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Sample records for linear evolution equations

  1. Fast wavelet based algorithms for linear evolution equations

    NASA Technical Reports Server (NTRS)

    Engquist, Bjorn; Osher, Stanley; Zhong, Sifen

    1992-01-01

    A class was devised of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent. The method draws on the work of Beylkin, Coifman, and Rokhlin which they applied to general Calderon-Zygmund type integral operators. A modification of their idea is applied to linear hyperbolic and parabolic equations, with spatially varying coefficients. A significant speedup over standard methods is obtained when applied to hyperbolic equations in one space dimension and parabolic equations in multidimensions.

  2. Evolution equation for non-linear cosmological perturbations

    SciTech Connect

    Brustein, Ram; Riotto, Antonio E-mail: Antonio.Riotto@cern.ch

    2011-11-01

    We present a novel approach, based entirely on the gravitational potential, for studying the evolution of non-linear cosmological matter perturbations. Starting from the perturbed Einstein equations, we integrate out the non-relativistic degrees of freedom of the cosmic fluid and obtain a single closed equation for the gravitational potential. We then verify the validity of the new equation by comparing its approximate solutions to known results in the theory of non-linear cosmological perturbations. First, we show explicitly that the perturbative solution of our equation matches the standard perturbative solutions. Next, using the mean field approximation to the equation, we show that its solution reproduces in a simple way the exponential suppression of the non-linear propagator on small scales due to the velocity dispersion. Our approach can therefore reproduce the main features of the renormalized perturbation theory and (time)-renormalization group approaches to the study of non-linear cosmological perturbations, with some possibly important differences. We conclude by a preliminary discussion of the nature of the full solutions of the equation and their significance.

  3. Molecular representation of molar domain (volume), evolution equations, and linear constitutive relations for volume transport.

    PubMed

    Eu, Byung Chan

    2008-09-01

    In the traditional theories of irreversible thermodynamics and fluid mechanics, the specific volume and molar volume have been interchangeably used for pure fluids, but in this work we show that they should be distinguished from each other and given distinctive statistical mechanical representations. In this paper, we present a general formula for the statistical mechanical representation of molecular domain (volume or space) by using the Voronoi volume and its mean value that may be regarded as molar domain (volume) and also the statistical mechanical representation of volume flux. By using their statistical mechanical formulas, the evolution equations of volume transport are derived from the generalized Boltzmann equation of fluids. Approximate solutions of the evolution equations of volume transport provides kinetic theory formulas for the molecular domain, the constitutive equations for molar domain (volume) and volume flux, and the dissipation of energy associated with volume transport. Together with the constitutive equation for the mean velocity of the fluid obtained in a previous paper, the evolution equations for volume transport not only shed a fresh light on, and insight into, irreversible phenomena in fluids but also can be applied to study fluid flow problems in a manner hitherto unavailable in fluid dynamics and irreversible thermodynamics. Their roles in the generalized hydrodynamics will be considered in the sequel. PMID:19044872

  4. Linear Equations: Equivalence = Success

    ERIC Educational Resources Information Center

    Baratta, Wendy

    2011-01-01

    The ability to solve linear equations sets students up for success in many areas of mathematics and other disciplines requiring formula manipulations. There are many reasons why solving linear equations is a challenging skill for students to master. One major barrier for students is the inability to interpret the equals sign as anything other than…

  5. Overdetermined Systems of Linear Equations.

    ERIC Educational Resources Information Center

    Williams, Gareth

    1990-01-01

    Explored is an overdetermined system of linear equations to find an appropriate least squares solution. A geometrical interpretation of this solution is given. Included is a least squares point discussion. (KR)

  6. Mode decomposition evolution equations

    PubMed Central

    Wang, Yang; Wei, Guo-Wei; Yang, Siyang

    2011-01-01

    Partial differential equation (PDE) based methods have become some of the most powerful tools for exploring the fundamental problems in signal processing, image processing, computer vision, machine vision and artificial intelligence in the past two decades. The advantages of PDE based approaches are that they can be made fully automatic, robust for the analysis of images, videos and high dimensional data. A fundamental question is whether one can use PDEs to perform all the basic tasks in the image processing. If one can devise PDEs to perform full-scale mode decomposition for signals and images, the modes thus generated would be very useful for secondary processing to meet the needs in various types of signal and image processing. Despite of great progress in PDE based image analysis in the past two decades, the basic roles of PDEs in image/signal analysis are only limited to PDE based low-pass filters, and their applications to noise removal, edge detection, segmentation, etc. At present, it is not clear how to construct PDE based methods for full-scale mode decomposition. The above-mentioned limitation of most current PDE based image/signal processing methods is addressed in the proposed work, in which we introduce a family of mode decomposition evolution equations (MoDEEs) for a vast variety of applications. The MoDEEs are constructed as an extension of a PDE based high-pass filter (Europhys. Lett., 59(6): 814, 2002) by using arbitrarily high order PDE based low-pass filters introduced by Wei (IEEE Signal Process. Lett., 6(7): 165, 1999). The use of arbitrarily high order PDEs is essential to the frequency localization in the mode decomposition. Similar to the wavelet transform, the present MoDEEs have a controllable time-frequency localization and allow a perfect reconstruction of the original function. Therefore, the MoDEE operation is also called a PDE transform. However, modes generated from the present approach are in the spatial or time domain and can be

  7. Linear superposition in nonlinear equations.

    PubMed

    Khare, Avinash; Sukhatme, Uday

    2002-06-17

    Several nonlinear systems such as the Korteweg-de Vries (KdV) and modified KdV equations and lambda phi(4) theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions. PMID:12059300

  8. Supersymmetric fifth order evolution equations

    SciTech Connect

    Tian, K.; Liu, Q. P.

    2010-03-08

    This paper considers supersymmetric fifth order evolution equations. Within the framework of symmetry approach, we give a list containing six equations, which are (potentially) integrable systems. Among these equations, the most interesting ones include a supersymmetric Sawada-Kotera equation and a novel supersymmetric fifth order KdV equation. For the latter, we supply some properties such as a Hamiltonian structures and a possible recursion operator.

  9. Equating Scores from Adaptive to Linear Tests

    ERIC Educational Resources Information Center

    van der Linden, Wim J.

    2006-01-01

    Two local methods for observed-score equating are applied to the problem of equating an adaptive test to a linear test. In an empirical study, the methods were evaluated against a method based on the test characteristic function (TCF) of the linear test and traditional equipercentile equating applied to the ability estimates on the adaptive test…

  10. Successfully Transitioning to Linear Equations

    ERIC Educational Resources Information Center

    Colton, Connie; Smith, Wendy M.

    2014-01-01

    The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…

  11. Lie algebras and linear differential equations.

    NASA Technical Reports Server (NTRS)

    Brockett, R. W.; Rahimi, A.

    1972-01-01

    Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

  12. Evolutions equations in computational anatomy.

    PubMed

    Younes, Laurent; Arrate, Felipe; Miller, Michael I

    2009-03-01

    One of the main purposes in computational anatomy is the measurement and statistical study of anatomical variations in organs, notably in the brain or the heart. Over the last decade, our group has progressively developed several approaches for this problem, all related to the Riemannian geometry of groups of diffeomorphisms and the shape spaces on which these groups act. Several important shape evolution equations that are now used routinely in applications have emerged over time. Our goal in this paper is to provide an overview of these equations, placing them in their theoretical context, and giving examples of applications in which they can be used. We introduce the required theoretical background before discussing several classes of equations of increasingly complexity. These equations include energy minimizing evolutions deriving from Riemannian gradient descent, geodesics, parallel transport and Jacobi fields. PMID:19059343

  13. Transformation matrices between non-linear and linear differential equations

    NASA Technical Reports Server (NTRS)

    Sartain, R. L.

    1983-01-01

    In the linearization of systems of non-linear differential equations, those systems which can be exactly transformed into the second order linear differential equation Y"-AY'-BY=0 where Y, Y', and Y" are n x 1 vectors and A and B are constant n x n matrices of real numbers were considered. The 2n x 2n matrix was used to transform the above matrix equation into the first order matrix equation X' = MX. Specially the matrix M and the conditions which will diagonalize or triangularize M were studied. Transformation matrices P and P sub -1 were used to accomplish this diagonalization or triangularization to return to the solution of the second order matrix differential equation system from the first order system.

  14. Solution Methods for Certain Evolution Equations

    NASA Astrophysics Data System (ADS)

    Vega-Guzman, Jose Manuel

    Solution methods for certain linear and nonlinear evolution equations are presented in this dissertation. Emphasis is placed mainly on the analytical treatment of nonautonomous differential equations, which are challenging to solve despite the existent numerical and symbolic computational software programs available. Ideas from the transformation theory are adopted allowing one to solve the problems under consideration from a non-traditional perspective. First, the Cauchy initial value problem is considered for a class of nonautonomous and inhomogeneous linear diffusion-type equation on the entire real line. Explicit transformations are used to reduce the equations under study to their corresponding standard forms emphasizing on natural relations with certain Riccati(and/or Ermakov)-type systems. These relations give solvability results for the Cauchy problem of the parabolic equation considered. The superposition principle allows to solve formally this problem from an unconventional point of view. An eigenfunction expansion approach is also considered for this general evolution equation. Examples considered to corroborate the efficacy of the proposed solution methods include the Fokker-Planck equation, the Black-Scholes model and the one-factor Gaussian Hull-White model. The results obtained in the first part are used to solve the Cauchy initial value problem for certain inhomogeneous Burgers-type equation. The connection between linear (the Diffusion-type) and nonlinear (Burgers-type) parabolic equations is stress in order to establish a strong commutative relation. Traveling wave solutions of a nonautonomous Burgers equation are also investigated. Finally, it is constructed explicitly the minimum-uncertainty squeezed states for quantum harmonic oscillators. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. It is shown that the product of the variances attains the required minimum value

  15. Symbolic Solution of Linear Differential Equations

    NASA Technical Reports Server (NTRS)

    Feinberg, R. B.; Grooms, R. G.

    1981-01-01

    An algorithm for solving linear constant-coefficient ordinary differential equations is presented. The computational complexity of the algorithm is discussed and its implementation in the FORMAC system is described. A comparison is made between the algorithm and some classical algorithms for solving differential equations.

  16. Congeneric Models and Levine's Linear Equating Procedures.

    ERIC Educational Resources Information Center

    Brennan, Robert L.

    In 1955, R. Levine introduced two linear equating procedures for the common-item non-equivalent populations design. His procedures make the same assumptions about true scores; they differ in terms of the nature of the equating function used. In this paper, two parameterizations of a classical congeneric model are introduced to model the variables…

  17. Simplified Linear Equation Solvers users manual

    SciTech Connect

    Gropp, W. ); Smith, B. )

    1993-02-01

    The solution of large sparse systems of linear equations is at the heart of many algorithms in scientific computing. The SLES package is a set of easy-to-use yet powerful and extensible routines for solving large sparse linear systems. The design of the package allows new techniques to be used in existing applications without any source code changes in the applications.

  18. Systems of Linear Equations on a Spreadsheet.

    ERIC Educational Resources Information Center

    Bosch, William W.; Strickland, Jeff

    1998-01-01

    The Optimizer in Quattro Pro and the Solver in Excel software programs make solving linear and nonlinear optimization problems feasible for business mathematics students. Proposes ways in which the Optimizer or Solver can be coaxed into solving systems of linear equations. (ASK)

  19. Evolution equation for quantum coherence

    PubMed Central

    Hu, Ming-Liang; Fan, Heng

    2016-01-01

    The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures. PMID:27382933

  20. Evolution equation for quantum coherence

    NASA Astrophysics Data System (ADS)

    Hu, Ming-Liang; Fan, Heng

    2016-07-01

    The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures.

  1. Evolution equation for quantum coherence.

    PubMed

    Hu, Ming-Liang; Fan, Heng

    2016-01-01

    The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures. PMID:27382933

  2. The non-linear MSW equation

    NASA Astrophysics Data System (ADS)

    Thomson, Mark J.; McKellar, Bruce H. J.

    1991-04-01

    A simple, non-linear generalization of the MSW equation is presented and its analytic solution is outlined. The orbits of the polarization vector are shown to be periodic, and to lie on a sphere. Their non-trivial flow patterns fall into two topological categories, the more complex of which can become chaotic if perturbed.

  3. Synthesizing Strategies Creatively: Solving Linear Equations

    ERIC Educational Resources Information Center

    Ponce, Gregorio A.; Tuba, Imre

    2015-01-01

    New strategies can ignite teachers' imagination to create new lessons or adapt lessons created by others. In this article, the authors present the experience of an algebra teacher and his students solving linear and literal equations and explain how the use of ideas found in past NCTM journals helped bring this lesson to life. The…

  4. Observed Score Linear Equating with Covariates

    ERIC Educational Resources Information Center

    Branberg, Kenny; Wiberg, Marie

    2011-01-01

    This paper examined observed score linear equating in two different data collection designs, the equivalent groups design and the nonequivalent groups design, when information from covariates (i.e., background variables correlated with the test scores) was included. The main purpose of the study was to examine the effect (i.e., bias, variance, and…

  5. From Arithmetic Sequences to Linear Equations

    ERIC Educational Resources Information Center

    Matsuura, Ryota; Harless, Patrick

    2012-01-01

    The first part of the article focuses on deriving the essential properties of arithmetic sequences by appealing to students' sense making and reasoning. The second part describes how to guide students to translate their knowledge of arithmetic sequences into an understanding of linear equations. Ryota Matsuura originally wrote these lessons for…

  6. Procedural Embodiment and Magic in Linear Equations

    ERIC Educational Resources Information Center

    de Lima, Rosana Nogueira; Tall, David

    2008-01-01

    How do students think about algebra? Here we consider a theoretical framework which builds from natural human functioning in terms of embodiment--perceiving the world, acting on it and reflecting on the effect of the actions--to shift to the use of symbolism to solve linear equations. In the main, the students involved in this study do not…

  7. Stochastic thermodynamics for linear kinetic equations

    NASA Astrophysics Data System (ADS)

    Van den Broeck, C.; Toral, R.

    2015-07-01

    Stochastic thermodynamics is formulated for variables that are odd under time reversal. The invariance under spatial rotation of the collision rates due to the isotropy of the heat bath is shown to be a crucial ingredient. An alternative detailed fluctuation theorem is derived, expressed solely in terms of forward statistics. It is illustrated for a linear kinetic equation with kangaroo rates.

  8. Optical systolic solutions of linear algebraic equations

    NASA Technical Reports Server (NTRS)

    Neuman, C. P.; Casasent, D.

    1984-01-01

    The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

  9. Hypocoercivity of linear degenerately dissipative kinetic equations

    NASA Astrophysics Data System (ADS)

    Duan, Renjun

    2011-08-01

    In this paper we develop a general approach of studying the hypocoercivity for a class of linear kinetic equations with both transport and degenerately dissipative terms. As concrete examples, the relaxation operator, Fokker-Planck operator and linearized Boltzmann operator are considered when the spatial domain takes the whole space or torus and when there is a confining force or not. The key part of the developed approach is to construct some equivalent temporal energy functionals for obtaining time rates of the solution trending towards equilibrium in some Hilbert spaces. The result in the case of the linear Boltzmann equation with confining forces is new. The proof mainly makes use of the macro-micro decomposition combined with Kawashima's argument on dissipation of the hyperbolic-parabolic system. At the end, a Korn-type inequality with probability measure is provided to deal with dissipation of momentum components.

  10. Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics.

    PubMed

    García de Soria, María Isabel; Maynar, Pablo; Schehr, Grégory; Barrat, Alain; Trizac, Emmanuel

    2008-05-01

    We study the nonequilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of ballistic annihilation therefore constantly loses particles. The dynamics of perturbations around the free decay regime is investigated using the spectral properties of the linearized Boltzmann operator, which characterize linear excitations on all time scales. The linearized Boltzmann equation is solved in the hydrodynamic limit by a projection technique, which yields the evolution equations for the relevant coarse-grained fields and expressions for the transport coefficients. We finally present the results of molecular dynamics simulations that validate the theoretical predictions. PMID:18643046

  11. Localization of the eigenvalues of linear integral equations with applications to linear ordinary differential equations.

    NASA Technical Reports Server (NTRS)

    Sloss, J. M.; Kranzler, S. K.

    1972-01-01

    The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.

  12. Linear stochastic degenerate Sobolev equations and applications†

    NASA Astrophysics Data System (ADS)

    Liaskos, Konstantinos B.; Pantelous, Athanasios A.; Stratis, Ioannis G.

    2015-12-01

    In this paper, a general class of linear stochastic degenerate Sobolev equations with additive noise is considered. This class of systems is the infinite-dimensional analogue of linear descriptor systems in finite dimensions. Under appropriate assumptions, the mild and strong well-posedness for the initial value problem are studied using elements of the semigroup theory and properties of the stochastic convolution. The final value problem is also examined and it is proved that this is uniquely strongly solvable and the solution is continuously dependent on the final data. Based on the results of the forward and backward problem, the conditions for the exact controllability are investigated for a special but important class of these equations. The abstract results are illustrated by applications in complex media electromagnetics, in the one-dimensional stochastic Dirac equation in the non-relativistic limit and in a potential application in input-output analysis in economics. Dedicated to Professor Grigoris Kalogeropoulos on the occasion of his seventieth birthday.

  13. Double distributions and evolution equations

    SciTech Connect

    A.V. Radyushkin

    1998-05-01

    Applications of perturbative QCD to deeply virtual Compton scattering and hard exclusive meson electroproduction processes require a generalization of usual parton distributions for the case when long-distance information is accumulated in nonforward matrix elements < p{prime} {vert_bar}O(0,z){vert_bar}p > of quark and gluon light-cone operators. In their previous papers the authors used two types of nonperturbative functions parameterizing such matrix elements: double distributions F(x,y;t) and nonforward distribution functions F{sub {zeta}}(X;t). Here they discuss in more detail the double distributions (DD's) and evolution equations which they satisfy. They propose simple models for F(x,y;t=0) DD's with correct spectral and symmetry properties which also satisfy the reduction relations connecting them to the usual parton densities f(x). In this way, they obtain self-consistent models for the {zeta}-dependence of nonforward distributions. They show that, for small {zeta}, one can easily obtain nonforward distributions (in the X > {zeta} region) from the parton densities: F{sub {zeta}} (X;t=0) {approx} f(X{minus}{zeta}/2).

  14. Optimal trajectories based on linear equations

    NASA Technical Reports Server (NTRS)

    Carter, Thomas E.

    1990-01-01

    The Principal results of a recent theory of fuel optimal space trajectories for linear differential equations are presented. Both impulsive and bounded-thrust problems are treated. A new form of the Lawden Primer vector is found that is identical for both problems. For this reason, starting iteratives from the solution of the impulsive problem are highly effective in the solution of the two-point boundary-value problem associated with bounded thrust. These results were applied to the problem of fuel optimal maneuvers of a spacecraft near a satellite in circular orbit using the Clohessy-Wiltshire equations. For this case two-point boundary-value problems were solved using a microcomputer, and optimal trajectory shapes displayed. The results of this theory can also be applied if the satellite is in an arbitrary Keplerian orbit through the use of the Tschauner-Hempel equations. A new form of the solution of these equations has been found that is identical for elliptical, parabolic, and hyperbolic orbits except in the way that a certain integral is evaluated. For elliptical orbits this integral is evaluated through the use of the eccentric anomaly. An analogous evaluation is performed for hyperbolic orbits.

  15. Non-Markovian stochastic evolution equations

    NASA Astrophysics Data System (ADS)

    Costanza, G.

    2014-05-01

    Non-Markovian continuum stochastic and deterministic equations are derived from a set of discrete stochastic and deterministic evolution equations. Examples are given of discrete evolution equations whose updating rules depend on two or more previous time steps. Among them, the continuum stochastic evolution equation of the Newton second law, the stochastic evolution equation of a wave equation, the stochastic evolution equation for the scalar meson field, etc. are obtained as special cases. Extension to systems of evolution equations and other extensions are considered and examples are given. The concept of isomorphism and almost isomorphism are introduced in order to compare the coefficients of the continuum evolution equations of two different smoothing procedures that arise from two different approaches. Usually these discrepancies arising from two sources: On the one hand, the use of different representations of the generalized functions appearing in the models and, on the other hand, the different approaches used to describe the models. These new concept allows to overcome controversies that were appearing during decades in the literature.

  16. Equations of motion in the linear approximation.

    NASA Technical Reports Server (NTRS)

    Robinson, I.; Robinson, J. R.

    1972-01-01

    Attempt to develop a gauge-invariant theory of the motion of singularities in an n-dimensional Riemannian space. A gauge-invariant theory is developed for a nongeodetic particle represented by a time-like world line in the linearized field theory. The solution of the basic algebraic and differential equations is to be constructed from the source line, the future null cones emanating from it, a scale factor, and nothing else, and the leading term of the solution is the Schwarzschild field. A class of possible solutions is examined, and it is shown that this class contains just one acceptable member - namely, a particle in constant acceleration.

  17. Modified non-linear Burgers' equations and cosmic ray shocks

    NASA Technical Reports Server (NTRS)

    Zank, G. P.; Webb, G. M.; Mckenzie, J. F.

    1988-01-01

    A reductive perturbation scheme is used to derive a generalized non-linear Burgers' equation, which includes the effects of dispersion, in the long wavelength regime for the two-fluid hydrodynamical model used to describe cosmic ray acceleration by the first-order Fermi process in astrophysical shocks. The generalized Burger's equation is derived for both relativistic and non-relativistic cosmic ray shocks, and describes the time evolution of weak shocks in the theory of diffusive shock acceleration. The inclusion of dispersive effects modifies the phase velocity of the shock obtained from the lower order non-linear Burger's equation through the introduction of higher order terms from the long wavelength dispersion equation. The travelling wave solution of the generalized Burgers' equation for a single shock shows that larger cosmic ray pressures result in broader shock transitions. The results for relativistic shocks show a steepening of the shock as the shock speed approaches the relativistic cosmic ray sound speed. The dependence of the shock speed on the cosmic ray pressure is also discussed.

  18. On Solving Non-Autonomous Linear Difference Equations with Applications

    ERIC Educational Resources Information Center

    Abu-Saris, Raghib M.

    2006-01-01

    An explicit formula is established for the general solution of the homogeneous non-autonomous linear difference equation. The formula developed is then used to characterize globally periodic linear difference equations with constant coefficients.

  19. Convergence of Galerkin approximations for operator Riccati equations: A nonlinear evolution equation approach

    NASA Technical Reports Server (NTRS)

    Rosen, I. G.

    1988-01-01

    An approximation and convergence theory was developed for Galerkin approximations to infinite dimensional operator Riccati differential equations formulated in the space of Hilbert-Schmidt operators on a separable Hilbert space. The Riccati equation was treated as a nonlinear evolution equation with dynamics described by a nonlinear monotone perturbation of a strongly coercive linear operator. A generic approximation result was proven for quasi-autonomous nonlinear evolution system involving accretive operators which was then used to demonstrate the Hilbert-Schmidt norm convergence of Galerkin approximations to the solution of the Riccati equation. The application of the results was illustrated in the context of a linear quadratic optimal control problem for a one dimensional heat equation.

  20. On quintic equations with a linear window

    NASA Astrophysics Data System (ADS)

    Rosenau, Philip

    2016-01-01

    We study a quintic dispersive equation ut =[ au2 + b (uuxx + β ux2) + c (uu4x + 2q1uxu3x +q2uxx2) ]x and show that if β =q1 = -q2, it may be cast into vt = [ vLω u ]x, where v =uω, ω = 2 β + 1 and Lω is a fourth order linear operator. This enables to construct traveling patterns via superposition of solutions. A plethora of bell-shaped, multi-humped and asymmetric compacton, is found. Their interaction ranges from being almost elastic to a noisy one, including fusion of bell-shaped compactons and anti-compactons into robust asymmetric structures. A stationary, zero-mass, doublet-like compacton is found to be an attractor of topologically similar, zero-mass, excitations.

  1. Dielectric polarization evolution equations and relaxation times

    SciTech Connect

    Baker-Jarvis, James; Riddle, Bill; Janezic, Michael D.

    2007-05-15

    In this paper we develop dielectric polarization evolution equations, and the resulting frequency-domain expressions, and relationships for the resulting frequency dependent relaxation times. The model is based on a previously developed equation that was derived using statistical-mechanical theory. We extract relaxation times from dielectric data and give illustrative examples for the harmonic oscillator and derive expressions for the frequency-dependent relaxation times and a time-domain integrodifferential equation for the Cole-Davidson model.

  2. Stability of Linear Equations--Algebraic Approach

    ERIC Educational Resources Information Center

    Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.

    2012-01-01

    This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…

  3. Spectrum Analysis of the Linearized Relativistic Landau Equation

    NASA Astrophysics Data System (ADS)

    Luo, Lan; Yu, Hongjun

    2016-05-01

    In this work we prove the complete spectrum structure of the linearized relativistic Landau equation in L^2 by using the semigroup theory and the linear operator perturbation theory. Our results include the physical interesting Coulombic interaction.

  4. Hot atom populations in the terrestrial atmosphere. A comparison of the nonlinear and linearized Boltzmann equations

    NASA Astrophysics Data System (ADS)

    Sospedra-Alfonso, Reinel; Shizgal, Bernie D.

    2012-11-01

    We use a finite difference discretization method to solve the space homogeneous, isotropic nonlinear Boltzmann equation. We study the time evolution of the distribution function in relation to the solution of the linearized Boltzmann equation for three different initial conditions. The relaxation process is described in terms of the Laguerre moments and the spectral properties of the linearized collision operator. The motivation is the need to include self-collisions in the study of suprathermal oxygen atoms in the terrestrial exosphere.

  5. A Cognitive Approach to Solving Systems of Linear Equations

    ERIC Educational Resources Information Center

    Ramirez, Ariel A.

    2009-01-01

    Systems of linear equations are used in a variety of fields. Exposure to the concept of systems of equations initially occurs at the high school level and continues through college. Attempts to unearth what students understand about the solutions of linear systems have been limited. Gaps exist in our knowledge of how students understand systems…

  6. Stochastic differential equations for non-linear hydrodynamics

    NASA Astrophysics Data System (ADS)

    Español, Pep

    1998-02-01

    We formulate the stochastic differential equations for non-linear hydrodynamic fluctuations. The equations incorporate the random forces through a random stres tensor and random heat flux as in the Landau and Lifshitz theory. However, the equations are non-linear and the random forces are non-Gaussian. We provide explicit expressions for these random quantities in terms of the well-defined increments of the Wienner process.

  7. Solving Systems of Linear Equations by Ratio and Proportion

    ERIC Educational Resources Information Center

    Katsaras, Vasilios J.

    1978-01-01

    The author describes and gives two illustrations of a method for solving a system of two linear equations. The ratio of left members is equated to the ratio of right members, the ratio of the two variables is solved for, and the resultant ratio is substituted into an original equation. (MN)

  8. A General Linear Method for Equating with Small Samples

    ERIC Educational Resources Information Center

    Albano, Anthony D.

    2015-01-01

    Research on equating with small samples has shown that methods with stronger assumptions and fewer statistical estimates can lead to decreased error in the estimated equating function. This article introduces a new approach to linear observed-score equating, one which provides flexible control over how form difficulty is assumed versus estimated…

  9. Non-Linear Spring Equations and Stability

    ERIC Educational Resources Information Center

    Fay, Temple H.; Joubert, Stephan V.

    2009-01-01

    We discuss the boundary in the Poincare phase plane for boundedness of solutions to spring model equations of the form [second derivative of]x + x + epsilonx[superscript 2] = Fcoswt and the [second derivative of]x + x + epsilonx[superscript 3] = Fcoswt and report the results of a systematic numerical investigation on the global stability of…

  10. Evolution of linear perturbations in spherically symmetric dust spacetimes

    NASA Astrophysics Data System (ADS)

    February, S.; Larena, J.; Clarkson, C.; Pollney, D.

    2014-09-01

    We present results from a numerical code implementing a new method to solve the master equations describing the evolution of linear perturbations in a spherically symmetric but inhomogeneous background. This method can be used to simulate several configurations of physical interest, such as relativistic corrections to structure formation, the lensing of gravitational waves (GWs) and the evolution of perturbations in a cosmological void model. This paper focuses on the latter problem, i.e. structure formation in a Hubble scale void in the linear regime. This is considerably more complicated than linear perturbations of a homogeneous and isotropic background because the inhomogeneous background leads to coupling between density perturbations and rotational modes of the spacetime geometry, as well as GWs. Previous analyses of this problem ignored this coupling in the hope that the approximation does not affect the overall dynamics of structure formation in such models. We show that for a giga-parsec void, the evolution of the density contrast is well approximated by the previously studied decoupled evolution only for very large-scale modes. However, the evolution of the gravitational potentials within the void is inaccurate at more than the 10% level, and is even worse on small scales.

  11. Technology, Linear Equations, and Buying a Car.

    ERIC Educational Resources Information Center

    Sandefur, James T.

    1992-01-01

    Discusses the use of technology in solving compound interest-rate problems that can be modeled by linear relationships. Uses a graphing calculator to solve the specific problem of determining the amount of money that can be borrowed to buy a car for a given monthly payment and interest rate. (MDH)

  12. Eulerian action principles for linearized reduced dynamical equations

    NASA Astrophysics Data System (ADS)

    Brizard, Alain

    1994-08-01

    New Eulerian action principles for the linearized gyrokinetic Maxwell-Vlasov equations and the linearized kinetic-magnetohydrodynamic (kinetic-MHD) equations are presented. The variational fields for the linearized gyrokinetic Vlasov-Maxwell equations are the perturbed electromagnetic potentials (φ1,A1) and the gyroangle-independent gyrocenter (gy) function Sgy, while the variational fields for the linearized kinetic-MHD equations are the ideal MHD fluid displacement ξ and the gyroangle-independent drift-kinetic (dk) function Sdk (defined as the drift-kinetic limit of Sgy). According to the Lie-transform approach to Vlasov perturbation theory, Sgy generates first-order perturbations in the gyrocenter distribution F1≡{Sgy, F0}gc, where F1 satisfies the linearized gyrokinetic Vlasov equation and {, }gc denotes the unperturbed guiding-center (gc) Poisson bracket. Previous quadratic variational forms were constructed ad hoc from the linearized equations, and required the linearized gyrokinetic (or drift-kinetic) Vlasov equation to be solved a priori (e.g., by integration along an unperturbed guiding-center orbit) through the use of the normal-mode and ballooning-mode representations. The presented action principles ignore these requirements and, thus, apply to more general perturbations.

  13. Evolution equation for entanglement of assistance

    SciTech Connect

    Li Zongguo; Liu, W. M.; Zhao Mingjing; Fei Shaoming

    2010-04-15

    We investigate the time evolution of the entanglement of assistance when one subsystem undergoes the action of local noisy channels. A general factorization law is presented for the evolution equation of entanglement of assistance. Our results demonstrate that the dynamics of the entanglement of assistance is determined by the action of a noisy channel on the pure maximally entangled state, in which the entanglement reduction turns out to be universal for all quantum states entering the channel. This single quantity will make it easy to characterize the entanglement dynamics of entanglement of assistance under unknown channels in the experimental process of producing entangled states by assisted entanglement.

  14. Algebraic methods for the solution of some linear matrix equations

    NASA Technical Reports Server (NTRS)

    Djaferis, T. E.; Mitter, S. K.

    1979-01-01

    The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.

  15. An evolution equation modeling inversion of tulip flames

    SciTech Connect

    Dold, J.W.; Joulin, G.

    1995-02-01

    The authors attempt to reduce the number of physical ingredients needed to model the phenomenon of tulip-flame inversion to a bare minimum. This is achieved by synthesizing the nonlinear, first-order Michelson-Sivashinsky (MS) equation with the second order linear dispersion relation of Landau and Darrieus, which adds only one extra term to the MS equation without changing any of its stationary behavior and without changing its dynamics in the limit of small density change when the MS equation is asymptotically valid. However, as demonstrated by spectral numerical solutions, the resulting second-order nonlinear evolution equation is found to describe the inversion of tulip flames in good qualitative agreement with classical experiments on the phenomenon. This shows that the combined influences of front curvature, geometric nonlinearity and hydrodynamic instability (including its second-order, or inertial effects, which are an essential result of vorticity production at the flame front) are sufficient to reproduce the inversion process.

  16. On an evolution equation in a cell motility model

    NASA Astrophysics Data System (ADS)

    Mizuhara, Matthew S.; Berlyand, Leonid; Rybalko, Volodymyr; Zhang, Lei

    2016-04-01

    This paper deals with the evolution equation of a curve obtained as the sharp interface limit of a non-linear system of two reaction-diffusion PDEs. This system was introduced as a phase-field model of (crawling) motion of eukaryotic cells on a substrate. The key issue is the evolution of the cell membrane (interface curve) which involves shape change and net motion. This issue can be addressed both qualitatively and quantitatively by studying the evolution equation of the sharp interface limit for this system. However, this equation is non-linear and non-local and existence of solutions presents a significant analytical challenge. We establish existence of solutions for a wide class of initial data in the so-called subcritical regime. Existence is proved in a two step procedure. First, for smooth (H2) initial data we use a regularization technique. Second, we consider non-smooth initial data that are more relevant from the application point of view. Here, uniform estimates on the time when solutions exist rely on a maximum principle type argument. We also explore the long time behavior of the model using both analytical and numerical tools. We prove the nonexistence of traveling wave solutions with nonzero velocity. Numerical experiments show that presence of non-linearity and asymmetry of the initial curve results in a net motion which distinguishes it from classical volume preserving curvature motion. This is done by developing an algorithm for efficient numerical resolution of the non-local term in the evolution equation.

  17. Evolution equations for multi-time wavefunctions

    NASA Astrophysics Data System (ADS)

    Petrat, Soren Philipp

    Multi-time wavefunctions are of particular interest in relativistic quantum mechanics. A multi-time wavefunction has separate time-variables for each particle; this makes it a manifestly Lorentz-invariant object. The time-evolution equations are systems of Schrodinger equations; one for each particle's time variable and each with a certain Hamiltonian. We derive conditions under which these systems of equations have a common solution. Also, we derive three main results about concrete multi-time models. First we show that a model proposed by Durr and Tumulka in 2001 is inconsistent. The second result is a consistent model for a constant number of particles with a cutoff pair potential. The third result is a consistent theory for a simple quantum field theoretic model with creation and annihilation of particles. Existence and uniqueness of solutions is proven for both models.

  18. Linear systems of equations solved using mathematical algorithms

    NASA Technical Reports Server (NTRS)

    Bareiss, E. H.

    1968-01-01

    New mathematical algorithm solves linear systems of equations, AX equals B, and preserves the integer properties of the coefficients. The algorithms presented can also be used for the efficient evaluation of determinates and their leading minors.

  19. Out-of-Core Solutions of Complex Sparse Linear Equations

    NASA Technical Reports Server (NTRS)

    Yip, E. L.

    1982-01-01

    ETCLIB is library of subroutines for obtaining out-of-core solutions of complex sparse linear equations. Routines apply to dense and sparse matrices too large to be stored in core. Useful for solving any set of linear equations, but particularly useful in cases where coefficient matrix has no special properties that guarantee convergence with any of interative processes. The only assumption made is that coefficient matrix is not singular.

  20. Exact solution of some linear matrix equations using algebraic methods

    NASA Technical Reports Server (NTRS)

    Djaferis, T. E.; Mitter, S. K.

    1977-01-01

    A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.

  1. On the Fundamental Solution of a Linearized Homogeneous Coagulation Equation

    NASA Astrophysics Data System (ADS)

    Escobedo, Miguel; Velázquez, J. J. L.

    2010-08-01

    We describe the fundamental solution of the equation that is obtained by linearization of the coagulation equation with kernel K( x, y) = ( xy)λ/2, around the steady state f( x) = x -(3+λ)/2 with {λ in (1, 2)} . Detailed estimates on its asymptotics are obtained. Some consequences are deduced for the flux properties of the particles distributions described by such models.

  2. Liapunov functions for non-linear difference equation stability analysis.

    NASA Technical Reports Server (NTRS)

    Park, K. E.; Kinnen, E.

    1972-01-01

    Liapunov functions to determine the stability of non-linear autonomous difference equations can be developed through the use of auxiliary exact difference equations. For this purpose definitions are introduced for the gradient of an implicit function of a discrete variable, a principal sum, a definite sum and an exact difference equation, and a theorem for exactness of a difference form is proved. Examples illustrate the procedure.

  3. Semigroup theory and numerical approximation for equations in linear viscoelasticity

    NASA Technical Reports Server (NTRS)

    Fabiano, R. H.; Ito, K.

    1990-01-01

    A class of abstract integrodifferential equations used to model linear viscoelastic beams is investigated analytically, applying a Hilbert-space approach. The basic equation is rewritten as a Cauchy problem, and its well-posedness is demonstrated. Finite-dimensional subspaces of the state space and an estimate of the state operator are obtained; approximation schemes for the equations are constructed; and the convergence is proved using the Trotter-Kato theorem of linear semigroup theory. The actual convergence behavior of different approximations is demonstrated in numerical computations, and the results are presented in tables.

  4. Rational approximations to solutions of linear differential equations

    PubMed Central

    Chudnovsky, D. V.; Chudnovsky, G. V.

    1983-01-01

    Rational approximations of Padé and Padé type to solutions of differential equations are considered. One of the main results is a theorem stating that a simultaneous approximation to arbitrary solutions of linear differential equations over C(x) cannot be “better” than trivial ones implied by the Dirichlet box principle. This constitutes, in particular, the solution in the linear case of Kolchin's problem that the “Roth's theorem” holds for arbitrary solutions of algebraic differential equations. Complete effective proofs for several valuations are presented based on the Wronskian methods and graded subrings of Picard-Vessiot extensions. PMID:16593357

  5. Evolution equations: Frobenius integrability, conservation laws and travelling waves

    NASA Astrophysics Data System (ADS)

    Prince, Geoff; Tehseen, Naghmana

    2015-10-01

    We give new results concerning the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. In particular, we give a powerful result which explains the extraordinary integrability of some of these equations. We also discuss ‘local’ conservations laws for evolution equations in general and demonstrate all the results for the Korteweg-de Vries equation.

  6. Problems with the linear q-Fokker Planck equation

    NASA Astrophysics Data System (ADS)

    Yano, Ryosuke

    2015-05-01

    In this letter, we discuss the linear q-Fokker Planck equation, whose solution follows Tsallis distribution, from the viewpoint of kinetic theory. Using normal definitions of moments, we can expand the distribution function with infinite moments for 0 ⩽ q < 1, whereas we cannot expand the distribution function with infinite moments for 1 < q owing to emergences of characteristic points in moments. From Grad's 13 moment equations for the linear q-Fokker Planck equation, the dissipation rate of the heat flux via the linear q-Fokker Planck equation diverges at 0 ⩽ q < 2/3. In other words, the thermal conductivity, which defines the heat flux with the spatial gradient of the temperature and the thermal conductivity, which defines the heat flux with the spacial gradient of the density, jumps to zero at q = 2/3, discontinuously.

  7. Evolution equation for geometric quantum correlation measures

    NASA Astrophysics Data System (ADS)

    Hu, Ming-Liang; Fan, Heng

    2015-05-01

    A simple relation is established for the evolution equation of quantum-information-processing protocols such as quantum teleportation, remote state preparation, Bell-inequality violation, and particularly the dynamics of geometric quantum correlation measures. This relation shows that when the system traverses the local quantum channel, various figures of merit of the quantum correlations for different protocols demonstrate a factorization decay behavior for dynamics. We identified the family of quantum states for different kinds of quantum channels under the action of which the relation holds. This relation simplifies the assessment of many quantum tasks.

  8. Solutions to Class of Linear and Nonlinear Fractional Differential Equations

    NASA Astrophysics Data System (ADS)

    Abdel-Salam, Emad A.-B.; Hassan, Gamal F.

    2016-02-01

    In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional KdV equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the (3+1)-space-time fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.

  9. Solutions to Class of Linear and Nonlinear Fractional Differential Equations

    NASA Astrophysics Data System (ADS)

    Emad A-B., Abdel-Salam; Gamal, F. Hassan

    2016-02-01

    In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional KdV equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the (3+1)-space-time fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag-Leffler function methods. The obtained results recover the well-know solutions when α = 1.

  10. Monitoring derivation of the quantum linear Boltzmann equation

    SciTech Connect

    Hornberger, Klaus; Vacchini, Bassano

    2008-02-15

    We show how the effective equation of motion for a distinguished quantum particle in an ideal gas environment can be obtained by means of the monitoring approach introduced by Hornberger [EPL 77, 50007 (2007)]. The resulting Lindblad master equation accounts for the quantum effects of the scattering dynamics in a nonperturbative fashion and it describes decoherence and dissipation in a unified framework. It incorporates various established equations as limiting cases and reduces to the classical linear Boltzmann equation once the state is diagonal in momentum.

  11. An integrable shallow water equation with linear and nonlinear dispersion.

    PubMed

    Dullin, H R; Gottwald, G A; Holm, D D

    2001-11-01

    We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still preserves complete integrability via the inverse scattering transform method. Its traveling wave solutions contain both the KdV solitons and the CH peakons as limiting cases. PMID:11690414

  12. New Equating Methods and Their Relationships with Levine Observed Score Linear Equating under the Kernel Equating Framework

    ERIC Educational Resources Information Center

    Chen, Haiwen; Holland, Paul

    2010-01-01

    In this paper, we develop a new curvilinear equating for the nonequivalent groups with anchor test (NEAT) design under the assumption of the classical test theory model, that we name curvilinear Levine observed score equating. In fact, by applying both the kernel equating framework and the mean preserving linear transformation of…

  13. Dark soliton solutions of (N+1)-dimensional nonlinear evolution equations

    NASA Astrophysics Data System (ADS)

    Demiray, Seyma Tuluce; Bulut, Hasan

    2016-06-01

    In this study, we investigate exact solutions of (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation by using generalized Kudryashov method. (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation can be returned to nonlinear ordinary differential equation by suitable transformation. Then, generalized Kudryashov method has been used to seek exact solutions of the (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation. Also, we obtain dark soliton solutions for these (N+1)-dimensional nonlinear evolution equations. Finally, we denote that this method can be applied to solve other nonlinear evolution equations.

  14. Construction of the wave operator for non-linear dispersive equations

    NASA Astrophysics Data System (ADS)

    Tsuruta, Kai Erik

    In this thesis, we will study non-linear dispersive equations. The primary focus will be on the construction of the positive-time wave operator for such equations. The positive-time wave operator problem arises in the study of the asymptotics of a partial differential equation. It is a map from a space of initial data X into itself, and is loosely defined as follows: Suppose that for a solution ψlin to the dispersive equation with no non-linearity and initial data ψ +, there exists a unique solution ψ to the non-linear equation with initial data ψ0 such that ψ behaves as ψ lin as t → infinity. Then the wave operator is the map W+ that takes ψ + to ψ0. By its definition, W+ is injective. An important additional question is whether or not the map is also surjective. If so, then every non-linear solution emanating from X behaves, in some sense, linearly as it evolves (this is known as asymptotic completeness). Thus, there is some justification for treating these solutions as their much simpler linear counterparts. The main results presented in this thesis revolve around the construction of the wave operator(s) at critical non-linearities. We will study the "semi-relativistic" Schrodinger equation as well as the Klein-Gordon-Schrodinger system on R2 . In both cases, we will impose fairly general quadratic non-linearities for which conservation laws cannot be relied upon. These non-linearities fall below the scaling required to employ such tools as the Strichartz estimates. We instead adapt the "first iteration method" of Jang, Li, and Zhang to our setting which depends crucially on the critical decay of the non-linear interaction of the linear evolution. To see the critical decay in our problem, careful analysis is needed to treat the regime where one has spatial and/or time resonance.

  15. An Object Oriented, Finite Element Framework for Linear Wave Equations

    SciTech Connect

    Koning, J M

    2004-08-12

    This dissertation documents an object oriented framework which can be used to solve any linear wave equation. The linear wave equations are expressed in the differential forms language. This differential forms expression allows a strict discrete interpretation of the system. The framework is implemented using the Galerkin Finite Element Method to define the discrete differential forms and operators. Finite element basis functions including standard scalar Nodal and vector Nedelec basis functions are used to implement the discrete differential forms resulting in a mixed finite element system. Discretizations of scalar and vector wave equations in the time and frequency domains will be demonstrated in both differential forms and vector calculi. This framework conserves energy, maintains physical continuity, is valid on unstructured grids, conditionally stable and second order accurate. Examples including linear electrodynamics, acoustics, elasticity and magnetohydrodynamics are demonstrated.

  16. Improved local linearization algorithm for solving the quaternion equations

    NASA Technical Reports Server (NTRS)

    Yen, K.; Cook, G.

    1980-01-01

    The objective of this paper is to develop a new and more accurate local linearization algorithm for numerically solving sets of linear time-varying differential equations. Of special interest is the application of this algorithm to the quaternion rate equations. The results are compared, both analytically and experimentally, with previous results using local linearization methods. The new algorithm requires approximately one-third more calculations per step than the previously developed local linearization algorithm; however, this disadvantage could be reduced by using parallel implementation. For some cases the new algorithm yields significant improvement in accuracy, even with an enlarged sampling interval. The reverse is true in other cases. The errors depend on the values of angular velocity, angular acceleration, and integration step size. One important result is that for the worst case the new algorithm can guarantee eigenvalues nearer the region of stability than can the previously developed algorithm.

  17. Solitary pulses in linearly coupled Ginzburg-Landau equations.

    PubMed

    Malomed, Boris A

    2007-09-01

    This article presents a brief review of dynamical models based on systems of linearly coupled complex Ginzburg-Landau (CGL) equations. In the simplest case, the system features linear gain, cubic nonlinearity (possibly combined with cubic loss), and group-velocity dispersion (GVD) in one equation, while the other equation is linear, featuring only intrinsic linear loss. The system models a dual-core fiber laser, with a parallel-coupled active core and an additional stabilizing passive (lossy) one. The model gives rise to exact analytical solutions for stationary solitary pulses (SPs). The article presents basic results concerning stability of the SPs; interactions between pulses are also considered, as are dark solitons (holes). In the case of the anomalous GVD, an unstable stationary SP may transform itself, via the Hopf bifurcation, into a stable localized breather. Various generalizations of the basic system are briefly reviewed too, including a model with quadratic (second-harmonic-generating) nonlinearity and a recently introduced model of a different but related type, based on linearly coupled CGL equations with cubic-quintic nonlinearity. The latter system features spontaneous symmetry breaking of stationary SPs, and also the formation of stable breathers. PMID:17903024

  18. Monte Carlo studies of multiwavelength pyrometry using linearized equations

    NASA Astrophysics Data System (ADS)

    Gathers, G. R.

    1992-03-01

    Multiwavelength pyrometry has been advertised as giving significant improvement in precision by overdetermining the solution with extra wavelengths and using least squares methods. Hiernaut et al. [1] have described a six-wavelength pyrometer for measurements in the range 2000 to 5000 K. They use the Wien approximation and model the logarithm of the emissivity as a linear function of wavelength in order to produce linear equations. The present work examines the measurement errors associated with their technique.

  19. Linear Equating for the NEAT Design: Parameter Substitution Models and Chained Linear Relationship Models

    ERIC Educational Resources Information Center

    Kane, Michael T.; Mroch, Andrew A.; Suh, Youngsuk; Ripkey, Douglas R.

    2009-01-01

    This paper analyzes five linear equating models for the "nonequivalent groups with anchor test" (NEAT) design with internal anchors (i.e., the anchor test is part of the full test). The analysis employs a two-dimensional framework. The first dimension contrasts two general approaches to developing the equating relationship. Under a "parameter…

  20. A Comparison of Chained Linear and Poststratification Linear Equating under Different Testing Conditions

    ERIC Educational Resources Information Center

    Puhan, Gautam

    2010-01-01

    In this study I compared results of chained linear, Tucker, and Levine-observed score equatings under conditions where the new and old forms samples were similar in ability and also when they were different in ability. The length of the anchor test was also varied to examine its effect on the three different equating methods. The three equating…

  1. Exact solution of some linear matrix equations using algebraic methods

    NASA Technical Reports Server (NTRS)

    Djaferis, T. E.; Mitter, S. K.

    1979-01-01

    Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.

  2. Time-harmonic Maxwell equations with asymptotically linear polarization

    NASA Astrophysics Data System (ADS)

    Qin, Dongdong; Tang, Xianhua

    2016-06-01

    This paper is concerned with the following time-harmonic semilinear Maxwell equation: nabla× (nabla× u)+λ u=f(x,u), &in Ω ν × u=0, &on partialΩ, where {Ωsubset {R}3} is a bounded, convex domain and {ν : partial Ωto {R}3} is the exterior normal. Motivated by recent work of Bartsch and Mederski and based on some observations and new techniques, we study above equation by developing the generalized Nehari manifold method. Particularly, existence of ground-state solutions of Nehari-Pankov type for the equation is established with asymptotically linear nonlinearity.

  3. Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE's. [ordinary differential equations

    NASA Technical Reports Server (NTRS)

    Geddes, K. O.

    1977-01-01

    If a linear ordinary differential equation with polynomial coefficients is converted into integrated form then the formal substitution of a Chebyshev series leads to recurrence equations defining the Chebyshev coefficients of the solution function. An explicit formula is presented for the polynomial coefficients of the integrated form in terms of the polynomial coefficients of the differential form. The symmetries arising from multiplication and integration of Chebyshev polynomials are exploited in deriving a general recurrence equation from which can be derived all of the linear equations defining the Chebyshev coefficients. Procedures for deriving the general recurrence equation are specified in a precise algorithmic notation suitable for translation into any of the languages for symbolic computation. The method is algebraic and it can therefore be applied to differential equations containing indeterminates.

  4. Regularized 13 moment equations for hard sphere molecules: Linear bulk equations

    NASA Astrophysics Data System (ADS)

    Struchtrup, Henning; Torrilhon, Manuel

    2013-05-01

    The regularized 13 moment equations of rarefied gas dynamics are derived for a monatomic hard sphere gas in the linear regime. The equations are based on an extended Grad-type moment system, which is systematically reduced by means of the Order of Magnitude Method [H. Struchtrup, "Stable transport equations for rarefied gases at high orders in the Knudsen number," Phys. Fluids 16(11), 3921-3934 (2004)], 10.1063/1.1782751. Chapman-Enskog expansion of the final equations yields the linear Burnett and super-Burnett equations. While the Burnett coefficients agree with literature values, this seems to be the first time that super-Burnett coefficients are computed for a hard sphere gas. As a first test of the equations the dispersion and damping of sound waves is considered.

  5. Effects of Classroom Instruction on Student Performance on, and Understanding of, Linear Equations and Linear Inequalities

    ERIC Educational Resources Information Center

    Vaiyavutjamai, Pongchawee; Clements, M. A.

    2006-01-01

    Two-hundred and thirty-one students in 6 Grade 9 classes in 2 secondary schools in Thailand attempted 54 pencil-and-paper tasks related to linear equations and linear inequalities immediately before and after they participated in 13 lessons on those topics. Students' written responses, and transcripts of pre- and postteaching interviews with 18…

  6. Linear System of Equations, Matrix Inversion, and Linear Programming Using MS Excel

    ERIC Educational Resources Information Center

    El-Gebeily, M.; Yushau, B.

    2008-01-01

    In this note, we demonstrate with illustrations two different ways that MS Excel can be used to solve Linear Systems of Equation, Linear Programming Problems, and Matrix Inversion Problems. The advantage of using MS Excel is its availability and transparency (the user is responsible for most of the details of how a problem is solved). Further, we…

  7. Constructive Development of the Solutions of Linear Equations in Introductory Ordinary Differential Equations

    ERIC Educational Resources Information Center

    Mallet, D. G.; McCue, S. W.

    2009-01-01

    The solution of linear ordinary differential equations (ODEs) is commonly taught in first-year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognizing what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to…

  8. Teaching Linear Equations: Case Studies from Finland, Flanders and Hungary

    ERIC Educational Resources Information Center

    Andrews, Paul; Sayers, Judy

    2012-01-01

    In this paper we compare how three teachers, one from each of Finland, Flanders and Hungary, introduce linear equations to grade 8 students. Five successive lessons were videotaped and analysed qualitatively to determine how teachers, each of whom was defined against local criteria as effective, addressed various literature-derived…

  9. Effective vs. Efficient: Teaching Methods of Solving Linear Equations

    ERIC Educational Resources Information Center

    Ivey, Kathy M. C.

    2003-01-01

    The choice of teaching an effective method--one that most students can master--or an efficient method--one that takes the fewest steps--occurs daily in Algebra I classrooms. This decision may not be made in the abstract, however, but rather in a ready-to-hand mode. This study examines how teachers solve linear equations when the purpose is…

  10. Insights into the School Mathematics Tradition from Solving Linear Equations

    ERIC Educational Resources Information Center

    Buchbinder, Orly; Chazan, Daniel; Fleming, Elizabeth

    2015-01-01

    In this article, we explore how the solving of linear equations is represented in English­-language algebra text books from the early nineteenth century when schooling was becoming institutionalized, and then survey contemporary teachers. In the text books, we identify the increasing presence of a prescribed order of steps (a canonical method) for…

  11. Pseudospectral Methods of Solution of the Linear and Linearized Boltzmann Equations; Transport and Relaxation

    NASA Astrophysics Data System (ADS)

    Shizgal, Bernie D.

    2011-05-01

    The study of the solution of the linearized Boltzmann equation has a very long history arising from the classic work by Chapman and Cowling. For small departures from a Maxwellian, the nonlinear Boltzmann equation can be linearized and the transport coefficients calculated with the Chapman-Enskog approach. This procedure leads to a set of linear integral equations which are generally solved with the expansion of the departure from Maxwellian in Sonine polynomials. The method has been used successfully for many decades to compare experimental transport data in atomic gases with theory generally carried out for realistic atom-atom differential cross sections. There are alternate pseudospectral methods which involve the discretization of the distribution function on a discrete grid. This paper considers a pseudospectral method of solution of the linearized hard sphere Boltzmann equation for the viscosity in a simple gas. The relaxation of a small departure from a Maxwellian is also considered for the linear test particle problem with unit mass ratio which is compared with the relaxation for the linearized one component Boltzmann equation.

  12. Parallel linear equation solvers for finite element computations

    NASA Technical Reports Server (NTRS)

    Ortega, James M.; Poole, Gene; Vaughan, Courtenay; Cleary, Andrew; Averick, Brett

    1989-01-01

    The overall objective of this research is to develop efficient methods for the solution of linear and nonlinear systems of equations on parallel and supercomputers, and to apply these methods to the solution of problems in structural analysis. Attention has been given so far only to linear equations. The methods considered for the solution of the stiffness equation Kx=f have been Choleski factorization and the conjugate gradient iteration with SSOR and Incomplete Choleski preconditioning. More detail on these methods will be given on subsequent slides. These methods have been used to solve for the static displacements for the mast and panel focus problems in conjunction with the CSM testbed system based on NICE/SPAR.

  13. The Linearized Kinetic Equation -- A Functional Analytic Approach

    NASA Astrophysics Data System (ADS)

    Brinkmann, Ralf Peter

    2009-10-01

    Kinetic models of plasma phenomena are difficult to address for two reasons. They i) are given as systems of nonlinear coupled integro-differential equations, and ii) involve generally six-dimensional distribution functions f(r,v,t). In situations which can be addressed in a linear regime, the first difficulty disappears, but the second one still poses considerable practical problems. This contribution presents an abstract approach to linearized kinetic theory which employs the methods of functional analysis. A kinetic electron equation with elastic electron-neutral interaction is studied in the electrostatic approximation. Under certain boundary conditions, a nonlinear functional, the kinetic free energy, exists which has the properties of a Lyapunov functional. In the linear regime, the functional becomes a quadratic form which motivates the definition of a bilinear scalar product, turning the space of all distribution functions into a Hilbert space. The linearized kinetic equation can then be described in terms of dynamical operators with well-defined properties. Abstract solutions can be constructed which have mathematically plausible properties. As an example, the formalism is applied to the example of the multipole resonance probe (MRP). Under the assumption of a Maxwellian background distribution, the kinetic model of that diagnostics device is compared to a previously investigated fluid model.

  14. Nonsingular big bounces and the evolution of linear fluctuations

    NASA Astrophysics Data System (ADS)

    Hwang, Jai-Chan; Noh, Hyerim

    2002-06-01

    We consider the evolutions of linear fluctuations as the background Friedmann world model goes from contracting to expanding phases through smooth and nonsingular bouncing phases. As long as gravity dominates over the pressure gradient in the perturbation equation, the growing mode in the expanding phase is characterized by a conserved amplitude; we call this a C mode. In spherical geometry with a pressureless medium, we show that there exists a special gauge-invariant combination Φ which stays constant throughout the evolution from the big bang to the big crunch, with the same value even after the bounce: it characterizes the coefficient of the C mode. We show this result by using a bounce model where the pressure gradient term is negligible during the bounce; this requires the additional presence of exotic matter. In such a bounce, even in more general situations for the equation of state before and after the bounce, the C mode in the expanding phase is affected only by the C mode in the contracting phase; thus the growing mode in the contracting phase decays away as the world model enters the expanding phase. When the background curvature plays a significant role during the bounce, the pressure gradient term becomes important and we cannot trace the C mode in the expanding phase to the one before the bounce. In such situations, perturbations in a fluid bounce model show exponential instability, whereas perturbations in a scalar field bounce model show oscillatory behavior.

  15. Novel algorithm of large-scale simultaneous linear equations.

    PubMed

    Fujiwara, T; Hoshi, T; Yamamoto, S; Sogabe, T; Zhang, S-L

    2010-02-24

    We review our recently developed methods of solving large-scale simultaneous linear equations and applications to electronic structure calculations both in one-electron theory and many-electron theory. This is the shifted COCG (conjugate orthogonal conjugate gradient) method based on the Krylov subspace, and the most important issue for applications is the shift equation and the seed switching method, which greatly reduce the computational cost. The applications to nano-scale Si crystals and the double orbital extended Hubbard model are presented. PMID:21386384

  16. An analytically solvable eigenvalue problem for the linear elasticity equations.

    SciTech Connect

    Day, David Minot; Romero, Louis Anthony

    2004-07-01

    Analytic solutions are useful for code verification. Structural vibration codes approximate solutions to the eigenvalue problem for the linear elasticity equations (Navier's equations). Unfortunately the verification method of 'manufactured solutions' does not apply to vibration problems. Verification books (for example [2]) tabulate a few of the lowest modes, but are not useful for computations of large numbers of modes. A closed form solution is presented here for all the eigenvalues and eigenfunctions for a cuboid solid with isotropic material properties. The boundary conditions correspond physically to a greased wall.

  17. On identifying transfer functions and state equations for linear systems.

    NASA Technical Reports Server (NTRS)

    Shieh, L. S.; Chen, C. F.; Huang, C. J.

    1972-01-01

    Two methods are established for identifying constant-coefficient, C to the 2n power type of noise-free linear systems if the time response data of the input-output or of all states are known. 2n response data are required to identify an nth-order transfer function or state equation for an unknown linear system. The order of the unknown system can be identified by checking a sequence of determinants. The Z transform and its inversion are mainly used.

  18. What happens to linear properties as we move from the Klein-Gordon equation to the sine-Gordon equation

    SciTech Connect

    Kovalyov, Mikhail

    2010-06-15

    In this article the sets of solutions of the sine-Gordon equation and its linearization the Klein-Gordon equation are discussed and compared. It is shown that the set of solutions of the sine-Gordon equation possesses a richer structure which partly disappears during linearization. Just like the solutions of the Klein-Gordon equation satisfy the linear superposition principle, the solutions of the sine-Gordon equation satisfy a nonlinear superposition principle.

  19. Parallel iterative methods for sparse linear and nonlinear equations

    NASA Technical Reports Server (NTRS)

    Saad, Youcef

    1989-01-01

    As three-dimensional models are gaining importance, iterative methods will become almost mandatory. Among these, preconditioned Krylov subspace methods have been viewed as the most efficient and reliable, when solving linear as well as nonlinear systems of equations. There has been several different approaches taken to adapt iterative methods for supercomputers. Some of these approaches are discussed and the methods that deal more specifically with general unstructured sparse matrices, such as those arising from finite element methods, are emphasized.

  20. Approximate solutions for non-linear iterative fractional differential equations

    NASA Astrophysics Data System (ADS)

    Damag, Faten H.; Kiliçman, Adem; Ibrahim, Rabha W.

    2016-06-01

    This paper establishes approximate solution for non-linear iterative fractional differential equations: d/γv (s ) d sγ =ℵ (s ,v ,v (v )), where γ ∈ (0, 1], s ∈ I := [0, 1]. Our method is based on some convergence tools for analytic solution in a connected region. We show that the suggested solution is unique and convergent by some well known geometric functions.

  1. Affine Vertex Operator Algebras and Modular Linear Differential Equations

    NASA Astrophysics Data System (ADS)

    Arike, Yusuke; Kaneko, Masanobu; Nagatomo, Kiyokazu; Sakai, Yuichi

    2016-05-01

    In this paper, we list all affine vertex operator algebras of positive integral levels whose dimensions of spaces of characters are at most 5 and show that a basis of the space of characters of each affine vertex operator algebra in the list gives a fundamental system of solutions of a modular linear differential equation. Further, we determine the dimensions of the spaces of characters of affine vertex operator algebras whose numbers of inequivalent simple modules are not exceeding 20.

  2. An interpretation and solution of ill-conditioned linear equations

    NASA Technical Reports Server (NTRS)

    Ojalvo, I. U.; Ting, T.

    1989-01-01

    Data insufficiency, poorly conditioned matrices and singularities in equations occur regularly in complex optimization, correlation, and interdisciplinary model studies. This work concerns itself with two methods of obtaining certain physically realistic solutions to ill-conditioned or singular algebraic systems of linear equations arising from such studies. Two efficient computational solution procedures that generally lead to locally unique solutions are presented when there is insufficient data to completely define the model, or a least-squares error formulation of this system results in an ill-conditioned system of equations. If it is assumed that a reasonable estimate of the uncertain data is available in both cases cited above, then we shall show how to obtain realistic solutions efficiently, in spite of the insufficiency of independent data. The proposed methods of solution are more efficient than singular-value decomposition for dealing with such systems, since they do not require solutions for all the non-zero eigenvalues of the coefficient matrix.

  3. Runge-Kutta Methods for Linear Ordinary Differential Equations

    NASA Technical Reports Server (NTRS)

    Zingg, David W.; Chisholm, Todd T.

    1997-01-01

    Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODES) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of the Runge-Kutta method must satisfy. This freedom is used to develop methods which are more efficient than conventional Runge-Kutta methods. A fourth-order method is presented which uses only two memory locations per dependent variable, while the classical fourth-order Runge-Kutta method uses three. This method is an excellent choice for simulations of linear wave phenomena if memory is a primary concern. In addition, fifth- and sixth-order methods are presented which require five and six stages, respectively, one fewer than their conventional counterparts, and are therefore more efficient. These methods are an excellent option for use with high-order spatial discretizations.

  4. Nonlinear Riccati equations as a unifying link between linear quantum mechanics and other fields of physics

    NASA Astrophysics Data System (ADS)

    Schuch, Dieter

    2014-04-01

    Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. I claim that linearity in quantum mechanics is not as essential as it apparently seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated and compared with similar space-dependent Riccati equations in supersymmetric quantum mechanics. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, like statistical thermodynamics and cosmology.

  5. Derivation of the Linear Landau Equation and Linear Boltzmann Equation from the Lorentz Model with Magnetic Field

    NASA Astrophysics Data System (ADS)

    Marcozzi, M.; Nota, A.

    2016-03-01

    We consider a test particle moving in a random distribution of obstacles in the plane, under the action of a uniform magnetic field, orthogonal to the plane. We show that, in a weak coupling limit, the particle distribution behaves according to the linear Landau equation with a magnetic transport term. Moreover, we show that, in a low density regime, when each obstacle generates an inverse power law potential, the particle distribution behaves according to the linear Boltzmann equation with a magnetic transport term. We provide an explicit control of the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. We compare these results with those ones obtained for a system of hard disks in Bobylev et al. (Phys Rev Lett 75:2, 1995), which show instead that the memory effects are not negligible in the Boltzmann-Grad limit.

  6. Obtaining General Relativity's N-body non-linear Lagrangian from iterative, linear algebraic scaling equations

    NASA Astrophysics Data System (ADS)

    Nordtvedt, K.

    2015-11-01

    A local system of bodies in General Relativity whose exterior metric field asymptotically approaches the Minkowski metric effaces any effects of the matter distribution exterior to its Minkowski boundary condition. To enforce to all orders this property of gravity which appears to hold in nature, a method using linear algebraic scaling equations is developed which generates by an iterative process an N-body Lagrangian expansion for gravity's motion-independent potentials which fulfills exterior effacement along with needed metric potential expansions. Then additional properties of gravity - interior effacement and Lorentz time dilation and spatial contraction - produce additional iterative, linear algebraic equations for obtaining the full non-linear and motion-dependent N-body gravity Lagrangian potentials as well.

  7. Charged anisotropic matter with linear or nonlinear equation of state

    SciTech Connect

    Varela, Victor; Rahaman, Farook; Ray, Saibal; Chakraborty, Koushik; Kalam, Mehedi

    2010-08-15

    Ivanov pointed out substantial analytical difficulties associated with self-gravitating, static, isotropic fluid spheres when pressure explicitly depends on matter density. Simplifications achieved with the introduction of electric charge were noticed as well. We deal with self-gravitating, charged, anisotropic fluids and get even more flexibility in solving the Einstein-Maxwell equations. In order to discuss analytical solutions we extend Krori and Barua's method to include pressure anisotropy and linear or nonlinear equations of state. The field equations are reduced to a system of three algebraic equations for the anisotropic pressures as well as matter and electrostatic energy densities. Attention is paid to compact sources characterized by positive matter density and positive radial pressure. Arising solutions satisfy the energy conditions of general relativity. Spheres with vanishing net charge contain fluid elements with unbounded proper charge density located at the fluid-vacuum interface. Notably the electric force acting on these fluid elements is finite, although the acting electric field is zero. Net charges can be huge (10{sup 19}C) and maximum electric field intensities are very large (10{sup 23}-10{sup 24} statvolt/cm) even in the case of zero net charge. Inward-directed fluid forces caused by pressure anisotropy may allow equilibrium configurations with larger net charges and electric field intensities than those found in studies of charged isotropic fluids. Links of these results with charged strange quark stars as well as models of dark matter including massive charged particles are highlighted. The van der Waals equation of state leading to matter densities constrained by cubic polynomial equations is briefly considered. The fundamental question of stability is left open.

  8. Exact traveling wave solutions for system of nonlinear evolution equations.

    PubMed

    Khan, Kamruzzaman; Akbar, M Ali; Arnous, Ahmed H

    2016-01-01

    In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis. PMID:27347461

  9. Chaotic dynamics and diffusion in a piecewise linear equation

    SciTech Connect

    Shahrear, Pabel; Glass, Leon; Edwards, Rod

    2015-03-15

    Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.

  10. Chaotic dynamics and diffusion in a piecewise linear equation

    NASA Astrophysics Data System (ADS)

    Shahrear, Pabel; Glass, Leon; Edwards, Rod

    2015-03-01

    Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.

  11. Some approximations in the linear dynamic equations of thin cylinders

    NASA Technical Reports Server (NTRS)

    El-Raheb, M.; Babcock, C. D., Jr.

    1981-01-01

    Theoretical analysis is performed on the linear dynamic equations of thin cylindrical shells to find the error committed by making the Donnell assumption and the neglect of in-plane inertia. At first, the effect of these approximations is studied on a shell with classical simply supported boundary condition. The same approximations are then investigated for other boundary conditions from a consistent approximate solution of the eigenvalue problem. The Donnell assumption is valid at frequencies high compared with the ring frequencies, for finite length thin shells. The error in the eigenfrequencies from omitting tangential inertia is appreciable for modes with large circumferential and axial wavelengths, independent of shell thickness and boundary conditions.

  12. A Solution to the Fundamental Linear Fractional Order Differential Equation

    NASA Technical Reports Server (NTRS)

    Hartley, Tom T.; Lorenzo, Carl F.

    1998-01-01

    This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.

  13. The Linear KdV Equation with an Interface

    NASA Astrophysics Data System (ADS)

    Deconinck, Bernard; Sheils, Natalie E.; Smith, David A.

    2016-07-01

    The interface problem for the linear Korteweg-de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of compatibility conditions at the boundary are imposed. We provide an explicit characterization of sufficient interface conditions for the construction of a solution using Fokas's Unified Transform Method. The problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives.

  14. Invariant tori for a class of nonlinear evolution equations

    SciTech Connect

    Kolesov, A Yu; Rozov, N Kh

    2013-06-30

    The paper looks at quite a wide class of nonlinear evolution equations in a Banach space, including the typical boundary value problems for the main wave equations in mathematical physics (the telegraph equation, the equation of a vibrating beam, various equations from the elastic stability and so on). For this class of equations a unified approach to the bifurcation of invariant tori of arbitrary finite dimension is put forward. Namely, the problem of the birth of such tori from the zero equilibrium is investigated under the assumption that in the stability problem for this equilibrium the situation arises close to an infinite-dimensional degeneracy. Bibliography: 28 titles.

  15. The condition of regular degeneration for singularly perturbed systems of linear differential-difference equations.

    NASA Technical Reports Server (NTRS)

    Cooke, K. L.; Meyer, K. R.

    1966-01-01

    Extension of problem of singular perturbation for linear scalar constant coefficient differential- difference equation with single retardation to several retardations, noting degenerate equation solution

  16. The relativistic equations of stellar structure and evolution

    NASA Technical Reports Server (NTRS)

    Thorne, K. S.

    1977-01-01

    The general-relativistic equations of stellar structure and evolution are reformulated in a notation which makes easy contact with Newtonian theory. Also, a general-relativistic version of the mixing-length formalism for convection is presented.

  17. Standard Errors of Equating for the Percentile Rank-Based Equipercentile Equating with Log-Linear Presmoothing

    ERIC Educational Resources Information Center

    Wang, Tianyou

    2009-01-01

    Holland and colleagues derived a formula for analytical standard error of equating using the delta-method for the kernel equating method. Extending their derivation, this article derives an analytical standard error of equating procedure for the conventional percentile rank-based equipercentile equating with log-linear smoothing. This procedure is…

  18. Unpacking the Complexity of Linear Equations from a Cognitive Load Theory Perspective

    ERIC Educational Resources Information Center

    Ngu, Bing Hiong; Phan, Huy P.

    2016-01-01

    The degree of element interactivity determines the complexity and therefore the intrinsic cognitive load of linear equations. The unpacking of linear equations at the level of operational and relational lines allows the classification of linear equations in a hierarchical level of complexity. Mapping similar operational and relational lines across…

  19. Landscape evolution models: A review of their fundamental equations

    NASA Astrophysics Data System (ADS)

    Chen, Alex; Darbon, Jérôme; Morel, Jean-Michel

    2014-08-01

    This paper reviews the main physical laws proposed in landscape evolution models (LEMs). It discusses first the main partial differential equations involved in these models and their variants. These equations govern water runoff, stream incision, regolith-bedrock interaction, hillslope evolution, and sedimentation. A synthesis of existing LEMs is proposed. It proposes three models with growing complexity and with a growing number of components: two-equation models with only two components, governing water and bedrock evolution; three-equation models with three components where water, bedrock, and sediment interact; and finally models with four equations and four interacting components, namely water, bedrock, suspended sediment, and regolith. This analysis is not a mere compilation of existing LEMs. It attempts at giving the simplest and most general physically consistent set of equations, coping with all requirements stated in LEMs and LEM software. Three issues are in particular addressed and hopefully resolved. The first one is a correct formulation of the water transport equation down slopes. A general formulation for this equation is proposed, coping not only with the simplest form computing the drainage area but also with a sound energy dissipation argument associated with the Saint-Venant shallow water equations. The second issue arises from the coexistence of two competing modes, namely the detachment-limited erosion mode on hillslopes, and the transport-limited sediment transport on river beds. The third issue (linked to the second) is the fact that no conservation law is available for material in these two modes. A simple solution proposed to resolve these issues is the introduction, as suggested by several authors, of an additional variable for suspended sediment load in water. With only three variables and three equations, the above-mentioned contradictions seem to be eliminated. Several numerical experiments on real digital elevation models (DEMs

  20. On the solutions of some linear complex quaternionic equations.

    PubMed

    Bolat, Cennet; İpek, Ahmet

    2014-01-01

    Some complex quaternionic equations in the type AX - XB = C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained. PMID:25101318

  1. Transport equations for linear surface waves with random underlying flows

    NASA Astrophysics Data System (ADS)

    Bal, Guillaume; Chou, Tom

    1999-11-01

    We define the Wigner distribution and use it to develop equations for linear surface capillary-gravity wave propagation in the transport regime. The energy density a(r, k) contained in waves propagating with wavevector k at field point r is given by dota(r,k) + nabla_k[U_⊥(r,z=0) \\cdotk + Ω(k)]\\cdotnabla_ra [13pt] \\: hspace1in - (nabla_r\\cdotU_⊥)a - nabla_r(k\\cdotU_⊥)\\cdotnabla_ka = Σ(δU^2) where U_⊥(r, z=0) is a slowly varying surface current, and Ω(k) = √(k^3+k)tanh kh is the free capillary-gravity dispersion relation. Note that nabla_r\\cdotU_⊥(r,z=0) neq 0, and that the surface currents exchange energy density with the propagating waves. When an additional weak random current √\\varepsilon δU(r/\\varepsilon) varying on the scale of k-1 is included, we find an additional scattering term Σ(δU^2) as a function of correlations in δU. Our results can be applied to the study of surface wave energy transport over a turbulent ocean.

  2. A Unified Approach to Linear Equating for the Nonequivalent Groups Design

    ERIC Educational Resources Information Center

    von Davier, Alina A.; Kong, Nan

    2005-01-01

    This article describes a new, unified framework for linear equating in a non-equivalent groups anchor test (NEAT) design. The authors focus on three methods for linear equating in the NEAT design--Tucker, Levine observed-score, and chain--and develop a common parameterization that shows that each particular equating method is a special case of the…

  3. Linear Spatial Evolution Formulation of Two-Dimensional Waves on Liquid Films Under Evaporating/Isothermal/Condensing Conditions

    SciTech Connect

    Xuemin Ye; Chunxi Li; Weiping Yan

    2002-07-01

    The linear spatial evolution formulation of the two-dimensional waves of the evaporating or isothermal or condensing liquid films falling down an inclined wall is established for the film thickness with the collocation method based on the boundary layer theory and complete boundary conditions. The evolution equation indicates that there are two different modes of waves in spatial evolution. And the flow stability is highly dependent on the evaporation or condensation, thermo-capillarity, surface tension, inclination angle and Reynolds number. (authors)

  4. New Results on the Linear Equating Methods for the Non-Equivalent-Groups Design

    ERIC Educational Resources Information Center

    von Davier, Alina A.

    2008-01-01

    The two most common observed-score equating functions are the linear and equipercentile functions. These are often seen as different methods, but von Davier, Holland, and Thayer showed that any equipercentile equating function can be decomposed into linear and nonlinear parts. They emphasized the dominant role of the linear part of the nonlinear…

  5. A Procedure to Construct Conservation Laws of Nonlinear Evolution Equations

    NASA Astrophysics Data System (ADS)

    Yaşar, Emrullah; San, Sait

    2016-05-01

    In this article, we established abundant local conservation laws to some nonlinear evolution equations by a new combined approach, which is a union of multiplier and Ibragimov's new conservation theorem method. One can conclude that the solutions of the adjoint equations corresponding to the new conservation theorem can be obtained via multiplier functions. Many new families of conservation laws of the Pochammer-Chree (PC) equation and the Kaup-Boussinesq type of coupled KdV system are successfully obtained. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations. The conserved vectors obtained here can be important for the explanation of some practical physical problems, reductions, and solutions of the underlying equations.

  6. Fokker-Planck equation of Schramm-Loewner evolution.

    PubMed

    Najafi, M N

    2015-08-01

    In this paper we statistically analyze the Fokker-Planck (FP) equation of Schramm-Loewner evolution (SLE) and its variant SLE(κ,ρc). After exploring the derivation and the properties of the Langevin equation of the tip of the SLE trace, we obtain the long- and short-time behaviors of the chordal SLE traces. We analyze the solutions of the FP and the corresponding Langevin equations and connect it to the conformal field theory (CFT) and present some exact results. We find the perturbative FP equation of the SLE(κ,ρc) traces and show that it is related to the higher-order correlation functions. Using the Langevin equation we find the long-time behaviors in this case. The CFT correspondence of this case is established and some exact results are presented. PMID:26382350

  7. Wave packet dynamics for a non-linear Schrödinger equation describing continuous position measurements

    NASA Astrophysics Data System (ADS)

    Zander, C.; Plastino, A. R.; Díaz-Alonso, J.

    2015-11-01

    We investigate time-dependent solutions for a non-linear Schrödinger equation recently proposed by Nassar and Miret-Artés (NM) to describe the continuous measurement of the position of a quantum particle (Nassar, 2013; Nassar and Miret-Artés, 2013). Here we extend these previous studies in two different directions. On the one hand, we incorporate a potential energy term in the NM equation and explore the corresponding wave packet dynamics, while in the previous works the analysis was restricted to the free-particle case. On the other hand, we investigate time-dependent solutions while previous studies focused on a stationary one. We obtain exact wave packet solutions for linear and quadratic potentials, and approximate solutions for the Morse potential. The free-particle case is also revisited from a time-dependent point of view. Our analysis of time-dependent solutions allows us to determine the stability properties of the stationary solution considered in Nassar (2013), Nassar and Miret-Artés (2013). On the basis of these results we reconsider the Bohmian approach to the NM equation, taking into account the fact that the evolution equation for the probability density ρ =| ψ | 2 is not a continuity equation. We show that the effect of the source term appearing in the evolution equation for ρ has to be explicitly taken into account when interpreting the NM equation from a Bohmian point of view.

  8. The relativistic equations of stellar structure and evolution

    NASA Technical Reports Server (NTRS)

    Thorne, K. S.

    1975-01-01

    The general relativistic equations of stellar structure and evolution are reformulated in a notation which makes easy contact with Newtonian theory. A general relativistic version of the mixing-length formalism for convection is presented. It is argued that in work on spherical systems, general relativity theorists have identified the wrong quantity as total mass-energy inside radius r.

  9. Nonlinear evolution of two fast-particle-driven modes near the linear stability threshold

    NASA Astrophysics Data System (ADS)

    Zaleśny, Jarosław; Galant, Grzegorz; Lisak, Mietek; Marczyński, Sławomir; Berczyński, Paweł; Gałkowski, Andrzej; Berczyński, Stefan

    2011-06-01

    A system of two coupled integro-differential equations is derived and solved for the non-linear evolution of two waves excited by the resonant interaction with fast ions just above the linear instability threshold. The effects of a resonant particle source and classical relaxation processes represented by the Krook, diffusion, and dynamical friction collision operators are included in the model, which exhibits different nonlinear evolution regimes, mainly depending on the type of relaxation process that restores the unstable distribution function of fast ions. When the Krook collisions or diffusion dominate, the wave amplitude evolution is characterized by modulation and saturation. However, when the dynamical friction dominates, the wave amplitude is in the explosive regime. In addition, it is found that the finite separation in the phase velocities of the two modes weakens the interaction strength between the modes.

  10. Symmetry Lie algebras and properties of linear ordinary differential equations with maximal dimension

    NASA Astrophysics Data System (ADS)

    Folly-Gbetoula, Mensah; Kara, A. H.

    2015-04-01

    Solutions of linear iterative equations and expressions for these solutions in terms of the parameters of the first-order source equation are obtained. Based on certain properties of iterative equations, finding the solutions is reduced to finding solutions of the second-order source equation. We have therefore found classes of solutions to the source equations by letting the parameters of the source equation be functions of a specific type such as monomials, functions of exponential and logarithmic type.

  11. Variations in the Solution of Linear First-Order Differential Equations. Classroom Notes

    ERIC Educational Resources Information Center

    Seaman, Brian; Osler, Thomas J.

    2004-01-01

    A special project which can be given to students of ordinary differential equations is described in detail. Students create new differential equations by changing the dependent variable in the familiar linear first-order equation (dv/dx)+p(x)v=q(x) by means of a substitution v=f(y). The student then creates a table of the new equations and…

  12. Equivalence groupoids of classes of linear ordinary differential equations and their group classification

    NASA Astrophysics Data System (ADS)

    Boyko, Vyacheslav M.; Popovych, Roman O.; Shapoval, Nataliya M.

    2015-06-01

    Admissible point transformations of classes of rth order linear ordinary differential equations (in particular, the whole class of such equations and its subclasses of equations in the rational form, the Laguerre-Forsyth form, the first and second Arnold forms) are exhaustively described. Using these results, the group classification of such equations is carried out within the algebraic approach in three different ways.

  13. Encouraging Students to Think Strategically when Learning to Solve Linear Equations

    ERIC Educational Resources Information Center

    Robson, Daphne; Abell, Walt; Boustead, Therese

    2012-01-01

    Students who are preparing to study science and engineering need to understand equation solving but adult students returning to study can find this difficult. In this paper, the design of an online resource, Equations2go, for helping students learn to solve linear equations is investigated. Students learning to solve equations need to consider…

  14. Soliton evolution and radiation loss for the Korteweg--de Vries equation

    SciTech Connect

    Kath, W.L.; Smyth, N.F. Department of Mathematics and Statistics, University of Edinburgh, The King's Buildings, Mayfield Road, Edinburgh EH93JZ, Scotland )

    1995-01-01

    The time-dependent behavior of solutions of the Korteweg--de Vries (KdV) equation for nonsoliton initial conditions is considered. While the exact solution of the KdV equation can in principle be obtained using the inverse scattering transform, in practice it can be extremely difficult to obtain information about a solution's transient evolution by this method. As an alternative, we present here an approximate method for investigating this transient evolution which is based upon the conservation laws associated with the KdV equation. Initial conditions which form one or two solitons are considered, and the resulting approximate evolution is found to be in good agreement with the numerical solution of the KdV equation. Justification for the approximations employed is also given by way of the linearized inverse scattering solution of the KdV equation. In addition, the final soliton state determined from the approximate equations agrees very well with the final state determined from the exact inverse scattering transform solution.

  15. Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

    SciTech Connect

    Caraballo, Tomas Kloeden, Peter E. Schmalfuss, Bjoern

    2004-10-15

    We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of anon-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semilinear stochastic partial differential equations with Lipschitz continuous non-linearities.

  16. High resolution numerical simulation of the linearized Euler equations in conservation law form

    NASA Technical Reports Server (NTRS)

    Sreenivas, Kidambi; Whitfield, David L.; Huff, Dennis L.

    1993-01-01

    A linearized Euler solver based on a high resolution numerical scheme is presented. The approach is to linearize the flux vector as opposed to carrying through the complete linearization analysis with the dependent variable vector written as a sum of the mean and the perturbed flow. This allows the linearized equations to be maintained in conservation law form. The linearized equations are used to compute unsteady flows in turbomachinery blade rows arising due to blade vibrations. Numerical solutions are compared to theoretical results (where available) and to numerical solutions of the nonlinear Euler equations.

  17. Multi-soliton rational solutions for some nonlinear evolution equations

    NASA Astrophysics Data System (ADS)

    Osman, Mohamed S.

    2016-01-01

    The Korteweg-de Vries equation (KdV) and the (2+ 1)-dimensional Nizhnik-Novikov-Veselov system (NNV) are presented. Multi-soliton rational solutions of these equations are obtained via the generalized unified method. The analysis emphasizes the power of this method and its capability of handling completely (or partially) integrable equations. Compared with Hirota's method and the inverse scattering method, the proposed method gives more general exact multi-wave solutions without much additional effort. The results show that, by virtue of symbolic computation, the generalized unified method may provide us with a straightforward and effective mathematical tool for seeking multi-soliton rational solutions for solving many nonlinear evolution equations arising in different branches of sciences.

  18. A Harnack's inequality for mixed type evolution equations

    NASA Astrophysics Data System (ADS)

    Paronetto, Fabio

    2016-03-01

    We define a homogeneous parabolic De Giorgi classes of order 2 which suits a mixed type class of evolution equations whose simplest example is μ (x)∂ u/∂ t - Δu = 0 where μ can be positive, null and negative, so in particular elliptic-parabolic and forward-backward parabolic equations are included. For functions belonging to this class we prove local boundedness and show a Harnack inequality which, as by-products, gives Hölder-continuity, in particular in the interface I where μ changes sign, and a maximum principle.

  19. Operator Factorization and the Solution of Second-Order Linear Ordinary Differential Equations

    ERIC Educational Resources Information Center

    Robin, W.

    2007-01-01

    The theory and application of second-order linear ordinary differential equations is reviewed from the standpoint of the operator factorization approach to the solution of ordinary differential equations (ODE). Using the operator factorization approach, the general second-order linear ODE is solved, exactly, in quadratures and the resulting…

  20. Supporting Students' Understanding of Linear Equations with One Variable Using Algebra Tiles

    ERIC Educational Resources Information Center

    Saraswati, Sari; Putri, Ratu Ilma Indra; Somakim

    2016-01-01

    This research aimed to describe how algebra tiles can support students' understanding of linear equations with one variable. This article is a part of a larger research on learning design of linear equations with one variable using algebra tiles combined with balancing method. Therefore, it will merely discuss one activity focused on how students…

  1. Cauchy problem for non-linear systems of equations in the critical case

    NASA Astrophysics Data System (ADS)

    Kaikina, E. I.; Naumkin, P. I.; Shishmarev, I. A.

    2004-12-01

    The large-time asymptotic behaviour is studied for a system of non-linear evolution dissipative equations \\displaystyle u_t+\\mathscr N(u,u)+\\mathscr Lu=0, \\qquad x\\in\\mathbb R^n, \\quad t>0, \\displaystyle u(0,x)=\\widetilde u(x), \\qquad x\\in\\mathbb R^n, where \\mathscr L is a linear pseudodifferential operator \\mathscr Lu=\\overline{\\mathscr F}_{\\xi\\to x}(L(\\xi)\\widehat u(\\xi)) and the non-linearity \\mathscr N is a quadratic pseudodifferential operator \\displaystyle \\mathscr N(u,u)=\\overline{\\mathscr F}_{\\xi\\to x}\\sum_{k,l=1}^m\\int_{\\mathbb R^n}A^{kl}(t,\\xi,y)\\widehat u_k(t,\\xi-y)\\widehat u_l(t,y)\\,dy,where \\widehat u\\equiv\\mathscr F_{x\\to\\xi}u is the Fourier transform. Under the assumptions that the initial data \\widetilde u\\in\\mathbf H^{\\beta,0}\\cap\\mathbf H^{0,\\beta}, \\beta>n/2 are sufficiently small, where \\displaystyle \\mathbf H^{n,m}=\\{\\phi\\in\\mathbf L^2:\\Vert\\langle x\\rangle^m\\lang......\\phi(x)\\Vert _{\\mathbf L^2}<\\infty\\}, \\qquad \\langle x\\rangle=\\sqrt{1+x^2}\\,,is a Sobolev weighted space, and that the total mass vector \\displaystyle M=\\int\\widetilde u(x)\\,dx\

  2. An iterative method for indefinite systems of linear equations

    NASA Technical Reports Server (NTRS)

    Ito, K.

    1984-01-01

    An iterative method for solving nonsymmetric indefinite linear systems is proposed. The method involves the successive use of a modified version of the conjugate residual method. A numerical example is given to illustrate the method.

  3. Symmetry classification and joint invariants for the scalar linear (1 + 1) elliptic equation

    NASA Astrophysics Data System (ADS)

    Mahomed, F. M.; Johnpillai, A. G.; Aslam, A.

    2015-08-01

    The equations for the classification of symmetries of the scalar linear (1 + 1) elliptic partial differential equation (PDE) are obtained in terms of Cotton's invariants. New joint differential invariants of the scalar linear elliptic (1 + 1) PDE in two independent variables are derived in terms of Cotton's invariants by application of the infinitesimal method. Joint differential invariants of the scalar linear elliptic equation are also deduced from the basis of the joint differential invariants of the scalar linear (1 + 1) hyperbolic equation under the application of the complex linear transformation. We also find a basis of joint differential invariants for such type of equations by utilization of the operators of invariant differentiation. The other invariants are functions of the basis elements and their invariant derivatives. Examples are given to illustrate our results.

  4. Linear equations in general purpose codes for stiff ODEs

    SciTech Connect

    Shampine, L. F.

    1980-02-01

    It is noted that it is possible to improve significantly the handling of linear problems in a general-purpose code with very little trouble to the user or change to the code. In such situations analytical evaluation of the Jacobian is a lot cheaper than numerical differencing. A slight change in the point at which the Jacobian is evaluated results in a more accurate Jacobian in linear problems. (RWR)

  5. Fast and local non-linear evolution of steep wave-groups on deep water: A comparison of approximate models to fully non-linear simulations

    NASA Astrophysics Data System (ADS)

    Adcock, T. A. A.; Taylor, P. H.

    2016-01-01

    The non-linear Schrödinger equation and its higher order extensions are routinely used for analysis of extreme ocean waves. This paper compares the evolution of individual wave-packets modelled using non-linear Schrödinger type equations with packets modelled using fully non-linear potential flow models. The modified non-linear Schrödinger Equation accurately models the relatively large scale non-linear changes to the shape of wave-groups, with a dramatic contraction of the group along the mean propagation direction and a corresponding extension of the width of the wave-crests. In addition, as extreme wave form, there is a local non-linear contraction of the wave-group around the crest which leads to a localised broadening of the wave spectrum which the bandwidth limited non-linear Schrödinger Equations struggle to capture. This limitation occurs for waves of moderate steepness and a narrow underlying spectrum.

  6. Application of variational and Galerkin equations to linear and nonlinear finite element analysis

    NASA Technical Reports Server (NTRS)

    Yu, Y.-Y.

    1974-01-01

    The paper discusses the application of the variational equation to nonlinear finite element analysis. The problem of beam vibration with large deflection is considered. The variational equation is shown to be flexible in both the solution of a general problem and in the finite element formulation. Difficulties are shown to arise when Galerkin's equations are used in the consideration of the finite element formulation of two-dimensional linear elasticity and of the linear classical beam.

  7. Accurate numerical solution of compressible, linear stability equations

    NASA Technical Reports Server (NTRS)

    Malik, M. R.; Chuang, S.; Hussaini, M. Y.

    1982-01-01

    The present investigation is concerned with a fourth order accurate finite difference method and its application to the study of the temporal and spatial stability of the three-dimensional compressible boundary layer flow on a swept wing. This method belongs to the class of compact two-point difference schemes discussed by White (1974) and Keller (1974). The method was apparently first used for solving the two-dimensional boundary layer equations. Attention is given to the governing equations, the solution technique, and the search for eigenvalues. A general purpose subroutine is employed for solving a block tridiagonal system of equations. The computer time can be reduced significantly by exploiting the special structure of two matrices.

  8. Integrability of the Wong Equations in the Class of Linear Integrals of Motion

    NASA Astrophysics Data System (ADS)

    Magazev, A. A.

    2016-04-01

    The Wong equations, which describe the motion of a classical charged particle with isospin in an external gauge field, are considered. The structure of the Lie algebra of the linear integrals of motion of these equations is investigated. An algebraic condition for integrability of the Wong equations is formulated. Some examples are considered.

  9. Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

    NASA Astrophysics Data System (ADS)

    Varlamov, Vladimir

    2007-03-01

    Rayleigh functions [sigma]l([nu]) are defined as series in inverse powers of the Bessel function zeros [lambda][nu],n[not equal to]0, where ; [nu] is the index of the Bessel function J[nu](x) and n=1,2,... is the number of the zeros. Convolutions of Rayleigh functions with respect to the Bessel index, are needed for constructing global-in-time solutions of semi-linear evolution equations in circular domains [V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain, Nonlinear Anal. 46 (2001) 699-725; V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424]. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424], where the properties of R1(m) were investigated. In the present work a general representation of Rl(m) in terms of [sigma]l([nu]) is deduced. On the basis of this a representation for the function R2(m) is obtained in terms of the [psi]-function. An asymptotic expansion is computed for R2(m) as m-->[infinity]. Such asymptotics are needed for establishing function spaces for solutions of semi-linear equations in bounded domains with periodicity conditions in one coordinate. As an example of application of Rl(m) a forced Boussinesq equationutt-2b[Delta]ut=-[alpha][Delta]2u+[Delta]u+[beta][Delta](u2)+f with [alpha],b=const>0 and [beta]=const[set membership, variant]R is considered in a unit disc with homogeneous boundary and initial data. Construction of its global-in-time solutions involves the use of the functions R1(m) and R2(m) which are responsible for the nonlinear smoothing effect.

  10. Parameterization and Monte Carlo solutions to PDF evolution equations

    NASA Astrophysics Data System (ADS)

    Suciu, Nicolae; Schüler, Lennart; Attinger, Sabine; Knabner, Peter

    2015-04-01

    The probability density function (PDF) of the chemical species concentrations transported in random environments is governed by unclosed evolution equations. The PDF is transported in the physical space by drift and diffusion processes described by coefficients derived by standard upscaling procedures. Its transport in the concentration space is described by a drift determined by reaction rates, in a closed form, as well as a term accounting for the sub-grid mixing process due to molecular diffusion and local scale hydrodynamic dispersion. Sub-grid mixing processes are usually described by models of the conditionally averaged diffusion flux or models of the conditional dissipation rate. We show that in certain situations mixing terms can also be derived, in the form of an Itô process, from simulated or measured concentration time series. Monte Carlo solutions to PDF evolution equations are usually constructed with systems of computational particles, which are well suited for highly dimensional advection-dominated problems. Such solutions require the fulfillment of specific consistency conditions relating the statistics of the random concentration field, function of both space and time, to that of the time random function describing an Itô process in physical and concentration spaces which governs the evolution of the system of particles. We show that the solution of the Fokker-Planck equation for the concentration-position PDF of the Itô process coincides with the solution of the PDF equation only for constant density flows in spatially statistically homogeneous systems. We also find that the solution of the Fokker-Planck equation is still equivalent to the solution of the PDF equation weighted by the variable density or by other conserved scalars. We illustrate the parameterization of the sub-grid mixing by time series and the Monte Carlo solution for a problem of contaminant transport in groundwater. The evolution of the system of computational particles whose

  11. On the relationship between ODE solvers and iterative solvers for linear equations

    SciTech Connect

    Lorber, A.; Joubert, W.; Carey, G.F.

    1994-12-31

    The connection between the solution of linear systems of equations by both iterative methods and explicit time stepping techniques is investigated. Based on the similarities, a suite of Runge-Kutta time integration schemes with extended stability domains are developed using Chebyshev iteration polynomials. These Runge-Kutta schemes are applied to linear and non-linear systems arising from the numerical solution of PDE`s containing either physical or artificial transient terms. Specifically, the solutions of model linear convection and convection-diffusion equations are presented, as well as the solution of a representative non-linear Navier-Stokes fluid flow problem. Included are results of parallel computations.

  12. Evolution of Linear Mitochondrial Genomes in Medusozoan Cnidarians

    PubMed Central

    Kayal, Ehsan; Bentlage, Bastian; Collins, Allen G.; Pirro, Stacy; Lavrov, Dennis V.

    2012-01-01

    In nearly all animals, mitochondrial DNA (mtDNA) consists of a single circular molecule that encodes several subunits of the protein complexes involved in oxidative phosphorylation as well as part of the machinery for their expression. By contrast, mtDNA in species belonging to Medusozoa (one of the two major lineages in the phylum Cnidaria) comprises one to several linear molecules. Many questions remain on the ubiquity of linear mtDNA in medusozoans and the mechanisms responsible for its evolution, replication, and transcription. To address some of these questions, we determined the sequences of nearly complete linear mtDNA from 24 species representing all four medusozoan classes: Cubozoa, Hydrozoa, Scyphozoa, and Staurozoa. All newly determined medusozoan mitochondrial genomes harbor the 17 genes typical for cnidarians and map as linear molecules with a high degree of gene order conservation relative to the anthozoans. In addition, two open reading frames (ORFs), polB and ORF314, are identified in cubozoan, schyphozoan, staurozoan, and trachyline hydrozoan mtDNA. polB belongs to the B-type DNA polymerase gene family, while the product of ORF314 may act as a terminal protein that binds telomeres. We posit that these two ORFs are remnants of a linear plasmid that invaded the mitochondrial genomes of the last common ancestor of Medusozoa and are responsible for its linearity. Hydroidolinan hydrozoans have lost the two ORFs and instead have duplicated cox1 at each end of their mitochondrial chromosome(s). Fragmentation of mtDNA occurred independently in Cubozoa and Hydridae (Hydrozoa, Hydroidolina). Our broad sampling allows us to reconstruct the evolutionary history of linear mtDNA in medusozoans. PMID:22113796

  13. Collection of parallel linear equations routines for the Denelcor HEP

    SciTech Connect

    Dongarra, J.J.; Hiromoto, R.E.

    1983-09-01

    This note describes the implementation and performance results for a few standard linear algebra routines on the Denelcor HEP computer. The algorithms used here are based on high-level modules which facilitate portability and perform efficiently in a wide range of environments.

  14. Who Needs Linear Equating under the NEAT Design?

    ERIC Educational Resources Information Center

    Maris, Gunter; Schmittmann, Verena D.; Borsboom, Denny

    2010-01-01

    Test equating under the NEAT design is, at best, a necessary evil. At bottom, the procedure aims to reach a conclusion on what a tested person would have done, if he or she were administered a set of items that were in fact never administered. It is not possible to infer such a conclusion from the data, because one simply has not made the required…

  15. Mixed finite elements for the Richards' equation: linearization procedure

    NASA Astrophysics Data System (ADS)

    Pop, I. S.; Radu, F.; Knabner, P.

    2004-07-01

    We consider mixed finite element discretization for a class of degenerate parabolic problems including the Richards' equation. After regularization, time discretization is achieved by an Euler implicit scheme, while mixed finite elements are employed for the discretization in space. Based on the results obtained in (Radu et al. RANA Preprint 02-06, Eindhoven University of Technology, 2002), this paper considers a simple iterative scheme to solve the emerging nonlinear elliptic problems.

  16. Tidal evolution of the Galilean satellites - A linearized theory

    NASA Technical Reports Server (NTRS)

    Greenberg, R.

    1981-01-01

    The Laplace resonance among the Galilean satellites Io, Europa, and Ganymede is traditionally reduced to a pendulum-like dynamical problem by neglecting short-period variations of several orbital elements. However, some of these variations that can now be neglected may once have had longer periods, comparable to the 'pendulum' period, if the system was formerly in deep resonance (pairs of periods even closer to the ratio 2:1 than they are now). In that case, the dynamical system cannot be reduced to fewer than nine dimensions. The nine-dimensional system is linearized here in order to study small variations about equilibrium. When tidal effects are included, the resulting evolution is substantially the same as was indicated by the pendulum approach, except that evolution out of deep resonance is found to be somewhat slower than suggested by extrapolation of the pendulum results. This slower rate helps support the hypothesis that the system may have evolved from deep resonance.

  17. A quasi-linear kinetic equation for cosmic rays in the interplanetary medium

    NASA Technical Reports Server (NTRS)

    Luhmann, J. G.

    1976-01-01

    A kinetic equation for interplanetary cosmic rays is set up with the aid of weak-plasma-turbulence theory for an idealized radially symmetric model of the interplanetary magnetic field. As a starting point, this treatment invokes the Vlasov equation instead of the traditional Fokker-Planck equation. Quasi-linear theory is applied to obtain a momentum diffusion equation for the heliocentric frame of reference which describes the interaction of cosmic rays with convecting magnetic irregularities in the solar-wind plasma. Under restricted conditions, the well-known equation of solar modulation can be obtained from this kinetic equation.

  18. Comparing the full time-dependent Bogoliubov-de-Gennes equations to their linear approximation: a numerical investigation

    NASA Astrophysics Data System (ADS)

    Hainzl, Christian; Seyrich, Jonathan

    2016-05-01

    In this paper we report on the results of a numerical study of the nonlinear time-dependent Bardeen-Cooper-Schrieffer (BCS) equations, often also denoted as Bogoliubov-de-Gennes (BdG) equations, for a one-dimensional system of fermions with contact interaction. We show that, even above the critical temperature, the full equations and their linear approximation give rise to completely different evolutions. In contrast to its linearization, the full nonlinear equation does not show any diffusive behavior in the order parameter. This means that the order parameter does not follow a Ginzburg-Landau-type of equation, in accordance with a recent theoretical result in [R.L. Frank, C. Hainzl, B. Schlein, R. Seiringer, to appear in Lett. Math. Phys., arXiv:1504.05885 (2016)]. We include a full description on the numerical implementation of the partial differential BCS/BdG equations.

  19. Special functions arising in the study of semi-linear equations in circular domains

    NASA Astrophysics Data System (ADS)

    Varlamov, Vladimir

    2007-05-01

    Rayleigh functions are defined by the formulawhere are zeros of the Bessel function J[nu](x) and n=1,2,3,..., is the number of the zero. These functions appear in the classical problems of vibrating circular membranes, heat conduction in cylinders and diffraction through circular apertures. In the present paper it is shown that a new family of special functions, convolutions of Rayleigh functions with respect to the Bessel index,arises in constructing solutions of semi-linear evolution equations in circular domains (see also [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424]). As an example of its application a forced Cahn-Hilliard equation is considered in a unit disc with homogeneous boundary and initial conditions. Construction of its global-in-time solutions involves the use of R1(m) and R2(m). A general representation of Rl(m) is deduced and on the basis of that a particular result for R2(m) is obtained convenient for computing its asymptotics as m-->[infinity]. The latter issue is important for establishing a function space to which a solution of the corresponding problem belongs.

  20. Finding linear dependencies in integration-by-parts equations: A Monte Carlo approach

    NASA Astrophysics Data System (ADS)

    Kant, Philipp

    2014-05-01

    The reduction of a large number of scalar integrals to a small set of master integrals via Laporta’s algorithm is common practice in multi-loop calculations. It is also a major bottleneck in terms of running time and memory consumption. It involves solving a large set of linear equations where many of the equations are linearly dependent. We propose a simple algorithm that eliminates all linearly dependent equations from a given system, reducing the time and space requirements of a subsequent run of Laporta’s algorithm.

  1. Modeling taper charge with a non-linear equation

    NASA Technical Reports Server (NTRS)

    Mcdermott, P. P.

    1985-01-01

    Work aimed at modeling the charge voltage and current characteristics of nickel-cadmium cells subject to taper charge is presented. Work reported at previous NASA Battery Workshops has shown that the voltage of cells subject to constant current charge and discharge can be modeled very accurately with the equation: voltage = A + (B/(C-X)) + De to the -Ex where A, B, D, and E are fit parameters and x is amp-hr of charge removed during discharge or returned during charge. In a constant current regime, x is also equivalent to time on charge or discharge.

  2. Integrals and integral equations in linearized wing theory

    NASA Technical Reports Server (NTRS)

    Lomax, Harvard; Heaslet, Max A; Fuller, Franklyn B

    1951-01-01

    The formulas of subsonic and supersonic wing theory for source, doublet, and vortex distributions are reviewed and a systematic presentation is provided which relates these distributions to the pressure and to the vertical induced velocity in the plane of the wing. It is shown that care must be used in treating the singularities involved in the analysis and that the order of integration is not always reversible. Concepts suggested by the irreversibility of order of integration are shown to be useful in the inversion of singular integral equations when operational techniques are used. A number of examples are given to illustrate the methods presented, attention being directed to supersonic flight speed.

  3. Evolution equation for soft physics at high energy

    NASA Astrophysics Data System (ADS)

    Brogueira, P.; Dias de Deus, J.

    2010-07-01

    Based on the nonlinear logistic equation we study, in a qualitative and semi-quantitative way, the evolution with energy and saturation of the elastic differential cross-section in pp(\\bar{p}p) collisions at high energy. Geometrical scaling occurs at the black disc limit, and scaling develops first for small values of the scaling variable |t|σtot.. Our prediction for dσ/dt at LHC, with two zeros and a minimum at large |t| differs, as far as we know, from all existing ones.

  4. Clutter locus equation for more general linear array orientation

    NASA Astrophysics Data System (ADS)

    Bickel, Douglas L.

    2011-06-01

    The clutter locus is an important concept in space-time adaptive processing (STAP) for ground moving target indicator (GMTI) radar systems. The clutter locus defines the expected ground clutter location in the angle-Doppler domain. Typically in literature, the clutter locus is presented as a line, or even a set of ellipsoids, under certain assumptions about the geometry of the array. Most often, the array is assumed to be in the horizontal plane containing the velocity vector. This paper will give a more general 3-dimensional interpretation of the clutter locus for a general linear array orientation.

  5. Prediction of Undsteady Flows in Turbomachinery Using the Linearized Euler Equations on Deforming Grids

    NASA Technical Reports Server (NTRS)

    Clark, William S.; Hall, Kenneth C.

    1994-01-01

    A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. The model accounts for blade loading, blade geometry, shock motion, and wake motion. Assuming that the unsteadiness in the flow is small relative to the nonlinear mean solution, the unsteady Euler equations can be linearized about the mean flow. This yields a set of linear variable coefficient equations that describe the small amplitude harmonic motion of the fluid. These linear equations are then discretized on a computational grid and solved using standard numerical techniques. For transonic flows, however, one must use a linear discretization which is a conservative linearization of the non-linear discretized Euler equations to ensure that shock impulse loads are accurately captured. Other important features of this analysis include a continuously deforming grid which eliminates extrapolation errors and hence, increases accuracy, and a new numerically exact, nonreflecting far-field boundary condition treatment based on an eigenanalysis of the discretized equations. Computational results are presented which demonstrate the computational accuracy and efficiency of the method and demonstrate the effectiveness of the deforming grid, far-field nonreflecting boundary conditions, and shock capturing techniques. A comparison of the present unsteady flow predictions to other numerical, semi-analytical, and experimental methods shows excellent agreement. In addition, the linearized Euler method presented requires one or two orders-of-magnitude less computational time than traditional time marching techniques making the present method a viable design tool for aeroelastic analyses.

  6. Promoting Understanding of Linear Equations with the Median-Slope Algorithm

    ERIC Educational Resources Information Center

    Edwards, Michael Todd

    2005-01-01

    The preliminary findings resulting when invented algorithm is used with entry-level students while introducing linear equations is described. As calculations are accessible, the algorithm is preferable to more rigorous statistical procedures in entry-level classrooms.

  7. Analytic solutions of the Rayleigh equation for linear density profiles

    NASA Astrophysics Data System (ADS)

    Cherfils, C.; Lafitte, O.

    2000-08-01

    We consider the Rayleigh-Taylor instability in linear density profiles and we derive the exact analytic expressions of the growth rates and associated eigenfunctions. We study the behavior of the multiple eigenvalues in both the short- and the long-wavelength limit. As the largest eigenvalue γmax reduces to the classical Rayleigh growth rate; the other eigenvalues vanish as the front thickness tends to zero. Furthermore, the simple expression of γmax exact to first order in the long-wavelength limit differs from the widely used estimate Akg/(1+AkL0), where g is the acceleration, A the Atwood number, k the wave number of the perturbation, and L0 the minimum density gradient scale length.

  8. Perturbations of linear delay differential equations at the verge of instability

    NASA Astrophysics Data System (ADS)

    Lingala, N.; Namachchivaya, N. Sri

    2016-06-01

    The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. This paper considers linear DDEs that are on the verge of instability, i.e., a pair of roots of the characteristic equation lies on the imaginary axis of the complex plane and all other roots have negative real parts. It is shown that when small noise perturbations are present, the probability distribution of the dynamics can be approximated by the probability distribution of a certain one-dimensional stochastic differential equation (SDE) without delay. This is advantageous because equations without delay are easier to simulate and one-dimensional SDEs are analytically tractable. When the perturbations are also linear, it is shown that the stability depends on a specific complex number. The theory is applied to study oscillators with delayed feedback. Some errors in other articles that use multiscale approach are pointed out.

  9. Polynomial elimination theory and non-linear stability analysis for the Euler equations

    NASA Technical Reports Server (NTRS)

    Kennon, S. R.; Dulikravich, G. S.; Jespersen, D. C.

    1986-01-01

    Numerical methods are presented that exploit the polynomial properties of discretizations of the Euler equations. It is noted that most finite difference or finite volume discretizations of the steady-state Euler equations produce a polynomial system of equations to be solved. These equations are solved using classical polynomial elimination theory, with some innovative modifications. This paper also presents some preliminary results of a new non-linear stability analysis technique. This technique is applicable to determining the stability of polynomial iterative schemes. Results are presented for applying the elimination technique to a one-dimensional test case. For this test case, the exact solution is computed in three iterations. The non-linear stability analysis is applied to determine the optimal time step for solving Burgers' equation using the MacCormack scheme. The estimated optimal time step is very close to the time step that arises from a linear stability analysis.

  10. On the compatibility equations of nonlinear and linear elasticity in the presence of boundary conditions

    NASA Astrophysics Data System (ADS)

    Angoshtari, Arzhang; Yavari, Arash

    2015-12-01

    We use Hodge-type orthogonal decompositions for studying the compatibility equations of the displacement gradient and the linear strain with prescribed boundary displacements. We show that the displacement gradient is compatible if and only if for any equilibrated virtual first Piola-Kirchhoff stress tensor field, the virtual work done by the displacement gradient is equal to the virtual work done by the prescribed boundary displacements. This condition is very similar to the classical compatibility equations for the linear strain. Since these compatibility equations for linear and nonlinear strains involve infinite-dimensional spaces and consequently are not easy to use in practice, we derive alternative compatibility equations, which are written in terms of some finite-dimensional spaces and are more useful in practice. Using these new compatibility equations, we present some non-trivial examples that show that compatible strains may become incompatible in the presence of prescribed boundary displacements.

  11. Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations

    SciTech Connect

    Schüler, D.; Alonso, S.; Bär, M.; Torcini, A.

    2014-12-15

    Pattern formation often occurs in spatially extended physical, biological, and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue, we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular, revealing for the Swift-Hohenberg equations, a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of a weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.

  12. Trajectory approach to the Schrödinger-Langevin equation with linear dissipation for ground states

    NASA Astrophysics Data System (ADS)

    Chou, Chia-Chun

    2015-11-01

    The Schrödinger-Langevin equation with linear dissipation is integrated by propagating an ensemble of Bohmian trajectories for the ground state of quantum systems. Substituting the wave function expressed in terms of the complex action into the Schrödinger-Langevin equation yields the complex quantum Hamilton-Jacobi equation with linear dissipation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation is simultaneously integrated with the trajectory guidance equation. Then, the computational method is applied to the harmonic oscillator, the double well potential, and the ground vibrational state of methyl iodide. The excellent agreement between the computational and the exact results for the ground state energies and wave functions shows that this study provides a synthetic trajectory approach to the ground state of quantum systems.

  13. Boundedness of solutions for non-linear quasi-periodic differential equations with Liouvillean frequency

    NASA Astrophysics Data System (ADS)

    Wang, Jing; You, Jiangong

    2016-07-01

    We study the boundedness of solutions for non-linear quasi-periodic differential equations with Liouvillean frequencies. We proved that if the forcing is quasi-periodic in time with two frequencies which is not super-Liouvillean, then all solutions of the equation are bounded. The proof is based on action-angle variables and modified KAM theory.

  14. A block iterative LU solver for weakly coupled linear systems. [in fluid dynamics equations

    NASA Technical Reports Server (NTRS)

    Cooke, C. H.

    1977-01-01

    A hybrid technique, called the block iterative LU solver, is proposed for solving the linear equations resulting from a finite element numerical analysis of certain fluid dynamics problems where the equations are weakly coupled between distinct sets of variables. Either the block Jacobi iterative method or the block Gauss-Seidel iterative solver is combined with LU decomposition.

  15. On group classification of normal systems of linear second-order ordinary differential equations

    NASA Astrophysics Data System (ADS)

    Meleshko, S. V.; Moyo, S.

    2015-05-01

    In this paper we study the general group classification of systems of linear second-order ordinary differential equations inspired from earlier works and recent results on the group classification of such systems. Some interesting results and subsequent theorem arising from this particular study are discussed here. This paper considers the study of irreducible systems of second-order ordinary differential equations.

  16. Stochastic differential equation model for linear growth birth and death processes with immigration and emigration

    SciTech Connect

    Granita; Bahar, A.

    2015-03-09

    This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found.

  17. Stochastic differential equation model for linear growth birth and death processes with immigration and emigration

    NASA Astrophysics Data System (ADS)

    Granita, Bahar, A.

    2015-03-01

    This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found.

  18. Linear Ordinary Differential Equations with Constant Coefficients. Revisiting the Impulsive Response Method Using Factorization

    ERIC Educational Resources Information Center

    Camporesi, Roberto

    2011-01-01

    We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of…

  19. Chesterton and Mathematics: The Three Riders of Apocalypse (Introduction to Systems of Linear Equations, Workshop).

    ERIC Educational Resources Information Center

    Ramirez, Rene; Flores, Homero

    This paper takes G.K. Chesterton's short story, "The Three Horsemen of Apocalypse," as a motivating introduction to the study of linear equations systems, as well as a review of the concept of linear function. The guide has three objectives: (1) to illustrate how to use non-mathematical sources to create math problems; (2) to use the graphing…

  20. A Classroom Note on: An Alternative Method for Solving Linear Equations

    ERIC Educational Resources Information Center

    Klikovac, Ida; Riedinger, Michael

    2011-01-01

    The method of "Double False Position" is an arithmetic approach to solving linear equations that pre-dates current algebraic methods by more than 3,000 years. The method applies to problems that, in algebraic notation, would be expressed as y = L(x), where L(x) is a linear function of x. Double False Position works by evaluating the described…

  1. Fractional hereditariness of lipid membranes: Instabilities and linearized evolution.

    PubMed

    Deseri, L; Pollaci, P; Zingales, M; Dayal, K

    2016-05-01

    In this work lipid ordering phase changes arising in planar membrane bilayers is investigated both accounting for elasticity alone and for effective viscoelastic response of such assemblies. The mechanical response of such membranes is studied by minimizing the Gibbs free energy which penalizes perturbations of the changes of areal stretch and their gradients only (Deseri and Zurlo, 2013). As material instabilities arise whenever areal stretches characterizing homogeneous configurations lie inside the spinoidal zone of the free energy density, bifurcations from such configurations are shown to occur as oscillatory perturbations of the in-plane displacement. Experimental observations (Espinosa et al., 2011) show a power-law in-plane viscous behavior of lipid structures allowing for an effective viscoelastic behavior of lipid membranes, which falls in the framework of Fractional Hereditariness. A suitable generalization of the variational principle invoked for the elasticity is applied in this case, and the corresponding Euler-Lagrange equation is found together with a set of boundary and initial conditions. Separation of variables allows for showing how Fractional Hereditariness owes bifurcated modes with a larger number of spatial oscillations than the corresponding elastic analog. Indeed, the available range of areal stresses for material instabilities is found to increase with respect to the purely elastic case. Nevertheless, the time evolution of the perturbations solving the Euler-Lagrange equation above exhibits time-decay and the large number of spatial oscillation slowly relaxes, thereby keeping the features of a long-tail type time-response. PMID:26897568

  2. CFORM- LINEAR CONTROL SYSTEM DESIGN AND ANALYSIS: CLOSED FORM SOLUTION AND TRANSIENT RESPONSE OF THE LINEAR DIFFERENTIAL EQUATION

    NASA Technical Reports Server (NTRS)

    Jamison, J. W.

    1994-01-01

    CFORM was developed by the Kennedy Space Center Robotics Lab to assist in linear control system design and analysis using closed form and transient response mechanisms. The program computes the closed form solution and transient response of a linear (constant coefficient) differential equation. CFORM allows a choice of three input functions: the Unit Step (a unit change in displacement); the Ramp function (step velocity); and the Parabolic function (step acceleration). It is only accurate in cases where the differential equation has distinct roots, and does not handle the case for roots at the origin (s=0). Initial conditions must be zero. Differential equations may be input to CFORM in two forms - polynomial and product of factors. In some linear control analyses, it may be more appropriate to use a related program, Linear Control System Design and Analysis (KSC-11376), which uses root locus and frequency response methods. CFORM was written in VAX FORTRAN for a VAX 11/780 under VAX VMS 4.7. It has a central memory requirement of 30K. CFORM was developed in 1987.

  3. Reconciling the eigenmode analysis with the Maxwell Bloch equations approach to superradiance in the linear regime

    NASA Astrophysics Data System (ADS)

    Friedberg, Richard; Manassah, Jamal T.

    2008-07-01

    The superradiance from a slab of inverted two-level atoms is theoretically analyzed in the linear regime from both the perspective of the expansion in eigenfunctions of the integral equation with the Lienard-Wiechert potential as kernel, and that of linearizing the Maxwell-Bloch equations. We show the equivalence of both approaches. We show that the so-called Reduced Maxwell-Bloch equations do not yield even approximately the correct solution when applied in the obvious way, but that they can be made to give the correct solution by adding a fictitious input field.

  4. Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics

    NASA Astrophysics Data System (ADS)

    Mirzazadeh, Mohammad; Ekici, Mehmet; Sonmezoglu, Abdullah; Ortakaya, Sami; Eslami, Mostafa; Biswas, Anjan

    2016-05-01

    This paper studies a few nonlinear evolution equations that appear with fractional temporal evolution and fractional spatial derivatives. These are Benjamin-Bona-Mahoney equation, dispersive long wave equation and Nizhnik-Novikov-Veselov equation. The extended Jacobi's elliptic function expansion method is implemented to obtain soliton and other periodic singular solutions to these equations. In the limiting case, when the modulus of ellipticity approaches zero or unity, these doubly periodic functions approach solitary waves or shock waves or periodic singular solutions emerge.

  5. Approximating electronically excited states with equation-of-motion linear coupled-cluster theory

    SciTech Connect

    Byrd, Jason N. Rishi, Varun; Perera, Ajith; Bartlett, Rodney J.

    2015-10-28

    A new perturbative approach to canonical equation-of-motion coupled-cluster theory is presented using coupled-cluster perturbation theory. A second-order Møller-Plesset partitioning of the Hamiltonian is used to obtain the well known equation-of-motion many-body perturbation theory equations and two new equation-of-motion methods based on the linear coupled-cluster doubles and linear coupled-cluster singles and doubles wavefunctions. These new methods are benchmarked against very accurate theoretical and experimental spectra from 25 small organic molecules. It is found that the proposed methods have excellent agreement with canonical equation-of-motion coupled-cluster singles and doubles state for state orderings and relative excited state energies as well as acceptable quantitative agreement for absolute excitation energies compared with the best estimate theory and experimental spectra.

  6. Central equation of state in spherical characteristic evolutions

    SciTech Connect

    Barreto, W.; Castillo, L.; Barrios, E.

    2009-10-15

    We study the evolution of a perfect-fluid sphere coupled to a scalar radiation field. By ensuring a Ricci invariant regularity as a conformally flat spacetime at the central world line we find that the fluid coupled to the scalar field satisfies the equation of state {rho}{sub c}+3p{sub c}=const at the center of the sphere, where the energy {rho}{sub c} density and the pressure p{sub c} do not necessarily contain the scalar field contribution. The fluid can be interpreted as anisotropic and radiant because of the scalar field, but it becomes perfect and nonradiative at the center of the sphere. These results are currently being considered to build up a numerical relativistic hydrodynamic solver.

  7. Initial-value problem for a linear ordinary differential equation of noninteger order

    SciTech Connect

    Pskhu, Arsen V

    2011-04-30

    An initial-value problem for a linear ordinary differential equation of noninteger order with Riemann-Liouville derivatives is stated and solved. The initial conditions of the problem ensure that (by contrast with the Cauchy problem) it is uniquely solvable for an arbitrary set of parameters specifying the orders of the derivatives involved in the equation; these conditions are necessary for the equation under consideration. The problem is reduced to an integral equation; an explicit representation of the solution in terms of the Wright function is constructed. As a consequence of these results, necessary and sufficient conditions for the solvability of the Cauchy problem are obtained. Bibliography: 7 titles.

  8. Entropy production and the geometry of dissipative evolution equations.

    PubMed

    Reina, Celia; Zimmer, Johannes

    2015-11-01

    Purely dissipative evolution equations are often cast as gradient flow structures, z ̇=K(z)DS(z), where the variable z of interest evolves towards the maximum of a functional S according to a metric defined by an operator K. While the functional often follows immediately from physical considerations (e.g., the thermodynamic entropy), the operator K and the associated geometry does not necessarily do so (e.g., Wasserstein geometry for diffusion). In this paper, we present a variational statement in the sense of maximum entropy production that directly delivers a relationship between the operator K and the constraints of the system. In particular, the Wasserstein metric naturally arises here from the conservation of mass or energy, and depends on the Onsager resistivity tensor, which, itself, may be understood as another metric, as in the steepest entropy ascent formalism. This variational principle is exemplified here for the simultaneous evolution of conserved and nonconserved quantities in open systems. It thus extends the classical Onsager flux-force relationships and the associated variational statement to variables that do not have a flux associated to them. We further show that the metric structure K is intimately linked to the celebrated Freidlin-Wentzell theory of stochastically perturbed gradient flows, and that the proposed variational principle encloses an infinite-dimensional fluctuation-dissipation statement. PMID:26651657

  9. Calculation of unsteady flows in turbomachinery using the linearized Euler equations

    NASA Astrophysics Data System (ADS)

    Hall, Kenneth C.; Crawley, Edward F.

    1989-06-01

    A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem).

  10. The non-linear coupled spin 2-spin 3 Cotton equation in three dimensions

    NASA Astrophysics Data System (ADS)

    Linander, Hampus; Nilsson, Bengt E. W.

    2016-07-01

    In the context of three-dimensional conformal higher spin theory we derive, in the frame field formulation, the full non-linear spin 3 Cotton equation coupled to spin 2. This is done by solving the corresponding Chern-Simons gauge theory system of equations, that is, using F = 0 to eliminate all auxiliary fields and thus expressing the Cotton equation in terms of just the spin 3 frame field and spin 2 covariant derivatives and tensors (Schouten). In this derivation we neglect the spin 4 and higher spin sectors and approximate the star product commutator by a Poisson bracket. The resulting spin 3 Cotton equation is complicated but can be related to linearized versions in the metric formulation obtained previously by other authors. The expected symmetry (spin 3 "translation", "Lorentz" and "dilatation") properties are verified for Cotton and other relevant tensors but some perhaps unexpected features emerge in the process, in particular in relation to the non-linear equations. We discuss the structure of this non-linear spin 3 Cotton equation but its explicit form is only presented here, in an exact but not completely refined version, in appended files obtained by computer algebra methods. Both the frame field and metric formulations are provided.

  11. Determining mixed linear-nonlinear coupled differential equations from multivariate discrete time series sequences

    NASA Astrophysics Data System (ADS)

    Irving, A. D.; Dewson, T.

    1997-02-01

    A new method is described for extracting mixed linear-nonlinear coupled differential equations from multivariate discrete time series data. It is assumed in the present work that the solution of the coupled ordinary differential equations can be represented as a multivariate Volterra functional expansion. A tractable hierarchy of moment equations is generated by operating on a suitably truncated Volterra functional expansion. The hierarchy facilitates the calculation of the coefficients of the coupled differential equations. In order to demonstrate the method's ability to accurately estimate the coefficients of the governing differential equations, it is applied to data derived from the numerical solution of the Lorenz equations with additive noise. The method is then used to construct a dynamic global mid- and high-magnetic latitude ionospheric model where nonlinear phenomena such as period doubling and quenching occur. It is shown that the estimated inhomogeneous coupled second-order differential equation model for the ionospheric foF2 peak plasma density can accurately forecast the future behaviour of a set of ionosonde stations which encompass the earth. Finally, the method is used to forecast the future behaviour of a portfolio of Japanese common stock prices. The hierarchy method can be used to characterise the observed behaviour of a wide class of coupled linear and mixed linear-nonlinear phenomena.

  12. Regularized moment equations for binary gas mixtures: Derivation and linear analysis

    NASA Astrophysics Data System (ADS)

    Gupta, Vinay Kumar; Struchtrup, Henning; Torrilhon, Manuel

    2016-04-01

    The applicability of the order of magnitude method [H. Struchtrup, "Stable transport equations for rarefied gases at high orders in the Knudsen number," Phys. Fluids 16, 3921-3934 (2004)] is extended to binary gas mixtures in order to derive various sets of equations—having minimum number of moments at a given order of accuracy in the Knudsen number—for binary mixtures of monatomic-inert-ideal gases interacting with the Maxwell interaction potential. For simplicity, the equations are derived in the linear regime up to third order accuracy in the Knudsen number. At zeroth order, the method produces the Euler equations; at first order, it results into the Fick, Navier-Stokes, and Fourier equations; at second order, it yields a set of 17 moment equations; and at third order, it leads to the regularized 17-moment equations. The transport coefficients in the Fick, Navier-Stokes, and Fourier equations obtained through order of magnitude method are compared with those obtained through the classical Chapman-Enskog expansion method. It is established that the different temperatures of different constituents do not play a role up to second order accurate theories in the Knudsen number, whereas they do contribute to third order accurate theory in the Knudsen number. Furthermore, it is found empirically that the zeroth, first, and second order accurate equations are linearly stable for all binary gas mixtures; however, although the third order accurate regularized 17-moment equations are linearly stable for most of the mixtures, they are linearly unstable for mixtures having extreme difference in molecular masses.

  13. A note on improved F-expansion method combined with Riccati equation applied to nonlinear evolution equations

    PubMed Central

    Islam, Md. Shafiqul; Khan, Kamruzzaman; Akbar, M. Ali; Mastroberardino, Antonio

    2014-01-01

    The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering. PMID:26064530

  14. Construction of rogue wave and lump solutions for nonlinear evolution equations

    NASA Astrophysics Data System (ADS)

    Lü, Zhuosheng; Chen, Yinnan

    2015-07-01

    Based on symbolic computation and an ansatz, we present a constructive algorithm to seek rogue wave and lump solutions for nonlinear evolution equations. As illustrative examples, we consider the potential-YTSF equation and a variable coefficient KP equation, and obtain nonsingular rational solutions of the two equations. The solutions can be rogue wave or lump solutions under different parameter conditions. We also present graphic illustration of some special solutions which would help better understand the evolution of solution waves.

  15. Non-linear evolution of the cosmic neutrino background

    SciTech Connect

    Villaescusa-Navarro, Francisco; Viel, Matteo; Peña-Garay, Carlos E-mail: spb@ias.edu E-mail: viel@oats.inaf.it

    2013-03-01

    We investigate the non-linear evolution of the relic cosmic neutrino background by running large box-size, high resolution N-body simulations which incorporate cold dark matter (CDM) and neutrinos as independent particle species. Our set of simulations explore the properties of neutrinos in a reference ΛCDM model with total neutrino masses between 0.05-0.60 eV in cold dark matter haloes of mass 10{sup 11}−10{sup 15} h{sup −1}M{sub s}un, over a redshift range z = 0−2. We compute the halo mass function and show that it is reasonably well fitted by the Sheth-Tormen formula, once the neutrino contribution to the total matter is removed. More importantly, we focus on the CDM and neutrino properties of the density and peculiar velocity fields in the cosmological volume, inside and in the outskirts of virialized haloes. The dynamical state of the neutrino particles depends strongly on their momentum: whereas neutrinos in the low velocity tail behave similarly to CDM particles, neutrinos in the high velocity tail are not affected by the clustering of the underlying CDM component. We find that the neutrino (linear) unperturbed momentum distribution is modified and mass and redshift dependent deviations from the expected Fermi-Dirac distribution are in place both in the cosmological volume and inside haloes. The neutrino density profiles around virialized haloes have been carefully investigated and a simple fitting formula is provided. The neutrino profile, unlike the cold dark matter one, is found to be cored with core size and central density that depend on the neutrino mass, redshift and mass of the halo, for halos of masses larger than ∼ 10{sup 13.5}h{sup −1}M{sub s}un. For lower masses the neutrino profile is best fitted by a simple power-law relation in the range probed by the simulations. The results we obtain are numerically converged in terms of neutrino profiles at the 10% level for scales above ∼ 200 h{sup −1}kpc at z = 0, and are stable with

  16. Anisotropic static spheres with linear equation of state in isotropic coordinates

    NASA Astrophysics Data System (ADS)

    Govender, M.; Thirukkanesh, S.

    2015-08-01

    In this paper we present a general framework for generating exact solutions to the Einstein field equations for static, anisotropic fluid spheres in comoving, isotropic coordinates obeying a linear equation of state of the form . We show that all possible solutions can be obtained via a single generating function defined in terms of one of the gravitational potentials. The physical viability of our solution-generating method is illustrated by modeling a static fluid sphere describing a strange star.

  17. Exact solutions and linear stability analysis for two-dimensional Ablowitz-Ladik equation

    NASA Astrophysics Data System (ADS)

    Zhang, Jin-Liang; Wang, Hong-Xian

    2014-04-01

    The Ablowitz-Ladik equation is a very important model in nonlinear mathematical physics. In this paper, the hyperbolic function solitary wave solutions, the trigonometric function periodic wave solutions, and the rational wave solutions with more arbitrary parameters of two-dimensional Ablowitz-Ladik equation are derived by using the (G'/G)-expansion method, and the effects of the parameters (including the coupling constant and other parameters) on the linear stability of the exact solutions are analysed and numerically simulated.

  18. Master equation solutions in the linear regime of characteristic formulation of general relativity

    NASA Astrophysics Data System (ADS)

    Cedeño M., C. E.; de Araujo, J. C. N.

    2015-12-01

    From the field equations in the linear regime of the characteristic formulation of general relativity, Bishop, for a Schwarzschild's background, and Mädler, for a Minkowski's background, were able to show that it is possible to derive a fourth order ordinary differential equation, called master equation, for the J metric variable of the Bondi-Sachs metric. Once β , another Bondi-Sachs potential, is obtained from the field equations, and J is obtained from the master equation, the other metric variables are solved integrating directly the rest of the field equations. In the past, the master equation was solved for the first multipolar terms, for both the Minkowski's and Schwarzschild's backgrounds. Also, Mädler recently reported a generalisation of the exact solutions to the linearised field equations when a Minkowski's background is considered, expressing the master equation family of solutions for the vacuum in terms of Bessel's functions of the first and the second kind. Here, we report new solutions to the master equation for any multipolar moment l , with and without matter sources in terms only of the first kind Bessel's functions for the Minkowski, and in terms of the Confluent Heun's functions (Generalised Hypergeometric) for radiative (nonradiative) case in the Schwarzschild's background. We particularize our families of solutions for the known cases for l =2 reported previously in the literature and find complete agreement, showing the robustness of our results.

  19. An efficient parallel algorithm for the solution of a tridiagonal linear system of equations

    NASA Technical Reports Server (NTRS)

    Stone, H. S.

    1971-01-01

    Tridiagonal linear systems of equations are solved on conventional serial machines in a time proportional to N, where N is the number of equations. The conventional algorithms do not lend themselves directly to parallel computations on computers of the ILLIAC IV class, in the sense that they appear to be inherently serial. An efficient parallel algorithm is presented in which computation time grows as log sub 2 N. The algorithm is based on recursive doubling solutions of linear recurrence relations, and can be used to solve recurrence relations of all orders.

  20. A Piecewise Linear Finite Element Discretization of the Diffusion Equation for Arbitrary Polyhedral Grids

    SciTech Connect

    Bailey, T S; Adams, M L; Yang, B; Zika, M R

    2005-07-15

    We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses piecewise linear weight and basis functions in the finite element approximation, and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We show that this new PWL method gives solutions comparable to those from Palmer's finite-volume method. However, since the PWL method produces a symmetric positive definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids.

  1. An efficient parallel algorithm for the solution of a tridiagonal linear system of equations.

    NASA Technical Reports Server (NTRS)

    Stone, H. S.

    1973-01-01

    Tridiagonal linear systems of equations can be solved on conventional serial machines in a time proportional to N, where N is the number of equations. The conventional algorithms do not lend themselves directly to parallel computation on computers of the Illiac IV class, in the sense that they appear to be inherently serial. An efficient parallel algorithm is presented in which computation time grows as log(sub-2) N. The algorithm is based on recursive doubling solutions of linear recurrence relations, and can be used to solve recurrence relations of all orders.

  2. Analysis and accurate numerical solutions of the integral equation derived from the linearized BGKW equation for the steady Couette flow

    NASA Astrophysics Data System (ADS)

    Jiang, Shidong; Luo, Li-Shi

    2016-07-01

    The integral equation for the flow velocity u (x ; k) in the steady Couette flow derived from the linearized Bhatnagar-Gross-Krook-Welander kinetic equation is studied in detail both theoretically and numerically in a wide range of the Knudsen number k between 0.003 and 100.0. First, it is shown that the integral equation is a Fredholm equation of the second kind in which the norm of the compact integral operator is less than 1 on Lp for any 1 ≤ p ≤ ∞ and thus there exists a unique solution to the integral equation via the Neumann series. Second, it is shown that the solution is logarithmically singular at the endpoints. More precisely, if x = 0 is an endpoint, then the solution can be expanded as a double power series of the form ∑n=0∞∑m=0∞cn,mxn(xln ⁡ x) m about x = 0 on a small interval x ∈ (0 , a) for some a > 0. And third, a high-order adaptive numerical algorithm is designed to compute the solution numerically to high precision. The solutions for the flow velocity u (x ; k), the stress Pxy (k), and the half-channel mass flow rate Q (k) are obtained in a wide range of the Knudsen number 0.003 ≤ k ≤ 100.0; and these solutions are accurate for at least twelve significant digits or better, thus they can be used as benchmark solutions.

  3. On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer

    NASA Technical Reports Server (NTRS)

    Hu, Fang Q.

    1995-01-01

    Recently, Berenger introduced a Perfectly Matched Layer (PML) technique for absorbing electromagnetic waves. In the present paper, a perfectly matched layer is proposed for absorbing out-going two-dimensional waves in a uniform mean flow, generated by linearized Euler equations. It is well known that the linearized Euler equations support acoustic waves, which travel with the speed of sound relative to the mean flow, and vorticity and entropy waves, which travel with the mean flow. The PML equations to be used at a region adjacent to the artificial boundary for absorbing these linear waves are defined. Plane waves solutions to the PML equations are developed and wave propagation and absorption properties are given. It is shown that the theoretical reflection coefficients at an interface between the Euler and PML domains are zero, independent of the angle of incidence and frequency of the waves. As such, the present study points out a possible alternative approach for absorbing out-going waves of the Euler equations with little or no reflection in computation. Numerical examples that demonstrate the validity of the proposed PML equations are also presented.

  4. Quasi-linear equation for magnetoplasma oscillations in the weakly relativistic approximation

    NASA Astrophysics Data System (ADS)

    Rizzato, F. B.

    Some limitations which are present in the dynamical equations for collisionless plasmas are discussed. Some elementary corrections to the linear theories are obtained in a heuristic form, which directly lead to the so-called quasi-linear theories in its non-relativistic and relativistic forms. The effect of the relativistic variation of the gyrofrequency on the diffusion coefficient is examined in a typically perturbative approximation.

  5. Green's function-stochastic methods framework for probing nonlinear evolution problems: Burger's equation, the nonlinear Schroedinger's equation, and hydrodynamic organization of near-molecular-scale vorticity

    SciTech Connect

    Keanini, R.G.

    2011-04-15

    Research Highlights: > Systematic approach for physically probing nonlinear and random evolution problems. > Evolution of vortex sheets corresponds to evolution of an Ornstein-Uhlenbeck process. > Organization of near-molecular scale vorticity mediated by hydrodynamic modes. > Framework allows calculation of vorticity evolution within random strain fields. - Abstract: A framework which combines Green's function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green's function and stochastic representative solutions of linear drift-diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems - Burgers' equation and the nonlinear Schroedinger's equation - are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole-Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger's vortex sheets. Here, the governing vorticity equation corresponds to the Fokker-Planck equation of an Ornstein-Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the motion

  6. Distribution of error in least-squares solution of an overdetermined system of linear simultaneous equations

    NASA Technical Reports Server (NTRS)

    Miller, C. D.

    1972-01-01

    Probability density functions were derived for errors in the evaluation of unknowns by the least squares method in system of nonhomogeneous linear equations. Coefficients of the unknowns were assumed correct and computational precision were also assumed. A vector space was used, with number of dimensions equal to the number of equations. An error vector was defined and assumed to have uniform distribution of orientation throughout the vector space. The density functions are shown to be insensitive to the biasing effects of the source of the system of equations.

  7. FAST TRACK COMMUNICATION: On the Liouvillian solution of second-order linear differential equations and algebraic invariant curves

    NASA Astrophysics Data System (ADS)

    Man, Yiu-Kwong

    2010-10-01

    In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle-Singer method for computing first integrals of differential equations. Some examples on using this approach are provided.

  8. The primer vector in linear, relative-motion equations. [spacecraft trajectory optimization

    NASA Technical Reports Server (NTRS)

    1980-01-01

    Primer vector theory is used in analyzing a set of linear, relative-motion equations - the Clohessy-Wiltshire equations - to determine the criteria and necessary conditions for an optimal, N-impulse trajectory. Since the state vector for these equations is defined in terms of a linear system of ordinary differential equations, all fundamental relations defining the solution of the state and costate equations, and the necessary conditions for optimality, can be expressed in terms of elementary functions. The analysis develops the analytical criteria for improving a solution by (1) moving any dependent or independent variable in the initial and/or final orbit, and (2) adding intermediate impulses. If these criteria are violated, the theory establishes a sufficient number of analytical equations. The subsequent satisfaction of these equations will result in the optimal position vectors and times of an N-impulse trajectory. The solution is examined for the specific boundary conditions of (1) fixed-end conditions, two-impulse, and time-open transfer; (2) an orbit-to-orbit transfer; and (3) a generalized rendezvous problem. A sequence of rendezvous problems is solved to illustrate the analysis and the computational procedure.

  9. First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity

    NASA Technical Reports Server (NTRS)

    Cai, Z.; Manteuffel, T. A.; McCormick, S. F.

    1996-01-01

    Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two- and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity in an H(exp 1) product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity with estimates that are uniform in the Lame constants.

  10. Validity and slopes of the linear equation of state for natural brines in salt lake systems

    NASA Astrophysics Data System (ADS)

    Kohfahl, C.; Post, V. E. A.; Hamann, E.; Prommer, H.; Simmons, C. T.

    2015-04-01

    Many density-dependent groundwater flow simulations rely on a linear equation of state that relates the fluid density to the total dissolved solute content (TDS). This approach ignores non-linear volume of mixing effects, as well as the impact of any chemical reactions. These effects can be considered by using geochemical codes that implement algorithms that calculate the density of a fluid based on the concentration of individual solute species. While in principle such algorithms could be used in-lieu of a linear equation of state in a groundwater model, the computational overhead is such that the use of a more simplified equation of state is preferred. This requires that the assumption of linearity as well as the appropriate value of the linear slope have to be determined. Here, published density measurements of 7 chemically-distinct salt lake brines are compared with densities calculated by PHREEQC-3, confirming the applicability of PHREEQC's algorithm to salt lake brines, as well as to seawater brines and artificial brines from laboratory experiments. Further, calculations with PHREEQC-3 are used to assess the impact of mineral precipitation reactions during evaporative concentration. Results show that the density-TDS relationship is essentially linear over a wide concentration range, and that slopes range between 0.64 and 0.75, with the upper end of the range applying to Na-CO3-Cl brines and the lower end to Na-Cl brines. Mineral precipitation of highly-soluble evaporate minerals such as halite and trona limit TDS, and may lead to considerable errors in coupled flow simulations based on a linear equation of state at high concentrations. Misrepresentation of the slope may lead to an error of up to 20% in the calculated length of the brine nose bordering a salt lake, or of the Rayleigh number, which indicates if a density stratification is stable or not.

  11. Solving Linear Equations: A Comparison of Concrete and Virtual Manipulatives in Middle School Mathematics

    ERIC Educational Resources Information Center

    Magruder, Robin Lee

    2012-01-01

    The purpose of this embedded quasi-experimental mixed methods research was to use solving simple linear equations as the lens for looking at the effectiveness of concrete and virtual manipulatives as compared to a control group using learning methods without manipulatives. Further, the researcher wanted to investigate unique benefits and drawbacks…

  12. Analysis of the linear stability of compressible boundary layers using the PSE. [parabolic stability equations

    NASA Technical Reports Server (NTRS)

    Bertolotti, F. P.; Herbert, TH.

    1991-01-01

    The application of linearized parabolic stability equations (PSE) to compressible flow is considered. The effect of mean-flow nonparallelism is found to be weak on 2D waves and strong on 3D waves. Results for a single choice of free-stream parameters that corresponds to the atmospheric conditions at 15,000 m above sea level are presented.

  13. Generalization of the relaxation method for the inverse solution of nonlinear and linear transfer equations

    NASA Technical Reports Server (NTRS)

    Chahine, M. T.

    1977-01-01

    A mapping transformation is derived for the inverse solution of nonlinear and linear integral equations of the types encountered in remote sounding studies. The method is applied to the solution of specific problems for the determination of the thermal and composition structure of planetary atmospheres from a knowledge of their upwelling radiance.

  14. A Graphical Method for Assessing the Identification of Linear Structural Equation Models

    ERIC Educational Resources Information Center

    Eusebi, Paolo

    2008-01-01

    A graphical method is presented for assessing the state of identifiability of the parameters in a linear structural equation model based on the associated directed graph. We do not restrict attention to recursive models. In the recent literature, methods based on graphical models have been presented as a useful tool for assessing the state of…

  15. The Multifaceted Variable Approach: Selection of Method in Solving Simple Linear Equations

    ERIC Educational Resources Information Center

    Tahir, Salma; Cavanagh, Michael

    2010-01-01

    This paper presents a comparison of the solution strategies used by two groups of Year 8 students as they solved linear equations. The experimental group studied algebra following a multifaceted variable approach, while the comparison group used a traditional approach. Students in the experimental group employed different solution strategies,…

  16. Effect of Coannular Flow on Linearized Euler Equation Predictions of Jet Noise

    NASA Technical Reports Server (NTRS)

    Hixon, R.; Shih, S.-H.; Mankbadi, Reda R.

    1997-01-01

    An improved version of a previously validated linearized Euler equation solver is used to compute the noise generated by coannular supersonic jets. Results for a single supersonic jet are compared to the results from both a normal velocity profile and an inverted velocity profile supersonic jet.

  17. Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients.

    PubMed

    Boyko, Vyacheslav M; Popovych, Roman O; Shapoval, Nataliya M

    2013-01-01

    Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach. PMID:23564972

  18. The Green's function for the three-dimensional linear Boltzmann equation via Fourier transform

    NASA Astrophysics Data System (ADS)

    Machida, Manabu

    2016-04-01

    The linear Boltzmann equation with constant coefficients in the three-dimensional infinite space is revisited. It is known that the Green's function can be calculated via the Fourier transform in the case of isotropic scattering. In this paper, we show that the three-dimensional Green's function can be computed with the Fourier transform even in the case of arbitrary anisotropic scattering.

  19. Futility of Log-Linear Smoothing When Equating with Unrepresentative Small Samples

    ERIC Educational Resources Information Center

    Puhan, Gautam

    2011-01-01

    The impact of log-linear presmoothing on the accuracy of small sample chained equipercentile equating was evaluated under two conditions. In the first condition the small samples differed randomly in ability from the target population. In the second condition the small samples were systematically different from the target population. Results…

  20. Solving System Of Linear Equations Using The Bimodal Optical Computer (Experimental Results)

    NASA Astrophysics Data System (ADS)

    Habli, M. A.; Abushagur, M. A. G.; Caulfield, H. J.

    1988-08-01

    Hardware and software design of the Bimodal Optical Computer (BOC) and its implementations are presented. Experimental results of the BOC for solving a system of linear equations Ax = b is reported. The effect of calibration, the convergence reliability of the BOC, and the convergence of problems with singular matrices are studied.

  1. Standard Error of Linear Observed-Score Equating for the NEAT Design with Nonnormally Distributed Data

    ERIC Educational Resources Information Center

    Zu, Jiyun; Yuan, Ke-Hai

    2012-01-01

    In the nonequivalent groups with anchor test (NEAT) design, the standard error of linear observed-score equating is commonly estimated by an estimator derived assuming multivariate normality. However, real data are seldom normally distributed, causing this normal estimator to be inconsistent. A general estimator, which does not rely on the…

  2. The Integration of Teacher's Pedagogical Content Knowledge Components in Teaching Linear Equation

    ERIC Educational Resources Information Center

    Yusof, Yusminah Mohd.; Effandi, Zakaria

    2015-01-01

    This qualitative research aimed to explore the integration of the components of pedagogical content knowledge (PCK) in teaching Linear Equation with one unknown. For the purpose of the study, a single local case study with multiple participants was used. The selection of the participants was made based on various criteria: having more than 5 years…

  3. Lines of Eigenvectors and Solutions to Systems of Linear Differential Equations

    ERIC Educational Resources Information Center

    Rasmussen, Chris; Keynes, Michael

    2003-01-01

    The purpose of this paper is to describe an instructional sequence where students invent a method for locating lines of eigenvectors and corresponding solutions to systems of two first order linear ordinary differential equations with constant coefficients. The significance of this paper is two-fold. First, it represents an innovative alternative…

  4. On the equivalence of a class of inverse decomposition algorithms for solving systems of linear equations

    NASA Technical Reports Server (NTRS)

    Tsao, Nai-Kuan

    1989-01-01

    A class of direct inverse decomposition algorithms for solving systems of linear equations is presented. Their behavior in the presence of round-off errors is analyzed. It is shown that under some mild restrictions on their implementation, the class of direct inverse decomposition algorithms presented are equivalent in terms of the error complexity measures.

  5. Flipping an Algebra Classroom: Analyzing, Modeling, and Solving Systems of Linear Equations

    ERIC Educational Resources Information Center

    Kirvan, Rebecca; Rakes, Christopher R.; Zamora, Regie

    2015-01-01

    The present study investigated whether flipping an algebra classroom led to a stronger focus on conceptual understanding and improved learning of systems of linear equations for 54 seventh- and eighth-grade students using teacher journal data and district-mandated unit exam items. Multivariate analysis of covariance was used to compare scores on…

  6. Using the Number Line to Investigate the Solving of Linear Equations

    ERIC Educational Resources Information Center

    Dickinson, Paul; Eade, Frank

    2004-01-01

    The curriculum for eleven-year old students in the United Kingdom, currently adopted by most schools, includes solving linear equations with the unknown on one side only before moving onto those with the unknown on both sides in later years. School textbooks struggle with the balance between developing algebraic understanding and training…

  7. The method of local linear approximation in the theory of nonlinear functional-differential equations

    SciTech Connect

    Slyusarchuk, Vasilii E

    2010-10-06

    Conditions for the existence of solutions to the nonlinear functional-differential equation (d{sup m}x(t))/dt{sup m} + (fx)(t)=h(t), t element of R in the space of functions bounded on the axes are obtained by using local linear approximation to the operator F. Bibliography: 21 items.

  8. An Empirical Comparison of Five Linear Equating Methods for the NEAT Design

    ERIC Educational Resources Information Center

    Suh, Youngsuk; Mroch, Andrew A.; Kane, Michael T.; Ripkey, Douglas R.

    2009-01-01

    In this study, a data base containing the responses of 40,000 candidates to 90 multiple-choice questions was used to mimic data sets for 50-item tests under the "nonequivalent groups with anchor test" (NEAT) design. Using these smaller data sets, we evaluated the performance of five linear equating methods for the NEAT design with five levels of…

  9. Solution of the Schrodinger Equation for a Diatomic Oscillator Using Linear Algebra: An Undergraduate Computational Experiment

    ERIC Educational Resources Information Center

    Gasyna, Zbigniew L.

    2008-01-01

    Computational experiment is proposed in which a linear algebra method is applied to the solution of the Schrodinger equation for a diatomic oscillator. Calculations of the vibration-rotation spectrum for the HCl molecule are presented and the results show excellent agreement with experimental data. (Contains 1 table and 1 figure.)

  10. The Use of Graphs in Specific Situations of the Initial Conditions of Linear Differential Equations

    ERIC Educational Resources Information Center

    Buendía, Gabriela; Cordero, Francisco

    2013-01-01

    In this article, we present a discussion on the role of graphs and its significance in the relation between the number of initial conditions and the order of a linear differential equation, which is known as the initial value problem. We propose to make a functional framework for the use of graphs that intends to broaden the explanations of the…

  11. A new approach to parton recombination in the QCD evolution equations

    NASA Astrophysics Data System (ADS)

    Wei Zhu

    1999-06-01

    Parton recombination is reconsidered in perturbation theory without using the AGK cutting rules in the leading order of the recombination. We use time-ordered perturbation theory to sum the cut diagrams, which are neglected in the GLR evolution equation. We present a set of new evolution equations including parton recombination.

  12. The cosmic microwave background bispectrum from the non-linear evolution of the cosmological perturbations

    SciTech Connect

    Pitrou, Cyril; Uzan, Jean-Philippe; Bernardeau, Francis E-mail: uzan@iap.fr

    2010-07-01

    This article presents the first computation of the complete bispectrum of the cosmic microwave background temperature anisotropies arising from the evolution of all cosmic fluids up to second order, including neutrinos. Gravitational couplings, electron density fluctuations and the second order Boltzmann equation are fully taken into account. Comparison to limiting cases that appeared previously in the literature are provided. These are regimes for which analytical insights can be given. The final results are expressed in terms of equivalent f{sub NL} for different configurations. It is found that for moments up to l{sub max} = 2000, the signal generated by non-linear effects is equivalent to f{sub NL} ≅ 5 for both local-type and equilateral-type primordial non-Gaussianity.

  13. The linear Boltzmann equation in slab geometry - Development and verification of a reliable and efficient solution

    NASA Technical Reports Server (NTRS)

    Stamnes, K.; Lie-Svendsen, O.; Rees, M. H.

    1991-01-01

    The linear Boltzmann equation can be cast in a form mathematically identical to the radiation-transport equation. A multigroup procedure is used to reduce the energy (or velocity) dependence of the transport equation to a series of one-speed problems. Each of these one-speed problems is equivalent to the monochromatic radiative-transfer problem, and existing software is used to solve this problem in slab geometry. The numerical code conserves particles in elastic collisions. Generic examples are provided to illustrate the applicability of this approach. Although this formalism can, in principle, be applied to a variety of test particle or linearized gas dynamics problems, it is particularly well-suited to study the thermalization of suprathermal particles interacting with a background medium when the thermal motion of the background cannot be ignored. Extensions of the formalism to include external forces and spherical geometry are also feasible.

  14. Analytical solution of boundary integral equations for 2-D steady linear wave problems

    NASA Astrophysics Data System (ADS)

    Chuang, J. M.

    2005-10-01

    Based on the Fourier transform, the analytical solution of boundary integral equations formulated for the complex velocity of a 2-D steady linear surface flow is derived. It has been found that before the radiation condition is imposed, free waves appear both far upstream and downstream. In order to cancel the free waves in far upstream regions, the eigensolution of a specific eigenvalue, which satisfies the homogeneous boundary integral equation, is found and superposed to the analytical solution. An example, a submerged vortex, is used to demonstrate the derived analytical solution. Furthermore, an analytical approach to imposing the radiation condition in the numerical solution of boundary integral equations for 2-D steady linear wave problems is proposed.

  15. Solution of a singularly perturbed Cauchy problem for linear systems of ordinary differential equations by the method of spectral decomposition

    NASA Astrophysics Data System (ADS)

    Shaldanbayev, Amir; Shomanbayeva, Manat; Kopzhassarova, Asylzat

    2016-08-01

    This paper proposes a fundamentally new method of investigation of a singularly perturbed Cauchy problem for a linear system of ordinary differential equations based on the spectral theory of equations with deviating argument.

  16. Equations of State: Gateway to Planetary Origin and Evolution (Invited)

    NASA Astrophysics Data System (ADS)

    Melosh, J.

    2013-12-01

    Research over the past decades has shown that collisions between solid bodies govern many crucial phases of planetary origin and evolution. The accretion of the terrestrial planets was punctuated by planetary-scale impacts that generated deep magma oceans, ejected primary atmospheres and probably created the moons of Earth and Pluto. Several extrasolar planetary systems are filled with silicate vapor and condensed 'tektites', probably attesting to recent giant collisions. Even now, long after the solar system settled down from its violent birth, a large asteroid impact wiped out the dinosaurs, while other impacts may have played a role in the origin of life on Earth and perhaps Mars, while maintaining a steady exchange of small meteorites between the terrestrial planets and our moon. Most of these events are beyond the scale at which experiments are possible, so that our main research tool is computer simulation, constrained by the laws of physics and the behavior of materials during high-speed impact. Typical solar system impact velocities range from a few km/s in the outer solar system to 10s of km/s in the inner system. Extrasolar planetary systems expand that range to 100s of km/sec typical of the tightly clustered planetary systems now observed. Although computer codes themselves are currently reaching a high degree of sophistication, we still rely on experimental studies to determine the Equations of State (EoS) of materials critical for the correct simulation of impact processes. The recent expansion of the range of pressures available for study, from a few 100 GPa accessible with light gas guns up to a few TPa from current high energy accelerators now opens experimental access to the full velocity range of interest in our solar system. The results are a surprise: several groups in both the USA and Japan have found that silicates and even iron melt and vaporize much more easily in an impact than previously anticipated. The importance of these findings is

  17. Linearized model collision operators for multiple ion species plasmas and gyrokinetic entropy balance equations

    SciTech Connect

    Sugama, H.; Watanabe, T.-H.; Nunami, M.

    2009-11-15

    Linearized model collision operators for multiple ion species plasmas are presented that conserve particles, momentum, and energy and satisfy adjointness relations and Boltzmann's H-theorem even for collisions between different particle species with unequal temperatures. The model collision operators are also written in the gyrophase-averaged form that can be applied to the gyrokinetic equation. Balance equations for the turbulent entropy density, the energy of electromagnetic fluctuations, the turbulent transport fluxes of particle and heat, and the collisional dissipation are derived from the gyrokinetic equation including the collision term and Maxwell equations. It is shown that, in the steady turbulence, the entropy produced by the turbulent transport fluxes is dissipated in part by collisions in the nonzonal-mode region and in part by those in the zonal-mode region after the nonlinear entropy transfer from nonzonal to zonal modes.

  18. On a hierarchy of nonlinearly dispersive generalized Korteweg - de Vries evolution equations

    SciTech Connect

    Christov, Ivan C.

    2015-08-20

    We propose a hierarchy of nonlinearly dispersive generalized Korteweg–de Vries (KdV) evolution equations based on a modification of the Lagrangian density whose induced action functional the KdV equation extremizes. Two recent nonlinear evolution equations describing wave propagation in certain generalized continua with an inherent material length scale are members of the proposed hierarchy. Like KdV, the equations from the proposed hierarchy possess Hamiltonian structure. Unlike KdV, the solutions to these equations can be compact (i.e., they vanish outside of some open interval) and, in addition, peaked. Implicit solutions for these peaked, compact traveling waves (“peakompactons”) are presented.

  19. Operational method of solution of linear non-integer ordinary and partial differential equations.

    PubMed

    Zhukovsky, K V

    2016-01-01

    We propose operational method with recourse to generalized forms of orthogonal polynomials for solution of a variety of differential equations of mathematical physics. Operational definitions of generalized families of orthogonal polynomials are used in this context. Integral transforms and the operational exponent together with some special functions are also employed in the solutions. The examples of solution of physical problems, related to such problems as the heat propagation in various models, evolutional processes, Black-Scholes-like equations etc. are demonstrated by the operational technique. PMID:26900541

  20. About simple nonlinear and linear superpositions of special exact solutions of Veselov-Novikov equation

    SciTech Connect

    Dubrovsky, V. G.; Topovsky, A. V.

    2013-03-15

    New exact solutions, nonstationary and stationary, of Veselov-Novikov (VN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u{sup (n)}, n= 1, Horizontal-Ellipsis , N are constructed via Zakharov and Manakov {partial_derivative}-dressing method. Simple nonlinear superpositions are represented up to a constant by the sums of solutions u{sup (n)} and calculated by {partial_derivative}-dressing on nonzero energy level of the first auxiliary linear problem, i.e., 2D stationary Schroedinger equation. It is remarkable that in the zero energy limit simple nonlinear superpositions convert to linear ones in the form of the sums of special solutions u{sup (n)}. It is shown that the sums u=u{sup (k{sub 1})}+...+u{sup (k{sub m})}, 1 Less-Than-Or-Slanted-Equal-To k{sub 1} < k{sub 2} < Horizontal-Ellipsis < k{sub m} Less-Than-Or-Slanted-Equal-To N of arbitrary subsets of these solutions are also exact solutions of VN equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions. By construction these exact solutions represent also new exact transparent potentials of 2D stationary Schroedinger equation and can serve as model potentials for electrons in planar structures of modern electronics.

  1. About simple nonlinear and linear superpositions of special exact solutions of Veselov-Novikov equation

    NASA Astrophysics Data System (ADS)

    Dubrovsky, V. G.; Topovsky, A. V.

    2013-03-01

    New exact solutions, nonstationary and stationary, of Veselov-Novikov (VN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u(n), n = 1, …, N are constructed via Zakharov and Manakov overline{partial }-dressing method. Simple nonlinear superpositions are represented up to a constant by the sums of solutions u(n) and calculated by overline{partial }-dressing on nonzero energy level of the first auxiliary linear problem, i.e., 2D stationary Schrödinger equation. It is remarkable that in the zero energy limit simple nonlinear superpositions convert to linear ones in the form of the sums of special solutions u(n). It is shown that the sums u= u^{(k_1)}+ldots + u^{(k_m)}, 1 ⩽ k1 < k2 < … < km ⩽ N of arbitrary subsets of these solutions are also exact solutions of VN equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions. By construction these exact solutions represent also new exact transparent potentials of 2D stationary Schrödinger equation and can serve as model potentials for electrons in planar structures of modern electronics.

  2. Solution of dense systems of linear equations in electromagnetic scattering calculations

    SciTech Connect

    Rahola, J.

    1994-12-31

    The discrete-dipole approximation (DDA) is a method for calculating the scattering of light by an irregular particle. The DDA has been used for example in calculations of optical properties of cosmic dust. In this method the particle is approximated by interacting electromagnetic dipoles. Computationally the DDA method includes the solution of large dense systems of linear equations where the coefficient matrix is complex symmetric. In the author`s work, the linear systems of equations are solved by various iterative methods such as the conjugate gradient method applied to the normal equations and QMR. The linear systems have rather low condition numbers due to which many iterative methods perform quite well even without any preconditioning. Some possible preconditioning strategies are discussed. Finally, some fast special methods for computing the matrix-vector product in the iterative methods are considered. In some cases, the matrix-vector product can be computed with the fast Fourier transform, which enables the author to solve dense linear systems of hundreds of thousands of unknowns.

  3. Solving large-scale sparse eigenvalue problems and linear systems of equations for accelerator modeling

    SciTech Connect

    Gene Golub; Kwok Ko

    2009-03-30

    The solutions of sparse eigenvalue problems and linear systems constitute one of the key computational kernels in the discretization of partial differential equations for the modeling of linear accelerators. The computational challenges faced by existing techniques for solving those sparse eigenvalue problems and linear systems call for continuing research to improve on the algorithms so that ever increasing problem size as required by the physics application can be tackled. Under the support of this award, the filter algorithm for solving large sparse eigenvalue problems was developed at Stanford to address the computational difficulties in the previous methods with the goal to enable accelerator simulations on then the world largest unclassified supercomputer at NERSC for this class of problems. Specifically, a new method, the Hemitian skew-Hemitian splitting method, was proposed and researched as an improved method for solving linear systems with non-Hermitian positive definite and semidefinite matrices.

  4. Second order non-linear equations of motion for spinning highly flexible line elements

    NASA Technical Reports Server (NTRS)

    Salama, M.; Trubert, M.; Essawi, M.; Utku, S.

    1982-01-01

    The second order nonlinear equations of motion are formulated for spinning line elements having little or no intrinsic structural stiffness. The derivation is based on the extended Hamilton's principle and includes the effect of initial geometric imperfections (axial, curvature, and twist) on the line element dynamics. For comparison with previous work, the nonlinear equations are reduced to a linearized form frequently found in the literature. The comparison revealed several new spin-stiffening terms that have not been previously identified and/or retained. They combine geometric imperfections, rotary inertia, Coriolis, and gyroscopic terms.

  5. Parallel FE Approximation of the Even/Odd Parity Form of the Linear Boltzmann Equation

    SciTech Connect

    Drumm, Clifton R.; Lorenz, Jens

    1999-07-21

    A novel solution method has been developed to solve the linear Boltzmann equation on an unstructured triangular mesh. Instead of tackling the first-order form of the equation, this approach is based on the even/odd-parity form in conjunction with the conventional mdtigroup discrete-ordinates approximation. The finite element method is used to treat the spatial dependence. The solution method is unique in that the space-direction dependence is solved simultaneously, eliminating the need for the conventional inner iterations, and the method is well suited for massively parallel computers.

  6. Non-classical symmetries and invariant solutions of non-linear Dirac equations

    NASA Astrophysics Data System (ADS)

    Rocha, P. M. M.; Khanna, F. C.; Rocha Filho, T. M.; Santana, A. E.

    2015-09-01

    We apply Lie and non-classical symmetry methods to partial differential equations in order to derive solutions of the non-linear Dirac equation corresponding to the Gross-Neveu model in d = (1 + 1) and d = (2 + 1) space-time dimensions. For each d, we first identify sub-algebras of the Poincaré-Lie algebra and for each such sub-algebra, we calculate the invariant solution. Non-classical symmetries are also determined and used to derive new solutions for the Gross-Neveu model.

  7. Analytical solutions of linear and nonlinear Klein-Fock-Gordon equation

    NASA Astrophysics Data System (ADS)

    Khan, Najeeb Alam; Rasheed, Sajida

    2015-03-01

    In this paper, we deal with some linear and nonlinear Klein-Fock-Gordon (KFG) equations, which is a relativistic version of the Schrödinger equation. The approximate analytical solutions are obtained by using the homotopy analysis method (HAM). The efficiency of the HAM is that it provides a practical way to control the convergence region of series solutions by introducing an auxiliary parameter }. Analytical results presented are in agreement with the existing results in open literature, which confirm the effectiveness of this method.

  8. On exponential stability of linear Levin-Nohel integro-differential equations

    NASA Astrophysics Data System (ADS)

    Tien Dung, Nguyen

    2015-02-01

    The aim of this paper is to investigate the exponential stability for linear Levin-Nohel integro-differential equations with time-varying delays. To the best of our knowledge, the exponential stability for such equations has not yet been discussed. In addition, since we do not require that the kernel and delay are continuous, our results improve those obtained in Becker and Burton [Proc. R. Soc. Edinburgh, Sect. A: Math. 136, 245-275 (2006)]; Dung [J. Math. Phys. 54, 082705 (2013)]; and Jin and Luo [Comput. Math. Appl. 57(7), 1080-1088 (2009)].

  9. A bivariate Chebyshev spectral collocation quasilinearization method for nonlinear evolution parabolic equations.

    PubMed

    Motsa, S S; Magagula, V M; Sibanda, P

    2014-01-01

    This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature. PMID:25254252

  10. A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations

    PubMed Central

    Motsa, S. S.; Magagula, V. M.; Sibanda, P.

    2014-01-01

    This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature. PMID:25254252