (DFHM) with uncertain parameters.A simple nonlinear state feedback controller is designed for this purpose.By revised Marotto theorem,it is proven that the chaos generated by this controller satisfies the Li-Yorke definition.An example is presented to demonstrate the effectiveness of the approach.
Directory of Open Access Journals (Sweden)
Omholt Stig W
2011-06-01
Full Text Available Abstract Background Deterministic dynamic models of complex biological systems contain a large number of parameters and state variables, related through nonlinear differential equations with various types of feedback. A metamodel of such a dynamic model is a statistical approximation model that maps variation in parameters and initial conditions (inputs to variation in features of the trajectories of the state variables (outputs throughout the entire biologically relevant input space. A sufficiently accurate mapping can be exploited both instrumentally and epistemically. Multivariate regression methodology is a commonly used approach for emulating dynamic models. However, when the input-output relations are highly nonlinear or non-monotone, a standard linear regression approach is prone to give suboptimal results. We therefore hypothesised that a more accurate mapping can be obtained by locally linear or locally polynomial regression. We present here a new method for local regression modelling, Hierarchical Cluster-based PLS regression (HC-PLSR, where fuzzy C-means clustering is used to separate the data set into parts according to the structure of the response surface. We compare the metamodelling performance of HC-PLSR with polynomial partial least squares regression (PLSR and ordinary least squares (OLS regression on various systems: six different gene regulatory network models with various types of feedback, a deterministic mathematical model of the mammalian circadian clock and a model of the mouse ventricular myocyte function. Results Our results indicate that multivariate regression is well suited for emulating dynamic models in systems biology. The hierarchical approach turned out to be superior to both polynomial PLSR and OLS regression in all three test cases. The advantage, in terms of explained variance and prediction accuracy, was largest in systems with highly nonlinear functional relationships and in systems with positive feedback
Towards optical sensing with hyperbolic metamaterials
Mackay, Tom G
2015-01-01
A possible means of optical sensing, based on a porous hyperbolic material which is infiltrated by a fluid containing an analyte to be sensed, was investigated theoretically. The sensing mechanism relies on the observation that extraordinary plane waves propagate in the infiltrated hyperbolic material only in directions enclosed by a cone aligned with the optic axis of the infiltrated hyperbolic material. The angle this cone subtends to the plane perpendicular to the optic axis is $\\theta_c$. The sensitivity of $\\theta_c$ to changes in refractive index of the infiltrating fluid, namely $n_b$, was explored; also considered were the permittivity parameters and porosity of the hyperbolic material, as well as the shape and size of its pores. Sensitivity was gauged by the derivative $d \\theta_c / d n_b$. In parametric numerical studies, values of $d \\theta_c / d n_b$ in excess of 500 degrees per refractive index unit were computed, depending upon the constitutive parameters of the porous hyperbolic material and in...
IDENTIFICATION PECULIARITIES OF HYPERBOLE AND EUPHEMISMS
Directory of Open Access Journals (Sweden)
E. A. Kupriianycheva
2016-01-01
Full Text Available The article represents a research carried out within a cognitive-discursive paradigm of modern linguistics. The study represents an attempt to develop a method for hyperbole and euphemism identification as special cases of a metaphor. The Authors of article use the following determinations of tropes. Hyperbole is an expression that is more extreme than justified given its ontological referent [1, p. 163]. Euphemiya (greek eu – "good", phemi – "say" are the mitigations promoting effect indirect names substitutes of terrible, shameful or odious which brought to life by moral or religious motives [2]. As a basis for new method we use the Hyperbole Identification Procedure developed by research group from Vrije Universiteit Amsterdam under the leadership of G. Steen. The detailed analysis of hyperbolization and euphemization allows revealing and describing specifics of processes. The amplification of the existing sign is characteristic of the hyperbolization. The lexical unit with negative meaning becomes more expressional with additional negative connotations. The positive sign amplifies with addition of new positive meanings. In the euphemiya the positive connotations mitigate the negative meaning of lexical unit; sometimes it is possible full replacement negative on positive.
Curvature-based Hyperbolic Systems for General Relativity
Choquet-Bruhat, Y; Anderson, A; Choquet-Bruhat, Yvonne; York, James W.; Anderson, Arlen
1998-01-01
We review curvature-based hyperbolic forms of the evolution part of the Cauchy problem of General Relativity that we have obtained recently. We emphasize first order symmetrizable hyperbolic systems possessing only physical characteristics.
Multilayer cladding with hyperbolic dispersion for plasmonic waveguides
DEFF Research Database (Denmark)
Babicheva, Viktoriia; Shalaginov, Mikhail Y.; Ishii, Satoshi;
2015-01-01
We study the properties of plasmonic waveguides with a dielectric core and multilayer metal-dielectric claddings that possess hyperbolic dispersion. The waveguides hyperbolic multilayer claddings show better performance in comparison to conventional plasmonic waveguides. © OSA 2015....
Purcell effect in Hyperbolic Metamaterial Resonators
Slobozhanyuk, Alexey P; Powell, David A; Iorsh, Ivan; Shalin, Alexander S; Segovia, Paulina; Krasavin, Alexey V; Wurtz, Gregory A; Podolskiy, Viktor A; Belov, Pavel A; Zayats, Anatoly V
2015-01-01
The radiation dynamics of optical emitters can be manipulated by properly designed material structures providing high local density of photonic states, a phenomenon often referred to as the Purcell effect. Plasmonic nanorod metamaterials with hyperbolic dispersion of electromagnetic modes are believed to deliver a significant Purcell enhancement with both broadband and non-resonant nature. Here, we have investigated finite-size cavities formed by nanorod metamaterials and shown that the main mechanism of the Purcell effect in these hyperbolic resonators originates from the cavity hyperbolic modes, which in a microscopic description stem from the interacting cylindrical surface plasmon modes of the finite number of nanorods forming the cavity. It is found that emitters polarized perpendicular to the nanorods exhibit strong decay rate enhancement, which is predominantly influenced by the rod length. We demonstrate that this enhancement originates from Fabry-Perot modes of the metamaterial cavity. The Purcell fa...
Lasing Action with Gold Nanorod Hyperbolic Metamaterials
Chandrasekar, Rohith; Meng, Xiangeng; Shalaginov, Mikhail Y; Lagutchev, Alexei; Kim, Young L; Wei, Alexander; Kildishev, Alexander V; Boltasseva, Alexandra; Shalaev, Vladimir M
2016-01-01
Coherent nanoscale photon sources are of paramount importance to achieving all-optical communication. Several nanolasers smaller than the diffraction limit have been theoretically proposed and experimentally demonstrated using plasmonic cavities to confine optical fields. Such compact cavities exhibit large Purcell factors, thereby enhancing spontaneous emission, which feeds into the lasing mode. However, most plasmonic nanolasers reported so far have employed resonant nanostructures and therefore had the lasing restricted to the proximity of the resonance wavelength. Here, we report on an approach based on gold nanorod hyperbolic metamaterials for lasing. Hyperbolic metamaterials provide broadband Purcell enhancement due to large photonic density of optical states, while also supporting surface plasmon modes to deliver optical feedback for lasing due to nonlocal effects in nanorod media. We experimentally demonstrate the advantage of hyperbolic metamaterials in achieving lasing action by its comparison with ...
Hyperbolic metamaterial lens with hydrodynamic nonlocal response.
Yan, Wei; Mortensen, N Asger; Wubs, Martijn
2013-06-17
We investigate the effects of hydrodynamic nonlocal response in hyperbolic metamaterials (HMMs), focusing on the experimentally realizable parameter regime where unit cells are much smaller than an optical wavelength but much larger than the wavelengths of the longitudinal pressure waves of the free-electron plasma in the metal constituents. We derive the nonlocal corrections to the effective material parameters analytically, and illustrate the noticeable nonlocal effects on the dispersion curves numerically. As an application, we find that the focusing characteristics of a HMM lens in the local-response approximation and in the hydrodynamic Drude model can differ considerably. In particular, the optimal frequency for imaging in the nonlocal theory is blueshifted with respect to that in the local theory. Thus, to detect whether nonlocal response is at work in a hyperbolic metamaterial, we propose to measure the near-field distribution of a hyperbolic metamaterial lens.
Domain Decomposition Methods for Hyperbolic Problems
Indian Academy of Sciences (India)
Pravir Dutt; Subir Singh Lamba
2009-04-01
In this paper a method is developed for solving hyperbolic initial boundary value problems in one space dimension using domain decomposition, which can be extended to problems in several space dimensions. We minimize a functional which is the sum of squares of the 2 norms of the residuals and a term which is the sum of the squares of the 2 norms of the jumps in the function across interdomain boundaries. To make the problem well posed the interdomain boundaries are made to move back and forth at alternate time steps with sufficiently high speed. We construct parallel preconditioners and obtain error estimates for the method. The Schwarz waveform relaxation method is often employed to solve hyperbolic problems using domain decomposition but this technique faces difficulties if the system becomes characteristic at the inter-element boundaries. By making the inter-element boundaries move faster than the fastest wave speed associated with the hyperbolic system we are able to overcome this problem.
Broad-band acoustic hyperbolic metamaterial
Shen, Chen; Sui, Ni; Wang, Wenqi; Cummer, Steven A; Jing, Yun
2015-01-01
Acoustic metamaterials (AMMs) are engineered materials, made from subwavelength structures, that exhibit useful or unusual constitutive properties. There has been intense research interest in AMMs since its first realization in 2000 by Liu et al. A number of functionalities and applications have been proposed and achieved using AMMs. Hyperbolic metamaterials are one of the most important types of metamaterials due to their extreme anisotropy and numerous possible applications, including negative refraction, backward waves, spatial filtering, and subwavelength imaging. Although the importance of acoustic hyperbolic metamaterials (AHMMs) as a tool for achieving full control of acoustic waves is substantial, the realization of a broad-band and truly hyperbolic AMM has not been reported so far. Here, we demonstrate the design and experimental characterization of a broadband AHMM that operates between 1.0 kHz and 2.5 kHz.
Near-Field Heat Transfer between Multilayer Hyperbolic Metamaterials
Biehs, Svend-Age; Ben-Abdallah, Philippe
2017-02-01
We review the near-field radiative heat flux between hyperbolic materials focusing on multilayer hyperbolic meta-materials. We discuss the formation of the hyperbolic bands, the impact of ordering of the multilayer slabs, as well as the impact of the first single layer on the heat transfer. Furthermore, we compare the contribution of surface modes to that of hyperbolic modes. Finally, we also compare the exact results with predictions from effective medium theory.
Near-field heat transfer between multilayer hyperbolic metamaterials
Energy Technology Data Exchange (ETDEWEB)
Biehs, Svend-Age [Oldenburg Univ. (Germany). Inst. fuer Physik; Ben-Abdallah, Philippe [Univ. Paris-Sud 11, Palaiseau (France). Lab. Charles Fabry; Univ. Sherbrooke, PQ (Canada). Dept. of Mechanical Engineering
2017-05-01
We review the near-field radiative heat flux between hyperbolic materials focusing on multilayer hyperbolic meta-materials. We discuss the formation of the hyperbolic bands, the impact of ordering of the multilayer slabs, as well as the impact of the first single layer on the heat transfer. Furthermore, we compare the contribution of surface modes to that of hyperbolic modes. Finally, we also compare the exact results with predictions from effective medium theory.
Discounting of Reward Sequences: a Test of Competing Formal Models of Hyperbolic Discounting
Directory of Open Access Journals (Sweden)
Noah eZarr
2014-03-01
Full Text Available Humans are known to discount future rewards hyperbolically in time. Nevertheless, a formal recursive model of hyperbolic discounting has been elusive until recently, with the introduction of the hyperbolically discounted temporal difference (HDTD model. Prior to that, models of learning (especially reinforcement learning have relied on exponential discounting, which generally provides poorer fits to behavioral data. Recently, it has been shown that hyperbolic discounting can also be approximated by a summed distribution of exponentially discounted values, instantiated in the µAgents model. The HDTD model and the µAgents model differ in one key respect, namely how they treat sequences of rewards. The µAgents model is a particular implementation of a parallel discounting model, which values sequences based on the summed value of the individual rewards whereas the HDTD model contains a nonlinear interaction. To discriminate among these models, we ascertained how subjects discounted a sequence of three rewards, and then we tested how well each candidate model fit the subject data. The results show that the parallel model generally provides a better fit to the human data.
Hamiltonian Optics of Hyperbolic Polaritons in Nanogranules.
Sun, Zhiyuan; Gutiérrez-Rubio, Á; Basov, D N; Fogler, M M
2015-07-08
Semiclassical quantization rules and numerical calculations are applied to study polariton modes of materials whose permittivity tensor has principal values of opposite sign (so-called hyperbolic materials). The spectra of volume- and surface-confined polaritons are computed for spheroidal nanogranules of hexagonal boron nitride, a natural hyperbolic crystal. The field distribution created by polaritons excited by an external dipole source is predicted to exhibit raylike patterns due to classical periodic orbits. Near-field infrared imaging and Purcell-factor measurements are suggested to test these predictions.
Spectral methods for partial differential equations
Hussaini, M. Y.; Streett, C. L.; Zang, T. A.
1984-01-01
Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid-dynamical applications are emphasized.
Hyperbolicity of the complement of plane algebraic curves
Dethloff, G E; Dethloff, Gerd; Schumacher, Georeg
1993-01-01
The paper is a contribution of the conjecture of Kobayashi that the complement o f a generic plain curve of degree at least five is hyperbolic. The main result is that the complement of a generic configuration of three quadr ics is hyperbolic and hyperbolically embedded as well as the complement of two q uadrics and a line.
The hyperbolic factor: A measure of time inconsistency
K.I.M. Rohde (Kirsten)
2010-01-01
textabstractMany studies have found that discounting is hyperbolic rather than constant. Hyperbolic discounting induces time-inconsistent behavior and is becoming increasingly popular in economic applications. Most studies that provide evidence in favor of hyperbolic discounting either are merely
The hyperbolic factor: A measure of time inconsistency
K.I.M. Rohde (Kirsten)
2010-01-01
textabstractMany studies have found that discounting is hyperbolic rather than constant. Hyperbolic discounting induces time-inconsistent behavior and is becoming increasingly popular in economic applications. Most studies that provide evidence in favor of hyperbolic discounting either are merely qu
Hyperbolic equations and frequency interactions
Caffarelli, Luis
1998-01-01
The research topic for this IAS/PCMI Summer Session was nonlinear wave phenomena. Mathematicians from the more theoretical areas of PDEs were brought together with those involved in applications. The goal was to share ideas, knowledge, and perspectives. How waves, or "frequencies", interact in nonlinear phenomena has been a central issue in many of the recent developments in pure and applied analysis. It is believed that wavelet theory-with its simultaneous localization in both physical and frequency space and its lacunarity-is and will be a fundamental new tool in the treatment of the phenomena. Included in this volume are write-ups of the "general methods and tools" courses held by Jeff Rauch and Ingrid Daubechies. Rauch's article discusses geometric optics as an asymptotic limit of high-frequency phenomena. He shows how nonlinear effects are reflected in the asymptotic theory. In the article "Harmonic Analysis, Wavelets and Applications" by Daubechies and Gilbert the main structure of the wavelet theory is...
A decoupled system of hyperbolic equations for linearized cosmological perturbations
Ramírez, J
2002-01-01
A decoupled system of hyperbolic partial differential equations for linear perturbations around spatially flat FRW universes is obtained for the first time. The two key ingredients in obtaining this system are i) the explicit decomposition of the perturbing energy momentum tensor into two pieces: intrinsic and free, which have definite and distinct mathematical properties and physical interpretation, and ii) the introduction of a new gauge which plays a similar role as harmonic gauge does for perturbations around Minkowski space-time. The proposed formalism could be highly relevant for physical cosmology, since our universe is most likely flat. We deal with classical perturbations, but our treatment, being covariant, is also very appropriate for the description of linearized quantum gravity around cosmological backgrounds.
Second-order hyperbolic Fuchsian systems and applications
Beyer, Florian
2010-01-01
We introduce a new class of singular partial differential equations, referred to as the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First, we establish a general existence theory of solutions with asymptotic behavior prescribed on the singularity, which relies on a new approximation scheme, suitable also for numerical purposes. Second, this theory is applied to the (vacuum) Einstein equations for Gowdy spacetimes, and allows us to recover, by more direct arguments, well-posedness results established earlier by Rendall and collaborators. Another main contribution in this paper is the proposed approximation scheme, which we refer to as the Fuchsian numerical algorithm and is shown to provide highly accurate numerical approximations to the singular initial value problem. For the class of Gowdy spacetimes, the numerical experiments presented here show the interest and efficiency of the proposed method and demonstrate t...
The Hyperbolic Sine Cardinal and the Catenary
Sanchez-Reyes, Javier
2012-01-01
The hyperbolic function sinh(x)/x receives scant attention in the literature. We show that it admits a clear geometric interpretation as the ratio between length and chord of a symmetric catenary segment. The inverse, together with the use of dimensionless parameters, furnishes a compact, explicit construction of a general catenary segment of…
Cuspidal discrete series for projective hyperbolic spaces
DEFF Research Database (Denmark)
Andersen, Nils Byrial; Flensted-Jensen, Mogens
2013-01-01
Abstract. We have in [1] proposed a definition of cusp forms on semisimple symmetric spaces G/H, involving the notion of a Radon transform and a related Abel transform. For the real non-Riemannian hyperbolic spaces, we showed that there exists an infinite number of cuspidal discrete series, and a...
The homogeneous geometries of real hyperbolic space
DEFF Research Database (Denmark)
Castrillón López, Marco; Gadea, Pedro Martínez; Swann, Andrew Francis
We describe the holonomy algebras of all canonical connections of homogeneous structures on real hyperbolic spaces in all dimensions. The structural results obtained then lead to a determination of the types, in the sense of Tricerri and Vanhecke, of the corresponding homogeneous tensors. We use ...
Hyperbolic L2-modules with Reproducing Kernels
Institute of Scientific and Technical Information of China (English)
David EELPODE; Frank SOMMEN
2006-01-01
Abstract In this paper, the Dirac operator on the Klein model for the hyperbolic space is considered. A function space containing L2-functions on the sphere Sm-1 in (R)m, which are boundary values of solutions for this operator, is defined, and it is proved that this gives rise to a Hilbert module with a reproducing kernel.
Hyperbolic spaces in Teichm\\"uller spaces
Leininger, Christopher J
2011-01-01
We prove, for any n, that there is a closed connected orientable surface S so that the hyperbolic space H^n almost-isometrically embeds into the Teichm\\"uller space of S, with quasi-convex image lying in the thick part. As a consequence, H^n quasi-isometrically embeds in the curve complex of S.
KAWA lecture notes on complex hyperbolic geometry
Rousseau, Erwan
2016-01-01
These lecture notes are based on a mini-course given at the fifth KAWA Winter School on March 24-29, 2014 at CIRM, Marseille. They provide an introduction to hyperbolicity of complex algebraic varieties namely the geometry of entire curves, and a description of some recent developments.
Mass Law Predicts Hyperbolic Hypoxic Ventilatory Response
Severinghaus, John W.
The hyperbolic hypoxic ventilatory response vs PaO2, HVRp, is interpreted as relecting a mass hyperbolic relationship of cytochrome PcO2 to cytochrome potential Ec, offset 32 torr by the constant diffusion gradient between arterial blood and cytochrome in CB at its constant metabolic rate dot VO_2 . Ec is taken to be a linear function of redox reduction and CB ventilatory drive. As Ec rises in hypoxia, the absolute potentials of each step in the citric acid cycle rises equally while the potential drop across each step remains constant because flux rate remains constant. A hypothetic HVRs ( dot VE vs SaO2) response curve computed from these assumptions is strikingly non linear. A hypothetic HVRp calculated from an assumed linear HVRs cannot be fit to the observed hyperbolic increase of ventilation in response to isocapnic hypoxia at PO2 less than 40 torr. The incompatibility of these results suggest that in future studies HVRs will not be found to be linear, especially below 80% SaO2 and HVRp will fail to be accurately hyperbolic.
Analytic vortex solutions on compact hyperbolic surfaces
Maldonado, R
2015-01-01
We construct, for the first time, Abelian-Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations.
Analytic vortex solutions on compact hyperbolic surfaces
Maldonado, Rafael; Manton, Nicholas S.
2015-06-01
We construct, for the first time, abelian Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations.
Volume of a doubly truncated hyperbolic tetrahedron
Kolpakov, Alexander
2012-01-01
The present paper regards the volume function of a doubly truncated hyperbolic tetrahedron. Starting from the previous results of J. Murakami, U. Yano and A. Ushijima, we have developed a unified approach to expressing the volume in different geometric cases by dilogarithm functions and to treat properly the many analytic strata of the latter. Finally, several numeric examples are given.
On the hyperbolicity condition in linear elasticity
Directory of Open Access Journals (Sweden)
Remigio Russo
1991-05-01
Full Text Available This talk, which is mainly expository and based on [2-5], discusses the hyperbolicity conditions in linear elastodynamics. Particular emphasis is devoted to the key role it plays in the uniqueness questions associated with the mixed boundary-initial value problem in unbounded domains.
Approximation properties of fine hyperbolic graphs
Indian Academy of Sciences (India)
Benyin Fu
2016-05-01
In this paper, we propose a definition of approximation property which is called the metric invariant translation approximation property for a countable discrete metric space. Moreover, we use the techniques of Ozawa’s to prove that a fine hyperbolic graph has the metric invariant translation approximation property.
Stability analysis and H∞ control of discrete T–S fuzzy hyperbolic systems
Directory of Open Access Journals (Sweden)
Duan Ruirui
2016-03-01
Full Text Available This paper focuses on the problem of constraint control for a class of discrete-time nonlinear systems. Firstly, a new discrete T–S fuzzy hyperbolic model is proposed to represent a class of discrete-time nonlinear systems. By means of the parallel distributed compensation (PDC method, a novel asymptotic stabilizing control law with the “soft” constraint property is designed. The main advantage is that the proposed control method may achieve a small control amplitude. Secondly, for an uncertain discrete T–S fuzzy hyperbolic system with external disturbances, by the proposed control method, the robust stability and H∞ performance are developed by using a Lyapunov function, and some sufficient conditions are established through seeking feasible solutions of some linear matrix inequalities (LMIs to obtain several positive diagonally dominant (PDD matrices. Finally, the validity and feasibility of the proposed schemes are demonstrated by a numerical example and a Van de Vusse one, and some comparisons of the discrete T–S fuzzy hyperbolic model with the discrete T–S fuzzy linear one are also given to illustrate the advantage of our approach.
Guided Waves in Asymmetric Hyperbolic Slab Waveguides. The TM Mode Case
Lyashko, Ekaterina I
2016-01-01
Nonlinear guided wave modes in an asymmetric slab waveguide formed by an isotropic dielectric layer placed on a linear or nonlinear substrate and covered by a hyperbolic material are investigated. Optical axis is normal to the slab plane. The dispersion relations for TM waves are found. It is shown that there are additional cut-off frequencies for each TM mode. The effects of the nonlinearity on the dispersion relations are investigated and discussed. There are the modes, which are corresponded with situation where the peak of electric field is localized in the nonlinear substrate. These modes are absent in the linear waveguide. To excite these modes the power must exceed certain threshold value.
Institute of Scientific and Technical Information of China (English)
苏道毕力格; 王晓民; 乌云莫日根
2014-01-01
In this paper, we study the application of the symmetry classification to the boundary value problem of nonli-near partial differential equations. Firstly, by using differential characteristic set algorithm for the complete symmetry classification of partial differential equations, the complete symmetry classification of a given boundary value problem of nonlinear partial differential equations is proposed. Secondly, by using an extended symmetry, the boundary value problem of nonlinear partial differential equations is reduced to an initial value problem of the original differential equations. Finally, we numerically solve the initial value problem of the original differential equations by using Runge-Kutta method.%研究了微分方程对称分类在非线性偏微分方程组边值问题中的应用。首先，利用偏微分方程(组)完全对称分类微分特征列集算法确定了给定非线性偏微分方程组边值问题的完全对称分类；其次，利用一个扩充对称将非线性偏微分方程组边值问题约化为常微分方程组初值问题；最后，利用龙格-库塔法求解了常微分方程组初值问题的数值解。
Partial Differential Equations of Physics
Geroch, Robert
1996-01-01
Apparently, all partial differential equations that describe physical phenomena in space-time can be cast into a universal quasilinear, first-order form. In this paper, we do two things. First, we describe some broad features of systems of differential equations so formulated. Examples of such features include hyperbolicity of the equations, constraints and their roles (e.g., in connection with the initial-value formulation), how diffeomorphism freedom is manifest, and how interactions betwee...
A Hyperbolic Decay of the Dst Index during the Recovery Phase of Intense Geomagnetic Storms
Aguado, J; Saiz, E; Cerrato, Y
2014-01-01
What one commonly considers for reproducing the recovery phase of magnetosphere, as seen by the Dst index, is exponential function. However, the magnetosphere recovers faster in the first hours than in the late recovery phase. The early steepness followed by the late smoothness in the magnetospheric response is a feature that leads to the proposal of a hyperbolic decay function to reproduce the recovery phase, instead of the exponential function. A superposed epoch analysis of recovery phases of intense storms from 1963-2006 was performed, categorizing the storms by their intensity into five subsets. The hyperbolic decay function reproduces experimental data better than what the exponential function does for any subset of storms, which indicates a non-linear coupling between dDst/dt and Dst. Moreover, this kind of mathematical function, where the degree of reduction of the Dst index depends on time, allows for explaining different lifetimes of the physical mechanisms involved in the recovery phase and provide...
Dispersionless gaps and cavity modes in photonic crystals containing hyperbolic metamaterials
Xue, Chun-hua; Ding, Yaqiong; Jiang, Hai-tao; Li, Yunhui; Wang, Zhan-shan; Zhang, Ye-wen; Chen, Hong
2016-03-01
We theoretically study dispersionless gaps and cavity modes in one-dimensional photonic crystals composed of hyperbolic metamaterials and dielectric. Bragg gaps in conventional all-dielectric photonic crystals are always dispersive because propagating phases in two kinds of dielectrics decrease with incident angle. Here, based on phase variation compensation between a hyperbolic metamaterial layer and an isotropic dielectric layer, the dispersion of the gap can be offset and thus a dispersionless gap can be realized. Moreover, the dispersionless property of such gap has a wide parameter space. The dispersionless gap can be used to realize a dispersionless cavity mode. The dispersionless gaps and cavity modes will possess significant applications for all-angle reflectors, high-Q filters excited with finite-sized sources, and nonlinear wave mixing processes.
Winicour, Jeffrey
2017-08-01
An algebraic-hyperbolic method for solving the Hamiltonian and momentum constraints has recently been shown to be well posed for general nonlinear perturbations of the initial data for a Schwarzschild black hole. This is a new approach to solving the constraints of Einstein’s equations which does not involve elliptic equations and has potential importance for the construction of binary black hole data. In order to shed light on the underpinnings of this approach, we consider its application to obtain solutions of the constraints for linearized perturbations of Minkowski space. In that case, we find the surprising result that there are no suitable Cauchy hypersurfaces in Minkowski space for which the linearized algebraic-hyperbolic constraint problem is well posed.
A remark on geometric desingularization of a non-hyperbolic point using hyperbolic space
Kuehn, Christian
2016-06-01
A steady state (or equilibrium point) of a dynamical system is hyperbolic if the Jacobian at the steady state has no eigenvalues with zero real parts. In this case, the linearized system does qualitatively capture the dynamics in a small neighborhood of the hyperbolic steady state. However, one is often forced to consider non-hyperbolic steady states, for example in the context of bifurcation theory. A geometric technique to desingularize non-hyperbolic points is the blow-up method. The classical case of the method is motivated by desingularization techniques arising in algebraic geometry. The idea is to blow up the steady state to a sphere or a cylinder. In the blown-up space, one is then often able to gain additional hyperbolicity at steady states. The method has also turned out to be a key tool to desingularize multiple time scale dynamical systems with singularities. In this paper, we discuss an explicit example of the blow-up method where we replace the sphere in the blow-up by hyperbolic space. It is shown that the calculations work in the hyperbolic space case as for the spherical case. This approach may be even slightly more convenient if one wants to work with directional charts. Hence, it is demonstrated that the sphere should be viewed as an auxiliary object in the blow-up construction. Other smooth manifolds are also natural candidates to be inserted at steady states. Furthermore, we conjecture several problems where replacing the sphere could be particularly useful, i.e., in the context of singularities of geometric flows, for avoiding compactification, and generating 'interior' steady states.
Absence of solutions of differential inequalities and systems of hyperbolic type in conic domains
Energy Technology Data Exchange (ETDEWEB)
Laptev, G G [Tula State University, Tula (Russian Federation)
2002-12-31
We establish conditions sufficient for the absence of global solutions of semilinear hyperbolic inequalities and systems in conic domains of the Euclidean space R{sup N}. We consider a model problem in a cone K: that given by the inequality {partial_derivative}{sup 2}u/{partial_derivative}t{sup 2}-{delta}u{>=}|u|{sup q}, (x,t) element of K x (0,{infinity}). The proof is based on the test-function method developed by Veron, Mitidieri, Pokhozhaev, and Tesei.
Tunable VO2/Au hyperbolic metamaterial
Prayakarao, S.; Mendoza, B.; Devine, A.; Kyaw, C.; van Dover, R. B.; Liberman, V.; Noginov, M. A.
2016-08-01
Vanadium dioxide (VO2) is known to have a semiconductor-to-metal phase transition at ˜68 °C. Therefore, it can be used as a tunable component of an active metamaterial. The lamellar metamaterial studied in this work is composed of subwavelength VO2 and Au layers and is designed to undergo a temperature controlled transition from the optical hyperbolic phase to the metallic phase. VO2 films and VO2/Au lamellar metamaterial stacks have been fabricated and studied in electrical conductivity and optical (transmission and reflection) experiments. The observed temperature-dependent changes in the reflection and transmission spectra of the metamaterials and VO2 thin films are in a good qualitative agreement with theoretical predictions. The demonstrated optical hyperbolic-to-metallic phase transition is a unique physical phenomenon with the potential to enable advanced control of light-matter interactions.
Unknotting tunnels in hyperbolic 3-manifolds
Adams, Colin
2012-01-01
An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic in a hyperbolic 3-manifold M, we find sufficient conditions for it to be an unknotting tunnel. In particular, if the vertical geodesic corresponds to a 4-bracelet, 5-bracelet or 6-bracelet in the universal cover and has short enough length, it must be an unknotting tunnel. Furthermore, we consider a vertical geodesic that satisfies the elder sibling property, which means that in the universal cover, every horoball except the one centered at infinity is connected to a larger horoball by a lift of the vertical geodesic. Such a vertical geodesic with length less than ln(2) is then shown to be an unknotting tunnel.
Is the Bianchi identity always hyperbolic?
Rácz, István
2014-01-01
We consider $n+1$ dimensional smooth Riemannian and Lorentzian spaces satisfying Einstein's equations. The base manifold is assumed to be smoothly foliated by a one-parameter family of hypersurfaces. In both cases---likewise it is usually done in the Lorentzian case---Einstein's equations may be split into `Hamiltonian' and `momentum' constraints and a `reduced' set of field equations. It is shown that regardless whether the primary space is Riemannian or Lorentzian whenever the foliating hypersurfaces are Riemannian the `Hamiltonian' and `momentum' type expressions are subject to a subsidiary first order symmetric hyperbolic system. Since this subsidiary system is linear and homogeneous in the `Hamiltonian' and `momentum' type expressions the hyperbolicity of the system implies that in both cases the solutions to the `reduced' set of field equations are also solutions to the full set of equations provided that the constraints hold on one of the hypersurfaces foliating the base manifold.
Hyperbolic metamaterial lens with hydrodynamic nonlocal response
DEFF Research Database (Denmark)
Yan, Wei; Mortensen, N. Asger; Wubs, Martijn
2013-01-01
in the local-response approximation and in the hydrodynamic Drude model can differ considerably. In particular, the optimal frequency for imaging in the nonlocal theory is blueshifted with respect to that in the local theory. Thus, to detect whether nonlocal response is at work in a hyperbolic metamaterial, we......We investigate the effects of hydrodynamic nonlocal response in hyperbolic metamaterials (HMMs), focusing on the experimentally realizable parameter regime where unit cells are much smaller than an optical wavelength but much larger than the wavelengths of the longitudinal pressure waves...... of the free-electron plasma in the metal constituents. We derive the nonlocal corrections to the effective material parameters analytically, and illustrate the noticeable nonlocal effects on the dispersion curves numerically. As an application, we find that the focusing characteristics of a HMM lens...
Emergent hyperbolic geometry of growing simplicial complexes
Bianconi, Ginestra
2016-01-01
A large variety of interacting complex systems, including brain functional networks, protein interactions and collaboration networks is characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such they are ideal structures to investigate the hidden geometry of complex networks and explore whether this geometry is hyperbolic. Here we show an hyperbolic network geometry emerges spontaneously from models of growing simplicial complexes. The statistical and geometrical properties of the growing simplicial complexes strongly depend on their dimensionality and display all the universal properties of real complex networks. Interestingly, when the network dynamics includes an heterogeneous fitness of the faces, the growing simplicial complex can undergo phase transitions that are reflected by dramatic changes in the n...
The remains of a spinning, hyperbolic encounter
De Vittori, Lorenzo; Gupta, Anuradha; Jetzer, Philippe
2014-01-01
We review a recently proposed approach to construct gravitational wave (GW) polarization states of unbound spinning compact binaries. Through this rather simple method, we are able to include corrections due to the dominant order spin-orbit interactions, in the quadrupolar approximation and in a semi-analytic way. We invoke the 1.5 post-Newtonian (PN) accurate quasi-Keplerian parametrization for the radial part of the dynamics and impose its temporal evolution in the PN accurate polarization states equations. Further, we compute 1PN accurate amplitude corrections for the polarization states of non-spinning compact binaries on hyperbolic orbits. As an interesting application, we perform comparisons with previously available results for both the GW signals in the case of non-spinning binaries and the theoretical prediction for the amplitude of the memory effect on the metric after the hyperbolic passage.
Design of hyperbolic metamaterials by genetic algorithm
Goforth, Ian A.; Alisafaee, Hossein; Fullager, Daniel B.; Rosenbury, Chris; Fiddy, Michael A.
2014-09-01
We explain the design of one dimensional Hyperbolic Metamaterials (HMM) using a genetic algorithm (GA) and provide sample applications including the realization of negative refraction. The design method is a powerful optimization approach to find the optimal performance of such structures, which "naturally" finds HMM structures that are globally optimized for specific applications. We explain how a fitness function can be incorporated into the GA for different metamaterial properties.
On hyperbolic fixed points in ultrametric dynamics
Lindahl, Karl-Olof; 10.1134/S2070046610030052
2011-01-01
Let K be a complete ultrametric field. We give lower and upper bounds for the size of linearization discs for power series over K near hyperbolic fixed points. These estimates are maximal in the sense that there exist examples where these estimates give the exact size of the corresponding linearization disc. In particular, at repelling fixed points, the linearization disc is equal to the maximal disc on which the power series is injective.
Studies in the Hyperbolic Circle Problem
DEFF Research Database (Denmark)
Cherubini, Giacomo
In this thesis we study the remainder term e(s) in the hyperbolic lattice point counting problem. Our main approach to this problem is that of the spectral theory of automorphic forms. We show that the function e(s) exhibits properties similar to those of almost periodic functions, and we study d...... distribution for certain integral versions of it. Finally we describe what results can be obtained by application of fractional calculus, especially fractional integration to small order, to the problem....
Mapped tent pitching schemes for hyperbolic systems
Gopalakrishnan, J; Schöberl, J.; Wintersteiger, C.
2016-01-01
A spacetime domain can be progressively meshed by tent shaped objects. Numerical methods for solving hyperbolic systems using such tent meshes to advance in time have been proposed previously. Such schemes have the ability to advance in time by different amounts at different spatial locations. This paper explores a technique by which standard discretizations, including explicit time stepping, can be used within tent-shaped spacetime domains. The technique transforms the equations within a spa...
Hyperbolic spaces are of strictly negative type
DEFF Research Database (Denmark)
Hjorth, Poul G.; Kokkendorff, Simon L.; Markvorsen, Steen
2002-01-01
We study finite metric spaces with elements picked from, and distances consistent with, ambient Riemannian manifolds. The concepts of negative type and strictly negative type are reviewed, and the conjecture that hyperbolic spaces are of strictly negative type is settled, in the affirmative....... The technique of the proof is subsequently applied to show that every compact manifold of negative type must have trivial fundamental group, and to obtain a necessary criterion for product manifolds to be of negative type....
Accuracy property of certain hyperbolic difference schemes
Energy Technology Data Exchange (ETDEWEB)
Hicks, D.L.; Madsen, M.M.
1976-12-01
An accuracy property called the CFL1 property is shared by several successful difference schemes. It appears to be a property at least as important as the property of higher-order accuracy for hyperbolic difference schemes when weak solutions (e.g., shocks) are sought. Investigation of this property leads to suggestions of ways to improve the accuracy in such wavecodes as WONDY, CHARTD, and THREEDY. 10 figures.
Modular realizations of hyperbolic Weyl groups
Kleinschmidt, Axel; Palmkvist, Jakob
2010-01-01
We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions and octonions. We outline how to construct and analyse automorphic forms for these groups; their structure depends on the underlying arithmetic properties of the integer domains. We also give a new realization of the Weyl group W(E8) in terms of unit octavians and their automorphism group.
Remarks on the notion of global hyperbolicity
Sánchez, Miguel
2007-01-01
Global hyperbolicity is a classical and well-known concept, which lies in the core of General Relativity. Here we discuss briefly five approaches to this concept. They yield different definitions which become natural in diverse contexts: the initial value problem, singularity theorems, existence of maximizing causal geodesics, possibility to split globally the spacetime, causal boundaries. The neat formulation and definitive equivalence between these definitions have been completed only recently. A very brief summary is presented.
Einstein's Equations and Equivalent Hyperbolic Dynamical Systems
Anderson, A; York, J W; Anderson, Arlen; Choquet-Bruhat, Yvonne
1999-01-01
We discuss several explicitly causal hyperbolic formulations of Einstein's dynamical 3+1 equations in a coherent way, emphasizing throughout the fundamental role of the ``slicing function,'' $\\alpha$---the quantity that relates the lapse $N$ to the determinant of the spatial metric $\\bar{g}$ through $N = \\bar{g}^{1/2} \\alpha$. The slicing function allows us to demonstrate explicitly that every foliation of spacetime by spatial time-slices can be used in conjunction with the causal hyperbolic forms of the dynamical Einstein equations. Specifically, the slicing function plays an essential role (1) in a clearer form of the canonical action principle and Hamiltonian dynamics for gravity and leads to a recasting (2) of the Bianchi identities evolution of the gravitational constraints in vacuum, and also (3) of evolution of the energy and momentum components of the stress tensor in the presence of matter, (4) in an explicit rendering of four hyperbolic formulations of Einstein's equations with only physical charact...