Time-course window estimator for ordinary differential equations linear in the parameters

NARCIS (Netherlands)

Vujacic, Ivan; Dattner, Itai; Gonzalez, Javier; Wit, Ernst

In many applications obtaining ordinary differential equation descriptions of dynamic processes is scientifically important. In both, Bayesian and likelihood approaches for estimating parameters of ordinary differential equations, the speed and the convergence of the estimation procedure may

A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

Science.gov (United States)

Camporesi, Roberto

2016-01-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order 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 the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The approach presented here can be used in a first course on differential equations for science and engineering majors.

Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

Directory of Open Access Journals (Sweden)

Rutwig Campoamor-Stursberg

2016-03-01

Full Text Available A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems.

A Unified Introduction to Ordinary Differential Equations

Science.gov (United States)

Lutzer, Carl V.

2006-01-01

This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)

Analytical exact solution of the non-linear Schroedinger equation

International Nuclear Information System (INIS)

Martins, Alisson Xavier; Rocha Filho, Tarcisio Marciano da

2011-01-01

Full text: In this work we present how to classify and obtain analytical solutions of the Schroedinger equation with a generic non-linearity in 1+1 dimensions. Our approach is based on the determination of Lie symmetry transformation mapping solutions into solutions, and non-classical symmetry transformations, mapping a given solution into itself. From these symmetries it is then possible to reduce the equation to a system of ordinary differential equations which can then be solved using standard methods. The generic non-linearity is handled by considering it as an additional unknown in the determining equations for the symmetry transformations. This results in an over-determined system of non-linear partial differential equations. Its solution can then be determined in some cases by reducing it to the so called involutive (triangular) form, and then solved. This reduction is very tedious and can only performed using a computer algebra system. Once the determining system is solved, we obtain the explicit form for the non-linearity admitting a Lie or non-classical symmetry. The analytical solutions are then derived by solving the reduced ordinary differential equations. The non-linear determining system for the non-classical symmetry transformations and Lie symmetry generators are obtaining using the computer algebra package SADE (symmetry analysis of differential equations), developed at our group. (author)

LSODE, 1. Order Stiff or Non-Stiff Ordinary Differential Equations System Initial Value Problems

International Nuclear Information System (INIS)

Hindmarsh, A.C.; Petzold, L.R.

2005-01-01

1 - Description of program or function: LSODE (Livermore Solver for Ordinary Differential Equations) solves stiff and non-stiff systems of the form dy/dt = f. In the stiff case, it treats the Jacobian matrix df/dy as either a dense (full) or a banded matrix, and as either user-supplied or internally approximated by difference quotients. It uses Adams methods (predictor-corrector) in the non-stiff case, and Backward Differentiation Formula (BDF) methods (the Gear methods) in the stiff case. The linear systems that arise are solved by direct methods (LU factor/solve). The LSODE source is commented extensively to facilitate modification. Both a single-precision version and a double-precision version are available. 2 - Methods: It is assumed that the ODEs are given explicitly, so that the system can be written in the form dy/dt = f(t,y), where y is the vector of dependent variables, and t is the independent variable. LSODE contains two variable-order, variable- step (with interpolatory step-changing) integration methods. The first is the implicit Adams or non-stiff method, of orders one through twelve. The second is the backward differentiation or stiff method (or BDF method, or Gear's method), of orders one through five. 3 - Restrictions on the complexity of the problem: The differential equations must be given in explicit form, i.e., dy/dt = f(y,t). Problems with intermittent high-speed transients may cause inefficient or unstable performance

Exploring inductive linearization for pharmacokinetic-pharmacodynamic systems of nonlinear ordinary differential equations.

Science.gov (United States)

Hasegawa, Chihiro; Duffull, Stephen B

2018-02-01

Pharmacokinetic-pharmacodynamic systems are often expressed with nonlinear ordinary differential equations (ODEs). While there are numerous methods to solve such ODEs these methods generally rely on time-stepping solutions (e.g. Runge-Kutta) which need to be matched to the characteristics of the problem at hand. The primary aim of this study was to explore the performance of an inductive approximation which iteratively converts nonlinear ODEs to linear time-varying systems which can then be solved algebraically or numerically. The inductive approximation is applied to three examples, a simple nonlinear pharmacokinetic model with Michaelis-Menten elimination (E1), an integrated glucose-insulin model and an HIV viral load model with recursive feedback systems (E2 and E3, respectively). The secondary aim of this study was to explore the potential advantages of analytically solving linearized ODEs with two examples, again E3 with stiff differential equations and a turnover model of luteinizing hormone with a surge function (E4). The inductive linearization coupled with a matrix exponential solution provided accurate predictions for all examples with comparable solution time to the matched time-stepping solutions for nonlinear ODEs. The time-stepping solutions however did not perform well for E4, particularly when the surge was approximated by a square wave. In circumstances when either a linear ODE is particularly desirable or the uncertainty in matching the integrator to the ODE system is of potential risk, then the inductive approximation method coupled with an analytical integration method would be an appropriate alternative.

Ordinary differential equations principles and applications

CERN Document Server

Nandakumaran, A K; George, Raju K

2017-01-01

Written in a clear, logical and concise manner, this comprehensive resource allows students to quickly understand the key principles, techniques and applications of ordinary differential equations. Important topics including first and second order linear equations, initial value problems and qualitative theory are presented in separate chapters. The concepts of two point boundary value problems, physical models and first order partial differential equations are discussed in detail. The text uses tools of calculus and real analysis to get solutions in explicit form. While discussing first order linear systems, linear algebra techniques are used. The real-life applications are interspersed throughout the book to invoke reader's interest. The methods and tricks to solve numerous mathematical problems with sufficient derivations and explanation are provided. The proofs of theorems are explained for the benefit of the readers.

Higher derivative discontinuous solutions to linear ordinary differential equations: a new route to complexity?

International Nuclear Information System (INIS)

Datta, Dhurjati Prasad; Bose, Manoj Kumar

2004-01-01

We present a new one parameter family of second derivative discontinuous solutions to the simplest scale invariant linear ordinary differential equation. We also point out how the construction could be extended to generate families of higher derivative discontinuous solutions as well. The discontinuity can occur only for a subset of even order derivatives, viz., 2nd, 4th, 8th, 16th,.... The solutions are shown to break the discrete parity (reflection) symmetry of the underlying equation. These results are expected to gain significance in the contemporary search of a new dynamical principle for understanding complex phenomena in nature

First order linear ordinary differential equations in associative algebras

Directory of Open Access Journals (Sweden)

Gordon Erlebacher

2004-01-01

Full Text Available In this paper, we study the linear differential equation $$ frac{dx}{dt}=sum_{i=1}^n a_i(t x b_i(t + f(t $$ in an associative but non-commutative algebra $mathcal{A}$, where the $b_i(t$ form a set of commuting $mathcal{A}$-valued functions expressed in a time-independent spectral basis consisting of mutually annihilating idempotents and nilpotents. Explicit new closed solutions are derived, and examples are presented to illustrate the theory.

Lie group classification of first-order delay ordinary differential equations

Science.gov (United States)

Dorodnitsyn, Vladimir A.; Kozlov, Roman; Meleshko, Sergey V.; Winternitz, Pavel

2018-05-01

A group classification of first-order delay ordinary differential equations (DODEs) accompanied by an equation for the delay parameter (delay relation) is presented. A subset of such systems (delay ordinary differential systems or DODSs), which consists of linear DODEs and solution-independent delay relations, have infinite-dimensional symmetry algebras—as do nonlinear ones that are linearizable by an invertible transformation of variables. Genuinely nonlinear DODSs have symmetry algebras of dimension n, . It is shown how exact analytical solutions of invariant DODSs can be obtained using symmetry reduction.

Generation and Identification of Ordinary Differential Equations of Maximal Symmetry Algebra

Directory of Open Access Journals (Sweden)

J. C. Ndogmo

2016-01-01

Full Text Available An effective method for generating linear ordinary differential equations of maximal symmetry in their most general form is found, and an explicit expression for the point transformation reducing the equation to its canonical form is obtained. New expressions for the general solution are also found, as well as several identification and other results and a direct proof of the fact that a linear ordinary differential equation is iterative if and only if it is reducible to the canonical form by a point transformation. New classes of solvable equations parameterized by an arbitrary function are also found, together with simple algebraic expressions for the corresponding general solution.

A course in ordinary differential equations

CERN Document Server

Swift, Randall J

2014-01-01

Praise for the First Edition:"A Course in Ordinary Differential Equations deserves to be on the MAA's Basic Library List … the book with its layout, is very student friendly-it is easy to read and understand; every chapter and explanations flow smoothly and coherently … the reviewer would recommend this book highly for undergraduate introductory differential equation courses." -Srabasti Dutta, College of Saint Elizabeth, MAA Online, July 2008"An important feature is that the exposition is richly accompanied by computer algebra code (equally distributed between MATLAB, Mathematica, and Maple). The major part of the book is devoted to classical theory (both for systems and higher order equations). The necessary material from linear algebra is also covered. More advanced topics include numerical methods, stability of equilibria, bifurcations, Laplace transforms, and the power series method."-EMS Newsletter, June 2007"This is a delightful textbook for a standard one-semester undergraduate course in ordinary d...

A textbook on ordinary differential equations

CERN Document Server

Ahmad, Shair

2015-01-01

This book offers readers a primer on the theory and applications of Ordinary Differential Equations. The style used is simple, yet thorough and rigorous. Each chapter ends with a broad set of exercises that range from the routine to the more challenging and thought-provoking. Solutions to selected exercises can be found at the end of the book. The book contains many interesting examples on topics such as electric circuits, the pendulum equation, the logistic equation, the Lotka-Volterra system, the Laplace Transform, etc., which introduce students to a number of interesting aspects of the theory and applications. The work is mainly intended for students of Mathematics, Physics, Engineering, Computer Science and other areas of the natural and social sciences that use ordinary differential equations, and who have a firm grasp of Calculus and a minimal understanding of the basic concepts used in Linear Algebra. It also studies a few more advanced topics, such as Stability Theory and Boundary Value Problems, whic...

Solving Ordinary Differential Equations

Science.gov (United States)

Krogh, F. T.

1987-01-01

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Algorithmic Verification of Linearizability for Ordinary Differential Equations

KAUST Repository

Lyakhov, Dmitry A.

2017-07-19

For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a linear one by a point transformation of the dependent and independent variables. The first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra. The second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. The implementation of both algorithms is discussed and their application is illustrated using several examples.

Numerical solution of two-dimensional non-linear partial differential ...

African Journals Online (AJOL)

linear partial differential equations using a hybrid method. The solution technique involves discritizing the non-linear system of partial differential equations (PDEs) to obtain a corresponding nonlinear system of algebraic difference equations to be ...

On differential operators generating iterative systems of linear ODEs of maximal symmetry algebra

Science.gov (United States)

Ndogmo, J. C.

2017-06-01

Although every iterative scalar linear ordinary differential equation is of maximal symmetry algebra, the situation is different and far more complex for systems of linear ordinary differential equations, and an iterative system of linear equations need not be of maximal symmetry algebra. We illustrate these facts by examples and derive families of vector differential operators whose iterations are all linear systems of equations of maximal symmetry algebra. Some consequences of these results are also discussed.

Ordinary differential equations

CERN Document Server

Miller, Richard K

1982-01-01

Ordinary Differential Equations is an outgrowth of courses taught for a number of years at Iowa State University in the mathematics and the electrical engineering departments. It is intended as a text for a first graduate course in differential equations for students in mathematics, engineering, and the sciences. Although differential equations is an old, traditional, and well-established subject, the diverse backgrounds and interests of the students in a typical modern-day course cause problems in the selection and method of presentation of material. In order to compensate for this diversity,

Algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations with the use of parallel computations

Energy Technology Data Exchange (ETDEWEB)

Moryakov, A. V., E-mail: sailor@orc.ru [National Research Centre Kurchatov Institute (Russian Federation)

2016-12-15

An algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations is presented. The algorithm for systems of first-order differential equations is implemented in the EDELWEISS code with the possibility of parallel computations on supercomputers employing the MPI (Message Passing Interface) standard for the data exchange between parallel processes. The solution is represented by a series of orthogonal polynomials on the interval [0, 1]. The algorithm is characterized by simplicity and the possibility to solve nonlinear problems with a correction of the operator in accordance with the solution obtained in the previous iterative process.

Dichotomies for generalized ordinary differential equations and applications

Science.gov (United States)

Bonotto, E. M.; Federson, M.; Santos, F. L.

2018-03-01

In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between generalized ordinary differential equations and other equations, we translate our results to measure differential equations and impulsive differential equations. The fact that we work in the framework of generalized ordinary differential equations allows us to manage functions with many discontinuities and of unbounded variation.

The Schroedinger equation for central power law potentials and the classical theory of ordinary linear differential equations of the second order

International Nuclear Information System (INIS)

Lima, M.L.; Mignaco, J.A.

1985-01-01

It is shown that the rational power law potentials in the two-body radial Schoedinger equation admit a systematic treatment available from the classical theory of ordinary linear differential equations of the second order. The admissible potentials come into families evolved from equations having a fixed number of elementary singularities. As a consequence, relations are found and discussed among the several potentials in a family. (Author) [pt

Ordinary differential equations

CERN Document Server

Greenberg, Michael D

2014-01-01

Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps

Linear measure functional differential equations with infinite delay

OpenAIRE

Monteiro, G. (Giselle Antunes); Slavík, A.

2014-01-01

We use the theory of generalized linear ordinary differential equations in Banach spaces to study linear measure functional differential equations with infinite delay. We obtain new results concerning the existence, uniqueness, and continuous dependence of solutions. Even for equations with a finite delay, our results are stronger than the existing ones. Finally, we present an application to functional differential equations with impulses.

The Schroedinger equation for central power law potentials and the classical theory of ordinary linear differential equations of the second order

International Nuclear Information System (INIS)

Lima, M.L.; Mignaco, J.A.

1985-01-01

It is shown that the rational power law potentials in the two-body radial Schrodinger equations admit a systematic treatment available from the classical theory of ordinary linear differential equations of the second order. The resulting potentials come into families evolved from equations having a fixed number of elementary regular singularities. As a consequence, relations are found and discussed among the several potentials in a family. (Author) [pt

Parallels between control PDE's (Partial Differential Equations) and systems of ODE's (Ordinary Differential Equations)

Science.gov (United States)

Hunt, L. R.; Villarreal, Ramiro

1987-01-01

System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

Hojman's theorem of the third-order ordinary differential equation

International Nuclear Information System (INIS)

Hong-Sheng, Lü; Hong-Bin, Zhang; Shu-Long, Gu

2009-01-01

This paper extends Hojman's conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results. (general)

VODE, Variable Coefficient Ordinary Differential Equations (ODE) Solver

International Nuclear Information System (INIS)

Brown, P.N.; Hindmarsh, A.C.; Byrne, G.D.

2002-01-01

1 - Description of program or function: VODE is a package of subroutines for the numerical solution of the initial-value problem for systems of first-order ordinary differential equations. The package can be used for either stiff or non-stiff systems. In the stiff case, the Jacobian matrix is treated as full or banded. An algorithm is included for saving and reusing the Jacobian matrix under certain conditions. If storage is limited, this option may be suppressed. 2 - Method of solution - VODE uses the variable-order, variable- coefficient Adams-Moulton method for non-stiff systems and the variable-order, fixed-leading-coefficient Backward Differentiation Formula (BDF) method for stiff systems

Unique solvability of a non-linear non-local boundary-value problem for systems of non-linear functional differential equations

Czech Academy of Sciences Publication Activity Database

Dilna, N.; Rontó, András

2010-01-01

Roč. 60, č. 3 (2010), s. 327-338 ISSN 0139-9918 R&D Projects: GA ČR(CZ) GA201/06/0254 Institutional research plan: CEZ:AV0Z10190503 Keywords : non-linear boundary value-problem * functional differential equation * non-local condition * unique solvability * differential inequality Subject RIV: BA - General Mathematics Impact factor: 0.316, year: 2010 http://link.springer.com/article/10.2478%2Fs12175-010-0015-9

Monograph - The Numerical Integration of Ordinary Differential Equations.

Science.gov (United States)

Hull, T. E.

The materials presented in this monograph are intended to be included in a course on ordinary differential equations at the upper division level in a college mathematics program. These materials provide an introduction to the numerical integration of ordinary differential equations, and they can be used to supplement a regular text on this…

Numerical integration of asymptotic solutions of ordinary differential equations

Science.gov (United States)

Thurston, Gaylen A.

1989-01-01

Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.

Solving polynomial differential equations by transforming them to linear functional-differential equations

OpenAIRE

Nahay, John Michael

2008-01-01

We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order Abel differential equation with two nonlinear terms in order to demonstrate in as much detail as possible the computations necessary for a complete solution. We mention in our section on further developments that the basic transformation idea can be generali...

Ordinary differential equations with applications in molecular biology.

Science.gov (United States)

Ilea, M; Turnea, M; Rotariu, M

2012-01-01

Differential equations are of basic importance in molecular biology mathematics because many biological laws and relations appear mathematically in the form of a differential equation. In this article we presented some applications of mathematical models represented by ordinary differential equations in molecular biology. The vast majority of quantitative models in cell and molecular biology are formulated in terms of ordinary differential equations for the time evolution of concentrations of molecular species. Assuming that the diffusion in the cell is high enough to make the spatial distribution of molecules homogenous, these equations describe systems with many participating molecules of each kind. We propose an original mathematical model with small parameter for biological phospholipid pathway. All the equations system includes small parameter epsilon. The smallness of epsilon is relative to the size of the solution domain. If we reduce the size of the solution region the same small epsilon will result in a different condition number. It is clear that the solution for a smaller region is less difficult. We introduce the mathematical technique known as boundary function method for singular perturbation system. In this system, the small parameter is an asymptotic variable, different from the independent variable. In general, the solutions of such equations exhibit multiscale phenomena. Singularly perturbed problems form a special class of problems containing a small parameter which may tend to zero. Many molecular biology processes can be quantitatively characterized by ordinary differential equations. Mathematical cell biology is a very active and fast growing interdisciplinary area in which mathematical concepts, techniques, and models are applied to a variety of problems in developmental medicine and bioengineering. Among the different modeling approaches, ordinary differential equations (ODE) are particularly important and have led to significant advances

POSITIVE SOLUTIONS TO SEMI-LINEAR SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS IN BANACH SPACE

Institute of Scientific and Technical Information of China (English)

无

2008-01-01

In this paper,we study the existence of positive periodic solution to some second- order semi-linear differential equation in Banach space.By the fixed point index theory, we prove that the semi-linear differential equation has two positive periodic solutions.

Inverse problems in ordinary differential equations and applications

CERN Document Server

Llibre, Jaume

2016-01-01

This book is dedicated to study the inverse problem of ordinary differential equations, that is it focuses in finding all ordinary differential equations that satisfy a given set of properties. The Nambu bracket is the central tool in developing this approach. The authors start characterizing the ordinary differential equations in R^N which have a given set of partial integrals or first integrals. The results obtained are applied first to planar polynomial differential systems with a given set of such integrals, second to solve the 16th Hilbert problem restricted to generic algebraic limit cycles, third for solving the inverse problem for constrained Lagrangian and Hamiltonian mechanical systems, fourth for studying the integrability of a constrained rigid body. Finally the authors conclude with an analysis on nonholonomic mechanics, a generalization of the Hamiltonian principle, and the statement an solution of the inverse problem in vakonomic mechanics.

Subroutine for series solutions of linear differential equations

International Nuclear Information System (INIS)

Tasso, H.; Steuerwald, J.

1976-02-01

A subroutine for Taylor series solutions of systems of ordinary linear differential equations is descriebed. It uses the old idea of Lie series but allows simple implementation and is time-saving for symbolic manipulations. (orig.) [de

Lie symmetries and differential galois groups of linear equations

NARCIS (Netherlands)

Oudshoorn, W.R.; Put, M. van der

2002-01-01

For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In

Conservation properties of numerical integration methods for systems of ordinary differential equations

Science.gov (United States)

Rosenbaum, J. S.

1976-01-01

If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.

The qualitative theory of ordinary differential equations an introduction

CERN Document Server

Brauer, Fred

1989-01-01

""This is a very good book ... with many well-chosen examples and illustrations."" - American Mathematical MonthlyThis highly regarded text presents a self-contained introduction to some important aspects of modern qualitative theory for ordinary differential equations. It is accessible to any student of physical sciences, mathematics or engineering who has a good knowledge of calculus and of the elements of linear algebra. In addition, algebraic results are stated as needed; the less familiar ones are proved either in the text or in appendixes.The topics covered in the first three chapters a

Non-instantaneous impulses in differential equations

CERN Document Server

Agarwal, Ravi; O'Regan, Donal

2017-01-01

This monograph is the first published book devoted to the theory of differential equations with non-instantaneous impulses. It aims to equip the reader with mathematical models and theory behind real life processes in physics, biology, population dynamics, ecology and pharmacokinetics. The authors examine a wide scope of differential equations with non-instantaneous impulses through three comprehensive chapters, providing an all-rounded and unique presentation on the topic, including: - Ordinary differential equations with non-instantaneous impulses (scalar and n-dimensional case) - Fractional differential equa tions with non-instantaneous impulses (with Caputo fractional derivatives of order q ϵ (0, 1)) - Ordinary differential equations with non-instantaneous impulses occurring at random moments (with exponential, Erlang, or Gamma distribution) Each chapter focuses on theory, proofs and examples, and contains numerous graphs to enrich the reader’s understanding. Additionally, a carefully selected bibliogr...

Integration of differential equations by the pseudo-linear (PL) approximation

International Nuclear Information System (INIS)

Bonalumi, Riccardo A.

1998-01-01

A new method of integrating differential equations was originated with the technique of approximately calculating the integrals called the pseudo-linear (PL) procedure: this method is A-stable. This article contains the following examples: 1st order ordinary differential equations (ODEs), 2nd order linear ODEs, stiff system of ODEs (neutron kinetics), one-dimensional parabolic (diffusion) partial differential equations. In this latter case, this PL method coincides with the Crank-Nicholson method

Numerical analysis of systems of ordinary and stochastic differential equations

CERN Document Server

Artemiev, S S

1997-01-01

This text deals with numerical analysis of systems of both ordinary and stochastic differential equations. It covers numerical solution problems of the Cauchy problem for stiff ordinary differential equations (ODE) systems by Rosenbrock-type methods (RTMs).

Ordinary differential equations

CERN Document Server

Cox, William

1995-01-01

Building on introductory calculus courses, this text provides a sound foundation in the underlying principles of ordinary differential equations. Important concepts, including uniqueness and existence theorems, are worked through in detail and the student is encouraged to develop much of the routine material themselves, thus helping to ensure a solid understanding of the fundamentals required.The wide use of exercises, problems and self-assessment questions helps to promote a deeper understanding of the material and it is developed in such a way that it lays the groundwork for further

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear ordinary differential equations

Institute of Scientific and Technical Information of China (English)

WANG; Shunjin; ZHANG; Hua

2006-01-01

The problem of preserving fidelity in numerical computation of nonlinear ordinary differential equations is studied in terms of preserving local differential structure and approximating global integration structure of the dynamical system.The ordinary differential equations are lifted to the corresponding partial differential equations in the framework of algebraic dynamics,and a new algorithm-algebraic dynamics algorithm is proposed based on the exact analytical solutions of the ordinary differential equations by the algebraic dynamics method.In the new algorithm,the time evolution of the ordinary differential system is described locally by the time translation operator and globally by the time evolution operator.The exact analytical piece-like solution of the ordinary differential equations is expressd in terms of Taylor series with a local convergent radius,and its finite order truncation leads to the new numerical algorithm with a controllable precision better than Runge Kutta Algorithm and Symplectic Geometric Algorithm.

Ordinary differential equation for local accumulation time.

Science.gov (United States)

Berezhkovskii, Alexander M

2011-08-21

Cell differentiation in a developing tissue is controlled by the concentration fields of signaling molecules called morphogens. Formation of these concentration fields can be described by the reaction-diffusion mechanism in which locally produced molecules diffuse through the patterned tissue and are degraded. The formation kinetics at a given point of the patterned tissue can be characterized by the local accumulation time, defined in terms of the local relaxation function. Here, we show that this time satisfies an ordinary differential equation. Using this equation one can straightforwardly determine the local accumulation time, i.e., without preliminary calculation of the relaxation function by solving the partial differential equation, as was done in previous studies. We derive this ordinary differential equation together with the accompanying boundary conditions and demonstrate that the earlier obtained results for the local accumulation time can be recovered by solving this equation. © 2011 American Institute of Physics

The non-ordinary Regge behavior of the K{sup *}{sub 0}(800) or κ-meson versus the ordinary K{sup *}{sub 0}(1430)

Energy Technology Data Exchange (ETDEWEB)

Pelaez, J.R.; Rodas, A. [Universidad Complutense de Madrid, Departamento de Fisica Teorica II and UPARCOS, Madrid (Spain)

2017-06-15

The Regge trajectory of an elastic resonance can be calculated from dispersion theory, instead of fitted phenomenologically, using only its pole parameters as input. This also provides a correct treatment of resonance widths in Regge trajectories, essential for very wide resonances. In this work we first calculate the K{sup *}{sub 0}(1430) Regge trajectory, finding the ordinary almost real and linear behavior, typical of q anti q resonances. In contrast, for the K{sup *}{sub 0}(800) meson, the resulting Regge trajectory is non-linear and has a much smaller slope than ordinary resonances, being remarkably similar to that of the f{sub 0}(500) or σ meson. The slope of these unusual Regge trajectories seems to scale with the meson masses rather than with scales typical of quark degrees of freedom. We also calculate the range of the interaction responsible for the formation of these resonances. Our results strongly support a non-ordinary, predominantly meson-meson-like, interpretation for the lightest strange and non-strange resonances. (orig.)

ODEPACK, Initial Value Problems of Ordinary Differential Equation System

International Nuclear Information System (INIS)

Hindmarsh, A.C.; Petzold, L.R.

2005-01-01

I - Description of program or function: ODEPACK is a collection of Fortran solvers for the initial value problem for ordinary differential equation systems. It consists of nine solvers, namely a basic solver called LSODE and eight variants of it -- LSODES, LSODA, LSODAR, LSODPK, LSODKR, LSODI, LSOIBT, and LSODIS. The collection is suitable for both stiff and non-stiff systems. It includes solvers for systems given in explicit form, dy/dt = f(t,y), and also solvers for systems given in linearly implicit form, A(t,y) dy/dt = g(t,y). Two of the solvers use general sparse matrix solvers for the linear systems that arise. Two others use iterative (preconditioned Krylov) methods instead of direct methods for these linear systems. The most recent addition is LSODIS, which solves implicit problems with general sparse treatment of all matrices involved. The ODEPACK solvers are written in standard Fortran 77, with a few exceptions, and with minimal machine dependencies. There are separate double and single precision versions of ODEPACK. The actual solver names are those given above with a prefix of D- or S- for the double or single precision version, respectively, i.e. DLSODE/SLSODE, etc. Each solver consists of a main driver subroutine having the same name as the solver and some number of subordinate routines. For each solver, there is also a demonstration program, which solves one or two simple problems in a somewhat self-checking manner. A. Solvers for explicitly given systems. For each of the following solvers, it is assumed that the ODEs are given explicitly, so that the system can be written in the form dy/dt = f(t,y), where y is the vector of dependent variables, and t is the independent variable. 1. LSODE (Livermore Solver for Ordinary Differential Equations) is the basic solver of the collection. It solves stiff and non-stiff systems of the form dy/dt = f. In the stiff case, it treats the Jacobian matrix df/dy as either a dense (full) or a banded matrix, and as

Non-cooperative stochastic differential game theory of generalized Markov jump linear systems

CERN Document Server

Zhang, Cheng-ke; Zhou, Hai-ying; Bin, Ning

2017-01-01

This book systematically studies the stochastic non-cooperative differential game theory of generalized linear Markov jump systems and its application in the field of finance and insurance. The book is an in-depth research book of the continuous time and discrete time linear quadratic stochastic differential game, in order to establish a relatively complete framework of dynamic non-cooperative differential game theory. It uses the method of dynamic programming principle and Riccati equation, and derives it into all kinds of existence conditions and calculating method of the equilibrium strategies of dynamic non-cooperative differential game. Based on the game theory method, this book studies the corresponding robust control problem, especially the existence condition and design method of the optimal robust control strategy. The book discusses the theoretical results and its applications in the risk control, option pricing, and the optimal investment problem in the field of finance and insurance, enriching the...

FORSIM, Solution of Ordinary or Partial Differential Equation with Initial Conditions

International Nuclear Information System (INIS)

Carver, M.B.

1985-01-01

1 - Description of problem or function: FORSIM is a FORTRAN oriented simulation program which automates the continuous transient solution of systems of ordinary and/or partial differential equations. The user writes his equations in a FORTRAN subroutine, following prescribed rules, and loads this routine along with the executive routines. The executive routines then read in initial data supplied by the user and proceed with the integration. 2 - Method of solution: Partial differential equations are converted to coupled ordinary differential equations by suitable discretization formulae. Integration is done by variable order, variable step-size error controlled algorithms. 3 - Restrictions on the complexity of the problem - Maximum of: 1000 ordinary differential equations

A textbook on ordinary differential equations

CERN Document Server

Ahmad, Shair

2014-01-01

The book is a primer of the theory of Ordinary Differential Equations. Each chapter is completed by a broad set of exercises; the reader will also find a set of solutions of selected exercises. The book contains many interesting examples as well (like the equations for the electric circuits, the pendium equation, the logistic equation, the Lotka-Volterra system, and many other) which introduce the reader to some interesting aspects of the theory and its applications. The work is mainly addressed to students of Mathematics, Physics, Engineering, Statistics, Computer Sciences, with  knowledge of Calculus and Linear Algebra, and contains more advanced topics for further developments, such as Laplace transform; Stability theory and existence of solutions to Boundary Value problems. The authors are preparing a complete solutions manual, containing solutions to all the exercises published in the book. The manual will be available Summer 2014. Instructors who wish to adopt the book may request the manual by writing...

Numerical solution of ordinary differential equations

CERN Document Server

Fox, L

1987-01-01

Nearly 20 years ago we produced a treatise (of about the same length as this book) entitled Computing methods for scientists and engineers. It was stated that most computation is performed by workers whose mathematical training stopped somewhere short of the 'professional' level, and that some books are therefore needed which use quite simple mathematics but which nevertheless communicate the essence of the 'numerical sense' which is exhibited by the real computing experts and which is surely needed, at least to some extent, by all who use modern computers and modern numerical software. In that book we treated, at no great length, a variety of computational problems in which the material on ordinary differential equations occupied about 50 pages. At that time it was quite common to find books on numerical analysis, with a little on each topic ofthat field, whereas today we are more likely to see similarly-sized books on each major topic: for example on numerical linear algebra, numerical approximation, numeri...

Numerical methods for the solution of ordinary differential equations

International Nuclear Information System (INIS)

Azeem, M.

1999-01-01

The ode 113 code solves non-stiff differential equations and is a fully variable step, variable order, PECE implementation in terms of modified divided differences of Adams-Bashforth-Moulton family of formulas of order 1-12. The main objectives of this project were to modify PECE mode of ode 113 into PEC mode, study the variable step size and variable order strategy of both the modes and finally, develop the switching strategy between both PECE and PEC modes to minimize the cost of solving the ordinary differential equations. Using some test problems (including stiff, mild stiff and non-stiff), it was found that the PEC mode was more efficient for non-stiff problems at crude and intermediate tolerances and the PECE mode for all problems at the stringent tolerance. An automatic switching strategy was developed using the results observed from the step size and order plots of all the test problems for both the modes and gave the optimum results. (author)

From the hypergeometric differential equation to a non-linear Schrödinger one

International Nuclear Information System (INIS)

Plastino, A.; Rocca, M.C.

2015-01-01

We show that the q-exponential function is a hypergeometric function. Accordingly, it obeys the hypergeometric differential equation. We demonstrate that this differential equation can be transformed into a non-linear Schrödinger equation (NLSE). This NLSE exhibits both similarities and differences vis-a-vis the Nobre–Rego-Monteiro–Tsallis one. - Highlights: • We show that the q-exponential is a hypergeometric function. • It thus obeys the hypergeometric differential equation (HDE). • We show that the HDE can be cast as a non-linear Schrödinger equation. • This is different from the Nobre, Rego-Monteiro, Tsallis one.

Differential equation analysis in biomedical science and engineering ordinary differential equation applications with R

CERN Document Server

Schiesser, William E

2014-01-01

Features a solid foundation of mathematical and computational tools to formulate and solve real-world ODE problems across various fields With a step-by-step approach to solving ordinary differential equations (ODEs), Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R successfully applies computational techniques for solving real-worldODE problems that are found in a variety of fields, including chemistry, physics, biology,and physiology. The book provides readers with the necessary knowledge to reproduce andextend the comp

Increase in speed of Wilkinson-type ADC and improvement of differential non-linearity

Energy Technology Data Exchange (ETDEWEB)

Kinbara, S [Japan Atomic Energy Research Inst., Tokai, Ibaraki. Tokai Research Establishment

1977-06-01

It is shown that the differential non-linearity of a Wilkinson-type analog-to-digital converter (ADC) is dominated by the unbalance of even-numbered periods caused by the action of interference resulting from operation of a channel scaler. To improve this situation, new methods were tested which allow such action of interference to be dispersed. Measurements show that a differential non-linearity value of +- 0.043% is attainable for a clock rate of 300 MHz.

Ordinary differential equations introduction to the theory of ordinary differential equations in the real domain

CERN Document Server

Kurzweil, J

1986-01-01

The author, Professor Kurzweil, is one of the world's top experts in the area of ordinary differential equations - a fact fully reflected in this book. Unlike many classical texts which concentrate primarily on methods of integration of differential equations, this book pursues a modern approach: the topic is discussed in full generality which, at the same time, permits us to gain a deep insight into the theory and to develop a fruitful intuition. The basic framework of the theory is expanded by considering further important topics like stability, dependence of a solution on a parameter, Car

GDTM-Padé technique for the non-linear differential-difference equation

Directory of Open Access Journals (Sweden)

Lu Jun-Feng

2013-01-01

Full Text Available This paper focuses on applying the GDTM-Padé technique to solve the non-linear differential-difference equation. The bell-shaped solitary wave solution of Belov-Chaltikian lattice equation is considered. Comparison between the approximate solutions and the exact ones shows that this technique is an efficient and attractive method for solving the differential-difference equations.

The Semianalytical Solutions for Stiff Systems of Ordinary Differential Equations by Using Variational Iteration Method and Modified Variational Iteration Method with Comparison to Exact Solutions

Directory of Open Access Journals (Sweden)

Mehmet Tarik Atay

2013-01-01

Full Text Available The Variational Iteration Method (VIM and Modified Variational Iteration Method (MVIM are used to find solutions of systems of stiff ordinary differential equations for both linear and nonlinear problems. Some examples are given to illustrate the accuracy and effectiveness of these methods. We compare our results with exact results. In some studies related to stiff ordinary differential equations, problems were solved by Adomian Decomposition Method and VIM and Homotopy Perturbation Method. Comparisons with exact solutions reveal that the Variational Iteration Method (VIM and the Modified Variational Iteration Method (MVIM are easier to implement. In fact, these methods are promising methods for various systems of linear and nonlinear stiff ordinary differential equations. Furthermore, VIM, or in some cases MVIM, is giving exact solutions in linear cases and very satisfactory solutions when compared to exact solutions for nonlinear cases depending on the stiffness ratio of the stiff system to be solved.

Non-linear second-order periodic systems with non-smooth potential

Indian Academy of Sciences (India)

In this paper we study second order non-linear periodic systems driven by the ordinary vector -Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on ...

Non-linear second-order periodic systems with non-smooth potential

Indian Academy of Sciences (India)

R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22

Abstract. In this paper we study second order non-linear periodic systems driven by the ordinary vector p-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth ...

(Ln-bar, g)-spaces. Ordinary and tensor differentials

International Nuclear Information System (INIS)

Manoff, S.; Dimitrov, B.

1998-01-01

Different types of differentials as special cases of differential operators acting on tensor fields over (L n bar, g)-spaces are considered. The ordinary differential, the covariant differential as a special case of the covariant differential operator, and the Lie differential as a special case of the Lie differential operator are investigated. The tensor differential and its special types (Covariant tensor differential, and Lie tensor differential) are determined and their properties are discussed. Covariant symmetric and antisymmetric (external) tensor differentials, Lie symmetric, and Lie antisymmetric (external) tensor differentials are determined and considered over (L n bar, g)-spaces

Network Reconstruction From High-Dimensional Ordinary Differential Equations.

Science.gov (United States)

Chen, Shizhe; Shojaie, Ali; Witten, Daniela M

2017-01-01

We consider the task of learning a dynamical system from high-dimensional time-course data. For instance, we might wish to estimate a gene regulatory network from gene expression data measured at discrete time points. We model the dynamical system nonparametrically as a system of additive ordinary differential equations. Most existing methods for parameter estimation in ordinary differential equations estimate the derivatives from noisy observations. This is known to be challenging and inefficient. We propose a novel approach that does not involve derivative estimation. We show that the proposed method can consistently recover the true network structure even in high dimensions, and we demonstrate empirical improvement over competing approaches. Supplementary materials for this article are available online.

Solving ordinary differential equations by electrical analogy: a multidisciplinary teaching tool

Science.gov (United States)

Sanchez Perez, J. F.; Conesa, M.; Alhama, I.

2016-11-01

Ordinary differential equations are the mathematical formulation for a great variety of problems in science and engineering, and frequently, two different problems are equivalent from a mathematical point of view when they are formulated by the same equations. Students acquire the knowledge of how to solve these equations (at least some types of them) using protocols and strict algorithms of mathematical calculation without thinking about the meaning of the equation. The aim of this work is that students learn to design network models or circuits in this way; with simple knowledge of them, students can establish the association of electric circuits and differential equations and their equivalences, from a formal point of view, that allows them to associate knowledge of two disciplines and promote the use of this interdisciplinary approach to address complex problems. Therefore, they learn to use a multidisciplinary tool that allows them to solve these kinds of equations, even students of first course of engineering, whatever the order, grade or type of non-linearity. This methodology has been implemented in numerous final degree projects in engineering and science, e.g., chemical engineering, building engineering, industrial engineering, mechanical engineering, architecture, etc. Applications are presented to illustrate the subject of this manuscript.

Linear differential equations to solve nonlinear mechanical problems: A novel approach

OpenAIRE

Nair, C. Radhakrishnan

2004-01-01

Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential equation is known. Using the known solution of the non-linear differential equation, linear differential equations are set up. The solutions of these linear differential equations are found using standard techniques. Then the solutions of the linear differe...

Sparse Additive Ordinary Differential Equations for Dynamic Gene Regulatory Network Modeling.

Science.gov (United States)

Wu, Hulin; Lu, Tao; Xue, Hongqi; Liang, Hua

2014-04-02

The gene regulation network (GRN) is a high-dimensional complex system, which can be represented by various mathematical or statistical models. The ordinary differential equation (ODE) model is one of the popular dynamic GRN models. High-dimensional linear ODE models have been proposed to identify GRNs, but with a limitation of the linear regulation effect assumption. In this article, we propose a sparse additive ODE (SA-ODE) model, coupled with ODE estimation methods and adaptive group LASSO techniques, to model dynamic GRNs that could flexibly deal with nonlinear regulation effects. The asymptotic properties of the proposed method are established and simulation studies are performed to validate the proposed approach. An application example for identifying the nonlinear dynamic GRN of T-cell activation is used to illustrate the usefulness of the proposed method.

Solutions manual to accompany Ordinary differential equations

CERN Document Server

Greenberg, Michael D

2014-01-01

Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps

Differential equations a dynamical systems approach ordinary differential equations

CERN Document Server

Hubbard, John H

1991-01-01

This is a corrected third printing of the first part of the text Differential Equations: A Dynamical Systems Approach written by John Hubbard and Beverly West. The authors' main emphasis in this book is on ordinary differential equations. The book is most appropriate for upper level undergraduate and graduate students in the fields of mathematics, engineering, and applied mathematics, as well as the life sciences, physics and economics. Traditional courses on differential equations focus on techniques leading to solutions. Yet most differential equations do not admit solutions which can be written in elementary terms. The authors have taken the view that a differential equations defines functions; the object of the theory is to understand the behavior of these functions. The tools the authors use include qualitative and numerical methods besides the traditional analytic methods. The companion software, MacMath, is designed to bring these notions to life.

Symmetries, Integrals and Solutions of Ordinary Differential ...

Indian Academy of Sciences (India)

Second-and third-order scalar ordinary differential equations of maximal symmetry in the traditional sense of point, respectively contact, symmetry are examined for the mappings they produce in solutions and fundamental first integrals. The properties of the `exceptional symmetries', i.e. those not considered to be generic to ...

A first course in ordinary differential equations analytical and numerical methods

CERN Document Server

Hermann, Martin

2014-01-01

This book presents a modern introduction to analytical and numerical techniques for solving ordinary differential equations (ODEs). Contrary to the traditional format—the theorem-and-proof format—the book is focusing on analytical and numerical methods. The book supplies a variety of problems and examples, ranging from the elementary to the advanced level, to introduce and study the mathematics of ODEs. The analytical part of the book deals with solution techniques for scalar first-order and second-order linear ODEs, and systems of linear ODEs—with a special focus on the Laplace transform, operator techniques and power series solutions. In the numerical part, theoretical and practical aspects of Runge-Kutta methods for solving initial-value problems and shooting methods for linear two-point boundary-value problems are considered. The book is intended as a primary text for courses on the theory of ODEs and numerical treatment of ODEs for advanced undergraduate and early graduate students. It is assumed t...

Random ordinary differential equations and their numerical solution

CERN Document Server

Han, Xiaoying

2017-01-01

This book is intended to make recent results on the derivation of higher order numerical schemes for random ordinary differential equations (RODEs) available to a broader readership, and to familiarize readers with RODEs themselves as well as the closely associated theory of random dynamical systems. In addition, it demonstrates how RODEs are being used in the biological sciences, where non-Gaussian and bounded noise are often more realistic than the Gaussian white noise in stochastic differential equations (SODEs).   RODEs are used in many important applications and play a fundamental role in the theory of random dynamical systems.  They can be analyzed pathwise with deterministic calculus, but require further treatment beyond that of classical ODE theory due to the lack of smoothness in their time variable. Although classical numerical schemes for ODEs can be used pathwise for RODEs, they rarely attain their traditional order since the solutions of RODEs do not have sufficient smoothness to have Taylor ...

A linearizing transformation for the Korteweg-de Vries equation; generalizations to higher-dimensional nonlinear partial differential equations

NARCIS (Netherlands)

Dorren, H.J.S.

1998-01-01

It is shown that the Korteweg–de Vries (KdV) equation can be transformed into an ordinary linear partial differential equation in the wave number domain. Explicit solutions of the KdV equation can be obtained by subsequently solving this linear differential equation and by applying a cascade of

Blow up of solutions to ordinary differential equations arising in nonlinear dispersive problems

Directory of Open Access Journals (Sweden)

Milena Dimova

2018-03-01

Full Text Available We study a new class of ordinary differential equations with blow up solutions. Necessary and sufficient conditions for finite blow up time are proved. Based on the new differential equation, a revised version of the concavity method of Levine is proposed. As an application we investigate the non-existence of global solutions to the Cauchy problem of Klein-Gordon, and to the double dispersive equations. We obtain necessary and sufficient condition for finite time blow up with arbitrary positive energy. A very general sufficient condition for blow up is also given.

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

Science.gov (United States)

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…

Numerical solutions of ordinary and partial differential equations in the frequency domain

International Nuclear Information System (INIS)

Hazi, G.; Por, G.

1997-01-01

Numerical problems during the noise simulation in a nuclear power plant are discussed. The solutions of ordinary and partial differential equations are studied in the frequency domain. Numerical methods by the transfer function method are applied. It is shown that the correctness of the numerical methods is limited for ordinary differential equations in the frequency domain. To overcome the difficulties, step-size selection is suggested. (author)

Nonlinear ordinary differential equations analytical approximation and numerical methods

CERN Document Server

Hermann, Martin

2016-01-01

The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs. There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs. The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march...

Solution of linear ordinary differential equations by means of the method of variation of arbitrary constants

DEFF Research Database (Denmark)

Mejlbro, Leif

1997-01-01

An alternative formula for the solution of linear differential equations of order n is suggested. When applicable, the suggested method requires fewer and simpler computations than the well-known method using Wronskians.......An alternative formula for the solution of linear differential equations of order n is suggested. When applicable, the suggested method requires fewer and simpler computations than the well-known method using Wronskians....

Oscillation and non-oscillation criterion for Riemann–Weber type half-linear differential equations

Directory of Open Access Journals (Sweden)

Petr Hasil

2016-08-01

Full Text Available By the combination of the modified half-linear Prüfer method and the Riccati technique, we study oscillatory properties of half-linear differential equations. Taking into account the transformation theory of half-linear equations and using some known results, we show that the analysed equations in the Riemann–Weber form with perturbations in both terms are conditionally oscillatory. Within the process, we identify the critical oscillation values of their coefficients and, consequently, we decide when the considered equations are oscillatory and when they are non-oscillatory. As a direct corollary of our main result, we solve the so-called critical case for a certain type of half-linear non-perturbed equations.

Linear determining equations for differential constraints

International Nuclear Information System (INIS)

Kaptsov, O V

1998-01-01

A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical determining equations used in the search for admissible Lie operators. As applications of this approach equations of an ideal incompressible fluid and non-linear heat equations are discussed

A new numerical approximation of the fractal ordinary differential equation

Science.gov (United States)

Atangana, Abdon; Jain, Sonal

2018-02-01

The concept of fractal medium is present in several real-world problems, for instance, in the geological formation that constitutes the well-known subsurface water called aquifers. However, attention has not been quite devoted to modeling for instance, the flow of a fluid within these media. We deem it important to remind the reader that the concept of fractal derivative is not to represent the fractal sharps but to describe the movement of the fluid within these media. Since this class of ordinary differential equations is highly complex to solve analytically, we present a novel numerical scheme that allows to solve fractal ordinary differential equations. Error analysis of the method is also presented. Application of the method and numerical approximation are presented for fractal order differential equation. The stability and the convergence of the numerical schemes are investigated in detail. Also some exact solutions of fractal order differential equations are presented and finally some numerical simulations are presented.

Numerical Integration of Stiff System of Ordinary Differential ...

African Journals Online (AJOL)

The goal of this work is to develop, analyse and implement a K-step Implicit Rational Runge-Kutta schemes for Integration of Stiff system of Ordinary differential Equations. Its development adopted Taylor and Binomial series expansion Techniques to generate its parameters. The analysis of its basic properties adopted ...

A Comparison of Two-Stage Approaches for Fitting Nonlinear Ordinary Differential Equation Models with Mixed Effects.

Science.gov (United States)

Chow, Sy-Miin; Bendezú, Jason J; Cole, Pamela M; Ram, Nilam

2016-01-01

Several approaches exist for estimating the derivatives of observed data for model exploration purposes, including functional data analysis (FDA; Ramsay & Silverman, 2005 ), generalized local linear approximation (GLLA; Boker, Deboeck, Edler, & Peel, 2010 ), and generalized orthogonal local derivative approximation (GOLD; Deboeck, 2010 ). These derivative estimation procedures can be used in a two-stage process to fit mixed effects ordinary differential equation (ODE) models. While the performance and utility of these routines for estimating linear ODEs have been established, they have not yet been evaluated in the context of nonlinear ODEs with mixed effects. We compared properties of the GLLA and GOLD to an FDA-based two-stage approach denoted herein as functional ordinary differential equation with mixed effects (FODEmixed) in a Monte Carlo (MC) study using a nonlinear coupled oscillators model with mixed effects. Simulation results showed that overall, the FODEmixed outperformed both the GLLA and GOLD across all the embedding dimensions considered, but a novel use of a fourth-order GLLA approach combined with very high embedding dimensions yielded estimation results that almost paralleled those from the FODEmixed. We discuss the strengths and limitations of each approach and demonstrate how output from each stage of FODEmixed may be used to inform empirical modeling of young children's self-regulation.

Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation

Directory of Open Access Journals (Sweden)

Hamidreza Rezazadeh

2014-05-01

Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.

A three operator split-step method covering a larger set of non-linear partial differential equations

Science.gov (United States)

Zia, Haider

2017-06-01

This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized derivative non-linear Schrödinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies to is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material.

Unsteady Solution of Non-Linear Differential Equations Using Walsh Function Series

Science.gov (United States)

Gnoffo, Peter A.

2015-01-01

Walsh functions form an orthonormal basis set consisting of square waves. The discontinuous nature of square waves make the system well suited for representing functions with discontinuities. The product of any two Walsh functions is another Walsh function - a feature that can radically change an algorithm for solving non-linear partial differential equations (PDEs). The solution algorithm of non-linear differential equations using Walsh function series is unique in that integrals and derivatives may be computed using simple matrix multiplication of series representations of functions. Solutions to PDEs are derived as functions of wave component amplitude. Three sample problems are presented to illustrate the Walsh function series approach to solving unsteady PDEs. These include an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the use of the Walsh function solution algorithms, exploiting Fast Walsh Transforms in multi-dimensions (O(Nlog(N))). Details of a Fast Walsh Reciprocal, defined here for the first time, enable inversion of aWalsh Symmetric Matrix in O(Nlog(N)) operations. Walsh functions have been derived using a fractal recursion algorithm and these fractal patterns are observed in the progression of pairs of wave number amplitudes in the solutions. These patterns are most easily observed in a remapping defined as a fractal fingerprint (FFP). A prolongation of existing solutions to the next highest order exploits these patterns. The algorithms presented here are considered a work in progress that provide new alternatives and new insights into the solution of non-linear PDEs.

Generalized Ordinary Differential Equation Models.

Science.gov (United States)

Miao, Hongyu; Wu, Hulin; Xue, Hongqi

2014-10-01

Existing estimation methods for ordinary differential equation (ODE) models are not applicable to discrete data. The generalized ODE (GODE) model is therefore proposed and investigated for the first time. We develop the likelihood-based parameter estimation and inference methods for GODE models. We propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving ODEs. The simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ODE model estimation approach and the extended smoothing-based (ESB) method.

Robust estimation for ordinary differential equation models.

Science.gov (United States)

Cao, J; Wang, L; Xu, J

2011-12-01

Applied scientists often like to use ordinary differential equations (ODEs) to model complex dynamic processes that arise in biology, engineering, medicine, and many other areas. It is interesting but challenging to estimate ODE parameters from noisy data, especially when the data have some outliers. We propose a robust method to address this problem. The dynamic process is represented with a nonparametric function, which is a linear combination of basis functions. The nonparametric function is estimated by a robust penalized smoothing method. The penalty term is defined with the parametric ODE model, which controls the roughness of the nonparametric function and maintains the fidelity of the nonparametric function to the ODE model. The basis coefficients and ODE parameters are estimated in two nested levels of optimization. The coefficient estimates are treated as an implicit function of ODE parameters, which enables one to derive the analytic gradients for optimization using the implicit function theorem. Simulation studies show that the robust method gives satisfactory estimates for the ODE parameters from noisy data with outliers. The robust method is demonstrated by estimating a predator-prey ODE model from real ecological data. © 2011, The International Biometric Society.

The numerical solution of linear multi-term fractional differential equations: systems of equations

Science.gov (United States)

Edwards, John T.; Ford, Neville J.; Simpson, A. Charles

2002-11-01

In this paper, we show how the numerical approximation of the solution of a linear multi-term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity. We begin by showing how our method applies to a simple class of problems and we give a convergence result. We solve the Bagley Torvik equation as an example. We show how the method can be applied to a general linear multi-term equation and give two further examples.

Dual solutions of three-dimensional flow and heat transfer over a non-linearly stretching/shrinking sheet

Science.gov (United States)

Naganthran, Kohilavani; Nazar, Roslinda; Pop, Ioan

2018-05-01

This study investigated the influence of the non-linearly stretching/shrinking sheet on the boundary layer flow and heat transfer. A proper similarity transformation simplified the system of partial differential equations into a system of ordinary differential equations. This system of similarity equations is then solved numerically by using the bvp4c function in the MATLAB software. The generated numerical results presented graphically and discussed in the relevance of the governing parameters. Dual solutions found as the sheet stretched and shrunk in the horizontal direction. Stability analysis showed that the first solution is physically realizable whereas the second solution is not practicable.

Local linearization methods for the numerical integration of ordinary differential equations: An overview

International Nuclear Information System (INIS)

Jimenez, J.C.

2009-06-01

Local Linearization (LL) methods conform a class of one-step explicit integrators for ODEs derived from the following primary and common strategy: the vector field of the differential equation is locally (piecewise) approximated through a first-order Taylor expansion at each time step, thus obtaining successive linear equations that are explicitly integrated. Hereafter, the LL approach may include some additional strategies to improve that basic affine approximation. Theoretical and practical results have shown that the LL integrators have a number of convenient properties. These include arbitrary order of convergence, A-stability, linearization preserving, regularity under quite general conditions, preservation of the dynamics of the exact solution around hyperbolic equilibrium points and periodic orbits, integration of stiff and high-dimensional equations, low computational cost, and others. In this paper, a review of the LL methods and their properties is presented. (author)

Searching fundamental information in ordinary differential equations. Nondimensionalization technique.

Science.gov (United States)

Sánchez Pérez, J F; Conesa, M; Alhama, I; Alhama, F; Cánovas, M

2017-01-01

Classical dimensional analysis and nondimensionalization are assumed to be two similar approaches in the search for dimensionless groups. Both techniques, simplify the study of many problems. The first approach does not need to know the mathematical model, being sufficient a deep understanding of the physical phenomenon involved, while the second one begins with the governing equations and reduces them to their dimensionless form by simple mathematical manipulations. In this work, a formal protocol is proposed for applying the nondimensionalization process to ordinary differential equations, linear or not, leading to dimensionless normalized equations from which the resulting dimensionless groups have two inherent properties: In one hand, they are physically interpreted as balances between counteracting quantities in the problem, and on the other hand, they are of the order of magnitude unity. The solutions provided by nondimensionalization are more precise in every case than those from dimensional analysis, as it is illustrated by the applications studied in this work.

Valuing a gas-fired power plant: A comparison of ordinary linear models, regime-switching approaches, and models with stochastic volatility

International Nuclear Information System (INIS)

Heydari, Somayeh; Siddiqui, Afzal

2010-01-01

Energy prices are often highly volatile with unexpected spikes. Capturing these sudden spikes may lead to more informed decision-making in energy investments, such as valuing gas-fired power plants, than ignoring them. In this paper, non-linear regime-switching models and models with mean-reverting stochastic volatility are compared with ordinary linear models. The study is performed using UK electricity and natural gas daily spot prices and suggests that with the aim of valuing a gas-fired power plant with and without operational flexibility, non-linear models with stochastic volatility, specifically for logarithms of electricity prices, provide better out-of-sample forecasts than both linear models and regime-switching models.

Nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order

International Nuclear Information System (INIS)

Pradeep, R Gladwin; Chandrasekar, V K; Senthilvelan, M; Lakshmanan, M

2011-01-01

In this paper, we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations (ODEs) of arbitrary order related to linear ODEs through nonlocal relations. The procedure makes use of the Lie point symmetries of the linear ODEs and the nonlocal connection to deduce the nonlocal symmetries of the corresponding nonlinear ODEs. Using these nonlocal symmetries, we obtain reduction transformations and reduced equations to specific examples. We find that the reduced equations can be explicitly integrated to deduce the general solutions for these cases. We also extend this procedure to coupled higher order nonlinear ODEs with specific reference to second-order nonlinear ODEs. (paper)

Schwarzian conditions for linear differential operators with selected differential Galois groups

International Nuclear Information System (INIS)

Abdelaziz, Y; Maillard, J-M

2017-01-01

We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions can be generalized to arbitrary order linear differential operators with polynomial coefficients having selected differential Galois groups. For order three and order four linear differential operators we show that this pullback invariance up to conjugation eventually reduces to symmetric powers of an underlying order-two operator. We give, precisely, the conditions to have modular correspondences solutions for such Schwarzian differential equations, which was an open question in a previous paper. We analyze in detail a pullbacked hypergeometric example generalizing modular forms, that ushers a pullback invariance up to operator homomorphisms. We finally consider the more general problem of the equivalence of two different order-four linear differential Calabi–Yau operators up to pullbacks and conjugation, and clarify the cases where they have the same Yukawa couplings. (paper)

Schwarzian conditions for linear differential operators with selected differential Galois groups

Science.gov (United States)

Abdelaziz, Y.; Maillard, J.-M.

2017-11-01

We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions can be generalized to arbitrary order linear differential operators with polynomial coefficients having selected differential Galois groups. For order three and order four linear differential operators we show that this pullback invariance up to conjugation eventually reduces to symmetric powers of an underlying order-two operator. We give, precisely, the conditions to have modular correspondences solutions for such Schwarzian differential equations, which was an open question in a previous paper. We analyze in detail a pullbacked hypergeometric example generalizing modular forms, that ushers a pullback invariance up to operator homomorphisms. We finally consider the more general problem of the equivalence of two different order-four linear differential Calabi-Yau operators up to pullbacks and conjugation, and clarify the cases where they have the same Yukawa couplings.

Non-monotone positive solutions of second-order linear differential equations: existence, nonexistence and criteria

Directory of Open Access Journals (Sweden)

Mervan Pašić

2016-10-01

Full Text Available We study non-monotone positive solutions of the second-order linear differential equations: $(p(tx'' + q(t x = e(t$, with positive $p(t$ and $q(t$. For the first time, some criteria as well as the existence and nonexistence of non-monotone positive solutions are proved in the framework of some properties of solutions $\\theta (t$ of the corresponding integrable linear equation: $(p(t\\theta''=e(t$. The main results are illustrated by many examples dealing with equations which allow exact non-monotone positive solutions not necessarily periodic. Finally, we pose some open questions.

KRYSI, Ordinary Differential Equations Solver with Sdirk Krylov Method

International Nuclear Information System (INIS)

Hindmarsh, A.C.; Norsett, S.P.

2001-01-01

1 - Description of program or function: KRYSI is a set of FORTRAN subroutines for solving ordinary differential equations initial value problems. It is suitable for both stiff and non-stiff systems. When solving the implicit stage equations in the stiff case, KRYSI uses a Krylov subspace iteration method called the SPIGMR (Scaled Preconditioned Incomplete Generalized Minimum Residual) method. No explicit Jacobian storage is required, except where used in pre- conditioning. A demonstration problem is included with a description of two pre-conditioners that are natural for its solution by KRYSI. 2 - Method of solution: KRYSI uses a three-stage, third-order singly diagonally implicit Runge-Kutta (SDIRK) method. In the stiff case, a preconditioned Krylov subspace iteration within a (so-called) inexact Newton iteration is used to solve the system of nonlinear algebraic equations

Non-diagonal processes of singlet and ordinary quark production

International Nuclear Information System (INIS)

Bejlin, V.A.; Vereshkov, G.M.; Kuksa, V.I.

1995-01-01

Non-diagonal processes of singlet and ordinary quark production are analyzed in the model where the down singlet quark mixes with the ordinary ones. The possibility of experimental selection of h-quark effects is demonstrated

Design of TIR collimating lens for ordinary differential equation of extended light source

Science.gov (United States)

Zhan, Qianjing; Liu, Xiaoqin; Hou, Zaihong; Wu, Yi

2017-10-01

The source of LED has been widely used in our daily life. The intensity angle distribution of single LED is lambert distribution, which does not satisfy the requirement of people. Therefore, we need to distribute light and change the LED's intensity angle distribution. The most commonly method to change its intensity angle distribution is the free surface. Generally, using ordinary differential equations to calculate free surface can only be applied in a point source, but it will lead to a big error for the expand light. This paper proposes a LED collimating lens based on the ordinary differential equation, combined with the LED's light distribution curve, and adopt the method of calculating the center gravity of the extended light to get the normal vector. According to the law of Snell, the ordinary differential equations are constructed. Using the runge-kutta method for solution of ordinary differential equation solution, the curve point coordinates are gotten. Meanwhile, the edge point data of lens are imported into the optical simulation software TracePro. Based on 1mm×1mm single lambert body for light conditions, The degrees of collimating light can be close to +/-3. Furthermore, the energy utilization rate is higher than 85%. In this paper, the point light source is used to calculate partial differential equation method and compared with the simulation of the lens, which improve the effect of 1 degree of collimation.

Antiperiodic Boundary Value Problems for Second-Order Impulsive Ordinary Differential Equations

Directory of Open Access Journals (Sweden)

2009-02-01

Full Text Available We consider a second-order ordinary differential equation with antiperiodic boundary conditions and impulses. By using Schaefer's fixed-point theorem, some existence results are obtained.

Ordinary differential equations a graduate text

CERN Document Server

Bhamra, K S

2015-01-01

ORDINARY DIFFERENTIAL EQUATIONS: A Graduate Text presents a systematic and comprehensive introduction to ODEs for graduate and postgraduate students. The systematic organized text on differential inequalities, Gronwall's inequality, Nagumo's theorems, Osgood's criteria and applications of different equations of first order is dealt with in a greater depth. The book discusses qualitative and quantitative aspects of the Strum - Liouville problems, Green's function, integral equations, Laplace transform and is supported by a number of worked-out examples in each lesson to make the concepts clear. A lot of stress on stability theory is laid down, especially on Lyapunov and Poincare stability theory. A numerous figures in various lessons (in particular lessons dealing with stability theory) have been added to clarify the key concepts in DE theory. Nonlinear oscillation in conservative systems and Hamiltonian systems highlights basic nature of the systems considered. Perturbation techniques lesson deals in fairly d...

Differential-algebraic solutions of the heat equation

OpenAIRE

Buchstaber, Victor M.; Netay, Elena Yu.

2014-01-01

In this work we introduce the notion of differential-algebraic ansatz for the heat equation and explicitly construct heat equation and Burgers equation solutions given a solution of a homogeneous non-linear ordinary differential equation of a special form. The ansatz for such solutions is called the $n$-ansatz, where $n+1$ is the order of the differential equation.

A novel algebraic procedure for solving non-linear evolution equations of higher order

International Nuclear Information System (INIS)

Huber, Alfred

2007-01-01

We report here a systematic approach that can easily be used for solving non-linear partial differential equations (nPDE), especially of higher order. We restrict the analysis to the so called evolution equations describing any wave propagation. The proposed new algebraic approach leads us to traveling wave solutions and moreover, new class of solution can be obtained. The crucial step of our method is the basic assumption that the solutions satisfy an ordinary differential equation (ODE) of first order that can be easily integrated. The validity and reliability of the method is tested by its application to some non-linear evolution equations. The important aspect of this paper however is the fact that we are able to calculate distinctive class of solutions which cannot be found in the current literature. In other words, using this new algebraic method the solution manifold is augmented to new class of solution functions. Simultaneously we would like to stress the necessity of such sophisticated methods since a general theory of nPDE does not exist. Otherwise, for practical use the algebraic construction of new class of solutions is of fundamental interest

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

Science.gov (United States)

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.

Expectation propagation for large scale Bayesian inference of non-linear molecular networks from perturbation data.

Science.gov (United States)

Narimani, Zahra; Beigy, Hamid; Ahmad, Ashar; Masoudi-Nejad, Ali; Fröhlich, Holger

2017-01-01

Inferring the structure of molecular networks from time series protein or gene expression data provides valuable information about the complex biological processes of the cell. Causal network structure inference has been approached using different methods in the past. Most causal network inference techniques, such as Dynamic Bayesian Networks and ordinary differential equations, are limited by their computational complexity and thus make large scale inference infeasible. This is specifically true if a Bayesian framework is applied in order to deal with the unavoidable uncertainty about the correct model. We devise a novel Bayesian network reverse engineering approach using ordinary differential equations with the ability to include non-linearity. Besides modeling arbitrary, possibly combinatorial and time dependent perturbations with unknown targets, one of our main contributions is the use of Expectation Propagation, an algorithm for approximate Bayesian inference over large scale network structures in short computation time. We further explore the possibility of integrating prior knowledge into network inference. We evaluate the proposed model on DREAM4 and DREAM8 data and find it competitive against several state-of-the-art existing network inference methods.

Description and use of LSODE, the Livermore Solver for Ordinary Differential Equations

Science.gov (United States)

Radhakrishnan, Krishnan; Hindmarsh, Alan C.

1993-01-01

LSODE, the Livermore Solver for Ordinary Differential Equations, is a package of FORTRAN subroutines designed for the numerical solution of the initial value problem for a system of ordinary differential equations. It is particularly well suited for 'stiff' differential systems, for which the backward differentiation formula method of orders 1 to 5 is provided. The code includes the Adams-Moulton method of orders 1 to 12, so it can be used for nonstiff problems as well. In addition, the user can easily switch methods to increase computational efficiency for problems that change character. For both methods a variety of corrector iteration techniques is included in the code. Also, to minimize computational work, both the step size and method order are varied dynamically. This report presents complete descriptions of the code and integration methods, including their implementation. It also provides a detailed guide to the use of the code, as well as an illustrative example problem.

Solving (2 + 1)-dimensional sine-Poisson equation by a modified variable separated ordinary differential equation method

International Nuclear Information System (INIS)

Ka-Lin, Su; Yuan-Xi, Xie

2010-01-01

By introducing a more general auxiliary ordinary differential equation (ODE), a modified variable separated ordinary differential equation method is presented for solving the (2 + 1)-dimensional sine-Poisson equation. As a result, many explicit and exact solutions of the (2 + 1)-dimensional sine-Poisson equation are derived in a simple manner by this technique. (general)

Robust optimal control design using a differential game approach for open-loop linear quadratic descriptor systems

NARCIS (Netherlands)

Musthofa, M.W.; Salmah, S.; Engwerda, Jacob; Suparwanto, A.

This paper studies the robust optimal control problem for descriptor systems. We applied differential game theory to solve the disturbance attenuation problem. The robust control problem was converted into a reduced ordinary zero-sum game. Within a linear quadratic setting, we solved the problem for

Ordinary Differential Equation Models for Adoptive Immunotherapy.

Science.gov (United States)

Talkington, Anne; Dantoin, Claudia; Durrett, Rick

2018-05-01

Modified T cells that have been engineered to recognize the CD19 surface marker have recently been shown to be very successful at treating acute lymphocytic leukemias. Here, we explore four previous approaches that have used ordinary differential equations to model this type of therapy, compare their properties, and modify the models to address their deficiencies. Although the four models treat the workings of the immune system in slightly different ways, they all predict that adoptive immunotherapy can be successful to move a patient from the large tumor fixed point to an equilibrium with little or no tumor.

NIMROD: a program for inference via a normal approximation of the posterior in models with random effects based on ordinary differential equations.

Science.gov (United States)

Prague, Mélanie; Commenges, Daniel; Guedj, Jérémie; Drylewicz, Julia; Thiébaut, Rodolphe

2013-08-01

Models based on ordinary differential equations (ODE) are widespread tools for describing dynamical systems. In biomedical sciences, data from each subject can be sparse making difficult to precisely estimate individual parameters by standard non-linear regression but information can often be gained from between-subjects variability. This makes natural the use of mixed-effects models to estimate population parameters. Although the maximum likelihood approach is a valuable option, identifiability issues favour Bayesian approaches which can incorporate prior knowledge in a flexible way. However, the combination of difficulties coming from the ODE system and from the presence of random effects raises a major numerical challenge. Computations can be simplified by making a normal approximation of the posterior to find the maximum of the posterior distribution (MAP). Here we present the NIMROD program (normal approximation inference in models with random effects based on ordinary differential equations) devoted to the MAP estimation in ODE models. We describe the specific implemented features such as convergence criteria and an approximation of the leave-one-out cross-validation to assess the model quality of fit. In pharmacokinetics models, first, we evaluate the properties of this algorithm and compare it with FOCE and MCMC algorithms in simulations. Then, we illustrate NIMROD use on Amprenavir pharmacokinetics data from the PUZZLE clinical trial in HIV infected patients. Copyright © 2013 Elsevier Ireland Ltd. All rights reserved.

Symmetries of th-Order Approximate Stochastic Ordinary Differential Equations

OpenAIRE

Fredericks, E.; Mahomed, F. M.

2012-01-01

Symmetries of $n$ th-order approximate stochastic ordinary differential equations (SODEs) are studied. The determining equations of these SODEs are derived in an Itô calculus context. These determining equations are not stochastic in nature. SODEs are normally used to model nature (e.g., earthquakes) or for testing the safety and reliability of models in construction engineering when looking at the impact of random perturbations.

Error estimation in the neural network solution of ordinary differential equations.

Science.gov (United States)

Filici, Cristian

2010-06-01

In this article a method of error estimation for the neural approximation of the solution of an Ordinary Differential Equation is presented. Some examples of the application of the method support the theory presented. Copyright 2010. Published by Elsevier Ltd.

Approximate analytical methods for solving ordinary differential equations

CERN Document Server

Radhika, TSL; Rani, T Raja

2015-01-01

Approximate Analytical Methods for Solving Ordinary Differential Equations (ODEs) is the first book to present all of the available approximate methods for solving ODEs, eliminating the need to wade through multiple books and articles. It covers both well-established techniques and recently developed procedures, including the classical series solution method, diverse perturbation methods, pioneering asymptotic methods, and the latest homotopy methods.The book is suitable not only for mathematicians and engineers but also for biologists, physicists, and economists. It gives a complete descripti

Hybrid modeling of the crosstalk between signaling and transcriptional networks using ordinary differential equations and multi-valued logic.

Science.gov (United States)

Khan, Faiz M; Schmitz, Ulf; Nikolov, Svetoslav; Engelmann, David; Pützer, Brigitte M; Wolkenhauer, Olaf; Vera, Julio

2014-01-01

A decade of successful results indicates that systems biology is the appropriate approach to investigate the regulation of complex biochemical networks involving transcriptional and post-transcriptional regulations. It becomes mandatory when dealing with highly interconnected biochemical networks, composed of hundreds of compounds, or when networks are enriched in non-linear motifs like feedback and feedforward loops. An emerging dilemma is to conciliate models of massive networks and the adequate description of non-linear dynamics in a suitable modeling framework. Boolean networks are an ideal representation of massive networks that are humble in terms of computational complexity and data demand. However, they are inappropriate when dealing with nested feedback/feedforward loops, structural motifs common in biochemical networks. On the other hand, models of ordinary differential equations (ODEs) cope well with these loops, but they require enormous amounts of quantitative data for a full characterization of the model. Here we propose hybrid models, composed of ODE and logical sub-modules, as a strategy to handle large scale, non-linear biochemical networks that include transcriptional and post-transcriptional regulations. We illustrate the construction of this kind of models using as example a regulatory network centered on E2F1, a transcription factor involved in cancer. The hybrid modeling approach proposed is a good compromise between quantitative/qualitative accuracy and scalability when considering large biochemical networks with a small highly interconnected core, and module of transcriptionally regulated genes that are not part of critical regulatory loops. This article is part of a Special Issue entitled: Computational Proteomics, Systems Biology & Clinical Implications. Guest Editor: Yudong Cai. Copyright © 2013 Elsevier B.V. All rights reserved.

From Ordinary Differential Equations to Structural Causal Models: the deterministic case

NARCIS (Netherlands)

Mooij, J.M.; Janzing, D.; Schölkopf, B.; Nicholson, A.; Smyth, P.

2013-01-01

We show how, and under which conditions, the equilibrium states of a first-order Ordinary Differential Equation (ODE) system can be described with a deterministic Structural Causal Model (SCM). Our exposition sheds more light on the concept of causality as expressed within the framework of

Solving Second-Order Ordinary Differential Equations without Using Complex Numbers

Science.gov (United States)

Kougias, Ioannis E.

2009-01-01

Ordinary differential equations (ODEs) is a subject with a wide range of applications and the need of introducing it to students often arises in the last year of high school, as well as in the early stages of tertiary education. The usual methods of solving second-order ODEs with constant coefficients, among others, rely upon the use of complex…

Numerical solution of stiff systems of ordinary differential equations with applications to electronic circuits

Science.gov (United States)

Rosenbaum, J. S.

1971-01-01

Systems of ordinary differential equations in which the magnitudes of the eigenvalues (or time constants) vary greatly are commonly called stiff. Such systems of equations arise in nuclear reactor kinetics, the flow of chemically reacting gas, dynamics, control theory, circuit analysis and other fields. The research reported develops an A-stable numerical integration technique for solving stiff systems of ordinary differential equations. The method, which is called the generalized trapezoidal rule, is a modification of the trapezoidal rule. However, the method is computationally more efficient than the trapezoidal rule when the solution of the almost-discontinuous segments is being calculated.

Bifurcation of rupture path by linear and cubic damping force

Science.gov (United States)

Dennis L. C., C.; Chew X., Y.; Lee Y., C.

2014-06-01

Bifurcation of rupture path is studied for the effect of linear and cubic damping. Momentum equation with Rayleigh factor was transformed into ordinary differential form. Bernoulli differential equation was obtained and solved by the separation of variables. Analytical or exact solutions yielded the bifurcation was visible at imaginary part when the wave was non dispersive. For the dispersive wave, bifurcation of rupture path was invisible.

EXISTENCE OF POSITIVE SOLUTION TO TWO-POINT BOUNDARY VALUE PROBLEM FOR A SYSTEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

Institute of Scientific and Technical Information of China (English)

无

2009-01-01

In this paper, we consider a two-point boundary value problem for a system of second order ordinary differential equations. Under some conditions, we show the existence of positive solution to the system of second order ordinary differential equa-tions.

Oscillatory properties of solutions to certain two-dimensional systems of non-linear ordinary differential equations

Czech Academy of Sciences Publication Activity Database

Dosoudilová, M.; Lomtatidze, Alexander; Šremr, Jiří

2015-01-01

Roč. 120, June (2015), s. 57-75 ISSN 0362-546X Institutional support: RVO:67985840 Keywords : half-linear system * Hartman -Wintner theorem * Kamenev theorem Subject RIV: BA - General Mathematics Impact factor: 1.125, year: 2015 http://www.sciencedirect.com/science/article/pii/S0362546X15000620

Non-linear feedback control of the p53 protein-mdm2 inhibitor system using the derivative-free non-linear Kalman filter.

Science.gov (United States)

Rigatos, Gerasimos G

2016-06-01

It is proven that the model of the p53-mdm2 protein synthesis loop is a differentially flat one and using a diffeomorphism (change of state variables) that is proposed by differential flatness theory it is shown that the protein synthesis model can be transformed into the canonical (Brunovsky) form. This enables the design of a feedback control law that maintains the concentration of the p53 protein at the desirable levels. To estimate the non-measurable elements of the state vector describing the p53-mdm2 system dynamics, the derivative-free non-linear Kalman filter is used. Moreover, to compensate for modelling uncertainties and external disturbances that affect the p53-mdm2 system, the derivative-free non-linear Kalman filter is re-designed as a disturbance observer. The derivative-free non-linear Kalman filter consists of the Kalman filter recursion applied on the linearised equivalent of the protein synthesis model together with an inverse transformation based on differential flatness theory that enables to retrieve estimates for the state variables of the initial non-linear model. The proposed non-linear feedback control and perturbations compensation method for the p53-mdm2 system can result in more efficient chemotherapy schemes where the infusion of medication will be better administered.

Non-symmetric forms of non-linear vibrations of flexible cylindrical panels and plates under longitudinal load and additive white noise

Science.gov (United States)

Krysko, V. A.; Awrejcewicz, J.; Krylova, E. Yu; Papkova, I. V.; Krysko, A. V.

2018-06-01

Parametric non-linear vibrations of flexible cylindrical panels subjected to additive white noise are studied. The governing Marguerre equations are investigated using the finite difference method (FDM) of the second-order accuracy and the Runge-Kutta method. The considered mechanical structural member is treated as a system of many/infinite number of degrees of freedom (DoF). The dependence of chaotic vibrations on the number of DoFs is investigated. Reliability of results is guaranteed by comparing the results obtained using two qualitatively different methods to reduce the problem of PDEs (partial differential equations) to ODEs (ordinary differential equations), i.e. the Faedo-Galerkin method in higher approximations and the 4th and 6th order FDM. The Cauchy problem obtained by the FDM is eventually solved using the 4th-order Runge-Kutta methods. The numerical experiment yielded, for a certain set of parameters, the non-symmetric vibration modes/forms with and without white noise. In particular, it has been illustrated and discussed that action of white noise on chaotic vibrations implies quasi-periodicity, whereas the previously non-symmetric vibration modes are closer to symmetric ones.

Using trees to compute approximate solutions to ordinary differential equations exactly

Science.gov (United States)

Grossman, Robert

1991-01-01

Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.

Extending the Constant Coefficient Solution Technique to Variable Coefficient Ordinary Differential Equations

Science.gov (United States)

Mohammed, Ahmed; Zeleke, Aklilu

2015-01-01

We introduce a class of second-order ordinary differential equations (ODEs) with variable coefficients whose closed-form solutions can be obtained by the same method used to solve ODEs with constant coefficients. General solutions for the homogeneous case are discussed.

Solvability conditions for non-local boundary value problems for two-dimensional half-linear differential systems

Czech Academy of Sciences Publication Activity Database

Kiguradze, I.; Šremr, Jiří

2011-01-01

Roč. 74, č. 17 (2011), s. 6537-6552 ISSN 0362-546X Institutional research plan: CEZ:AV0Z10190503 Keywords : half-linear differential system * non-local boundary value problem * solvability Subject RIV: BA - General Mathematics Impact factor: 1.536, year: 2011 http://www.sciencedirect.com/science/article/pii/S0362546X11004573

A Simple Method to Find out when an Ordinary Differential Equation Is Separable

Science.gov (United States)

Cid, Jose Angel

2009-01-01

We present an alternative method to that of Scott (D. Scott, "When is an ordinary differential equation separable?", "Amer. Math. Monthly" 92 (1985), pp. 422-423) to teach the students how to discover whether a differential equation y[prime] = f(x,y) is separable or not when the nonlinearity f(x, y) is not explicitly factorized. Our approach is…

Value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations

Directory of Open Access Journals (Sweden)

Luo Li-Qin

2016-01-01

Full Text Available In this paper, we investigate the value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations, and obtain the results on the relations between the order of the solutions and the convergence exponents of the zeros, poles, a-points and small function value points of the solutions, which show the relations in the case of non-homogeneous equations are sharper than the ones in the case of homogeneous equations.

Inverse Boundary Value Problem for Non-linear Hyperbolic Partial Differential Equations

OpenAIRE

Nakamura, Gen; Vashisth, Manmohan

2017-01-01

In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\\geq 3$. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear isotropic wave equation with the speed $\\sqrt{\\gamma(x)}$ at each point $x$ in a given spacial domain. For any small solution $u=u(t,x)$ of this non-linear equation, we have the linear isotr...

Identifying and Exploring Relationships between Contextual Situations and Ordinary Differential Equations

Science.gov (United States)

Camacho-Machín, M.; Guerrero-Ortiz, C.

2015-01-01

The aim of this paper is to present and discuss some of the evidence regarding the resources that students use when they establish relationships between a contextual situation and an ordinary differential equation (ODE). We present research results obtained from work by seven students in a graduate level course in mathematics education, where they…

Two-dimensional differential transform method for solving linear and non-linear Schroedinger equations

International Nuclear Information System (INIS)

Ravi Kanth, A.S.V.; Aruna, K.

2009-01-01

In this paper, we propose a reliable algorithm to develop exact and approximate solutions for the linear and nonlinear Schroedinger equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.

Consistency of direct integral estimator for partially observed systems of ordinary differential equations

NARCIS (Netherlands)

Vujačić, Ivan; Dattner, Itai

In this paper we use the sieve framework to prove consistency of the ‘direct integral estimator’ of parameters for partially observed systems of ordinary differential equations, which are commonly used for modeling dynamic processes.

Convergence of hybrid methods for solving non-linear partial ...

African Journals Online (AJOL)

This paper is concerned with the numerical solution and convergence analysis of non-linear partial differential equations using a hybrid method. The solution technique involves discretizing the non-linear system of PDE to obtain a corresponding non-linear system of algebraic difference equations to be solved at each time ...

Optimal Control Strategies in a Two Dimensional Differential Game Using Linear Equation under a Perturbed System

Directory of Open Access Journals (Sweden)

Musa Danjuma SHEHU

2008-06-01

Full Text Available This paper lays emphasis on formulation of two dimensional differential games via optimal control theory and consideration of control systems whose dynamics is described by a system of Ordinary Differential equation in the form of linear equation under the influence of two controls U(. and V(.. Base on this, strategies were constructed. Hence we determine the optimal strategy for a control say U(. under a perturbation generated by the second control V(. within a given manifold M.

Ordinary differential equations basics and beyond

CERN Document Server

Schaeffer, David G

2016-01-01

This book develops the theory of ordinary differential equations (ODEs), starting from an introductory level (with no prior experience in ODEs assumed) through to a graduate-level treatment of the qualitative theory, including bifurcation theory (but not chaos). While proofs are rigorous, the exposition is reader-friendly, aiming for the informality of face-to-face interactions. A unique feature of this book is the integration of rigorous theory with numerous applications of scientific interest. Besides providing motivation, this synthesis clarifies the theory and enhances scientific literacy. Other features include: (i) a wealth of exercises at various levels, along with commentary that explains why they matter; (ii) figures with consistent color conventions to identify nullclines, periodic orbits, stable and unstable manifolds; and (iii) a dedicated website with software templates, problem solutions, and other resources supporting the text. Given its many applications, the book may be used comfortably in sc...

Magnetohydrodynamic flow of Carreau fluid over a convectively heated surface in the presence of non-linear radiation

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Khan, Masood [Department of Mathematics, Quaid-i-Azam University, Islamabad 44000 (Pakistan); Hashim, E-mail: hashim_alik@yahoo.com [Department of Mathematics, Quaid-i-Azam University, Islamabad 44000 (Pakistan); Hussain, M. [Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Islamabad 44000 (Pakistan); Azam, M. [Department of Mathematics, Quaid-i-Azam University, Islamabad 44000 (Pakistan)

2016-08-15

This paper presents a study of the magnetohydrodynamic (MHD) boundary layer flow of a non-Newtonian Carreau fluid over a convectively heated surface. The analysis of heat transfer is further performed in the presence of non-linear thermal radiation. The appropriate transformations are employed to bring the governing equations into dimensionless form. The numerical solutions of the partially coupled non-linear ordinary differential equations are obtained by using the Runge-Kutta Fehlberg integration scheme. The influence of non-dimensional governing parameters on the velocity, temperature, local skin friction coefficient and local Nusselt number is studied and discussed with the help of graphs and tables. Results proved that there is significant decrease in the velocity and the corresponding momentum boundary layer thickness with the growth in the magnetic parameter. However, a quite the opposite is true for the temperature and the corresponding thermal boundary layer thickness. - Highlights: • We investigated the Magnetohydrodynamic flow of Carreau constitutive fluid model. • Impact of non-linear thermal radiation is further taken into account. • Runge-Kutta Fehlberg method is employed to obtain the numerical solutions. • Fluid velocity is higher in case of hydromagnetic flow in comparison with hydrodynamic flow. • The local Nusselt number is a decreasing function of the thermal radiation parameter.

Universal formats for nonlinear ordinary differential systems

International Nuclear Information System (INIS)

Kerner, E.H.

1981-01-01

It is shown that very general nonlinear ordinary differential systems (embracing all that arise in practice) may, first, be brought down to polynomial systems (where the nonlinearities occur only as polynomials in the dependent variables) by introducing suitable new variables into the original system; second, that polynomial systems are reducible to ''Riccati systems,'' where the nonlinearities are quadratic at most; third, that Riccati systems may be brought to elemental universal formats containing purely quadratic terms with simple arrays of coefficients that are all zero or unity. The elemental systems have representations as novel types of matrix Riccati equations. Different starting systems and their associated Riccati systems differ from one another, at the final elemental level, in order and in initial data, but not in format

A variational approach to parameter estimation in ordinary differential equations

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Kaschek Daniel

2012-08-01

Full Text Available Abstract Background Ordinary differential equations are widely-used in the field of systems biology and chemical engineering to model chemical reaction networks. Numerous techniques have been developed to estimate parameters like rate constants, initial conditions or steady state concentrations from time-resolved data. In contrast to this countable set of parameters, the estimation of entire courses of network components corresponds to an innumerable set of parameters. Results The approach presented in this work is able to deal with course estimation for extrinsic system inputs or intrinsic reactants, both not being constrained by the reaction network itself. Our method is based on variational calculus which is carried out analytically to derive an augmented system of differential equations including the unconstrained components as ordinary state variables. Finally, conventional parameter estimation is applied to the augmented system resulting in a combined estimation of courses and parameters. Conclusions The combined estimation approach takes the uncertainty in input courses correctly into account. This leads to precise parameter estimates and correct confidence intervals. In particular this implies that small motifs of large reaction networks can be analysed independently of the rest. By the use of variational methods, elements from control theory and statistics are combined allowing for future transfer of methods between the two fields.

A variational approach to parameter estimation in ordinary differential equations.

Science.gov (United States)

Kaschek, Daniel; Timmer, Jens

2012-08-14

Ordinary differential equations are widely-used in the field of systems biology and chemical engineering to model chemical reaction networks. Numerous techniques have been developed to estimate parameters like rate constants, initial conditions or steady state concentrations from time-resolved data. In contrast to this countable set of parameters, the estimation of entire courses of network components corresponds to an innumerable set of parameters. The approach presented in this work is able to deal with course estimation for extrinsic system inputs or intrinsic reactants, both not being constrained by the reaction network itself. Our method is based on variational calculus which is carried out analytically to derive an augmented system of differential equations including the unconstrained components as ordinary state variables. Finally, conventional parameter estimation is applied to the augmented system resulting in a combined estimation of courses and parameters. The combined estimation approach takes the uncertainty in input courses correctly into account. This leads to precise parameter estimates and correct confidence intervals. In particular this implies that small motifs of large reaction networks can be analysed independently of the rest. By the use of variational methods, elements from control theory and statistics are combined allowing for future transfer of methods between the two fields.

Linearization: Geometric, Complex, and Conditional

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Asghar Qadir

2012-01-01

Full Text Available Lie symmetry analysis provides a systematic method of obtaining exact solutions of nonlinear (systems of differential equations, whether partial or ordinary. Of special interest is the procedure that Lie developed to transform scalar nonlinear second-order ordinary differential equations to linear form. Not much work was done in this direction to start with, but recently there have been various developments. Here, first the original work of Lie (and the early developments on it, and then more recent developments based on geometry and complex analysis, apart from Lie’s own method of algebra (namely, Lie group theory, are reviewed. It is relevant to mention that much of the work is not linearization but uses the base of linearization.

On the removal of boundary errors caused by Runge-Kutta integration of non-linear partial differential equations

Science.gov (United States)

Abarbanel, Saul; Gottlieb, David; Carpenter, Mark H.

1994-01-01

It has been previously shown that the temporal integration of hyperbolic partial differential equations (PDE's) may, because of boundary conditions, lead to deterioration of accuracy of the solution. A procedure for removal of this error in the linear case has been established previously. In the present paper we consider hyperbolic (PDE's) (linear and non-linear) whose boundary treatment is done via the SAT-procedure. A methodology is present for recovery of the full order of accuracy, and has been applied to the case of a 4th order explicit finite difference scheme.

Exploring Students' Understanding of Ordinary Differential Equations Using Computer Algebraic System (CAS)

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Maat, Siti Mistima; Zakaria, Effandi

2011-01-01

Ordinary differential equations (ODEs) are one of the important topics in engineering mathematics that lead to the understanding of technical concepts among students. This study was conducted to explore the students' understanding of ODEs when they solve ODE questions using a traditional method as well as a computer algebraic system, particularly…

Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras.

Science.gov (United States)

Gainetdinova, A A; Gazizov, R K

2017-01-01

We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.

A Numerical Scheme for Ordinary Differential Equations Having Time Varying and Nonlinear Coefficients Based on the State Transition Matrix

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Bartels, Robert E.

2002-01-01

A variable order method of integrating initial value ordinary differential equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. While it is more complex than most other methods, it produces exact solutions at arbitrary time step size when the time variation of the system can be modeled exactly by a polynomial. Solutions to several nonlinear problems exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with an exact solution and with solutions obtained by established methods.

International Conference on Differential Equations and Mathematical Physics

CERN Document Server

Saitō, Yoshimi

1987-01-01

The meeting in Birmingham, Alabama, provided a forum for the discussion of recent developments in the theory of ordinary and partial differential equations, both linear and non-linear, with particular reference to work relating to the equations of mathematical physics. The meeting was attended by about 250 mathematicians from 22 countries. The papers in this volume all involve new research material, with at least outline proofs; some papers also contain survey material. Topics covered include: Schrödinger theory, scattering and inverse scattering, fluid mechanics (including conservative systems and inertial manifold theory attractors), elasticity, non-linear waves, and feedback control theory.

Estimation of Ordinary Differential Equation Parameters Using Constrained Local Polynomial Regression.

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Ding, A Adam; Wu, Hulin

2014-10-01

We propose a new method to use a constrained local polynomial regression to estimate the unknown parameters in ordinary differential equation models with a goal of improving the smoothing-based two-stage pseudo-least squares estimate. The equation constraints are derived from the differential equation model and are incorporated into the local polynomial regression in order to estimate the unknown parameters in the differential equation model. We also derive the asymptotic bias and variance of the proposed estimator. Our simulation studies show that our new estimator is clearly better than the pseudo-least squares estimator in estimation accuracy with a small price of computational cost. An application example on immune cell kinetics and trafficking for influenza infection further illustrates the benefits of the proposed new method.

Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations

International Nuclear Information System (INIS)

Khan, Junaid Ali; Raja, Muhammad Asif Zahoor; Qureshi, Ijaz Mansoor

2011-01-01

We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs). The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error. The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique. The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations. We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods. The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed. (general)

Linear measure functional differential equations with infinite delay

Czech Academy of Sciences Publication Activity Database

Monteiro, Giselle Antunes; Slavík, A.

2014-01-01

Roč. 287, 11-12 (2014), s. 1363-1382 ISSN 0025-584X Institutional support: RVO:67985840 Keywords : measure functional differential equations * generalized ordinary differential equations * Kurzweil-Stieltjes integral Subject RIV: BA - General Mathematics Impact factor: 0.683, year: 2014 http://onlinelibrary.wiley.com/doi/10.1002/mana.201300048/abstract

Numerical analysis of non-linear vibrations of a fractionally damped cylindrical shell under the conditions of combinational internal resonance

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Rossikhin Yury A.

2018-01-01

Full Text Available Non-linear damped vibrations of a cylindrical shell embedded into a fractional derivative medium are investigated for the case of the combinational internal resonance, resulting in modal interaction, using two different numerical methods with further comparison of the results obtained. The damping properties of the surrounding medium are described by the fractional derivative Kelvin-Voigt model utilizing the Riemann-Liouville fractional derivatives. Within the first method, the generalized displacements of a coupled set of nonlinear ordinary differential equations of the second order are estimated using numerical solution of nonlinear multi-term fractional differential equations by the procedure based on the reduction of the problem to a system of fractional differential equations. According to the second method, the amplitudes and phases of nonlinear vibrations are estimated from the governing nonlinear differential equations describing amplitude-and-phase modulations for the case of the combinational internal resonance. A good agreement in results is declared.

Parameters and Fractional Differentiation Orders Estimation for Linear Continuous-Time Non-Commensurate Fractional Order Systems

KAUST Repository

Belkhatir, Zehor; Laleg-Kirati, Taous-Meriem

2017-01-01

This paper proposes a two-stage estimation algorithm to solve the problem of joint estimation of the parameters and the fractional differentiation orders of a linear continuous-time fractional system with non-commensurate orders. The proposed algorithm combines the modulating functions and the first-order Newton methods. Sufficient conditions ensuring the convergence of the method are provided. An error analysis in the discrete case is performed. Moreover, the method is extended to the joint estimation of smooth unknown input and fractional differentiation orders. The performance of the proposed approach is illustrated with different numerical examples. Furthermore, a potential application of the algorithm is proposed which consists in the estimation of the differentiation orders of a fractional neurovascular model along with the neural activity considered as input for this model.

Parameters and Fractional Differentiation Orders Estimation for Linear Continuous-Time Non-Commensurate Fractional Order Systems

KAUST Repository

Belkhatir, Zehor

2017-05-31

This paper proposes a two-stage estimation algorithm to solve the problem of joint estimation of the parameters and the fractional differentiation orders of a linear continuous-time fractional system with non-commensurate orders. The proposed algorithm combines the modulating functions and the first-order Newton methods. Sufficient conditions ensuring the convergence of the method are provided. An error analysis in the discrete case is performed. Moreover, the method is extended to the joint estimation of smooth unknown input and fractional differentiation orders. The performance of the proposed approach is illustrated with different numerical examples. Furthermore, a potential application of the algorithm is proposed which consists in the estimation of the differentiation orders of a fractional neurovascular model along with the neural activity considered as input for this model.

Three dimensional rotating flow of Powell-Eyring nanofluid with non-Fourier's heat flux and non-Fick's mass flux theory

Science.gov (United States)

Ibrahim, Wubshet

2018-03-01

This article numerically examines three dimensional boundary layer flow of a rotating Powell-Eyring nanofluid. In modeling heat transfer processes, non-Fourier heat flux theory and for mass transfer non-Fick's mass flux theory are employed. This theory is recently re-initiated and it becomes the active research area to resolves some drawback associated with the famous Fourier heat flux and mass flux theory. The mathematical model of the flow problem is a system of non-linear partial differential equations which are obtained using the boundary layer analysis. The non-linear partial differential equations have been transformed into non-linear high order ordinary differential equations using similarity transformation. Employing bvp4c algorithm from matlab software routine, the numerical solution of the transformed ordinary differential equations is obtained. The governing equations are constrained by parameters such as rotation parameter λ , the non-Newtonian parameter N, dimensionless thermal relaxation and concentration relaxation parameters δt and δc . The impacts of these parameters have been discussed thoroughly and illustrated using graphs and tables. The findings show that thermal relaxation time δt reduces the thermal and concentration boundary layer thickness. Further, the results reveal that the rotational parameter λ has the effect of decreasing the velocity boundary layer thickness in both x and y directions. Further examination pinpoints that the skin friction coefficient along x-axis is an increasing and skin friction coefficient along y-axis is a decreasing function of rotation parameter λ . Furthermore, the non-Newtonian fluid parameter N has the characteristic of reducing the amount of local Nusselt numbers -f″ (0) and -g″ (0) both in x and y -directions.

Finite-dimensional linear algebra

CERN Document Server

Gockenbach, Mark S

2010-01-01

Some Problems Posed on Vector SpacesLinear equationsBest approximationDiagonalizationSummaryFields and Vector SpacesFields Vector spaces Subspaces Linear combinations and spanning sets Linear independence Basis and dimension Properties of bases Polynomial interpolation and the Lagrange basis Continuous piecewise polynomial functionsLinear OperatorsLinear operatorsMore properties of linear operatorsIsomorphic vector spaces Linear operator equations Existence and uniqueness of solutions The fundamental theorem; inverse operatorsGaussian elimination Newton's method Linear ordinary differential eq

The relationship among the solutions of two auxiliary ordinary differential equations

International Nuclear Information System (INIS)

Liu Xiaoping; Liu Chunping

2009-01-01

In a recent article [Phys. Lett. A 356 (2006) 124], Sirendaoreji extended their auxiliary equation method by introducing a new auxiliary ordinary differential equation (NAODE) and its 14 solutions. Then the author studied some nonlinear evolution equations (NLEEs) and got more exact travelling wave solutions. In this paper, we will show that the 14 solutions of the NAODE are actually the same as the solutions obtained by original auxiliary equation method, and they are only different in the form.

Numerical solutions for magnetohydrodynamic flow of nanofluid over a bidirectional non-linear stretching surface with prescribed surface heat flux boundary

International Nuclear Information System (INIS)

Mahanthesh, B.; Gireesha, B.J.; Gorla, R.S. Reddy; Abbasi, F.M.; Shehzad, S.A.

2016-01-01

Numerical solutions of three-dimensional flow over a non-linear stretching surface are developed in this article. An electrically conducting flow of viscous nanoliquid is considered. Heat transfer phenomenon is accounted under thermal radiation, Joule heating and viscous dissipation effects. We considered the variable heat flux condition at the surface of sheet. The governing mathematical equations are reduced to nonlinear ordinary differential systems through suitable dimensionless variables. A well-known shooting technique is implemented to obtain the results of dimensionless velocities and temperature. The obtained results are plotted for multiple values of pertinent parameters to discuss the salient features of these parameters on fluid velocity and temperature. The expressions of skin-friction coefficient and Nusselt number are computed and analyzed comprehensively through numerical values. A comparison of present results with the previous results in absence of nanoparticle volume fraction, mixed convection and magnetic field is computed and an excellent agreement noticed. We also computed the results for both linear and non-linear stretching sheet cases. - Highlights: • Hydromagnetic flow of nanofluid over a bidirectional non-linear stretching surface is examined. • Cu, Al 2 O3 and TiO 2 types nanoparticles are taken into account. • Numerical solutions have been computed and addressed. • The values of skin-friction and Nusselt number are presented.

On method of solving third-order ordinary differential equations directly using Bernstein polynomials

Science.gov (United States)

Khataybeh, S. N.; Hashim, I.

2018-04-01

In this paper, we propose for the first time a method based on Bernstein polynomials for solving directly a class of third-order ordinary differential equations (ODEs). This method gives a numerical solution by converting the equation into a system of algebraic equations which is solved directly. Some numerical examples are given to show the applicability of the method.

Comparative numerical solutions of stiff Ordinary differential equations using magnus series expansion method

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SURE KÖME

2014-12-01

Full Text Available In this paper, we investigated the effect of Magnus Series Expansion Method on homogeneous stiff ordinary differential equations with different stiffness ratios. A Magnus type integrator is used to obtain numerical solutions of two different examples of stiff problems and exact and approximate results are tabulated. Furthermore, absolute error graphics are demonstrated in detail.

Non-linear partial differential equations an algebraic view of generalized solutions

CERN Document Server

Rosinger, Elemer E

1990-01-01

A massive transition of interest from solving linear partial differential equations to solving nonlinear ones has taken place during the last two or three decades. The availability of better computers has often made numerical experimentations progress faster than the theoretical understanding of nonlinear partial differential equations. The three most important nonlinear phenomena observed so far both experimentally and numerically, and studied theoretically in connection with such equations have been the solitons, shock waves and turbulence or chaotical processes. In many ways, these phenomen

The Cauchy problem for non-linear Klein-Gordon equations

International Nuclear Information System (INIS)

Simon, J.C.H.; Taflin, E.

1993-01-01

We consider in R n+1 , n≥2, the non-linear Klein-Gordon equation. We prove for such an equation that there is neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincare covariant then the non-linear representation of the Poincare-Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincare group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincare group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra. (orig.)

High Order A-stable Continuous General Linear Methods for Solution of Systems of Initial Value Problems in ODEs

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Dauda GuliburYAKUBU

2012-12-01

Full Text Available Accurate solutions to initial value systems of ordinary differential equations may be approximated efficiently by Runge-Kutta methods or linear multistep methods. Each of these has limitations of one sort or another. In this paper we consider, as a middle ground, the derivation of continuous general linear methods for solution of stiff systems of initial value problems in ordinary differential equations. These methods are designed to combine the advantages of both Runge-Kutta and linear multistep methods. Particularly, methods possessing the property of A-stability are identified as promising methods within this large class of general linear methods. We show that the continuous general linear methods are self-starting and have more ability to solve the stiff systems of ordinary differential equations, than the discrete ones. The initial value systems of ordinary differential equations are solved, for instance, without looking for any other method to start the integration process. This desirable feature of the proposed approach leads to obtaining very high accuracy of the solution of the given problem. Illustrative examples are given to demonstrate the novelty and reliability of the methods.

A partial differential equation model and its reduction to an ordinary differential equation model for prostate tumor growth under intermittent hormone therapy.

Science.gov (United States)

Tao, Youshan; Guo, Qian; Aihara, Kazuyuki

2014-10-01

Hormonal therapy with androgen suppression is a common treatment for advanced prostate tumors. The emergence of androgen-independent cells, however, leads to a tumor relapse under a condition of long-term androgen deprivation. Clinical trials suggest that intermittent androgen suppression (IAS) with alternating on- and off-treatment periods can delay the relapse when compared with continuous androgen suppression (CAS). In this paper, we propose a mathematical model for prostate tumor growth under IAS therapy. The model elucidates initial hormone sensitivity, an eventual relapse of a tumor under CAS therapy, and a delay of a relapse under IAS therapy, which are due to the coexistence of androgen-dependent cells, androgen-independent cells resulting from reversible changes by adaptation, and androgen-independent cells resulting from irreversible changes by genetic mutations. The model is formulated as a free boundary problem of partial differential equations that describe the evolution of populations of the abovementioned three types of cells during on-treatment periods and off-treatment periods. Moreover, the model can be transformed into a piecewise linear ordinary differential equation model by introducing three new volume variables, and the study of the resulting model may help to devise optimal IAS schedules.

On periodic bounded and unbounded solutions of second order nonlinear ordinary differential equations

Czech Academy of Sciences Publication Activity Database

Lomtatidze, Alexander

2017-01-01

Roč. 24, č. 2 (2017), s. 241-263 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : nonlinear ordinary differential equations * periodic boundary value problem * solvability Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2017.24.issue-2/gmj-2017-0009/gmj-2017-0009. xml

On periodic bounded and unbounded solutions of second order nonlinear ordinary differential equations

Czech Academy of Sciences Publication Activity Database

Lomtatidze, Alexander

2017-01-01

Roč. 24, č. 2 (2017), s. 241-263 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : nonlinear ordinary differential equations * periodic boundary value problem * solvability Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2017.24.issue-2/gmj-2017-0009/gmj-2017-0009.xml

ANALYSIS OF MARANGONI CONVECTION OF NON-NEWTONIAN POWER LAW FLUIDS WITH LINEAR TEMPERATURE DISTRIBUTION

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Yan Zhang

2011-01-01

Full Text Available The problem of steady, laminar, thermal Marangoni convection flow of non-Newtonian power law fluid along a horizontal surface with variable surface temperature is studied. The partial differential equations are transformed into ordinary differential equations by using a suitable similarity transformation and analytical approximate solutions are obtained by an efficient transformation, asymptotic expansion and Padé approximants technique. The effects of power law index and Marangoni number on velocity and temperature profiles are examined and discussed.

Non linear thermal radiation effect on Williamson fluid with particle-liquid suspension past a stretching surface

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K. Ganesh Kumar

Full Text Available A mathematical analysis of two-phase boundary layer flow and heat transfer of a Williamson fluid with fluid particle suspension over a stretching sheet has been carried out in this paper. The region of temperature jump and nonlinear thermal radiation is considered in the energy transfer process. The principal equations of boundary layer flow and temperature transmission are reformed to a set of non-linear ordinary differential equations under suitable similarity transformations. The transfigured equalities are solved numerically with the help of RKF-45 order method. The effect of influencing parameters on velocity and temperature transfer of fluid is examined and deliberated by plotted graphs and tabulated values. Significances of the mass concentration of dust particle parameter play a key role in controlling flow and thermal behavior of non-Newtonian fluids. Further, the temperature and concern boundary layer girth are declines for increasing values of Williamson parameter. Keywords: Two-phase flow, Williamson fluid, Nonlinear thermal radiation, Magnetic field, Temperature jump

Numerical solution of ordinary differential equations. For classical, relativistic and nano systems

International Nuclear Information System (INIS)

Greenspan, D.

2006-01-01

An up-to-date survey on numerical solutions with theory, intuition and applications. Ordinary differential equations (ODE) play a significant role in mathematics, physics and engineering sciences, and thus are part of relevant college and university courses. Many problems, however, both traditional and modern, do not possess exact solutions, and must be treated numerically. Usually this is done with software packages, but for this to be efficient requires a sound understanding of the mathematics involved. This work meets the need for an affordable textbook that helps in understanding numerical solutions of ODE. Carefully structured by an experienced textbook author, it provides a survey of ODE for various applications, both classical and modern, including such special applications as relativistic and nano systems. The examples are carefully explained and compiled into an algorithm, each of which is presented generically, independent of a specific programming language, while each chapter is rounded off with exercises. The text meets the demands of MA200 courses and of the newly created Numerical Solution of Differential Equations courses, making it ideal for both students and lecturers in physics, mathematics, mechanical engineering, electrical engineering, as well as for physicists, mathematicians, engineers, and electrical engineers. From the Contents - Euler's Method - Runge-Kutta Methods - The Method of Taylor Expansions - Large Second Order Systems with Application to Nano Systems - Completely Conservative, Covariant Numerical Methodology - Instability - Numerical Solution of Tridiagonal Linear Algebraic Systems and Related Nonlinear Systems - Approximate Solution of Boundary Value Problems - Special Relativistic Motion - Special Topics - Appendix: Basic Matrix Operations - Bibliography. (orig.) (orig.)

SAHYB-2, Solution of Ordinary Differential Equation with User-Supplied Subroutine

International Nuclear Information System (INIS)

Hoop, H. d'; Monterosso, R.

1967-01-01

1 - Nature of physical problem solved: SAHYB-2 is a general purpose programme for the solution of ordinary differential equations. These are written in a user-supplied subroutine called DER, which uses notations very close to mathematical formulas. Special mathematical functions are included in the programme, as: Function generation, delay generation, steps, ramps and pulses, as well as a simplified standard output procedure - boundary value problems or parametric optimisation may be handled by iterations adding a subroutine called REPEAT. 2 - Method of solution: Integration is carried out by constant step fourth-order Runge-Kutta method, or by a fixed or variable step Adams-Moulton predictor corrector method. 3 - Restrictions on the complexity of the problem: Maximum 150 differential equations of the first order. Maximum 30 tables for function generator or delay lines

Phase integral approximation for coupled ordinary differential equations of the Schroedinger type

International Nuclear Information System (INIS)

Skorupski, Andrzej A.

2008-01-01

Four generalizations of the phase integral approximation (PIA) to sets of ordinary differential equations of Schroedinger type [u j '' (x)+Σ k=1 N R jk (x)u k (x)=0, j=1,2,...,N] are described. The recurrence relations for higher order corrections are given in a form valid to arbitrary order and for the matrix R(x)[≡(R jk (x))] either Hermitian or non-Hermitian. For Hermitian and negative definite R(x) matrices, a Wronskian conserving PIA theory is formulated, which generalizes Fulling's current conserving theory pertinent to positive definite R(x) matrices. The idea of a modification of the PIA, which is well known for one equation [u '' (x)+R(x)u(x)=0], is generalized to sets. A simplification of Wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections. If the PIA is generated by a nondegenerate eigenvalue of the R(x) matrix, the eliminated integration is the only one present. In that case, the simplified theory becomes fully algorithmic and is generalized to non-Hermitian R(x) matrices. The general theory is illustrated by a few examples automatically generated by using the author's program in MATHEMATICA published in e-print arXiv:0710.5406 [math-ph

Generalized non-linear Schroedinger hierarchy

International Nuclear Information System (INIS)

Aratyn, H.; Gomes, J.F.; Zimerman, A.H.

1994-01-01

The importance in studying the completely integrable models have became evident in the last years due to the fact that those models present an algebraic structure extremely rich, providing the natural scenery for solitons description. Those models can be described through non-linear differential equations, pseudo-linear operators (Lax formulation), or a matrix formulation. The integrability implies in the existence of a conservation law associated to each of degree of freedom. Each conserved charge Q i can be associated to a Hamiltonian, defining a time evolution related to to a time t i through the Hamilton equation ∂A/∂t i =[A,Q i ]. Particularly, for a two-dimensions field theory, infinite degree of freedom exist, and consequently infinite conservation laws describing the time evolution in space of infinite times. The Hamilton equation defines a hierarchy of models which present a infinite set of conservation laws. This paper studies the generalized non-linear Schroedinger hierarchy

Existence of entire solutions of some non-linear differential-difference equations.

Science.gov (United States)

Chen, Minfeng; Gao, Zongsheng; Du, Yunfei

2017-01-01

In this paper, we investigate the admissible entire solutions of finite order of the differential-difference equations [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] are two non-zero polynomials, [Formula: see text] is a polynomial and [Formula: see text]. In addition, we investigate the non-existence of entire solutions of finite order of the differential-difference equation [Formula: see text], where [Formula: see text], [Formula: see text] are two non-constant polynomials, [Formula: see text], m , n are positive integers and satisfy [Formula: see text] except for [Formula: see text], [Formula: see text].

On the coupling of systems of hyperbolic conservation laws with ordinary differential equations

International Nuclear Information System (INIS)

Borsche, Raul; Colombo, Rinaldo M; Garavello, Mauro

2010-01-01

Motivated by applications to the piston problem, to a manhole model, to blood flow and to supply chain dynamics, this paper deals with a system of conservation laws coupled with a system of ordinary differential equations. The former is defined on a domain with boundary and the coupling is provided by the boundary condition. For each of the examples considered, numerical integrations are provided

Fluid-structure interactions in one-dimensional linear cases

International Nuclear Information System (INIS)

Schumann, U.

1979-01-01

The interaction of pressure waves in a pipe with an elastic endwall (piston) is analyzed using a linear ('acoustic') model. Two transient and two periodic cases are investigated. In the transient cases the motions are initiated by either a sudden pressure drop at the opeen end (breaking membrane) or by a sudden release of the piston from a non-equilibrium position ('snapback'); in the latter case the other end of the pipe is closed. In the periodic cases harmonic oscillations of the piston and the fluid are investigated with the other end of the pipe being either closed or open (kept at constant pressure). The problem is characterized by three non-dimensional numbers (e.g.: Mach-, Strouhal-, and an interaction-number). The solution of the wave equation for the pressure accounting for the coupling to the structure can be reduced analytically to the problem of integrating one ordinary differential equation of second order in time. This differential equation in turn can be integrated analytically at least for a certain time period. At later times this ordinary differential equation is integrated numerically. For the periodic cases eigenvalue-problems arise with an infinite number of solutions. The first few eigensolutions are given. (orig./RW) [de

A Non-linear Stochastic Model for an Office Building with Air Infiltration

DEFF Research Database (Denmark)

Thavlov, Anders; Madsen, Henrik

2015-01-01

This paper presents a non-linear heat dynamic model for a multi-room office building with air infiltration. Several linear and non-linear models, with and without air infiltration, are investigated and compared. The models are formulated using stochastic differential equations and the model...

Optical solver for a system of ordinary differential equations based on an external feedback assisted microring resonator.

Science.gov (United States)

Hou, Jie; Dong, Jianji; Zhang, Xinliang

2017-06-15

Systems of ordinary differential equations (SODEs) are crucial for describing the dynamic behaviors in various systems such as modern control systems which require observability and controllability. In this Letter, we propose and experimentally demonstrate an all-optical SODE solver based on the silicon-on-insulator platform. We use an add/drop microring resonator to construct two different ordinary differential equations (ODEs) and then introduce two external feedback waveguides to realize the coupling between these ODEs, thus forming the SODE solver. A temporal coupled mode theory is used to deduce the expression of the SODE. A system experiment is carried out for further demonstration. For the input 10 GHz NRZ-like pulses, the measured output waveforms of the SODE solver agree well with the calculated results.

Numerical solutions for magnetohydrodynamic flow of nanofluid over a bidirectional non-linear stretching surface with prescribed surface heat flux boundary

Energy Technology Data Exchange (ETDEWEB)

Mahanthesh, B., E-mail: bmanths@gmail.com [Department of Mathematics, AIMS Institutes, Peenya, 560058 Bangalore (India); Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta, 577451 Shimoga, Karnataka (India); Gireesha, B.J., E-mail: bjgireesu@rediffmail.com [Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta, 577451 Shimoga, Karnataka (India); Department of Mechanical Engineering, Cleveland State University, Cleveland, OH (United States); Gorla, R.S. Reddy, E-mail: r.gorla@csuohio.edu [Department of Mechanical Engineering, Cleveland State University, Cleveland, OH (United States); Abbasi, F.M., E-mail: abbasisarkar@gmail.com [Department of Mathematics, Comsats Institute of Information Technology, Islamabad 44000 (Pakistan); Shehzad, S.A., E-mail: ali_qau70@yahoo.com [Department of Mathematics, Comsats Institute of Information Technology, Sahiwal 57000 (Pakistan)

2016-11-01

Numerical solutions of three-dimensional flow over a non-linear stretching surface are developed in this article. An electrically conducting flow of viscous nanoliquid is considered. Heat transfer phenomenon is accounted under thermal radiation, Joule heating and viscous dissipation effects. We considered the variable heat flux condition at the surface of sheet. The governing mathematical equations are reduced to nonlinear ordinary differential systems through suitable dimensionless variables. A well-known shooting technique is implemented to obtain the results of dimensionless velocities and temperature. The obtained results are plotted for multiple values of pertinent parameters to discuss the salient features of these parameters on fluid velocity and temperature. The expressions of skin-friction coefficient and Nusselt number are computed and analyzed comprehensively through numerical values. A comparison of present results with the previous results in absence of nanoparticle volume fraction, mixed convection and magnetic field is computed and an excellent agreement noticed. We also computed the results for both linear and non-linear stretching sheet cases. - Highlights: • Hydromagnetic flow of nanofluid over a bidirectional non-linear stretching surface is examined. • Cu, Al{sub 2}O3 and TiO{sub 2} types nanoparticles are taken into account. • Numerical solutions have been computed and addressed. • The values of skin-friction and Nusselt number are presented.

Causal structure of oscillations in gene regulatory networks: Boolean analysis of ordinary differential equation attractors.

Science.gov (United States)

Sun, Mengyang; Cheng, Xianrui; Socolar, Joshua E S

2013-06-01

A common approach to the modeling of gene regulatory networks is to represent activating or repressing interactions using ordinary differential equations for target gene concentrations that include Hill function dependences on regulator gene concentrations. An alternative formulation represents the same interactions using Boolean logic with time delays associated with each network link. We consider the attractors that emerge from the two types of models in the case of a simple but nontrivial network: a figure-8 network with one positive and one negative feedback loop. We show that the different modeling approaches give rise to the same qualitative set of attractors with the exception of a possible fixed point in the ordinary differential equation model in which concentrations sit at intermediate values. The properties of the attractors are most easily understood from the Boolean perspective, suggesting that time-delay Boolean modeling is a useful tool for understanding the logic of regulatory networks.

Compact tunable silicon photonic differential-equation solver for general linear time-invariant systems.

Science.gov (United States)

Wu, Jiayang; Cao, Pan; Hu, Xiaofeng; Jiang, Xinhong; Pan, Ting; Yang, Yuxing; Qiu, Ciyuan; Tremblay, Christine; Su, Yikai

2014-10-20

We propose and experimentally demonstrate an all-optical temporal differential-equation solver that can be used to solve ordinary differential equations (ODEs) characterizing general linear time-invariant (LTI) systems. The photonic device implemented by an add-drop microring resonator (MRR) with two tunable interferometric couplers is monolithically integrated on a silicon-on-insulator (SOI) wafer with a compact footprint of ~60 μm × 120 μm. By thermally tuning the phase shifts along the bus arms of the two interferometric couplers, the proposed device is capable of solving first-order ODEs with two variable coefficients. The operation principle is theoretically analyzed, and system testing of solving ODE with tunable coefficients is carried out for 10-Gb/s optical Gaussian-like pulses. The experimental results verify the effectiveness of the fabricated device as a tunable photonic ODE solver.

Differential equations and integrable models: the SU(3) case

International Nuclear Information System (INIS)

Dorey, Patrick; Tateo, Roberto

2000-01-01

We exhibit a relationship between the massless a 2 (2) integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schroedinger equation. This forms part of a more general correspondence involving A 2 -related Bethe ansatz systems and third-order differential equations. A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the non-linear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators phi 12 , phi 21 and phi 15 . This is checked against previous results obtained using the thermodynamic Bethe ansatz

Effects of collisions on linear and non-linear spectroscopic line shapes

International Nuclear Information System (INIS)

Berman, P.R.

1978-01-01

A fundamental physical problem is the determination of atom-atom, atom-molecule and molecule-molecule differential and total scattering cross sections. In this work, a technique for studying atomic and molecular collisions using spectroscopic line shape analysis is discussed. Collisions occurring within an atomic or molecular sample influence the sample's absorptive or emissive properties. Consequently the line shapes associated with the linear or non-linear absorption of external fields by an atomic system reflect the collisional processes occurring in the gas. Explicit line shape expressions are derived characterizing linear or saturated absorption by two-or three-level 'active' atoms which are undergoing collisions with perturber atoms. The line shapes may be broadened, shifted, narrowed, or distorted as a result of collisions which may be 'phase-interrupting' or 'velocity-changing' in nature. Systematic line shape studies can be used to obtain information on both the differential and total active atom-perturber scattering cross sections. (Auth.)

Analytical approaches for the approximate solution of a nonlinear fractional ordinary differential equation

International Nuclear Information System (INIS)

Basak, K C; Ray, P C; Bera, R K

2009-01-01

The aim of the present analysis is to apply the Adomian decomposition method and He's variational method for the approximate analytical solution of a nonlinear ordinary fractional differential equation. The solutions obtained by the above two methods have been numerically evaluated and presented in the form of tables and also compared with the exact solution. It was found that the results obtained by the above two methods are in excellent agreement with the exact solution. Finally, a surface plot of the approximate solutions of the fractional differential equation by the above two methods is drawn for 0≤t≤2 and 1<α≤2.

On the existence of eigenmodes of linear quasi-periodic differential equations and their relation to the MHD continuum

International Nuclear Information System (INIS)

Salat, A.

1981-12-01

The existence of quasi-periodic eigensolutions of a linear second order ordinary differential equation with quasi-periodic coefficient f(ω 1 t,ω 2 t) is investigated numerically and graphically. For sufficiently incommensurate frequencies ω 1 , ω 2 a doubly indexed infinite sequence of eigenvalues and eigenmodes is obtained. The equation considered is a model for the magneto-hydrodynamic 'continuum' in general toroidal geometry. The result suggests that continuum modes exist at least on sufficiently irrational magnetic surfaces. (orig.)

Response of Non-Linear Systems to Renewal Impulses by Path Integration

DEFF Research Database (Denmark)

Nielsen, Søren R.K.; Iwankiewicz, R.

The cell-to-cell mapping (path integration) technique has been devised for MDOF non-linear and non-hysteretic systems subjected to random trains of impulses driven by an ordinary renewal point process with gamma-distributed integer parameter interarrival times (an Erlang process). Since the renewal...... point process has not independent increments the state vector of the system, consisting of the generalized displacements and velocities, is not a Markov process. Initially it is shown how the indicated systems can be converted to an equivalent Poisson driven system at the expense of introducing...... additional discrete-valued state variables for which the stochastic equations are also formulated....

approximate controllability of a non-autonomous differential equation

Indian Academy of Sciences (India)

53

for a non-autonomous functional differential equation using the theory of linear ... approximate controllability of various functional differential equations in abstract ...... the operator A(t) and into the requirement that x(t) ∈ D(A) for all t ≥ 0.

A canonical form of the equation of motion of linear dynamical systems

Science.gov (United States)

Kawano, Daniel T.; Salsa, Rubens Goncalves; Ma, Fai; Morzfeld, Matthias

2018-03-01

The equation of motion of a discrete linear system has the form of a second-order ordinary differential equation with three real and square coefficient matrices. It is shown that, for almost all linear systems, such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement. This canonical form of the equation of motion is unique up to an equivalence class for non-defective systems. As an important by-product, a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system.

On the classical theory of ordinary linear differential equations of the second order and the Schroedinger equation for power law potentials

International Nuclear Information System (INIS)

Lima, M.L.; Mignaco, J.A.

1983-01-01

The power law potentials in the Schroedinger equation solved recently are shown to come from the classical treatment of the singularities of a linear, second order differential equation. This allows to enlarge the class of solvable power law potentials. (Author) [pt

On a kind of symmetric weakly non-linear ordinary differential systems

Czech Academy of Sciences Publication Activity Database

Fečkan, M.; Rontó, András; Dilna, N.

2016-01-01

Roč. 140, č. 2 (2016), s. 188-230 ISSN 0007-4497 Institutional support: RVO:67985840 Keywords : symmetric solution * symmetry * periodic solution Subject RIV: BA - General Mathematics Impact factor: 0.750, year: 2016 http://www.sciencedirect.com/science/article/pii/S0007449715000895

Rigorous Integration of Non-Linear Ordinary Differential Equations in Chebyshev Basis

Czech Academy of Sciences Publication Activity Database

Dzetkulič, Tomáš

2015-01-01

Roč. 69, č. 1 (2015), s. 183-205 ISSN 1017-1398 R&D Projects: GA MŠk OC10048; GA ČR GD201/09/H057 Institutional research plan: CEZ:AV0Z10300504 Keywords : Initial value problem * Rigorous integration * Taylor model * Chebyshev basis Subject RIV: IN - Informatics, Computer Science Impact factor: 1.366, year: 2015

Feedback nash equilibria for linear quadratic descriptor differential games

NARCIS (Netherlands)

Engwerda, J.C.; Salmah, S.

2012-01-01

In this paper, we consider the non-cooperative linear feedback Nash quadratic differential game with an infinite planning horizon for descriptor systems of index one. The performance function is assumed to be indefinite. We derive both necessary and sufficient conditions under which this game has a

Feedback Nash Equilibria for Linear Quadratic Descriptor Differential Games

NARCIS (Netherlands)

Engwerda, J.C.; Salmah, Y.

2010-01-01

In this note we consider the non-cooperative linear feedback Nash quadratic differential game with an infinite planning horizon for descriptor systems of index one. The performance function is assumed to be indefinite. We derive both necessary and sufficient conditions under which this game has a

Quantifying uncertainty, variability and likelihood for ordinary differential equation models

LENUS (Irish Health Repository)

Weisse, Andrea Y

2010-10-28

Abstract Background In many applications, ordinary differential equation (ODE) models are subject to uncertainty or variability in initial conditions and parameters. Both, uncertainty and variability can be quantified in terms of a probability density function on the state and parameter space. Results The partial differential equation that describes the evolution of this probability density function has a form that is particularly amenable to application of the well-known method of characteristics. The value of the density at some point in time is directly accessible by the solution of the original ODE extended by a single extra dimension (for the value of the density). This leads to simple methods for studying uncertainty, variability and likelihood, with significant advantages over more traditional Monte Carlo and related approaches especially when studying regions with low probability. Conclusions While such approaches based on the method of characteristics are common practice in other disciplines, their advantages for the study of biological systems have so far remained unrecognized. Several examples illustrate performance and accuracy of the approach and its limitations.

MHD flow of Powell-Eyring nanofluid over a non-linear stretching sheet with variable thickness

Directory of Open Access Journals (Sweden)

T. Hayat

Full Text Available This research explores the magnetohydrodynamic (MHD boundary layer flow of Powell-Eyring nanofluid past a non-linear stretching sheet of variable thickness. An electrically conducting fluid is considered under the characteristics of magnetic field applied transverse to the sheet. The mathematical expressions are accomplished via boundary layer access, Brownian motion and thermophoresis phenomena. The flow analysis is subjected to a recently established conditions requiring zero nanoparticles mass flux. Adequate transformations are implemented for the reduction of partial differential systems to the ordinary differential systems. Series solutions for the governing nonlinear flow of momentum, temperature and nanoparticles concentration have been executed. Physical interpretation of numerous parameters is assigned by graphical illustrations and tabular values. Moreover the numerical data of drag coefficient and local heat transfer rate are executed and discussed. It is investigated that higher wall thickness parameter results in the reduction of velocity distribution. Effects of thermophoresis parameter on temperature and concentration profiles are qualitatively similar. Both the temperature and concentration profiles are enhanced for higher values of thermophoresis parameter. Keywords: MHD, Variable thicked surface, Powell-Eyring nanofluid, Zero mass flux conditions

Differential equations methods and applications

CERN Document Server

Said-Houari, Belkacem

2015-01-01

This book presents a variety of techniques for solving ordinary differential equations analytically and features a wealth of examples. Focusing on the modeling of real-world phenomena, it begins with a basic introduction to differential equations, followed by linear and nonlinear first order equations and a detailed treatment of the second order linear equations. After presenting solution methods for the Laplace transform and power series, it lastly presents systems of equations and offers an introduction to the stability theory. To help readers practice the theory covered, two types of exercises are provided: those that illustrate the general theory, and others designed to expand on the text material. Detailed solutions to all the exercises are included. The book is excellently suited for use as a textbook for an undergraduate class (of all disciplines) in ordinary differential equations. .

Cause and cure of sloppiness in ordinary differential equation models.

Science.gov (United States)

Tönsing, Christian; Timmer, Jens; Kreutz, Clemens

2014-08-01

Data-based mathematical modeling of biochemical reaction networks, e.g., by nonlinear ordinary differential equation (ODE) models, has been successfully applied. In this context, parameter estimation and uncertainty analysis is a major task in order to assess the quality of the description of the system by the model. Recently, a broadened eigenvalue spectrum of the Hessian matrix of the objective function covering orders of magnitudes was observed and has been termed as sloppiness. In this work, we investigate the origin of sloppiness from structures in the sensitivity matrix arising from the properties of the model topology and the experimental design. Furthermore, we present strategies using optimal experimental design methods in order to circumvent the sloppiness issue and present nonsloppy designs for a benchmark model.

Cause and cure of sloppiness in ordinary differential equation models

Science.gov (United States)

Tönsing, Christian; Timmer, Jens; Kreutz, Clemens

2014-08-01

Data-based mathematical modeling of biochemical reaction networks, e.g., by nonlinear ordinary differential equation (ODE) models, has been successfully applied. In this context, parameter estimation and uncertainty analysis is a major task in order to assess the quality of the description of the system by the model. Recently, a broadened eigenvalue spectrum of the Hessian matrix of the objective function covering orders of magnitudes was observed and has been termed as sloppiness. In this work, we investigate the origin of sloppiness from structures in the sensitivity matrix arising from the properties of the model topology and the experimental design. Furthermore, we present strategies using optimal experimental design methods in order to circumvent the sloppiness issue and present nonsloppy designs for a benchmark model.

Optimal overlapping of waveform relaxation method for linear differential equations

International Nuclear Information System (INIS)

Yamada, Susumu; Ozawa, Kazufumi

2000-01-01

Waveform relaxation (WR) method is extremely suitable for solving large systems of ordinary differential equations (ODEs) on parallel computers, but the convergence of the method is generally slow. In order to accelerate the convergence, the methods which decouple the system into many subsystems with overlaps some of the components between the adjacent subsystems have been proposed. The methods, in general, converge much faster than the ones without overlapping, but the computational cost per iteration becomes larger due to the increase of the dimension of each subsystem. In this research, the convergence of the WR method for solving constant coefficients linear ODEs is investigated and the strategy to determine the number of overlapped components which minimizes the cost of the parallel computations is proposed. Numerical experiments on an SR2201 parallel computer show that the estimated number of the overlapped components by the proposed strategy is reasonable. (author)

Dynamic Response of Non-Linear Inelsatic Systems to Poisson-Driven Stochastic Excitations

DEFF Research Database (Denmark)

Nielsen, Søren R. K.; Iwankiewicz, R.

of an equivalent linearization techni que and substituting the non-analytical non-linearity in the original system by the cubic form in the pertinent state variables. The response moments are evaluated for the equivalent systems with the help of a generalized Ito's differential rule. The analytical results...

Lie algebras and linear differential equations.

Science.gov (United States)

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.

Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions

Science.gov (United States)

Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em

2017-12-01

Starting with the asymptotic expansion of the error equation of the shifted Gr\\"{u}nwald--Letnikov formula, we derive a new modified weighted shifted Gr\\"{u}nwald--Letnikov (WSGL) formula by introducing appropriate correction terms. We then apply one special case of the modified WSGL formula to solve multi-term fractional ordinary and partial differential equations, and we prove the linear stability and second-order convergence for both smooth and non-smooth solutions. We show theoretically and numerically that numerical solutions up to certain accuracy can be obtained with only a few correction terms. Moreover, the correction terms can be tuned according to the fractional derivative orders without explicitly knowing the analytical solutions. Numerical simulations verify the theoretical results and demonstrate that the new formula leads to better performance compared to other known numerical approximations with similar resolution.

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

Science.gov (United States)

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.

The geometric approach to sets of ordinary differential equations and Hamiltonian dynamics

Science.gov (United States)

Estabrook, F. B.; Wahlquist, H. D.

1975-01-01

The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field V in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along V: scalar fields, forms, vectors, and integrals over subspaces. It is shown that to any field V can be associated a Hamiltonian structure of forms if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added. Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton's variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms must include the characteristics of the set.

Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells.

Science.gov (United States)

Sundnes, J; Lines, G T; Tveito, A

2001-08-01

The contraction of the heart is preceded and caused by a cellular electro-chemical reaction, causing an electrical field to be generated. Performing realistic computer simulations of this process involves solving a set of partial differential equations, as well as a large number of ordinary differential equations (ODEs) characterizing the reactive behavior of the cardiac tissue. Experiments have shown that the solution of the ODEs contribute significantly to the total work of a simulation, and there is thus a strong need to utilize efficient solution methods for this part of the problem. This paper presents how an efficient implicit Runge-Kutta method may be adapted to solve a complicated cardiac cell model consisting of 31 ODEs, and how this solver may be coupled to a set of PDE solvers to provide complete simulations of the electrical activity.

Differentiability of Palmer's linearization Theorem and converse result for density functions

OpenAIRE

Castañeda, Alvaro; Robledo, Gonzalo

2014-01-01

We study differentiability properties in a particular case of the Palmer's linearization Theorem, which states the existence of an homeomorphism $H$ between the solutions of a linear ODE system having exponential dichotomy and a quasilinear system. Indeed, if the linear system is uniformly asymptotically stable, sufficient conditions ensuring that $H$ is a $C^{2}$ preserving orientation diffeomorphism are given. As an application, we generalize a converse result of density functions for a non...

Finding higher order Darboux polynomials for a family of rational first order ordinary differential equations

Science.gov (United States)

Avellar, J.; Claudino, A. L. G. C.; Duarte, L. G. S.; da Mota, L. A. C. P.

2015-10-01

For the Darbouxian methods we are studying here, in order to solve first order rational ordinary differential equations (1ODEs), the most costly (computationally) step is the finding of the needed Darboux polynomials. This can be so grave that it can render the whole approach unpractical. Hereby we introduce a simple heuristics to speed up this process for a class of 1ODEs.

Bayesian analysis of non-linear differential equation models with application to a gut microbial ecosystem.

Science.gov (United States)

Lawson, Daniel J; Holtrop, Grietje; Flint, Harry

2011-07-01

Process models specified by non-linear dynamic differential equations contain many parameters, which often must be inferred from a limited amount of data. We discuss a hierarchical Bayesian approach combining data from multiple related experiments in a meaningful way, which permits more powerful inference than treating each experiment as independent. The approach is illustrated with a simulation study and example data from experiments replicating the aspects of the human gut microbial ecosystem. A predictive model is obtained that contains prediction uncertainty caused by uncertainty in the parameters, and we extend the model to capture situations of interest that cannot easily be studied experimentally. Copyright © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Stability of non-linear constitutive formulations for viscoelastic fluids

CERN Document Server

Siginer, Dennis A

2014-01-01

Stability of Non-linear Constitutive Formulations for Viscoelastic Fluids provides a complete and up-to-date view of the field of constitutive equations for flowing viscoelastic fluids, in particular on their non-linear behavior, the stability of these constitutive equations that is their predictive power, and the impact of these constitutive equations on the dynamics of viscoelastic fluid flow in tubes. This book gives an overall view of the theories and attendant methodologies developed independently of thermodynamic considerations as well as those set within a thermodynamic framework to derive non-linear rheological constitutive equations for viscoelastic fluids. Developments in formulating Maxwell-like constitutive differential equations as well as single integral constitutive formulations are discussed in the light of Hadamard and dissipative type of instabilities.

Non-linear self-reinforced growth of tearing modes with multiple rational surfaces

International Nuclear Information System (INIS)

Maschke, E.K.; Persson, M.; Dewar, R.L.; Australian National Univ., Canberra, ACT

1993-06-01

The non-linear evolution of tearing modes with multiple rational surfaces is discussed. It is demonstrated that, in the presence of small differential rotation, the non-linear growth might be faster than exponential. This growth occurs as the rotation frequencies of the plasma at the different rational surfaces go into equilibrium

Flow and heat transfer of MHD graphene oxide-water nanofluid between two non-parallel walls

Directory of Open Access Journals (Sweden)

Azimi Mohammadreza

2017-01-01

Full Text Available The steady 2-D heat transfer and flow between two non-parallel walls of a graphene oxide nanofluid in presence of uniform magnetic field are investigated in this paper. The analytical solution of the non-linear problem is obtained by Galerkin optimal homotopy asymptotic method. At first a similarity transformation is used to reduce the partial differential equations modeling the flow and heat transfer to ordinary non-linear differential equation systems containing the semi angle between the plate’s parameter, Reynolds number, the magnetic field strength, nanoparticle volume fraction, Eckert and Prandtl numbers. Finally, the obtained analytical results have been compared with results achieved from previous works in some cases.

A Hierarchical Linear Model for Estimating Gender-Based Earnings Differentials.

Science.gov (United States)

Haberfield, Yitchak; Semyonov, Moshe; Addi, Audrey

1998-01-01

Estimates of gender earnings inequality in data from 116,431 Jewish workers were compared using a hierarchical linear model (HLM) and ordinary least squares model. The HLM allows estimation of the extent to which earnings inequality depends on occupational characteristics. (SK)

The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu Transform Method

Directory of Open Access Journals (Sweden)

Hasan Bulut

2013-01-01

Full Text Available We introduce the rudiments of fractional calculus and the consequent applications of the Sumudu transform on fractional derivatives. Once this connection is firmly established in the general setting, we turn to the application of the Sumudu transform method (STM to some interesting nonhomogeneous fractional ordinary differential equations (FODEs. Finally, we use the solutions to form two-dimensional (2D graphs, by using the symbolic algebra package Mathematica Program 7.

Collocation methods for the solution of eigenvalue problems for singular ordinary differential equations

Directory of Open Access Journals (Sweden)

Winfried Auzinger

2006-01-01

Full Text Available We demonstrate that eigenvalue problems for ordinary differential equations can be recast in a formulation suitable for the solution by polynomial collocation. It is shown that the well-posedness of the two formulations is equivalent in the regular as well as in the singular case. Thus, a collocation code equipped with asymptotically correct error estimation and adaptive mesh selection can be successfully applied to compute the eigenvalues and eigenfunctions efficiently and with reliable control of the accuracy. Numerical examples illustrate this claim.

Assessment of available integration algorithms for initial value ordinary differential equations

International Nuclear Information System (INIS)

Carver, M.B.; Stewart, D.G.

1979-11-01

There exists an extremely large number of algorithms designed for the ordinary differential equation initial value problem. The integration is normally done by a finite sum at time intervals which are chosen dynamically to satisfy an imposed error tolerance. This report describes the basic logistics of the integration process, identifies common areas of difficulty, and establishes a comprehensive test profile for integration algorithms. A number of algorithms are described, and selected published subroutines are evaluated using the test profile. It concludes that an effective library for general use need have only two such routines. The two selected are versions of the well-known Gear and Runge-Kutta-Fehlberg algorithms. Full documentation and listings are included. (auth)

Basic linear partial differential equations

CERN Document Server

Treves, Francois

1975-01-01

Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students features most of the basic classical results. The methods, however, are decidedly nontraditional: in practically every instance, they tend toward a high level of abstraction. This approach recalls classical material to contemporary analysts in a language they can understand, as well as exploiting the field's wealth of examples as an introduction to modern theories.The four-part treatment covers the basic examples of linear partial differential equations and their

3D Laser Scanning Assisted by Ordinary Plane Mirror for Non-direct Viewing Area

Directory of Open Access Journals (Sweden)

ZHANG Fan

2017-12-01

Full Text Available Terrestrial 3D laser scanning is one of principal methods to get the geometric information of object surface,and the integrity of the scanned object is a basic requirement in data acquisition. In order to solve the missing point cloud problem due to the scanning dead angle caused by confined working space,this paper proposes a method using ordinary plane mirror to obtain laser scanning data for non-direct viewing area according to the plane mirror reflection principle,analyzes the influence mechanism of the ordinary plane mirror on the propagation path and distance of laser beam,deduces the coordinate equation of the object point corresponding to the image point reflected by ordinary plane mirror in laser scanning. Given the laser scanning characteristic,this paper introduces a mirror reflection system included target balls and ordinary plane mirror,and expounds the system construction,system calibration and constructing method of system coordinate system. The feasibility and precision of the method are verified by experiments.

Spectral theories for linear differential equations

International Nuclear Information System (INIS)

Sell, G.R.

1976-01-01

The use of spectral analysis in the study of linear differential equations with constant coefficients is not only a fundamental technique but also leads to far-reaching consequences in describing the qualitative behaviour of the solutions. The spectral analysis, via the Jordan canonical form, will not only lead to a representation theorem for a basis of solutions, but will also give a rather precise statement of the (exponential) growth rates of various solutions. Various attempts have been made to extend this analysis to linear differential equations with time-varying coefficients. The most complete such extensions is the Floquet theory for equations with periodic coefficients. For time-varying linear differential equations with aperiodic coefficients several authors have attempted to ''extend'' the Foquet theory. The precise meaning of such an extension is itself a problem, and we present here several attempts in this direction that are related to the general problem of extending the spectral analysis of equations with constant coefficients. The main purpose of this paper is to introduce some problems of current research. The primary problem we shall examine occurs in the context of linear differential equations with almost periodic coefficients. We call it ''the Floquet problem''. (author)

(2,0)-Super-Yang-Mills coupled to non-linear {sigma}-model

Energy Technology Data Exchange (ETDEWEB)

Goes-Negrao, M.S.; Penna-Firme, A.B. [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil); Negrao, M.R. [Universidade Federal, Rio de Janeiro, RJ (Brazil). Inst. de Fisica

1999-07-01

Considering a class of (2,0)-super yang-Mills multiplets that accommodate a pair of independent gauge potentials in connection with a single symmetry group, we present here their coupling to ordinary matter to non-linear {sigma}-models in (2,0)-superspace. The dynamics and the coupling of the gauge potentials are discussed and the interesting feature that comes out is a sort of chirality for one of the gauge potentials are discussed and the interesting feature that comes out is a sort of chirality for one of the gauge potentials once light-cone coordinates are chosen. (author)

(2,0)-Super-Yang-Mills coupled to non-linear σ-model

International Nuclear Information System (INIS)

Goes-Negrao, M.S.; Penna-Firme, A.B.; Negrao, M.R.

1999-07-01

Considering a class of (2,0)-super yang-Mills multiplets that accommodate a pair of independent gauge potentials in connection with a single symmetry group, we present here their coupling to ordinary matter to non-linear σ-models in (2,0)-superspace. The dynamics and the coupling of the gauge potentials are discussed and the interesting feature that comes out is a sort of chirality for one of the gauge potentials are discussed and the interesting feature that comes out is a sort of chirality for one of the gauge potentials once light-cone coordinates are chosen. (author)

Nonchaoticity of Ordinary Differential Equations Describing Autonomous Transcriptional Regulatory Circuits

International Nuclear Information System (INIS)

Li Pengfei; Hu Gang; Chen Runsheng

2008-01-01

Gene transcriptional regulation (TR) processes are often described by coupled nonlinear ordinary differential equations (ODEs). When the dimension of TR circuits is high (e.g. n ≥ 3) the motions of the corresponding ODEs may, very probably, show self-sustained oscillations and chaos. On the other hand, chaoticity may be harmful for the normal biological functions of TR processes. In this letter we numerically study the dynamics of 3-gene TR ODEs in great detail, and investigate many 4-, 5-, and 10-gene TR systems by randomly choosing figures and parameters in the conventionally accepted ranges. And we find that oscillations are very seldom and no chaotic motion is observed, even if the dimension of systems is sufficiently high (n ≥ 3). It is argued that the observation of nonchaoticity of these ODEs agrees with normal functions of actual TR processes

An accurate scheme by block method for third order ordinary ...

African Journals Online (AJOL)

problems of ordinary differential equations is presented in this paper. The approach of collocation approximation is adopted in the derivation of the scheme and then the scheme is applied as simultaneous integrator to special third order initial value problem of ordinary differential equations. This implementation strategy is ...

Non-linear wave equations:Mathematical techniques

International Nuclear Information System (INIS)

1978-01-01

An account of certain well-established mathematical methods, which prove useful to deal with non-linear partial differential equations is presented. Within the strict framework of Functional Analysis, it describes Semigroup Techniques in Banach Spaces as well as variational approaches towards critical points. Detailed proofs are given of the existence of local and global solutions of the Cauchy problem and of the stability of stationary solutions. The formal approach based upon invariance under Lie transformations deserves attention due to its wide range of applicability, even if the explicit solutions thus obtained do not allow for a deep analysis of the equations. A compre ensive introduction to the inverse scattering approach and to the solution concept for certain non-linear equations of physical interest are also presented. A detailed discussion is made about certain convergence and stability problems which arise in importance need not be emphasized. (author) [es

The Bessel polynomials and their differential operators

International Nuclear Information System (INIS)

Onyango Otieno, V.P.

1987-10-01

Differential operators associated with the ordinary and the generalized Bessel polynomials are defined. In each case the commutator bracket is constructed and shows that the differential operators associated with the Bessel polynomials and their generalized form are not commutative. Some applications of these operators to linear differential equations are also discussed. (author). 4 refs

Supersymmetric Yang-Mills fields as an integrable system and connections with other non-linear systems

International Nuclear Information System (INIS)

Chau, L.L.

1983-01-01

Integrable properties, i.e., existence of linear systems, infinite number of conservation laws, Reimann-Hilbert transforms, affine Lie algebra of Kac-Moody, and Bianchi-Baecklund transformation, are discussed for the constraint equations of the supersymmetric Yang-Mills fields. For N greater than or equal to 3 these constraint equations give equations of motion of the fields. These equations of motion reduce to the ordinary Yang-Mills equations as the spinor and scalar fields are eliminated. These understandings provide a possible method to solve the full Yang-Mills equations. Connections with other non-linear systems are also discussed. 53 references

Non-linear analysis of wave progagation using transform methods and plates and shells using integral equations

Science.gov (United States)

Pipkins, Daniel Scott

Two diverse topics of relevance in modern computational mechanics are treated. The first involves the modeling of linear and non-linear wave propagation in flexible, lattice structures. The technique used combines the Laplace Transform with the Finite Element Method (FEM). The procedure is to transform the governing differential equations and boundary conditions into the transform domain where the FEM formulation is carried out. For linear problems, the transformed differential equations can be solved exactly, hence the method is exact. As a result, each member of the lattice structure is modeled using only one element. In the non-linear problem, the method is no longer exact. The approximation introduced is a spatial discretization of the transformed non-linear terms. The non-linear terms are represented in the transform domain by making use of the complex convolution theorem. A weak formulation of the resulting transformed non-linear equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solution of the linear part of the transformed governing differential equation. Numerical results are presented for both linear and non-linear systems. The linear systems modeled are longitudinal and torsional rods and Bernoulli-Euler and Timoshenko beams. For non-linear systems, a viscoelastic rod and Von Karman type beam are modeled. The second topic is the analysis of plates and shallow shells under-going finite deflections by the Field/Boundary Element Method. Numerical results are presented for two plate problems. The first is the bifurcation problem associated with a square plate having free boundaries which is loaded by four, self equilibrating corner forces. The results are compared to two existing numerical solutions of the problem which differ substantially. non-linear model are compared to those

A new class of scale free solutions to linear ordinary differential equations and the universality of the golden mean (Radical radicand 5 -1)/2=0.618033...

CERN Document Server

Datta, D P

2003-01-01

A new class of finitely differentiable scale free solutions to the simplest class of ordinary differential equations is presented. Consequently, the real number set gets replaced by an extended physical set, each element of which is endowed with an equivalence class of infinitesimally separated neighbours in the form of random fluctuations. We show how a sense of time and evolution is intrinsically defined by the infinite continued fraction of the golden mean irrational number (Radical radicand 5 -1)/2, which plays a key role in this extended SL(2,R) formalism of calculus analogous to El Naschie's theory of E sup ( supinfinity sup ) spacetime manifold. Time may thereby undergo random inversions generating well defined random scales, thus allowing a dynamical system to evolve self similarly over the set of multiple scales. The late time stochastic fluctuations of a dynamical system enjoys the generic 1/f spectrum. A universal form of the related probability density is also derived. We prove that the golden mea...

Maximum principles for boundary-degenerate linear parabolic differential operators

OpenAIRE

Feehan, Paul M. N.

2013-01-01

We develop weak and strong maximum principles for boundary-degenerate, linear, parabolic, second-order partial differential operators, $Lu := -u_t-\\tr(aD^2u)-\\langle b, Du\\rangle + cu$, with \\emph{partial} Dirichlet boundary conditions. The coefficient, $a(t,x)$, is assumed to vanish along a non-empty open subset, $\\mydirac_0!\\sQ$, called the \\emph{degenerate boundary portion}, of the parabolic boundary, $\\mydirac!\\sQ$, of the domain $\\sQ\\subset\\RR^{d+1}$, while $a(t,x)$ may be non-zero at po...

Finiteness of Ricci flat supersymmetric non-linear sigma-models

International Nuclear Information System (INIS)

Alvarez-Gaume, L.; Ginsparg, P.

1985-01-01

Combining the constraints of Kaehler differential geometry with the universality of the normal coordinate expansion in the background field method, we study the ultraviolet behavior of 2-dimensional supersymmetric non-linear sigma-models with target space an arbitrary riemannian manifold M. We show that the constraint of N=2 supersymmetry requires that all counterterms to the metric beyond one-loop order are cohomologically trivial. It follows that such supersymmetric non-linear sigma-models defined on locally symmetric spaces are super-renormalizable and that N=4 models are on-shell ultraviolet finite to all orders of perturbation theory. (orig.)

On non-linear dynamics of a coupled electro-mechanical system

DEFF Research Database (Denmark)

Darula, Radoslav; Sorokin, Sergey

2012-01-01

Electro-mechanical devices are an example of coupled multi-disciplinary weakly non-linear systems. Dynamics of such systems is described in this paper by means of two mutually coupled differential equations. The first one, describing an electrical system, is of the first order and the second one...... excitation. The results are verified using a numerical model created in MATLAB Simulink environment. Effect of non-linear terms on dynamical response of the coupled system is investigated; the backbone and envelope curves are analyzed. The two phenomena, which exist in the electro-mechanical system: (a......, for mechanical system, is of the second order. The governing equations are coupled via linear and weakly non-linear terms. A classical perturbation method, a method of multiple scales, is used to find a steadystate response of the electro-mechanical system exposed to a harmonic close-resonance mechanical...

Some mathematical problems in non-linear Physics

International Nuclear Information System (INIS)

1983-01-01

The main results contained in this report are the following: I) A general analysis of non-autonomous conserved densities for simple linear evolution systems. II) Partial differential systems within a wide class are converted into Lagrange an form. III) Rigorous criteria for existence of integrating factor matrices. IV) Isolation of all third-order evolution equations with high order symmetries and conservation laws. (Author) 3 refs

Particle Swarm Optimization Based on Local Attractors of Ordinary Differential Equation System

Directory of Open Access Journals (Sweden)

Wenyu Yang

2014-01-01

Full Text Available Particle swarm optimization (PSO is inspired by sociological behavior. In this paper, we interpret PSO as a finite difference scheme for solving a system of stochastic ordinary differential equations (SODE. In this framework, the position points of the swarm converge to an equilibrium point of the SODE and the local attractors, which are easily defined by the present position points, also converge to the global attractor. Inspired by this observation, we propose a class of modified PSO iteration methods (MPSO based on local attractors of the SODE. The idea of MPSO is to choose the next update state near the present local attractor, rather than the present position point as in the original PSO, according to a given probability density function. In particular, the quantum-behaved particle swarm optimization method turns out to be a special case of MPSO by taking a special probability density function. The MPSO methods with six different probability density functions are tested on a few benchmark problems. These MPSO methods behave differently for different problems. Thus, our framework not only gives an interpretation for the ordinary PSO but also, more importantly, provides a warehouse of PSO-like methods to choose from for solving different practical problems.

Factorization of a class of almost linear second-order differential equations

International Nuclear Information System (INIS)

Estevez, P G; Kuru, S; Negro, J; Nieto, L M

2007-01-01

A general type of almost linear second-order differential equations, which are directly related to several interesting physical problems, is characterized. The solutions of these equations are obtained using the factorization technique, and their non-autonomous invariants are also found by means of scale transformations

Ordinary differential equations and Boolean networks in application to modelling of 6-mercaptopurine metabolism.

Science.gov (United States)

Lavrova, Anastasia I; Postnikov, Eugene B; Zyubin, Andrey Yu; Babak, Svetlana V

2017-04-01

We consider two approaches to modelling the cell metabolism of 6-mercaptopurine, one of the important chemotherapy drugs used for treating acute lymphocytic leukaemia: kinetic ordinary differential equations, and Boolean networks supplied with one controlling node, which takes continual values. We analyse their interplay with respect to taking into account ATP concentration as a key parameter of switching between different pathways. It is shown that the Boolean networks, which allow avoiding the complexity of general kinetic modelling, preserve the possibility of reproducing the principal switching mechanism.

Remark on zeros of solutions of second-order linear ordinary differential equations

Czech Academy of Sciences Publication Activity Database

Dosoudilová, M.; Lomtatidze, Alexander

2016-01-01

Roč. 23, č. 4 (2016), s. 571-577 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : second-order linear equation * zeros of solutions * periodic boundary value problem Subject RIV: BA - General Mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2016.23.issue-4/gmj-2016-0052/gmj-2016-0052. xml

Remark on zeros of solutions of second-order linear ordinary differential equations

Czech Academy of Sciences Publication Activity Database

Dosoudilová, M.; Lomtatidze, Alexander

2016-01-01

Roč. 23, č. 4 (2016), s. 571-577 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : second-order linear equation * zero s of solutions * periodic boundary value problem Subject RIV: BA - General Mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2016.23.issue-4/gmj-2016-0052/gmj-2016-0052.xml

Linear and non-linear optics of condensed matter

International Nuclear Information System (INIS)

McLean, T.P.

1977-01-01

Part I - Linear optics: 1. General introduction. 2. Frequency dependence of epsilon(ω, k vector). 3. Wave-vector dependence of epsilon(ω, k vector). 4. Tensor character of epsilon(ω, k vector). Part II - Non-linear optics: 5. Introduction. 6. A classical theory of non-linear response in one dimension. 7. The generalization to three dimensions. 8. General properties of the polarizability tensors. 9. The phase-matching condition. 10. Propagation in a non-linear dielectric. 11. Second harmonic generation. 12. Coupling of three waves. 13. Materials and their non-linearities. 14. Processes involving energy exchange with the medium. 15. Two-photon absorption. 16. Stimulated Raman effect. 17. Electro-optic effects. 18. Limitations of the approach presented here. (author)

A Line-Tau Collocation Method for Partial Differential Equations ...

African Journals Online (AJOL)

This paper deals with the numerical solution of second order linear partial differential equations with the use of the method of lines coupled with the tau collocation method. The method of lines is used to convert the partial differential equation (PDE) to a sequence of ordinary differential equations (ODEs) which is then ...

An analytical approximation scheme to two-point boundary value problems of ordinary differential equations

International Nuclear Information System (INIS)

Boisseau, Bruno; Forgacs, Peter; Giacomini, Hector

2007-01-01

A new (algebraic) approximation scheme to find global solutions of two-point boundary value problems of ordinary differential equations (ODEs) is presented. The method is applicable for both linear and nonlinear (coupled) ODEs whose solutions are analytic near one of the boundary points. It is based on replacing the original ODEs by a sequence of auxiliary first-order polynomial ODEs with constant coefficients. The coefficients in the auxiliary ODEs are uniquely determined from the local behaviour of the solution in the neighbourhood of one of the boundary points. The problem of obtaining the parameters of the global (connecting) solutions, analytic at one of the boundary points, reduces to find the appropriate zeros of algebraic equations. The power of the method is illustrated by computing the approximate values of the 'connecting parameters' for a number of nonlinear ODEs arising in various problems in field theory. We treat in particular the static and rotationally symmetric global vortex, the skyrmion, the Abrikosov-Nielsen-Olesen vortex, as well as the 't Hooft-Polyakov magnetic monopole. The total energy of the skyrmion and of the monopole is also computed by the new method. We also consider some ODEs coming from the exact renormalization group. The ground-state energy level of the anharmonic oscillator is also computed for arbitrary coupling strengths with good precision. (fast track communication)

Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator

Directory of Open Access Journals (Sweden)

Resat Yilmazer

2016-02-01

Full Text Available In this work; we present a method for solving the second-order linear ordinary differential equation of hypergeometric type. The solutions of this equation are given by the confluent hypergeometric functions (CHFs. Unlike previous studies, we obtain some different new solutions of the equation without using the CHFs. Therefore, we obtain new discrete fractional solutions of the homogeneous and non-homogeneous confluent hypergeometric differential equation (CHE by using a discrete fractional Nabla calculus operator. Thus, we obtain four different new discrete complex fractional solutions for these equations.

Pythagoras, Binomial, and de Moivre Revisited Through Differential Equations

OpenAIRE

Singh, Jitender; Bajaj, Renu

2018-01-01

The classical Pythagoras theorem, binomial theorem, de Moivre's formula, and numerous other deductions are made using the uniqueness theorem for the initial value problems in linear ordinary differential equations.

Differential equations

CERN Document Server

Barbu, Viorel

2016-01-01

This textbook is a comprehensive treatment of ordinary differential equations, concisely presenting basic and essential results in a rigorous manner. Including various examples from physics, mechanics, natural sciences, engineering and automatic theory, Differential Equations is a bridge between the abstract theory of differential equations and applied systems theory. Particular attention is given to the existence and uniqueness of the Cauchy problem, linear differential systems, stability theory and applications to first-order partial differential equations. Upper undergraduate students and researchers in applied mathematics and systems theory with a background in advanced calculus will find this book particularly useful. Supplementary topics are covered in an appendix enabling the book to be completely self-contained.

A local-global problem for linear differential equations

NARCIS (Netherlands)

Put, Marius van der; Reversat, Marc

An inhomogeneous linear differential equation Ly = f over a global differential field can have a formal solution for each place without having a global solution. The vector space lgl(L) measures this phenomenon. This space is interpreted in terms of cohomology of linear algebraic groups and is

A local-global problem for linear differential equations

NARCIS (Netherlands)

Put, Marius van der; Reversat, Marc

2008-01-01

An inhomogeneous linear differential equation Ly = f over a global differential field can have a formal solution for each place without having a global solution. The vector space lgl(L) measures this phenomenon. This space is interpreted in terms of cohomology of linear algebraic groups and is

Non-linear calculation of PCRV using dynamic relaxation

International Nuclear Information System (INIS)

Schnellenbach, G.

1979-01-01

A brief review is presented of a numerical method called the dynamic relaxation method for stress analysis of the concrete in prestressed concrete pressure vessels. By this method the three-dimensional elliptic differential equations of the continuum are changed into the four-dimensional hyperbolic differential equations known as wave equations. The boundary value problem of the static system is changed into an initial and boundary value problem for which a solution exists if the physical system is defined at time t=0. The effect of non-linear stress-strain behaviour of the material as well as creep and cracking are considered

EDITORIAL: Non-linear and non-Gaussian cosmological perturbations Non-linear and non-Gaussian cosmological perturbations

Science.gov (United States)

Sasaki, Misao; Wands, David

2010-06-01

In recent years there has been a resurgence of interest in the study of non-linear perturbations of cosmological models. This has been the result of both theoretical developments and observational advances. New theoretical challenges arise at second and higher order due to mode coupling and the need to develop new gauge-invariant variables beyond first order. In particular, non-linear interactions lead to deviations from a Gaussian distribution of primordial perturbations even if initial vacuum fluctuations are exactly Gaussian. These non-Gaussianities provide an important probe of models for the origin of structure in the very early universe. We now have a detailed picture of the primordial distribution of matter from surveys of the cosmic microwave background, notably NASA's WMAP satellite. The situation will continue to improve with future data from the ESA Planck satellite launched in 2009. To fully exploit these data cosmologists need to extend non-linear cosmological perturbation theory beyond the linear theory that has previously been sufficient on cosmological scales. Another recent development has been the realization that large-scale structure, revealed in high-redshift galaxy surveys, could also be sensitive to non-linearities in the primordial curvature perturbation. This focus section brings together a collection of invited papers which explore several topical issues in this subject. We hope it will be of interest to theoretical physicists and astrophysicists alike interested in understanding and interpreting recent developments in cosmological perturbation theory and models of the early universe. Of course it is only an incomplete snapshot of a rapidly developing field and we hope the reader will be inspired to read further work on the subject and, perhaps, fill in some of the missing pieces. This focus section is dedicated to the memory of Lev Kofman (1957-2009), an enthusiastic pioneer of inflationary cosmology and non-Gaussian perturbations.

The non-linear evolution of edge localized modes

International Nuclear Information System (INIS)

Wenninger, Ronald

2013-01-01

Edge localized modes (ELMs) are instabilities in the edge of tokamak plasmas in the high confinement regime (H-mode). Without them the edge transport in ordinary H-mode plasmas is too low to establish a stationary situation. However in a future device large unmitigated ELMs are believed to cause divertor power flux densities far in excess of tolerable material limits. Hence the size of energy loss per ELM and the resulting ELM frequency must be controlled. To proceed in understanding how the ELM size is determined and how ELM mitigation methods work it is necessary to characterize the non-linear evolution of pedestal erosion. In order to achieve this experimental data is compared to the results of ELM simulations with the code JOREK (reduced MHD, non-linear) applying a specially developed synthetic magnetic diagnostic. The experimental data are acquired by several fast sampling diagnostics at the experiments ASDEX Upgrade and TCV at a large number of toroidal/poloidal positions. A central element of the presented work is the detailed characterization of dominant magnetic perturbations during ELMs. These footprints of the instability can be observed most intensely in close temporal vicinity to the onset of pedestal erosion. Dominant magnetic perturbations are caused by current perturbations located at or inside the last closed flux surface. In ASDEX Upgrade under certain conditions dominant magnetic perturbations like other H-mode edge instabilities display a similarity to solitons. Furthermore - as expected - they are often observed to be correlated to a perturbation of electron temperature. In TCV it is possible to characterize the evolution of the toroidal structure of dominant magnetic perturbations. Between growing above the level of background fluctuations and the maximum perturbation level for all time instance a similar toroidal structure is observed. This rigid mode-structure is an indication for non-linear coupling. Most frequently the dominant toroidal

The non-linear evolution of edge localized modes

Energy Technology Data Exchange (ETDEWEB)

Wenninger, Ronald

2013-01-09

Edge localized modes (ELMs) are instabilities in the edge of tokamak plasmas in the high confinement regime (H-mode). Without them the edge transport in ordinary H-mode plasmas is too low to establish a stationary situation. However in a future device large unmitigated ELMs are believed to cause divertor power flux densities far in excess of tolerable material limits. Hence the size of energy loss per ELM and the resulting ELM frequency must be controlled. To proceed in understanding how the ELM size is determined and how ELM mitigation methods work it is necessary to characterize the non-linear evolution of pedestal erosion. In order to achieve this experimental data is compared to the results of ELM simulations with the code JOREK (reduced MHD, non-linear) applying a specially developed synthetic magnetic diagnostic. The experimental data are acquired by several fast sampling diagnostics at the experiments ASDEX Upgrade and TCV at a large number of toroidal/poloidal positions. A central element of the presented work is the detailed characterization of dominant magnetic perturbations during ELMs. These footprints of the instability can be observed most intensely in close temporal vicinity to the onset of pedestal erosion. Dominant magnetic perturbations are caused by current perturbations located at or inside the last closed flux surface. In ASDEX Upgrade under certain conditions dominant magnetic perturbations like other H-mode edge instabilities display a similarity to solitons. Furthermore - as expected - they are often observed to be correlated to a perturbation of electron temperature. In TCV it is possible to characterize the evolution of the toroidal structure of dominant magnetic perturbations. Between growing above the level of background fluctuations and the maximum perturbation level for all time instance a similar toroidal structure is observed. This rigid mode-structure is an indication for non-linear coupling. Most frequently the dominant toroidal

Calculus & ordinary differential equations

CERN Document Server

Pearson, David

1995-01-01

Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.

Differential calculus in normed linear spaces

CERN Document Server

Mukherjea, Kalyan

2007-01-01

This book presents Advanced Calculus from a geometric point of view: instead of dealing with partial derivatives of functions of several variables, the derivative of the function is treated as a linear transformation between normed linear spaces. Not only does this lead to a simplified and transparent exposition of "difficult" results like the Inverse and Implicit Function Theorems but also permits, without any extra effort, a discussion of the Differential Calculus of functions defined on infinite dimensional Hilbert or Banach spaces.The prerequisites demanded of the reader are modest: a sound understanding of convergence of sequences and series of real numbers, the continuity and differentiability properties of functions of a real variable and a little Linear Algebra should provide adequate background for understanding the book. The first two chapters cover much of the more advanced background material on Linear Algebra (like dual spaces, multilinear functions and tensor products.) Chapter 3 gives an ab ini...

PSsolver: A Maple implementation to solve first order ordinary differential equations with Liouvillian solutions

Science.gov (United States)

Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P.

2012-10-01

We present a set of software routines in Maple 14 for solving first order ordinary differential equations (FOODEs). The package implements the Prelle-Singer method in its original form together with its extension to include integrating factors in terms of elementary functions. The package also presents a theoretical extension to deal with all FOODEs presenting Liouvillian solutions. Applications to ODEs taken from standard references show that it solves ODEs which remain unsolved using Maple's standard ODE solution routines. New version program summary Program title: PSsolver Catalogue identifier: ADPR_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADPR_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2302 No. of bytes in distributed program, including test data, etc.: 31962 Distribution format: tar.gz Programming language: Maple 14 (also tested using Maple 15 and 16). Computer: Intel Pentium Processor P6000, 1.86 GHz. Operating system: Windows 7. RAM: 4 GB DDR3 Memory Classification: 4.3. Catalogue identifier of previous version: ADPR_v1_0 Journal reference of previous version: Comput. Phys. Comm. 144 (2002) 46 Does the new version supersede the previous version?: Yes Nature of problem: Symbolic solution of first order differential equations via the Prelle-Singer method. Solution method: The method of solution is based on the standard Prelle-Singer method, with extensions for the cases when the FOODE contains elementary functions. Additionally, an extension of our own which solves FOODEs with Liouvillian solutions is included. Reasons for new version: The program was not running anymore due to changes in the latest versions of Maple. Additionally, we corrected/changed some bugs/details that were hampering the smoother functioning of the routines. Summary

Approximate Forward Difference Equations for the Lower Order Non-Stationary Statistics of Geometrically Non-Linear Systems subject to Random Excitation

DEFF Research Database (Denmark)

Köylüoglu, H. U.; Nielsen, Søren R. K.; Cakmak, A. S.

Geometrically non-linear multi-degree-of-freedom (MDOF) systems subject to random excitation are considered. New semi-analytical approximate forward difference equations for the lower order non-stationary statistical moments of the response are derived from the stochastic differential equations...... of motion, and, the accuracy of these equations is numerically investigated. For stationary excitations, the proposed method computes the stationary statistical moments of the response from the solution of non-linear algebraic equations....

Conformable variational iteration method

Directory of Open Access Journals (Sweden)

Omer Acan

2017-02-01

Full Text Available In this study, we introduce the conformable variational iteration method based on new defined fractional derivative called conformable fractional derivative. This new method is applied two fractional order ordinary differential equations. To see how the solutions of this method, linear homogeneous and non-linear non-homogeneous fractional ordinary differential equations are selected. Obtained results are compared the exact solutions and their graphics are plotted to demonstrate efficiency and accuracy of the method.

Modelling and Inverse-Modelling: Experiences with O.D.E. Linear Systems in Engineering Courses

Science.gov (United States)

Martinez-Luaces, Victor

2009-01-01

In engineering careers courses, differential equations are widely used to solve problems concerned with modelling. In particular, ordinary differential equations (O.D.E.) linear systems appear regularly in Chemical Engineering, Food Technology Engineering and Environmental Engineering courses, due to the usefulness in modelling chemical kinetics,…

On radiative-magnetoconvective heat and mass transfer of a nanofluid past a non-linear stretching surface with Ohmic heating and convective surface boundary condition

Directory of Open Access Journals (Sweden)

Shweta Mishra

2016-12-01

Full Text Available In this paper magnetoconvective heat and mass transfer characteristics of a two-dimensional steady flow of a nanofluid over a non-linear stretching sheet in the presence of thermal radiation, Ohmic heating and viscous dissipation have been investigated numerically. The model used for the nanofluid incorporates the effects of the Brownian motion and the presence of nanoparticles in the base fluid. The governing equations are transformed into a system of nonlinear ordinary differential equations by using similarity transformation. The numerical solutions are obtained by using fifth order Runge–Kutta–Fehlberg method with shooting technique. The non-dimensional parameters on velocity, temperature and concentration profiles and also on local Nusselt number and Sherwood number are discussed. The results indicate that the local skin friction coefficient decreases as the value of the magnetic parameter increases whereas the Nusselt number and Sherwood number increase as the values of the Brownian motion parameter and magnetic parameter increase.

Fibonacci-like Differential Equations with a Polynomial Non-Homogeneous Part

NARCIS (Netherlands)

Asveld, P.R.J.

1989-01-01

We investigate non-homogeneous linear differential equations of the form $x''(t) + x'(t) - x(t) = p(t)$ where $p(t)$ is either a polynomial or a factorial polynomial in $t$. We express the solution of these differential equations in terms of the coefficients of $p(t)$, in the initial conditions, and

Linear non scarring alopecia of the scalp: A rare manifestation of lupus panniculitis

Directory of Open Access Journals (Sweden)

Sandhyarani Kshetrimayum

2016-01-01

Full Text Available Alopecia in a linear pattern is very rare with only a few cases reported in the medical literature. We report a case of linear non scarring alopecia involving the scalp in a 17-year-old boy with a histological diagnosis of lupus panniculitis. We report this case because of its rarity and also the inclusion of this entity as one of the rare differential of non scarring alopecia.

Non-linear optical materials

CERN Document Server

Saravanan, R

2018-01-01

Non-linear optical materials have widespread and promising applications, but the efforts to understand the local structure, electron density distribution and bonding is still lacking. The present work explores the structural details, the electron density distribution and the local bond length distribution of some non-linear optical materials. It also gives estimation of the optical band gap, the particle size, crystallite size, and the elemental composition from UV-Visible analysis, SEM, XRD and EDS of some non-linear optical materials respectively.

On the stability, the periodic solutions and the resolution of certain types of non linear equations, and of non linearly coupled systems of these equations, appearing in betatronic oscillations

International Nuclear Information System (INIS)

Valat, J.

1960-12-01

Universal stability diagrams have been calculated and experimentally checked for Hill-Meissner type equations with square-wave coefficients. The study of these equations in the phase-plane has then made it possible to extend the periodic solution calculations to the case of non-linear differential equations with periodic square-wave coefficients. This theory has been checked experimentally. For non-linear coupled systems with constant coefficients, a search was first made for solutions giving an algebraic motion. The elliptical and Fuchs's functions solve such motions. The study of non-algebraic motions is more delicate, apart from the study of nonlinear Lissajous's motions. A functional analysis shows that it is possible however in certain cases to decouple the system and to find general solutions. For non-linear coupled systems with periodic square-wave coefficients it is then possible to calculate the conditions leading to periodic solutions, if the two non-linear associated systems with constant coefficients fall into one of the categories of the above paragraph. (author) [fr

Linearity and Non-linearity of Photorefractive effect in Materials ...

African Journals Online (AJOL)

In this paper we have studied the Linearity and Non-linearity of Photorefractive effect in materials using the band transport model. For low light beam intensities the change in the refractive index is proportional to the electric field for linear optics while for non- linear optics the change in refractive index is directly proportional ...

Useful tools for non-linear systems: Several non-linear integral inequalities

Czech Academy of Sciences Publication Activity Database

Agahi, H.; Mohammadpour, A.; Mesiar, Radko; Vaezpour, M. S.

2013-01-01

Roč. 49, č. 1 (2013), s. 73-80 ISSN 0950-7051 R&D Projects: GA ČR GAP402/11/0378 Institutional support: RVO:67985556 Keywords : Monotone measure * Comonotone functions * Integral inequalities * Universal integral Subject RIV: BA - General Mathematics Impact factor: 3.058, year: 2013 http://library.utia.cas.cz/separaty/2013/E/mesiar-useful tools for non-linear systems several non-linear integral inequalities.pdf

Remark on periodic boundary-value problem for second-order linear ordinary differential equations

Czech Academy of Sciences Publication Activity Database

Dosoudilová, M.; Lomtatidze, Alexander

2018-01-01

Roč. 2018, č. 13 (2018), s. 1-7 ISSN 1072-6691 Institutional support: RVO:67985840 Keywords : second-order linear equation * periodic boundary value problem * unique solvability Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.954, year: 2016 https://ejde.math.txstate.edu/Volumes/2018/13/abstr.html

Dimensional analysis to transform the differential equations in partial derivates in the theory of heat transmission into ordinary ones

International Nuclear Information System (INIS)

Diaz Sanchidrian, C.

1989-01-01

The present paper applies dimensional analysis with spatial discrimination to transform the differential equations in partial derivatives developed in the theory of heat transmission into ordinary ones. The effectivity of the method is comparable to that methods based in transformations of uni or multiparametric groups, with the advantage of being more direct and simple. (Author)

Introduction to linear systems of differential equations

CERN Document Server

Adrianova, L Ya

1995-01-01

The theory of linear systems of differential equations is one of the cornerstones of the whole theory of differential equations. At its root is the concept of the Lyapunov characteristic exponent. In this book, Adrianova presents introductory material and further detailed discussions of Lyapunov exponents. She also discusses the structure of the space of solutions of linear systems. Classes of linear systems examined are from the narrowest to widest: 1)�autonomous, 2)�periodic, 3)�reducible to autonomous, 4)�nearly reducible to autonomous, 5)�regular. In addition, Adrianova considers the following: stability of linear systems and the influence of perturbations of the coefficients on the stability the criteria of uniform stability and of uniform asymptotic stability in terms of properties of the solutions several estimates of the growth rate of solutions of a linear system in terms of its coefficients How perturbations of the coefficients change all the elements of the spectrum of the system is defin...

Periodic linear differential stochastic processes

NARCIS (Netherlands)

Kwakernaak, H.

1975-01-01

Periodic linear differential processes are defined and their properties are analyzed. Equivalent representations are discussed, and the solutions of related optimal estimation problems are given. An extension is presented of Kailath and Geesey’s [1] results concerning the innovations representation

Non-Linear Dynamics of Saturn's Rings

Science.gov (United States)

Esposito, L. W.

2016-12-01

Non-linear processes can explain why Saturn's rings are so active and dynamic. Ring systems differ from simple linear systems in two significant ways: 1. They are systems of granular material: where particle-to-particle collisions dominate; thus a kinetic, not a fluid description needed. Stresses are strikingly inhomogeneous and fluctuations are large compared to equilibrium. 2. They are strongly forced by resonances: which drive a non-linear response, that push the system across thresholds that lead to persistent states. Some of this non-linearity is captured in a simple Predator-Prey Model: Periodic forcing from the moon causes streamline crowding; This damps the relative velocity. About a quarter phase later, the aggregates stir the system to higher relative velocity and the limit cycle repeats each orbit, with relative velocity ranging from nearly zero to a multiple of the orbit average. Summary of Halo Results: A predator-prey model for ring dynamics produces transient structures like `straw' that can explain the halo morphology and spectroscopy: Cyclic velocity changes cause perturbed regions to reach higher collision speeds at some orbital phases, which preferentially removes small regolith particles; surrounding particles diffuse back too slowly to erase the effect: this gives the halo morphology; this requires energetic collisions (v ≈ 10m/sec, with throw distances about 200km, implying objects of scale R ≈ 20km).Transform to Duffing Eqn : With the coordinate transformation, z = M2/3, the Predator-Prey equations can be combined to form a single second-order differential equation with harmonic resonance forcing.Ring dynamics and history implications: Moon-triggered clumping explains both small and large particles at resonances. We calculate the stationary size distribution using a cell-to-cell mapping procedure that converts the phase-plane trajectories to a Markov chain. Approximating it as an asymmetric random walk with reflecting boundaries

Uncertainty of pesticide residue concentration determined from ordinary and weighted linear regression curve.

Science.gov (United States)

Yolci Omeroglu, Perihan; Ambrus, Árpad; Boyacioglu, Dilek

2018-03-28

Determination of pesticide residues is based on calibration curves constructed for each batch of analysis. Calibration standard solutions are prepared from a known amount of reference material at different concentration levels covering the concentration range of the analyte in the analysed samples. In the scope of this study, the applicability of both ordinary linear and weighted linear regression (OLR and WLR) for pesticide residue analysis was investigated. We used 782 multipoint calibration curves obtained for 72 different analytical batches with high-pressure liquid chromatography equipped with an ultraviolet detector, and gas chromatography with electron capture, nitrogen phosphorus or mass spectrophotometer detectors. Quality criteria of the linear curves including regression coefficient, standard deviation of relative residuals and deviation of back calculated concentrations were calculated both for WLR and OLR methods. Moreover, the relative uncertainty of the predicted analyte concentration was estimated for both methods. It was concluded that calibration curve based on WLR complies with all the quality criteria set by international guidelines compared to those calculated with OLR. It means that all the data fit well with WLR for pesticide residue analysis. It was estimated that, regardless of the actual concentration range of the calibration, relative uncertainty at the lowest calibrated level ranged between 0.3% and 113.7% for OLR and between 0.2% and 22.1% for WLR. At or above 1/3 of the calibrated range, uncertainty of calibration curve ranged between 0.1% and 16.3% for OLR and 0% and 12.2% for WLR, and therefore, the two methods gave comparable results.

Genetic design of interpolated non-linear controllers for linear plants

International Nuclear Information System (INIS)

Ajlouni, N.

2000-01-01

The techniques of genetic algorithms are proposed as a means of designing non-linear PID control systems. It is shown that the use of genetic algorithms for this purpose results in highly effective non-linear PID control systems. These results are illustrated by using genetic algorithms to design a non-linear PID control system and contrasting the results with an optimally tuned linear PID controller. (author)

Photon induced non-linear quantized double layer charging in quaternary semiconducting quantum dots.

Science.gov (United States)

Nair, Vishnu; Ananthoju, Balakrishna; Mohapatra, Jeotikanta; Aslam, M

2018-03-15

Room temperature quantized double layer charging was observed in 2 nm Cu 2 ZnSnS 4 (CZTS) quantum dots. In addition to this we observed a distinct non-linearity in the quantized double layer charging arising from UV light modulation of double layer. UV light irradiation resulted in a 26% increase in the integral capacitance at the semiconductor-dielectric (CZTS-oleylamine) interface of the quantum dot without any change in its core size suggesting that the cause be photocapacitive. The increasing charge separation at the semiconductor-dielectric interface due to highly stable and mobile photogenerated carriers cause larger electrostatic forces between the quantum dot and electrolyte leading to an enhanced double layer. This idea was supported by a decrease in the differential capacitance possible due to an enhanced double layer. Furthermore the UV illumination enhanced double layer gives us an AC excitation dependent differential double layer capacitance which confirms that the charging process is non-linear. This ultimately illustrates the utility of a colloidal quantum dot-electrolyte interface as a non-linear photocapacitor. Copyright © 2017 Elsevier Inc. All rights reserved.

Mathematical Methods in Wave Propagation: Part 2--Non-Linear Wave Front Analysis

Science.gov (United States)

Jeffrey, Alan

1971-01-01

The paper presents applications and methods of analysis for non-linear hyperbolic partial differential equations. The paper is concluded by an account of wave front analysis as applied to the piston problem of gas dynamics. (JG)

Estimation of non-linear continuous time models for the heat exchange dynamics of building integrated photovoltaic modules

DEFF Research Database (Denmark)

Jimenez, M.J.; Madsen, Henrik; Bloem, J.J.

2008-01-01

This paper focuses on a method for linear or non-linear continuous time modelling of physical systems using discrete time data. This approach facilitates a more appropriate modelling of more realistic non-linear systems. Particularly concerning advanced building components, convective and radiati...... that a description of the non-linear heat transfer is essential. The resulting model is a non-linear first order stochastic differential equation for the heat transfer of the PV component....... heat interchanges are non-linear effects and represent significant contributions in a variety of components such as photovoltaic integrated facades or roofs and those using these effects as passive cooling strategies, etc. Since models are approximations of the physical system and data is encumbered...

Effect of heat radiation in a Walter’s liquid B fluid over a stretching sheet with non-uniform heat source/sink and elastic deformation

Directory of Open Access Journals (Sweden)

A.K. Abdul Hakeem

2014-07-01

Full Text Available In this present article heat transfer in a Walter’s liquid B fluid over an impermeable stretching sheet with non-uniform heat source/sink, elastic deformation and radiation are reported. The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The dimensionless governing equations for this investigation are solved analytically using hyper geometric functions. The results are carried out for prescribed surface temperature (PST and prescribed power law surface heat flux (PHF. The effects of viscous dissipation, Prandtl number, Eckert number, heat source/sink parameter with elastic deformation and radiation are shown in the several plots and addressed.

GLOBAL LINEARIZATION OF DIFFERENTIAL EQUATIONS WITH SPECIAL STRUCTURES

Institute of Scientific and Technical Information of China (English)

无

2011-01-01

This paper introduces the global linearization of the differential equations with special structures.The function in the differential equation is unbounded.We prove that the differential equation with unbounded function can be topologically linearlized if it has a special structure.

Mathematical Methods for Engineers and Scientists 2 Vector Analysis, Ordinary Differential Equations and Laplace Transforms

CERN Document Server

Tang, Kwong-Tin

2007-01-01

Pedagogical insights gained through 30 years of teaching applied mathematics led the author to write this set of student-oriented books. Topics such as complex analysis, matrix theory, vector and tensor analysis, Fourier analysis, integral transforms, ordinary and partial differential equations are presented in a discursive style that is readable and easy to follow. Numerous clearly stated, completely worked out examples together with carefully selected problem sets with answers are used to enhance students' understanding and manipulative skill. The goal is to make students comfortable and confident in using advanced mathematical tools in junior, senior, and beginning graduate courses.

On the asymptotic expansions of solutions of an nth order linear differential equation with power coefficients

International Nuclear Information System (INIS)

Paris, R.B.; Wood, A.D.

1984-11-01

The asymptotic expansions of solutions of a class of linear ordinary differential equations of arbitrary order n, containing a factor zsup(m) multiplying the lower order derivatives, are investigated for large values of z in the complex plane. Four classes of solutions are considered which exhibit the following behaviour as /z/ → infinity in certain sectors: (i) solutions whose behaviour is either exponentially large or algebraic (involving p ( < n) algebraic expansions), (ii) solutions which are exponentially small (iii) solutions with a single algebraic expansion and (iv) solutions which are even and odd functions of z whenever n+m is even. The asymptotic expansions of these solutions in a full neigbourhood of the point at infinity are obtained by means of the theory of the solutions in the case m=O developed in a previous paper

Could solitons be adiabatic invariants attached to certain non linear equations

International Nuclear Information System (INIS)

Lochak, P.

1984-01-01

Arguments are given to support the claim that solitons should be the adiabatic invariants associated to certain non linear partial differential equations; a precise mathematical form of this conjecture is then stated. As a particular case of the conjecture, the Korteweg-de Vries equation is studied. (Auth.)

Linear and non-linear simulation of joints contact surface using ...

African Journals Online (AJOL)

The joint modelling including non-linear effects needs accurate and precise study of their behaviors. When joints are under the dynamic loading, micro, macro- slip happens in contact surface which is non-linear reason of the joint contact surface. The non-linear effects of joint contact surface on total behavior of structure are ...

Rational approximations to solutions of linear differential equations.

Science.gov (United States)

Chudnovsky, D V; Chudnovsky, G V

1983-08-01

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.

Legendre-tau approximations for functional differential equations

Science.gov (United States)

Ito, K.; Teglas, R.

1986-01-01

The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.

Efficient Non Linear Loudspeakers

DEFF Research Database (Denmark)

Petersen, Bo R.; Agerkvist, Finn T.

2006-01-01

Loudspeakers have traditionally been designed to be as linear as possible. However, as techniques for compensating non linearities are emerging, it becomes possible to use other design criteria. This paper present and examines a new idea for improving the efficiency of loudspeakers at high levels...... by changing the voice coil layout. This deliberate non-linear design has the benefit that a smaller amplifier can be used, which has the benefit of reducing system cost as well as reducing power consumption....

A new class of scale free solutions to linear ordinary differential equations and the universality of the golden mean (Radical radicand 5 -1)/2=0.618033.

International Nuclear Information System (INIS)

Datta, Dhurjati Prasad

2003-01-01

A new class of finitely differentiable scale free solutions to the simplest class of ordinary differential equations is presented. Consequently, the real number set gets replaced by an extended physical set, each element of which is endowed with an equivalence class of infinitesimally separated neighbours in the form of random fluctuations. We show how a sense of time and evolution is intrinsically defined by the infinite continued fraction of the golden mean irrational number (Radical radicand 5 -1)/2, which plays a key role in this extended SL(2,R) formalism of calculus analogous to El Naschie's theory of E (∞) spacetime manifold. Time may thereby undergo random inversions generating well defined random scales, thus allowing a dynamical system to evolve self similarly over the set of multiple scales. The late time stochastic fluctuations of a dynamical system enjoys the generic 1/f spectrum. A universal form of the related probability density is also derived. We prove that the golden mean number is intrinsically random, letting all measurements in the physical universe fundamentally uncertain. The present analysis offers an explanation of the universal occurrence of the golden mean in diverse natural and biological processes as well as the mass spectrum of high energy particle physics

Analytical solutions of time-fractional models for homogeneous Gardner equation and non-homogeneous differential equations

Directory of Open Access Journals (Sweden)

Olaniyi Samuel Iyiola

2014-09-01

Full Text Available In this paper, we obtain analytical solutions of homogeneous time-fractional Gardner equation and non-homogeneous time-fractional models (including Buck-master equation using q-Homotopy Analysis Method (q-HAM. Our work displays the elegant nature of the application of q-HAM not only to solve homogeneous non-linear fractional differential equations but also to solve the non-homogeneous fractional differential equations. The presence of the auxiliary parameter h helps in an effective way to obtain better approximation comparable to exact solutions. The fraction-factor in this method gives it an edge over other existing analytical methods for non-linear differential equations. Comparisons are made upon the existence of exact solutions to these models. The analysis shows that our analytical solutions converge very rapidly to the exact solutions.

On modulated complex non-linear dynamical systems

International Nuclear Information System (INIS)

Mahmoud, G.M.; Mohamed, A.A.; Rauh, A.

1999-01-01

This paper is concerned with the development of an approximate analytical method to investigate periodic solutions and their stability in the case of modulated non-linear dynamical systems whose equation of motion is describe. Such differential equations appear, for example, in problems of colliding particle beams in high-energy accelerators or one-mass systems with two or more degrees of freedom, e.g. rotors. The significance of periodic solutions lies on the fact that all non-periodic responses, if convergent, would approach to periodic solutions at the steady-state conditions. The example shows a good agreement between numerical and analytical results for small values of ε. The effect of the periodic modulation on the stability of the 2π-periodic solutions is discussed

Non Linear Modelling and Control of Hydraulic Actuators

Directory of Open Access Journals (Sweden)

B. Šulc

2002-01-01

Full Text Available This paper deals with non-linear modelling and control of a differential hydraulic actuator. The nonlinear state space equations are derived from basic physical laws. They are more powerful than the transfer function in the case of linear models, and they allow the application of an object oriented approach in simulation programs. The effects of all friction forces (static, Coulomb and viscous have been modelled, and many phenomena that are usually neglected are taken into account, e.g., the static term of friction, the leakage between the two chambers and external space. Proportional Differential (PD and Fuzzy Logic Controllers (FLC have been applied in order to make a comparison by means of simulation. Simulation is performed using Matlab/Simulink, and some of the results are compared graphically. FLC is tuned in a such way that it produces a constant control signal close to its maximum (or minimum, where possible. In the case of PD control the occurrence of peaks cannot be avoided. These peaks produce a very high velocity that oversteps the allowed values.

High-order quantum algorithm for solving linear differential equations

International Nuclear Information System (INIS)

Berry, Dominic W

2014-01-01

Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms to general inhomogeneous sparse linear differential equations, which describe many classical physical systems. We examine the use of high-order methods (where the error over a time step is a high power of the size of the time step) to improve the efficiency. These provide scaling close to Δt 2 in the evolution time Δt. As with other algorithms of this type, the solution is encoded in amplitudes of the quantum state, and it is possible to extract global features of the solution. (paper)

ODEion--a software module for structural identification of ordinary differential equations.

Science.gov (United States)

Gennemark, Peter; Wedelin, Dag

2014-02-01

In the systems biology field, algorithms for structural identification of ordinary differential equations (ODEs) have mainly focused on fixed model spaces like S-systems and/or on methods that require sufficiently good data so that derivatives can be accurately estimated. There is therefore a lack of methods and software that can handle more general models and realistic data. We present ODEion, a software module for structural identification of ODEs. Main characteristic features of the software are: • The model space is defined by arbitrary user-defined functions that can be nonlinear in both variables and parameters, such as for example chemical rate reactions. • ODEion implements computationally efficient algorithms that have been shown to efficiently handle sparse and noisy data. It can run a range of realistic problems that previously required a supercomputer. • ODEion is easy to use and provides SBML output. We describe the mathematical problem, the ODEion system itself, and provide several examples of how the system can be used. Available at: http://www.odeidentification.org.

Scanning differential polarization microscope: Its use to image linear and circular differential scattering

International Nuclear Information System (INIS)

Mickols, W.; Maestre, M.F.

1988-01-01

A differential polarization microscope that couples the sensitivity of single-beam measurement of circular dichroism and circular differential scattering with the simultaneous measurement of linear dichroism and linear differential scattering has been developed. The microscope uses a scanning microscope stage and single-point illumination to give the very shallow depth of field found in confocal microscopy. This microscope can operate in the confocal mode as well as in the near confocal condition that can allow one to program the coherence and spatial resolution of the microscope. This microscope has been used to study the change in the structure of chromatin during the development of sperm in Drosophila

Stability problems for linear hyperbolic systems

International Nuclear Information System (INIS)

Eckhoff, K.S.

1975-05-01

The stability properties for the trivial solution of a general linear hyperbolic system of partial differential equations of the first order are studied. It is shown that results may be obtained by studying the stability properties of certain systems of ordinary differential equations which can be constructed from the hyperbolic system (the so-called transport equations). In some cases the associated stability problem for the transport equations can in fact be shown to be equivalent to the stability problem for the hyperbolic system, but in general the transport equations will only give the necessary conditions for stability. (Auth.)

Novel method of interpolation and extrapolation of functions by a linear initial value problem

CSIR Research Space (South Africa)

Shatalov, M

2008-09-01

Full Text Available A novel method of function approximation using an initial value, linear, ordinary differential equation (ODE) is presented. The main advantage of this method is to obtain the approximation expressions in a closed form. This technique can be taught...

Non-Linear Dynamics of Saturn’s Rings

Science.gov (United States)

Esposito, Larry W.

2015-11-01

Non-linear processes can explain why Saturn’s rings are so active and dynamic. Ring systems differ from simple linear systems in two significant ways: 1. They are systems of granular material: where particle-to-particle collisions dominate; thus a kinetic, not a fluid description needed. We find that stresses are strikingly inhomogeneous and fluctuations are large compared to equilibrium. 2. They are strongly forced by resonances: which drive a non-linear response, pushing the system across thresholds that lead to persistent states.Some of this non-linearity is captured in a simple Predator-Prey Model: Periodic forcing from the moon causes streamline crowding; This damps the relative velocity, and allows aggregates to grow. About a quarter phase later, the aggregates stir the system to higher relative velocity and the limit cycle repeats each orbit.Summary of Halo Results: A predator-prey model for ring dynamics produces transient structures like ‘straw’ that can explain the halo structure and spectroscopy: This requires energetic collisions (v ≈ 10m/sec, with throw distances about 200km, implying objects of scale R ≈ 20km).Transform to Duffing Eqn : With the coordinate transformation, z = M2/3, the Predator-Prey equations can be combined to form a single second-order differential equation with harmonic resonance forcing.Ring dynamics and history implications: Moon-triggered clumping at perturbed regions in Saturn’s rings creates both high velocity dispersion and large aggregates at these distances, explaining both small and large particles observed there. We calculate the stationary size distribution using a cell-to-cell mapping procedure that converts the phase-plane trajectories to a Markov chain. Approximating the Markov chain as an asymmetric random walk with reflecting boundaries allows us to determine the power law index from results of numerical simulations in the tidal environment surrounding Saturn. Aggregates can explain many dynamic aspects

Linear versus non-linear supersymmetry, in general

Energy Technology Data Exchange (ETDEWEB)

Ferrara, Sergio [Theoretical Physics Department, CERN,CH-1211 Geneva 23 (Switzerland); INFN - Laboratori Nazionali di Frascati,Via Enrico Fermi 40, I-00044 Frascati (Italy); Department of Physics and Astronomy, UniversityC.L.A.,Los Angeles, CA 90095-1547 (United States); Kallosh, Renata [SITP and Department of Physics, Stanford University,Stanford, California 94305 (United States); Proeyen, Antoine Van [Institute for Theoretical Physics, Katholieke Universiteit Leuven,Celestijnenlaan 200D, B-3001 Leuven (Belgium); Wrase, Timm [Institute for Theoretical Physics, Technische Universität Wien,Wiedner Hauptstr. 8-10, A-1040 Vienna (Austria)

2016-04-12

We study superconformal and supergravity models with constrained superfields. The underlying version of such models with all unconstrained superfields and linearly realized supersymmetry is presented here, in addition to the physical multiplets there are Lagrange multiplier (LM) superfields. Once the equations of motion for the LM superfields are solved, some of the physical superfields become constrained. The linear supersymmetry of the original models becomes non-linearly realized, its exact form can be deduced from the original linear supersymmetry. Known examples of constrained superfields are shown to require the following LM’s: chiral superfields, linear superfields, general complex superfields, some of them are multiplets with a spin.

Linear versus non-linear supersymmetry, in general

International Nuclear Information System (INIS)

Ferrara, Sergio; Kallosh, Renata; Proeyen, Antoine Van; Wrase, Timm

2016-01-01

We study superconformal and supergravity models with constrained superfields. The underlying version of such models with all unconstrained superfields and linearly realized supersymmetry is presented here, in addition to the physical multiplets there are Lagrange multiplier (LM) superfields. Once the equations of motion for the LM superfields are solved, some of the physical superfields become constrained. The linear supersymmetry of the original models becomes non-linearly realized, its exact form can be deduced from the original linear supersymmetry. Known examples of constrained superfields are shown to require the following LM’s: chiral superfields, linear superfields, general complex superfields, some of them are multiplets with a spin.

Algebraic Traveling Wave Solutions of a Non-local Hydrodynamic-type Model

International Nuclear Information System (INIS)

Chen, Aiyong; Zhu, Wenjing; Qiao, Zhijun; Huang, Wentao

2014-01-01

In this paper we consider the algebraic traveling wave solutions of a non-local hydrodynamic-type model. It is shown that algebraic traveling wave solutions exist if and only if an associated first order ordinary differential system has invariant algebraic curve. The dynamical behavior of the associated ordinary differential system is analyzed. Phase portraits of the associated ordinary differential system is provided under various parameter conditions. Moreover, we classify algebraic traveling wave solutions of the model. Some explicit formulas of smooth solitary wave and cuspon solutions are obtained

Third-order ordinary differential equations Y”' = f(x, y, y'', y′”) with ...

African Journals Online (AJOL)

dimensional symmetry algebra. Mathematics Subject Classication (2010): 34A05, 34A25, 53A55, 76M60. Key words: Linearization, third order ODEs, point transformation, contact transformation, Lie symmetries, relative differential invariants.

ON THE STABILIZATION OF THE LINEAR HYBRID SYSTEM STRUCTURE

Directory of Open Access Journals (Sweden)

Kirillov

2014-11-01

Full Text Available The linear control hybrid system, consisting of a fi- nite set of subsystems (modes having different dimensions, is considered. The moments of reset time are determined by some complementary function – evolutionary time. This function satisfies the special complementary ordinary differential equation. The mode stabilization problem is solved for some class of piecewise linear controls. The method of stabilization relies on the set of invariant planes, the existence of which is due to the special form of the hybrid system.

The choice among non-callable bonds and make whole, claw back and otherwise ordinary callable bonds

OpenAIRE

Booth, LD; Gounopoulos, D; Skinner, F

2012-01-01

This paper seeks to explain determinates of the choice and the pricing of various types of callable and non-callable bonds. We find that the popularity of different types of callable and non-callable bonds is significantly related to the economic environment. In addition, the popularity of claw back bonds appear to be driven by agency considerations, make whole bonds by the debt overhang problem, ordinary callable bonds by the need by banks to deal with interest rate changes and non-callable ...

On deformations of linear differential systems

NARCIS (Netherlands)

Gontsov, R.R.; Poberezhnyi, V.A.; Helminck, G.F.

2011-01-01

This article concerns deformations of meromorphic linear differential systems. Problems relating to their existence and classification are reviewed, and the global and local behaviour of solutions to deformation equations in a neighbourhood of their singular set is analysed. Certain classical

Linear-quadratic control and quadratic differential forms for multidimensional behaviors

NARCIS (Netherlands)

Napp, D.; Trentelman, H.L.

2011-01-01

This paper deals with systems described by constant coefficient linear partial differential equations (nD-systems) from a behavioral point of view. In this context we treat the linear-quadratic control problem where the performance functional is the integral of a quadratic differential form. We look

Non-linear finite element modeling

DEFF Research Database (Denmark)

Mikkelsen, Lars Pilgaard

The note is written for courses in "Non-linear finite element method". The note has been used by the author teaching non-linear finite element modeling at Civil Engineering at Aalborg University, Computational Mechanics at Aalborg University Esbjerg, Structural Engineering at the University...

Fast integration-based prediction bands for ordinary differential equation models.

Science.gov (United States)

Hass, Helge; Kreutz, Clemens; Timmer, Jens; Kaschek, Daniel

2016-04-15

To gain a deeper understanding of biological processes and their relevance in disease, mathematical models are built upon experimental data. Uncertainty in the data leads to uncertainties of the model's parameters and in turn to uncertainties of predictions. Mechanistic dynamic models of biochemical networks are frequently based on nonlinear differential equation systems and feature a large number of parameters, sparse observations of the model components and lack of information in the available data. Due to the curse of dimensionality, classical and sampling approaches propagating parameter uncertainties to predictions are hardly feasible and insufficient. However, for experimental design and to discriminate between competing models, prediction and confidence bands are essential. To circumvent the hurdles of the former methods, an approach to calculate a profile likelihood on arbitrary observations for a specific time point has been introduced, which provides accurate confidence and prediction intervals for nonlinear models and is computationally feasible for high-dimensional models. In this article, reliable and smooth point-wise prediction and confidence bands to assess the model's uncertainty on the whole time-course are achieved via explicit integration with elaborate correction mechanisms. The corresponding system of ordinary differential equations is derived and tested on three established models for cellular signalling. An efficiency analysis is performed to illustrate the computational benefit compared with repeated profile likelihood calculations at multiple time points. The integration framework and the examples used in this article are provided with the software package Data2Dynamics, which is based on MATLAB and freely available at http://www.data2dynamics.org helge.hass@fdm.uni-freiburg.de Supplementary data are available at Bioinformatics online. © The Author 2015. Published by Oxford University Press. All rights reserved. For Permissions, please e

Fertility Differentials and Educational Attainment in Portugal: A Non-Linear Relationship

Directory of Open Access Journals (Sweden)

Tiago de Oliveira, Isabel

2009-01-01

Full Text Available AbstractThis analysis of the Portuguese case shows a non-linear relationship betweenthe number of children and education in recent years. Using the data from tenyears before this hypothesis was confirmed, and we can see that the generaldecline in Portuguese fertility within the last decade was due to the fertilitydecrease of the less educated people, although partly attenuated by the fertilityincrease of the upper social groups. The reasons for a non-linear relationshipare discussed within the context of female employment rates and salarydifferentials by educational attainment. The main hypothesis is that differencesin fertility are related to an ‘education-work’ effect amongst those in the lesseducated groups and to an ‘education-income’ effect amongst the moreeducated.RésuméL’analyse de cas de la situation au Portugal démontre une relation non linéaireentre le nombre d’enfants et le niveau de scolarité au cours des dernièresannées. Les données recueillies pendant les dix dernières années ont étéétudiées avant de confirmer cette hypothèse ; nous avons pu voir que le déclingénéral dans le taux de fécondité au Portugal pendant la dernière décade étaitcausé par un déclin de fécondité chez les personnes moins éduquées ; ceci a étépartiellement atténué par une hausse dans le taux de fécondité dans les classessupérieures. Les raisons de cette relation non linéaire sont discutées dans lecontexte des taux d’emploi des femmes et les différentiels de salaire selon lesniveaux de scolarité. L’hypothèse majeure est que les différences dans les tauxde fécondité sont reliés à un effet « scolarité-travail » parmi les groupes moinséduqués et à un effet « scolarité-salaire » parmi les classes mieux éduqués.

INPUT-OUTPUT STRUCTURE OF LINEAR-DIFFERENTIAL ALGEBRAIC SYSTEMS

NARCIS (Netherlands)

KUIJPER, M; SCHUMACHER, JM

Systems of linear differential and algebraic equations occur in various ways, for instance, as a result of automated modeling procedures and in problems involving algebraic constraints, such as zero dynamics and exact model matching. Differential/algebraic systems may represent an input-output

Shooting method for third order simultaneous ordinary differential equations with application to magnetohydrodynamic boundary layer

International Nuclear Information System (INIS)

Srivastava, A.C.; Hazarika, G.C.

1990-01-01

An algorithm based on the shooting method has been developed for the solution of a two-point boundary value problem consisting of a system of third order simultaneous ordinary differential equations. The Falkner-Skan equations for electrically conducting viscous fluid with applied magnetic field has been solved by using this algorithm for various values of the wedge angle and magnetic parameters. The shooting method seems to be well convergent for a system as the results are in good agreement with those obtained by other methods. It is observed that both viscous boundary layer and magnetic boundary layer decrease while velocity as well as magnetic field increase with the increase of the wedge angle. (author). 6 tabs., 7 refs

Non-linear osmosis

Science.gov (United States)

Diamond, Jared M.

1966-01-01

1. The relation between osmotic gradient and rate of osmotic water flow has been measured in rabbit gall-bladder by a gravimetric procedure and by a rapid method based on streaming potentials. Streaming potentials were directly proportional to gravimetrically measured water fluxes. 2. As in many other tissues, water flow was found to vary with gradient in a markedly non-linear fashion. There was no consistent relation between the water permeability and either the direction or the rate of water flow. 3. Water flow in response to a given gradient decreased at higher osmolarities. The resistance to water flow increased linearly with osmolarity over the range 186-825 m-osM. 4. The resistance to water flow was the same when the gall-bladder separated any two bathing solutions with the same average osmolarity, regardless of the magnitude of the gradient. In other words, the rate of water flow is given by the expression (Om — Os)/[Ro′ + ½k′ (Om + Os)], where Ro′ and k′ are constants and Om and Os are the bathing solution osmolarities. 5. Of the theories advanced to explain non-linear osmosis in other tissues, flow-induced membrane deformations, unstirred layers, asymmetrical series-membrane effects, and non-osmotic effects of solutes could not explain the results. However, experimental measurements of water permeability as a function of osmolarity permitted quantitative reconstruction of the observed water flow—osmotic gradient curves. Hence non-linear osmosis in rabbit gall-bladder is due to a decrease in water permeability with increasing osmolarity. 6. The results suggest that aqueous channels in the cell membrane behave as osmometers, shrinking in concentrated solutions of impermeant molecules and thereby increasing membrane resistance to water flow. A mathematical formulation of such a membrane structure is offered. PMID:5945254

A Solution Method for Linear and Geometrically Nonlinear MDOF Systems with Random Properties subject to Random Excitation

DEFF Research Database (Denmark)

Micaletti, R. C.; Cakmak, A. S.; Nielsen, Søren R. K.

structural properties. The resulting state-space formulation is a system of ordinary stochastic differential equations with random coefficient and deterministic initial conditions which are subsequently transformed into ordinary stochastic differential equations with deterministic coefficients and random......A method for computing the lower-order moments of randomly-excited multi-degree-of-freedom (MDOF) systems with random structural properties is proposed. The method is grounded in the techniques of stochastic calculus, utilizing a Markov diffusion process to model the structural system with random...... initial conditions. This transformation facilitates the derivation of differential equations which govern the evolution of the unconditional statistical moments of response. Primary consideration is given to linear systems and systems with odd polynomial nonlinearities, for in these cases...

Periodic oscillations in linear continuous media coupled with nonlinear discrete systems

International Nuclear Information System (INIS)

Lupini, R.

1998-01-01

A general derivation of partial differential equations with boundary conditions in the form of ordinary differential equations is obtained using the principle of stationary action for a Lagrangian function composed of continuous plus discrete parts in interaction across the boundaries of a 1-dimensional medium. This approach leads directly to the theorem of energy conservation. For linear continuous medium, homogeneous Dirichlet condition at one boundary, and nonlinear oscillator at the other boundary, the entire differential problem reduces to a nonlinear differential-difference equation of neutral type and of the second order. The lag parameter is τ = l/c, where c is the phase speed, l the length of the continuum. The Author investigate the problem of the occurrence of periodic solutions of period integer multiple of the lag (super harmonic solutions) in the case of zero inertia of the boundary system. The problem for such oscillations is shown to reduce to systems of ordinary differential equations with matching conditions in a phase space of lower dimensionality: Phase-plane techniques are used to determine solutions of period 4τ, 8τ and 6τ

Solving non-linear Horn clauses using a linear Horn clause solver

DEFF Research Database (Denmark)

Kafle, Bishoksan; Gallagher, John Patrick; Ganty, Pierre

2016-01-01

In this paper we show that checking satisfiability of a set of non-linear Horn clauses (also called a non-linear Horn clause program) can be achieved using a solver for linear Horn clauses. We achieve this by interleaving a program transformation with a satisfiability checker for linear Horn...... clauses (also called a solver for linear Horn clauses). The program transformation is based on the notion of tree dimension, which we apply to a set of non-linear clauses, yielding a set whose derivation trees have bounded dimension. Such a set of clauses can be linearised. The main algorithm...... dimension. We constructed a prototype implementation of this approach and performed some experiments on a set of verification problems, which shows some promise....

Geometrically and material non-linear analysis of bubble condenser steel structure

International Nuclear Information System (INIS)

Gyoergyi, J.; Lenkei, P.

2003-01-01

In frame of the project funded by the European Commission (EC) through the Phare and Tacis Programmes experimentally investigate the behaviour of the bubble condenser system (BCS) during phenomena induced by postulated design basis accidents (DBA). The bubble condenser steel structure consists of 12 trays. To enable the Bubble Condenser Test Prototype to be representative of the majority of trays and sections, it was decided to model a typical tray. The test results demonstrate the integrity of the standard tray pressure retaining boundary (side wall, face wall, ceiling and bottom) against a differential pressure (30 kPa). The stability of the side wall and the face wall of tray level 12 was not assured for this differential pressure. The thermal-hydraulic tests demonstrate that the maximum differential pressure across the tray walls in the case of Large Break Loss of Coolant Accident (LBLOCA) is 20 kPa. We have got from the experiences the differential pressure in function of time. The results of the approximate calculations showed the effect of nonlinearly. In case of calculation by FEM model we have done the elastic and linear analyses, and calculated with the geometrically and material non-linearity. (author)

Linear matrix differential equations of higher-order and applications

Directory of Open Access Journals (Sweden)

Mustapha Rachidi

2008-07-01

Full Text Available In this article, we study linear differential equations of higher-order whose coefficients are square matrices. The combinatorial method for computing the matrix powers and exponential is adopted. New formulas representing auxiliary results are obtained. This allows us to prove properties of a large class of linear matrix differential equations of higher-order, in particular results of Apostol and Kolodner are recovered. Also illustrative examples and applications are presented.

Non-linear analysis of skew thin plate by finite difference method

International Nuclear Information System (INIS)

Kim, Chi Kyung; Hwang, Myung Hwan

2012-01-01

This paper deals with a discrete analysis capability for predicting the geometrically nonlinear behavior of skew thin plate subjected to uniform pressure. The differential equations are discretized by means of the finite difference method which are used to determine the deflections and the in-plane stress functions of plates and reduced to several sets of linear algebraic simultaneous equations. For the geometrically non-linear, large deflection behavior of the plate, the non-linear plate theory is used for the analysis. An iterative scheme is employed to solve these quasi-linear algebraic equations. Several problems are solved which illustrate the potential of the method for predicting the finite deflection and stress. For increasing lateral pressures, the maximum principal tensile stress occurs at the center of the plate and migrates toward the corners as the load increases. It was deemed important to describe the locations of the maximum principal tensile stress as it occurs. The load-deflection relations and the maximum bending and membrane stresses for each case are presented and discussed

Linear Algebraic Method for Non-Linear Map Analysis

International Nuclear Information System (INIS)

Yu, L.; Nash, B.

2009-01-01

We present a newly developed method to analyze some non-linear dynamics problems such as the Henon map using a matrix analysis method from linear algebra. Choosing the Henon map as an example, we analyze the spectral structure, the tune-amplitude dependence, the variation of tune and amplitude during the particle motion, etc., using the method of Jordan decomposition which is widely used in conventional linear algebra.

Lattice Designs in Standard and Simple Implicit Multi-linear Regression

OpenAIRE

Wooten, Rebecca D.

2016-01-01

Statisticians generally use ordinary least squares to minimize the random error in a subject response with respect to independent explanatory variable. However, Wooten shows illustrates how ordinary least squares can be used to minimize the random error in the system without defining a subject response. Using lattice design Wooten shows that non-response analysis is a superior alternative rotation of the pyramidal relationship between random variables and parameter estimates in multi-linear r...

Stabilization of chromium salt in ordinary portland cement

Indian Academy of Sciences (India)

Ordinary Portland cement (OPC) samples containing the chromium salt have been investigated using differential microcalorimetry, conductometry and Fourier transform infrared spectroscopic analysis. The effect of chromium on OPC hydration was evaluated by continuous observing of early hydration.

A non-differentiable solution for the local fractional telegraph equation

Directory of Open Access Journals (Sweden)

Li Jie

2017-01-01

Full Text Available In this paper, we consider the linear telegraph equations with local fractional derivative. The local fractional Laplace series expansion method is used to handle the local fractional telegraph equation. The analytical solution with the non-differentiable graphs is discussed in detail. The proposed method is efficient and accurate.

Piecewise-linear and bilinear approaches to nonlinear differential equations approximation problem of computational structural mechanics

OpenAIRE

Leibov Roman

2017-01-01

This paper presents a bilinear approach to nonlinear differential equations system approximation problem. Sometimes the nonlinear differential equations right-hand sides linearization is extremely difficult or even impossible. Then piecewise-linear approximation of nonlinear differential equations can be used. The bilinear differential equations allow to improve piecewise-linear differential equations behavior and reduce errors on the border of different linear differential equations systems ...

A meta-analysis of cambium phenology and growth: linear and non-linear patterns in conifers of the northern hemisphere.

Science.gov (United States)

Rossi, Sergio; Anfodillo, Tommaso; Cufar, Katarina; Cuny, Henri E; Deslauriers, Annie; Fonti, Patrick; Frank, David; Gricar, Jozica; Gruber, Andreas; King, Gregory M; Krause, Cornelia; Morin, Hubert; Oberhuber, Walter; Prislan, Peter; Rathgeber, Cyrille B K

2013-12-01

Ongoing global warming has been implicated in shifting phenological patterns such as the timing and duration of the growing season across a wide variety of ecosystems. Linear models are routinely used to extrapolate these observed shifts in phenology into the future and to estimate changes in associated ecosystem properties such as net primary productivity. Yet, in nature, linear relationships may be special cases. Biological processes frequently follow more complex, non-linear patterns according to limiting factors that generate shifts and discontinuities, or contain thresholds beyond which responses change abruptly. This study investigates to what extent cambium phenology is associated with xylem growth and differentiation across conifer species of the northern hemisphere. Xylem cell production is compared with the periods of cambial activity and cell differentiation assessed on a weekly time scale on histological sections of cambium and wood tissue collected from the stems of nine species in Canada and Europe over 1-9 years per site from 1998 to 2011. The dynamics of xylogenesis were surprisingly homogeneous among conifer species, although dispersions from the average were obviously observed. Within the range analysed, the relationships between the phenological timings were linear, with several slopes showing values close to or not statistically different from 1. The relationships between the phenological timings and cell production were distinctly non-linear, and involved an exponential pattern. The trees adjust their phenological timings according to linear patterns. Thus, shifts of one phenological phase are associated with synchronous and comparable shifts of the successive phases. However, small increases in the duration of xylogenesis could correspond to a substantial increase in cell production. The findings suggest that the length of the growing season and the resulting amount of growth could respond differently to changes in environmental conditions.

Complex centers of polynomial differential equations

Directory of Open Access Journals (Sweden)

Mohamad Ali M. Alwash

2007-07-01

Full Text Available We present some results on the existence and nonexistence of centers for polynomial first order ordinary differential equations with complex coefficients. In particular, we show that binomial differential equations without linear terms do not have complex centers. Classes of polynomial differential equations, with more than two terms, are presented that do not have complex centers. We also study the relation between complex centers and the Pugh problem. An algorithm is described to solve the Pugh problem for equations without complex centers. The method of proof involves phase plane analysis of the polar equations and a local study of periodic solutions.

Differential recurrence formulae for orthogonal polynomials

Directory of Open Access Journals (Sweden)

Anton L. W. von Bachhaus

1995-11-01

Full Text Available Part I - By combining a general 2nd-order linear homogeneous ordinary differential equation with the three-term recurrence relation possessed by all orthogonal polynomials, it is shown that sequences of orthogonal polynomials which satisfy a differential equation of the above mentioned type necessarily have a differentiation formula of the type: gn(xY'n(x=fn(xYn(x+Yn-1(x. Part II - A recurrence formula of the form: rn(xY'n(x+sn(xY'n+1(x+tn(xY'n-1(x=0, is derived using the result of Part I.

Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of the Second Order and Families of Two-Dimensional Autonomous Systems

Directory of Open Access Journals (Sweden)

M. P. Markakis

2010-01-01

Full Text Available Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind ((Ab-1 equations. Based on the results of a previous work, concerning a closed-form solution of a general (Ab-1 equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals (in closed or parametric form have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.

Self-oscillations of aircraft landing gear shock-strut at considerable non-linear friction

Directory of Open Access Journals (Sweden)

Б.М. Шифрин

2004-01-01

Full Text Available  The report considers self-oscillations at ε >1. The previous works were dedicated to the elastic frictional L.G. shock strut oscillations, the mathematical model of which is a non-linear differential equation with low ε parameter of its right-hand part.

Using system theory and energy methods to prove existence of non-linear PDE's

NARCIS (Netherlands)

Zwart, H.J.

2015-01-01

In this discussion paper we present an idea of combining techniques known from systems theory with energy estimates to show existence for a class of non-linear partial differential equations (PDE's). At the end of the paper a list of research questions with possible approaches is given.

Macroscopic and non-linear quantum games

International Nuclear Information System (INIS)

Aerts, D.; D'Hooghe, A.; Posiewnik, A.; Pykacz, J.

2005-01-01

Full text: We consider two models of quantum games. The first one is Marinatto and Weber's 'restricted' quantum game in which only the identity and the spin-flip operators are used. We show that this quantum game allows macroscopic mechanistic realization with the use of a version of the 'macroscopic quantum machine' described by Aerts already in 1980s. In the second model we use non-linear quantum state transformations which operate on points of spin-1/2 on the Bloch sphere and which can be used to distinguish optimally between two non-orthogonal states. We show that efficiency of these non-linear strategies out-perform any linear ones. Some hints on the possible theory of non-linear quantum games are given. (author)

Non-linear dielectric monitoring of biological suspensions

International Nuclear Information System (INIS)

Treo, E F; Felice, C J

2007-01-01

Non-linear dielectric spectroscopy as a tool for in situ monitoring of enzyme assumes a non-linear behavior of the sample when a sinusoidal voltage is applied to it. Even many attempts have been made to improve the original experiments, all of them had limited success. In this paper we present upgrades made to a non-linear dielectric spectrometer developed and the results obtained when using different cells. We emphasized on the electrode surface, characterizing the grinding and polishing procedure. We found that the biological medium does not behave as expected, and the non-linear response is generated in the electrode-electrolyte interface. The electrochemistry of this interface can bias unpredictably the measured non-linear response

The necessary and sufficient conditions of the optimality for hyperbolic systems with non-differentiable performance functional

International Nuclear Information System (INIS)

Kowalewski, A.

1982-11-01

In this paper an optimal control problem with non-differentiable cost function for distributed parameter system is solved. As an example an optimal control problem for system described by a linear partial differential of hyperbolic type with the Neuman's boundary condition is considered. By use of the Milutin-Dubovicki method, necessary and sufficient conditions of optimality with non-differentiable performance functional and constrained control are derived for Neuman's problem. (author)

Soil non-linearity and its effect on the dynamic behaviour of offshore platform foundations

Energy Technology Data Exchange (ETDEWEB)

Madshus, Christian

1997-07-01

This thesis focuses on non-linear soil response to the type of cyclic loading experienced under offshore gravity base platform foundations. These loads are dominated by a cyclic component around the main wave frequency, which may well mobilize soil non-linearity under severe sea-states. Superimposed on this main component are lower level higher frequency loads caused by resonant oscillations of the platform. The thesis presents results of specially designed triaxial tests to simulate this loading condition. The tests simultaneously applied two cyclic load components at different frequencies and amplitudes. The measured soil response to each component has been isolated through a frequency domain separation. It was found that the soil responds to the superimposed high frequency low level component as if the soil had a cyclically time-varying stiffness. If the superimposed component does not lead to load reversals, this stiffness variation is controlled by the frequency and amplitude of the main load component and by the hysteretic non-linearity of the soil. If the superimposed component causes reversals, the influence of the hysteretic non-linearity on the stiffness variation is reduced. The higher the degree of reversal, the more this influence it taken over by the variation in the instantaneous unloading-reloading stiffness of the soil. It was also found that this type of two-frequency cyclic soil testing is generally superior over conventional single-frequency testing in the way it enforces the soil to reveal several of its inherent properties not deducible from ordinary tests. Benefits of analyzing non-linear response in the frequency domain is demonstrated throughout this thesis. The ability of various theoretical soil models to simulate the observed soil behaviour under two-frequency cyclic loading has, been investigated through numerical analyses. It was found that only those models that are based on kinematic hardening are able to reproduce what was observed

Linear and non-linear energy barriers in systems of interacting single-domain ferromagnetic particles

International Nuclear Information System (INIS)

Petrila, Iulian; Bodale, Ilie; Rotarescu, Cristian; Stancu, Alexandru

2011-01-01

A comparative analysis between linear and non-linear energy barriers used for modeling statistical thermally-excited ferromagnetic systems is presented. The linear energy barrier is obtained by new symmetry considerations about the anisotropy energy and the link with the non-linear energy barrier is also presented. For a relevant analysis we compare the effects of linear and non-linear energy barriers implemented in two different models: Preisach-Neel and Ising-Metropolis. The differences between energy barriers which are reflected in different coercive field dependence of the temperature are also presented. -- Highlights: → The linear energy barrier is obtained from symmetry considerations. → The linear and non-linear energy barriers are calibrated and implemented in Preisach-Neel and Ising-Metropolis models. → The temperature and time effects of the linear and non-linear energy barriers are analyzed.

Penalized Nonlinear Least Squares Estimation of Time-Varying Parameters in Ordinary Differential Equations

KAUST Repository

Cao, Jiguo; Huang, Jianhua Z.; Wu, Hulin

2012-01-01

Ordinary differential equations (ODEs) are widely used in biomedical research and other scientific areas to model complex dynamic systems. It is an important statistical problem to estimate parameters in ODEs from noisy observations. In this article we propose a method for estimating the time-varying coefficients in an ODE. Our method is a variation of the nonlinear least squares where penalized splines are used to model the functional parameters and the ODE solutions are approximated also using splines. We resort to the implicit function theorem to deal with the nonlinear least squares objective function that is only defined implicitly. The proposed penalized nonlinear least squares method is applied to estimate a HIV dynamic model from a real dataset. Monte Carlo simulations show that the new method can provide much more accurate estimates of functional parameters than the existing two-step local polynomial method which relies on estimation of the derivatives of the state function. Supplemental materials for the article are available online.

Non-linear singular problems in p-adic analysis: associative algebras of p-adic distributions

International Nuclear Information System (INIS)

Albeverio, S; Khrennikov, A Yu; Shelkovich, V M

2005-01-01

We propose an algebraic theory which can be used for solving both linear and non-linear singular problems of p-adic analysis related to p-adic distributions (generalized functions). We construct the p-adic Colombeau-Egorov algebra of generalized functions, in which Vladimirov's pseudo-differential operator plays the role of differentiation. This algebra is closed under Fourier transformation and associative convolution. Pointvalues of generalized functions are defined, and it turns out that any generalized function is uniquely determined by its pointvalues. We also construct an associative algebra of asymptotic distributions, which is generated by the linear span of the set of associated homogeneous p-adic distributions. This algebra is embedded in the Colombeau-Egorov algebra as a subalgebra. In addition, a new technique for constructing weak asymptotics is developed

Non linear system become linear system

Directory of Open Access Journals (Sweden)

Petre Bucur

2007-01-01

Full Text Available The present paper refers to the theory and the practice of the systems regarding non-linear systems and their applications. We aimed the integration of these systems to elaborate their response as well as to highlight some outstanding features.

On the reduction of the degree of linear differential operators

International Nuclear Information System (INIS)

Bobieński, Marcin; Gavrilov, Lubomir

2011-01-01

Let L be a linear differential operator with coefficients in some differential field k of characteristic zero with algebraically closed field of constants. Let k a be the algebraic closure of k. For a solution y 0 , Ly 0 = 0, we determine the linear differential operator of minimal degree L-tilde and coefficients in k a , such that L-tilde y 0 =0. This result is then applied to some Picard–Fuchs equations which appear in the study of perturbations of plane polynomial vector fields of Lotka–Volterra type

Modeling non-linear kinetics of hyperpolarized [1-(13)C] pyruvate in the crystalloid-perfused rat heart

NARCIS (Netherlands)

Mariotti, E.; Orton, M. R.; Eerbeek, O.; Ashruf, J. F.; Zuurbier, C. J.; Southworth, R.; Eykyn, T. R.

2016-01-01

Hyperpolarized (13)C MR measurements have the potential to display non-linear kinetics. We have developed an approach to describe possible non-first-order kinetics of hyperpolarized [1-(13)C] pyruvate employing a system of differential equations that agrees with the principle of conservation of mass

Joint estimation of the fractional differentiation orders and the unknown input for linear fractional non-commensurate system

KAUST Repository

Belkhatir, Zehor

2015-11-05

This paper deals with the joint estimation of the unknown input and the fractional differentiation orders of a linear fractional order system. A two-stage algorithm combining the modulating functions with a first-order Newton method is applied to solve this estimation problem. First, the modulating functions approach is used to estimate the unknown input for a given fractional differentiation orders. Then, the method is combined with a first-order Newton technique to identify the fractional orders jointly with the input. To show the efficiency of the proposed method, numerical examples illustrating the estimation of the neural activity, considered as input of a fractional model of the neurovascular coupling, along with the fractional differentiation orders are presented in both noise-free and noisy cases.

Population stochastic modelling (PSM)--an R package for mixed-effects models based on stochastic differential equations.

Science.gov (United States)

Klim, Søren; Mortensen, Stig Bousgaard; Kristensen, Niels Rode; Overgaard, Rune Viig; Madsen, Henrik

2009-06-01

The extension from ordinary to stochastic differential equations (SDEs) in pharmacokinetic and pharmacodynamic (PK/PD) modelling is an emerging field and has been motivated in a number of articles [N.R. Kristensen, H. Madsen, S.H. Ingwersen, Using stochastic differential equations for PK/PD model development, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 109-141; C.W. Tornøe, R.V. Overgaard, H. Agersø, H.A. Nielsen, H. Madsen, E.N. Jonsson, Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations, Pharm. Res. 22 (August(8)) (2005) 1247-1258; R.V. Overgaard, N. Jonsson, C.W. Tornøe, H. Madsen, Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 85-107; U. Picchini, S. Ditlevsen, A. De Gaetano, Maximum likelihood estimation of a time-inhomogeneous stochastic differential model of glucose dynamics, Math. Med. Biol. 25 (June(2)) (2008) 141-155]. PK/PD models are traditionally based ordinary differential equations (ODEs) with an observation link that incorporates noise. This state-space formulation only allows for observation noise and not for system noise. Extending to SDEs allows for a Wiener noise component in the system equations. This additional noise component enables handling of autocorrelated residuals originating from natural variation or systematic model error. Autocorrelated residuals are often partly ignored in PK/PD modelling although violating the hypothesis for many standard statistical tests. This article presents a package for the statistical program R that is able to handle SDEs in a mixed-effects setting. The estimation method implemented is the FOCE(1) approximation to the population likelihood which is generated from the individual likelihoods that are approximated using the Extended Kalman Filter's one-step predictions.

A practical course in differential equations and mathematical modeling

CERN Document Server

Ibragimov , Nail H

2009-01-01

A Practical Course in Differential Equations and Mathematical Modelling is a unique blend of the traditional methods of ordinary and partial differential equations with Lie group analysis enriched by the author's own theoretical developments. The book which aims to present new mathematical curricula based on symmetry and invariance principles is tailored to develop analytic skills and working knowledge in both classical and Lie's methods for solving linear and nonlinear equations. This approach helps to make courses in differential equations, mathematical modelling, distributions and fundame

Lectures on differential Galois theory

CERN Document Server

Magid, Andy R

1994-01-01

Differential Galois theory studies solutions of differential equations over a differential base field. In much the same way that ordinary Galois theory is the theory of field extensions generated by solutions of (one variable) polynomial equations, differential Galois theory looks at the nature of the differential field extension generated by the solutions of differential equations. An additional feature is that the corresponding differential Galois groups (of automorphisms of the extension fixing the base and commuting with the derivation) are algebraic groups. This book deals with the differential Galois theory of linear homogeneous differential equations, whose differential Galois groups are algebraic matrix groups. In addition to providing a convenient path to Galois theory, this approach also leads to the constructive solution of the inverse problem of differential Galois theory for various classes of algebraic groups. Providing a self-contained development and many explicit examples, this book provides ...

Linearized gravity in terms of differential forms

Science.gov (United States)

Baykal, Ahmet; Dereli, Tekin

2017-01-01

A technique to linearize gravitational field equations is developed in which the perturbation metric coefficients are treated as second rank, symmetric, 1-form fields belonging to the Minkowski background spacetime by using the exterior algebra of differential forms.

Stability and Linear Quadratic Differential Games of Discrete-Time Markovian Jump Linear Systems with State-Dependent Noise

Directory of Open Access Journals (Sweden)

Huiying Sun

2014-01-01

Full Text Available We mainly consider the stability of discrete-time Markovian jump linear systems with state-dependent noise as well as its linear quadratic (LQ differential games. A necessary and sufficient condition involved with the connection between stochastic Tn-stability of Markovian jump linear systems with state-dependent noise and Lyapunov equation is proposed. And using the theory of stochastic Tn-stability, we give the optimal strategies and the optimal cost values for infinite horizon LQ stochastic differential games. It is demonstrated that the solutions of infinite horizon LQ stochastic differential games are concerned with four coupled generalized algebraic Riccati equations (GAREs. Finally, an iterative algorithm is presented to solve the four coupled GAREs and a simulation example is given to illustrate the effectiveness of it.

Certain non-linear differential polynomials sharing a non zero polynomial

Directory of Open Access Journals (Sweden)

Majumder Sujoy

2015-10-01

functions sharing a nonzero polynomial and obtain two results which improves and generalizes the results due to L. Liu [Uniqueness of meromorphic functions and differential polynomials, Comput. Math. Appl., 56 (2008, 3236-3245.] and P. Sahoo [Uniqueness and weighted value sharing of meromorphic functions, Applied. Math. E-Notes., 11 (2011, 23-32.].

Heat transfer in boundary layer stagnation-point flow towards a shrinking sheet with non-uniform heat flux

International Nuclear Information System (INIS)

Bhattacharyya Krishnendu

2013-01-01

In this paper, the effect of non-uniform heat flux on heat transfer in boundary layer stagnation-point flow over a shrinking sheet is studied. The variable boundary heat fluxes are considered of two types: direct power-law variation with the distance along the sheet and inverse power-law variation with the distance. The governing partial differential equations (PDEs) are transformed into non linear self-similar ordinary differential equations (ODEs) by similarity transformations, and then those are solved using very efficient shooting method. The direct variation and inverse variation of heat flux along the sheet have completely different effects on the temperature distribution. Moreover, the heat transfer characteristics in the presence of non-uniform heat flux for several values of physical parameters are also found to be interesting

Modified Differential Transform Method for Two Singular Boundary Values Problems

Directory of Open Access Journals (Sweden)

Yinwei Lin

2014-01-01

Full Text Available This paper deals with the two singular boundary values problems of second order. Two singular points are both boundary values points of the differential equation. The numerical solutions are developed by modified differential transform method (DTM for expanded point. Linear and nonlinear models are solved by this method to get more reliable and efficient numerical results. It can also solve ordinary differential equations where the traditional one fails. Besides, we give the convergence of this new method.

"Real-Time Optical Laboratory Linear Algebra Solution Of Partial Differential Equations"

Science.gov (United States)

Casasent, David; Jackson, James

1986-03-01

A Space Integrating (SI) Optical Linear Algebra Processor (OLAP) employing space and frequency-multiplexing, new partitioning and data flow, and achieving high accuracy performance with a non base-2 number system is described. Laboratory data on the performance of this system and the solution of parabolic Partial Differential Equations (PDEs) is provided. A multi-processor OLAP system is also described for the first time. It use in the solution of multiple banded matrices that frequently arise is then discussed. The utility and flexibility of this processor compared to digital systolic architectures should be apparent.

Computation of Optimal Monotonicity Preserving General Linear Methods

KAUST Repository

Ketcheson, David I.

2009-07-01

Monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution. We formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations, and propose an efficient algorithm for its solution. This algorithm reliably finds optimal methods even among classes involving very high order accuracy and that use many steps and/or stages. The optimality of some recently proposed methods is verified, and many more efficient methods are found. We use similar algorithms to find optimal strong stability preserving linear multistep methods of both explicit and implicit type, including methods for hyperbolic PDEs that use downwind-biased operators.

Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

Directory of Open Access Journals (Sweden)

S. Narayanamoorthy

2015-01-01

Full Text Available We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

Links among impossible differential, integral and zero correlation linear cryptanalysis

DEFF Research Database (Denmark)

Sun, Bing; Liu, Zhiqiang; Rijmen, Vincent

2015-01-01

is to fix this gap and establish links between impossible differential cryptanalysis and integral cryptanalysis. Firstly, by introducing the concept of structure and dual structure, we prove that a → b is an impossible differential of a structure E if and only if it is a zero correlation linear hull...... linear hull always indicates the existence of an integral distinguisher. With this observation we improve the number of rounds of integral distinguishers of Feistel structures, CAST-256, SMS4 and Camellia. Finally, we conclude that an r-round impossible differential of E always leads to an r...

Linearity and Non-linearity of Photorefractive effect in Materials ...

African Journals Online (AJOL)

Linearity and Non-linearity of Photorefractive effect in Materials using the Band transport ... For low light beam intensities the change in the refractive index is ... field is spatially phase shifted by /2 relative to the interference fringe pattern, which ...

Reduced differential transform method for partial differential equations within local fractional derivative operators

Directory of Open Access Journals (Sweden)

Hossein Jafari

2016-04-01

Full Text Available The non-differentiable solution of the linear and non-linear partial differential equations on Cantor sets is implemented in this article. The reduced differential transform method is considered in the local fractional operator sense. The four illustrative examples are given to show the efficiency and accuracy features of the presented technique to solve local fractional partial differential equations.

Correlations and Non-Linear Probability Models

DEFF Research Database (Denmark)

Breen, Richard; Holm, Anders; Karlson, Kristian Bernt

2014-01-01

the dependent variable of the latent variable model and its predictor variables. We show how this correlation can be derived from the parameters of non-linear probability models, develop tests for the statistical significance of the derived correlation, and illustrate its usefulness in two applications. Under......Although the parameters of logit and probit and other non-linear probability models are often explained and interpreted in relation to the regression coefficients of an underlying linear latent variable model, we argue that they may also be usefully interpreted in terms of the correlations between...... certain circumstances, which we explain, the derived correlation provides a way of overcoming the problems inherent in cross-sample comparisons of the parameters of non-linear probability models....

Pod systems: an equivariant ordinary differential equation approach to dynamical systems on a spatial domain

International Nuclear Information System (INIS)

Elmhirst, Toby; Stewart, Ian; Doebeli, Michael

2008-01-01

We present a class of systems of ordinary differential equations (ODEs), which we call 'pod systems', that offers a new perspective on dynamical systems defined on a spatial domain. Such systems are typically studied as partial differential equations, but pod systems bring the analytic techniques of ODE theory to bear on the problems, and are thus able to study behaviours and bifurcations that are not easily accessible to the standard methods. In particular, pod systems are specifically designed to study spatial dynamical systems that exhibit multi-modal solutions. A pod system is essentially a linear combination of parametrized functions in which the coefficients and parameters are variables whose dynamics are specified by a system of ODEs. That is, pod systems are concerned with the dynamics of functions of the form Ψ(s, t) = y 1 (t) φ(s; x 1 (t)) + ··· + y N (t) φ(s; x N (t)), where s in R n is the spatial variable and φ: R n × R d → R. The parameters x i in R d and coefficients y i in R are dynamic variables which evolve according to some system of ODEs, x-dot i = G i (x, y) and y-dot i = H i (x, y), for i = 1, ..., N. The dynamics of Ψ in function space can then be studied through the dynamics of the x and y in finite dimensions. A vital feature of pod systems is that the ODEs that specify the dynamics of the x and y variables are not arbitrary; restrictions on G i and H i are required to guarantee that the dynamics of Ψ in function space are well defined (that is, that trajectories are unique). One important restriction is symmetry in the ODEs which arises because Ψ is invariant under permutations of the indices of the (x i , y i ) pairs. However, this is not the whole story, and the primary goal of this paper is to determine the necessary structure of the ODEs explicitly to guarantee that the dynamics of Ψ are well defined

Hartman-Wintner growth results for sublinear functional differential equations

Directory of Open Access Journals (Sweden)

John A. D. Appleby

2017-01-01

Full Text Available This article determines the rate of growth to infinity of scalar autonomous nonlinear functional and Volterra differential equations. In these equations, the right-hand side is a positive continuous linear functional of f(x. We assume f grows sublinearly, leading to subexponential growth in the solutions. The main results show that the solution of the functional differential equations are asymptotic to that of an auxiliary autonomous ordinary differential equation with right-hand side proportional to f. This happens provided f grows more slowly than l(x=x/log(x. The linear-logarithmic growth rate is also shown to be critical: if f grows more rapidly than l, the ODE dominates the FDE; if f is asymptotic to a constant multiple of l, the FDE and ODE grow at the same rate, modulo a constant non-unit factor; if f grows more slowly than l, the ODE and FDE grow at exactly the same rate. A partial converse of the last result is also proven. In the case when the growth rate is slower than that of the ODE, sharp bounds on the growth rate are determined. The Volterra and finite memory equations can have differing asymptotic behaviour and we explore the source of these differences.

Growth of meromorphic solutions of higher-order linear differential equations

Directory of Open Access Journals (Sweden)

Wenjuan Chen

2009-01-01

Full Text Available In this paper, we investigate the higher-order linear differential equations with meromorphic coefficients. We improve and extend a result of M.S. Liu and C.L. Yuan, by using the estimates for the logarithmic derivative of a transcendental meromorphic function due to Gundersen, and the extended Winman-Valiron theory which proved by J. Wang and H.X. Yi. In addition, we also consider the nonhomogeneous linear differential equations.

Linear mode conversion in a toroidal plasma

International Nuclear Information System (INIS)

Hellsten, T.

1980-05-01

Linear mode conversion at the perpendicular ion cyclotron resonance has been treated for an axially symmetric toroidal plasma. The mode conversion appears between a fast electromagnetic wave and a slow-quasi electrostatic wave, due to finite electron inertia. The problem reduces to the Orr-Sommerfeld equation where the coefficients determining the reflectron, transmission and conversion are functions of the arc length along a poloidal intersection of the resonance surface. These coefficients can be determined from eigenfunctions of an ordinary differential equation. (author)

Fourier imaging of non-linear structure formation

Energy Technology Data Exchange (ETDEWEB)

Brandbyge, Jacob; Hannestad, Steen, E-mail: jacobb@phys.au.dk, E-mail: sth@phys.au.dk [Department of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, DK-8000 Aarhus C (Denmark)

2017-04-01

We perform a Fourier space decomposition of the dynamics of non-linear cosmological structure formation in ΛCDM models. From N -body simulations involving only cold dark matter we calculate 3-dimensional non-linear density, velocity divergence and vorticity Fourier realizations, and use these to calculate the fully non-linear mode coupling integrals in the corresponding fluid equations. Our approach allows for a reconstruction of the amount of mode coupling between any two wavenumbers as a function of redshift. With our Fourier decomposition method we identify the transfer of power from larger to smaller scales, the stable clustering regime, the scale where vorticity becomes important, and the suppression of the non-linear divergence power spectrum as compared to linear theory. Our results can be used to improve and calibrate semi-analytical structure formation models.

Fourier imaging of non-linear structure formation

International Nuclear Information System (INIS)

Brandbyge, Jacob; Hannestad, Steen

2017-01-01

We perform a Fourier space decomposition of the dynamics of non-linear cosmological structure formation in ΛCDM models. From N -body simulations involving only cold dark matter we calculate 3-dimensional non-linear density, velocity divergence and vorticity Fourier realizations, and use these to calculate the fully non-linear mode coupling integrals in the corresponding fluid equations. Our approach allows for a reconstruction of the amount of mode coupling between any two wavenumbers as a function of redshift. With our Fourier decomposition method we identify the transfer of power from larger to smaller scales, the stable clustering regime, the scale where vorticity becomes important, and the suppression of the non-linear divergence power spectrum as compared to linear theory. Our results can be used to improve and calibrate semi-analytical structure formation models.

Algorithms for non-linear M-estimation

DEFF Research Database (Denmark)

Madsen, Kaj; Edlund, O; Ekblom, H

1997-01-01

In non-linear regression, the least squares method is most often used. Since this estimator is highly sensitive to outliers in the data, alternatives have became increasingly popular during the last decades. We present algorithms for non-linear M-estimation. A trust region approach is used, where...

Mathematical modeling based on ordinary differential equations: A promising approach to vaccinology.

Science.gov (United States)

Bonin, Carla Rezende Barbosa; Fernandes, Guilherme Cortes; Dos Santos, Rodrigo Weber; Lobosco, Marcelo

2017-02-01

New contributions that aim to accelerate the development or to improve the efficacy and safety of vaccines arise from many different areas of research and technology. One of these areas is computational science, which traditionally participates in the initial steps, such as the pre-screening of active substances that have the potential to become a vaccine antigen. In this work, we present another promising way to use computational science in vaccinology: mathematical and computational models of important cell and protein dynamics of the immune system. A system of Ordinary Differential Equations represents different immune system populations, such as B cells and T cells, antigen presenting cells and antibodies. In this way, it is possible to simulate, in silico, the immune response to vaccines under development or under study. Distinct scenarios can be simulated by varying parameters of the mathematical model. As a proof of concept, we developed a model of the immune response to vaccination against the yellow fever. Our simulations have shown consistent results when compared with experimental data available in the literature. The model is generic enough to represent the action of other diseases or vaccines in the human immune system, such as dengue and Zika virus.

Evidence from the Semarkona ordinary chondrite for 26Al heating of small planets

International Nuclear Information System (INIS)

Hutcheon, I.D.

1989-01-01

We report the first observation of radiogenic 26 Mg in non-refractory meteoritic material, a plagio-clase-bearing, olivine-pyroxene clast chondrule in the Semarkona ordinary chondrite. The inferred initial abundance of 26 Al is sufficient to produce incipient melting in well insulated bodies of chondritic composition. We conclude that planetary accretion and differentiation must have begun on a timescale comparable to the half life of 26 Al and that, even if widespread melting did not occur, 26 Al heating played a significant role in thermal metamorphism on small planets. (author)

Non-linear realization of α0 -extended supersymmetry

International Nuclear Information System (INIS)

Nishino, Hitoshi

2000-01-01

As generalizations of the original Volkov-Akulov action in four-dimensions, actions are found for all space-time dimensions D invariant under N non-linear realized global supersymmetries. We also give other such actions invariant under the global non-linear supersymmetry. As an interesting consequence, we find a non-linear supersymmetric Born-Infeld action for a non-Abelian gauge group for arbitrary D and N , which coincides with the linearly supersymmetric Born-Infeld action in D=10 at the lowest order. For the gauge group U(N) for M(atrix)-theory, this model has N 2 -extended non-linear supersymmetries, so that its large N limit corresponds to the infinitely many (α 0 ) supersymmetries. We also perform a duality transformation from F μν into its Hodge dual N μ 1 ctdot μD-2 . We next point out that any Chern-Simons action for any (super)groups has the non-linear supersymmetry as a hidden symmetry. Subsequently, we present a superspace formulation for the component results. We further find that as long as superspace supergravity is consistent, this generalized Volkov-Akulov action can further accommodate such curved superspace backgrounds with local supersymmetry, as a super p -brane action with fermionic kappa-symmetry. We further elaborate these results to what we call 'simplified' (Supersymmetry) 2 -models, with both linear and non-linear representations of supersymmetries in superspace at the same time. Our result gives a proof that there is no restriction on D or N for global non-linear supersymmetry. We also see that the non-linear realization of supersymmetry in 'curved' space-time can be interpreted as 'non-perturbative' effect starting with the 'flat' space-time

Neural Networks for Non-linear Control

DEFF Research Database (Denmark)

Sørensen, O.

1994-01-01

This paper describes how a neural network, structured as a Multi Layer Perceptron, is trained to predict, simulate and control a non-linear process.......This paper describes how a neural network, structured as a Multi Layer Perceptron, is trained to predict, simulate and control a non-linear process....

Non-Linear Approximation of Bayesian Update

KAUST Repository

Litvinenko, Alexander

2016-01-01

We develop a non-linear approximation of expensive Bayesian formula. This non-linear approximation is applied directly to Polynomial Chaos Coefficients. In this way, we avoid Monte Carlo sampling and sampling error. We can show that the famous Kalman Update formula is a particular case of this update.

Non-Linear Approximation of Bayesian Update

KAUST Repository

Litvinenko, Alexander

2016-06-23

We develop a non-linear approximation of expensive Bayesian formula. This non-linear approximation is applied directly to Polynomial Chaos Coefficients. In this way, we avoid Monte Carlo sampling and sampling error. We can show that the famous Kalman Update formula is a particular case of this update.

The linear-non-linear frontier for the Goldstone Higgs

International Nuclear Information System (INIS)

Gavela, M.B.; Saa, S.; Kanshin, K.; Machado, P.A.N.

2016-01-01

The minimal SO(5)/SO(4) σ-model is used as a template for the ultraviolet completion of scenarios in which the Higgs particle is a low-energy remnant of some high-energy dynamics, enjoying a (pseudo) Nambu-Goldstone-boson ancestry. Varying the σ mass allows one to sweep from the perturbative regime to the customary non-linear implementations. The low-energy benchmark effective non-linear Lagrangian for bosons and fermions is obtained, determining as well the operator coefficients including linear corrections. At first order in the latter, three effective bosonic operators emerge which are independent of the explicit soft breaking assumed. The Higgs couplings to vector bosons and fermions turn out to be quite universal: the linear corrections are proportional to the explicit symmetry-breaking parameters. Furthermore, we define an effective Yukawa operator which allows a simple parametrization and comparison of different heavy-fermion ultraviolet completions. In addition, one particular fermionic completion is explored in detail, obtaining the corresponding leading low-energy fermionic operators. (orig.)

The linear-non-linear frontier for the Goldstone Higgs

Energy Technology Data Exchange (ETDEWEB)

Gavela, M.B.; Saa, S. [IFT-UAM/CSIC, Universidad Autonoma de Madrid, Departamento de Fisica Teorica y Instituto de Fisica Teorica, Madrid (Spain); Kanshin, K. [Universita di Padova, Dipartimento di Fisica e Astronomia ' G. Galilei' , Padua (Italy); INFN, Padova (Italy); Machado, P.A.N. [IFT-UAM/CSIC, Universidad Autonoma de Madrid, Departamento de Fisica Teorica y Instituto de Fisica Teorica, Madrid (Spain); Fermi National Accelerator Laboratory, Theoretical Physics Department, Batavia, IL (United States)

2016-12-15

The minimal SO(5)/SO(4) σ-model is used as a template for the ultraviolet completion of scenarios in which the Higgs particle is a low-energy remnant of some high-energy dynamics, enjoying a (pseudo) Nambu-Goldstone-boson ancestry. Varying the σ mass allows one to sweep from the perturbative regime to the customary non-linear implementations. The low-energy benchmark effective non-linear Lagrangian for bosons and fermions is obtained, determining as well the operator coefficients including linear corrections. At first order in the latter, three effective bosonic operators emerge which are independent of the explicit soft breaking assumed. The Higgs couplings to vector bosons and fermions turn out to be quite universal: the linear corrections are proportional to the explicit symmetry-breaking parameters. Furthermore, we define an effective Yukawa operator which allows a simple parametrization and comparison of different heavy-fermion ultraviolet completions. In addition, one particular fermionic completion is explored in detail, obtaining the corresponding leading low-energy fermionic operators. (orig.)

Linear versus non-linear structural information limit in high-resolution transmission electron microscopy

International Nuclear Information System (INIS)

Van Aert, S.; Chen, J.H.; Van Dyck, D.

2010-01-01

A widely used performance criterion in high-resolution transmission electron microscopy (HRTEM) is the information limit. It corresponds to the inverse of the maximum spatial object frequency that is linearly transmitted with sufficient intensity from the exit plane of the object to the image plane and is limited due to partial temporal coherence. In practice, the information limit is often measured from a diffractogram or from Young's fringes assuming a weak phase object scattering beyond the inverse of the information limit. However, for an aberration corrected electron microscope, with an information limit in the sub-angstrom range, weak phase objects are no longer applicable since they do not scatter sufficiently in this range. Therefore, one relies on more strongly scattering objects such as crystals of heavy atoms observed along a low index zone axis. In that case, dynamical scattering becomes important such that the non-linear and linear interaction may be equally important. The non-linear interaction may then set the experimental cut-off frequency observed in a diffractogram. The goal of this paper is to quantify both the linear and the non-linear information transfer in terms of closed form analytical expressions. Whereas the cut-off frequency set by the linear transfer can be directly related with the attainable resolution, information from the non-linear transfer can only be extracted using quantitative, model-based methods. In contrast to the historic definition of the information limit depending on microscope parameters only, the expressions derived in this paper explicitly incorporate their dependence on the structure parameters as well. In order to emphasize this dependence and to distinguish from the usual information limit, the expressions derived for the inverse cut-off frequencies will be referred to as the linear and non-linear structural information limit. The present findings confirm the well-known result that partial temporal coherence has

Non-linear soil-structure interaction

International Nuclear Information System (INIS)

Wolf, J.P.

1984-01-01

The basic equation of motion to analyse the interaction of a non-linear structure and an irregular soil with the linear unbounded soil is formulated in the time domain. The contribution of the unbounded soil involves convolution integrals of the dynamic-stiffness coefficients in the time domain and the corresponding motions. As another possibility, a flexibility formulation fot the contribution of the unbounded soil using the dynamic-flexibility coefficients in the time domain, together with the direct-stiffness method for the structure and the irregular soil can be applied. As an example of a non-linear soil-structure-interaction analysis, the partial uplift of the basemat of a structure is examined. (Author) [pt

In silico ordinary differential equation/partial differential equation hemodialysis model estimates methadone removal during dialysis

Science.gov (United States)

Linares, Oscar A; Schiesser, William E; Fudin, Jeffrey; Pham, Thien C; Bettinger, Jeffrey J; Mathew, Roy O; Daly, Annemarie L

2015-01-01

Background There is a need to have a model to study methadone’s losses during hemodialysis to provide informed methadone dose recommendations for the practitioner. Aim To build a one-dimensional (1-D), hollow-fiber geometry, ordinary differential equation (ODE) and partial differential equation (PDE) countercurrent hemodialyzer model (ODE/PDE model). Methodology We conducted a cross-sectional study in silico that evaluated eleven hemodialysis patients. Patients received a ceiling dose of methadone hydrochloride 30 mg/day. Outcome measures included: the total amount of methadone removed during dialysis; methadone’s overall intradialytic mass transfer rate coefficient, km; and, methadone’s removal rate, jME. Each metric was measured at dialysate flow rates of 250 mL/min and 800 mL/min. Results The ODE/PDE model revealed a significant increase in the change of methadone’s mass transfer with increased dialysate flow rate, %Δkm=18.56, P=0.02, N=11. The total amount of methadone mass transferred across the dialyzer membrane with high dialysate flow rate significantly increased (0.042±0.016 versus 0.052±0.019 mg/kg, P=0.02, N=11). This was accompanied by a small significant increase in methadone’s mass transfer rate (0.113±0.002 versus 0.014±0.002 mg/kg/h, P=0.02, N=11). The ODE/PDE model accurately predicted methadone’s removal during dialysis. The absolute value of the prediction errors for methadone’s extraction and throughput were less than 2%. Conclusion ODE/PDE modeling of methadone’s hemodialysis is a new approach to study methadone’s removal, in particular, and opioid removal, in general, in patients with end-stage renal disease on hemodialysis. ODE/PDE modeling accurately quantified the fundamental phenomena of methadone’s mass transfer during hemodialysis. This methodology may lead to development of optimally designed intradialytic opioid treatment protocols, and allow dynamic monitoring of outflow plasma opioid concentrations for model

In silico ordinary differential equation/partial differential equation hemodialysis model estimates methadone removal during dialysis.

Science.gov (United States)

Linares, Oscar A; Schiesser, William E; Fudin, Jeffrey; Pham, Thien C; Bettinger, Jeffrey J; Mathew, Roy O; Daly, Annemarie L

2015-01-01

There is a need to have a model to study methadone's losses during hemodialysis to provide informed methadone dose recommendations for the practitioner. To build a one-dimensional (1-D), hollow-fiber geometry, ordinary differential equation (ODE) and partial differential equation (PDE) countercurrent hemodialyzer model (ODE/PDE model). We conducted a cross-sectional study in silico that evaluated eleven hemodialysis patients. Patients received a ceiling dose of methadone hydrochloride 30 mg/day. Outcome measures included: the total amount of methadone removed during dialysis; methadone's overall intradialytic mass transfer rate coefficient, km ; and, methadone's removal rate, j ME. Each metric was measured at dialysate flow rates of 250 mL/min and 800 mL/min. The ODE/PDE model revealed a significant increase in the change of methadone's mass transfer with increased dialysate flow rate, %Δkm =18.56, P=0.02, N=11. The total amount of methadone mass transferred across the dialyzer membrane with high dialysate flow rate significantly increased (0.042±0.016 versus 0.052±0.019 mg/kg, P=0.02, N=11). This was accompanied by a small significant increase in methadone's mass transfer rate (0.113±0.002 versus 0.014±0.002 mg/kg/h, P=0.02, N=11). The ODE/PDE model accurately predicted methadone's removal during dialysis. The absolute value of the prediction errors for methadone's extraction and throughput were less than 2%. ODE/PDE modeling of methadone's hemodialysis is a new approach to study methadone's removal, in particular, and opioid removal, in general, in patients with end-stage renal disease on hemodialysis. ODE/PDE modeling accurately quantified the fundamental phenomena of methadone's mass transfer during hemodialysis. This methodology may lead to development of optimally designed intradialytic opioid treatment protocols, and allow dynamic monitoring of outflow plasma opioid concentrations for model predictive control during dialysis in humans.

Resummation of the 1/N-expansion of the non-linear σ-model by Dyson-Schwinger equations

International Nuclear Information System (INIS)

Drouffe, J.M.; Flyvbjerg, H.

1988-02-01

Dyson-Schwinger equations for the O(N)-symmetric non-linear σ-model are derived and expanded in 1/N. A closed set of equations is obtained by keeping only the leading term and the first correction term in this expansion. These equations are solved numerically in 2 dimensions on square lattices of sizes 50x50 and 100x100. Results for the magnetic susceptibility and the mass gap are compared with predictions of the ordinary 1/N-expansion and with Monte Carlo results. The results obtained with the Dyson-Schwinger equations show the same scaling behavior as found in the Monte Carlo results. This is not the behavior predicted by the perturbative renormalization group. (orig.)

FIFI 3: A digital computer code for the solution of sets of first order differential equations and the analysis of process plant dynamics

International Nuclear Information System (INIS)

Sumner, H.M.

1965-11-01

FIFI 3 is a FORTRAN Code embodying a technique for the analysis of process plant dynamics. As such, it is essentially a tool for the integration of sets of first order ordinary differential equations, either linear or non-linear; special provision is made for the inclusion of time-delayed variables in the mathematical model of the plant. The method of integration is new and is centred on a stable multistep predictor-corrector algorithm devised by the late Mr. F.G. Chapman, of the UKAEA, Winfrith. The theory on which the Code is based and detailed rules for using it are described in Parts I and II respectively. (author)

Quad-copter UAV BLDC Motor Control: Linear v/s non-linear control maps

Directory of Open Access Journals (Sweden)

Deep Parikh

2015-08-01

Full Text Available This paper presents some investigations and comparison of using linear versus non-linear static motor-control maps for the speed control of a BLDC (Brush Less Direct Current motors used in quad-copter UAV (Unmanned Aerial Vehicles. The motor-control map considered here is the inverse of the static map relating motor-speed output to motor-voltage input for a typical out-runner type Brushless DC Motors (BLDCM.  Traditionally, quad-copter BLDC motor speed control uses simple linear motor-control map defined by the motor-constant specification. However, practical BLDC motors show non-linear characteristic, particularly when operated across wide operating speed-range as is commonly required in quad-copter UAV flight operations. In this paper, our investigations to compare performance of linear versus non-linear motor-control maps are presented. The investigations cover simulation-based and experimental study of BLDC motor speed control systems for  quad-copter vehicle available. First the non-linear map relating rotor RPM to motor voltage for quad-copter BLDC motor is obtained experimentally using an optical speed encoder. The performance of the linear versus non-linear motor-control-maps for the speed control are studied. The investigations also cover study of time-responses for various standard test input-signals e.g. step, ramp and pulse inputs, applied as the reference speed-commands. Also, simple 2-degree of freedom test-bed is developed in our laboratory to help test the open-loop and closed-loop experimental investigations. The non-linear motor-control map is found to perform better in BLDC motor speed tracking control performance and thereby helping achieve better quad-copter roll-angle attitude control.

Singular Linear Differential Equations in Two Variables

NARCIS (Netherlands)

Braaksma, B.L.J.; Put, M. van der

2008-01-01

The formal and analytic classification of integrable singular linear differential equations has been studied among others by R. Gerard and Y. Sibuya. We provide a simple proof of their main result, namely: For certain irregular systems in two variables there is no Stokes phenomenon, i.e. there is no

SIVA/DIVA- INITIAL VALUE ORDINARY DIFFERENTIAL EQUATION SOLUTION VIA A VARIABLE ORDER ADAMS METHOD

Science.gov (United States)

Krogh, F. T.

1994-01-01

The SIVA/DIVA package is a collection of subroutines for the solution of ordinary differential equations. There are versions for single precision and double precision arithmetic. These solutions are applicable to stiff or nonstiff differential equations of first or second order. SIVA/DIVA requires fewer evaluations of derivatives than other variable order Adams predictor-corrector methods. There is an option for the direct integration of second order equations which can make integration of trajectory problems significantly more efficient. Other capabilities of SIVA/DIVA include: monitoring a user supplied function which can be separate from the derivative; dynamically controlling the step size; displaying or not displaying output at initial, final, and step size change points; saving the estimated local error; and reverse communication where subroutines return to the user for output or computation of derivatives instead of automatically performing calculations. The user must supply SIVA/DIVA with: 1) the number of equations; 2) initial values for the dependent and independent variables, integration stepsize, error tolerance, etc.; and 3) the driver program and operational parameters necessary for subroutine execution. SIVA/DIVA contains an extensive diagnostic message library should errors occur during execution. SIVA/DIVA is written in FORTRAN 77 for batch execution and is machine independent. It has a central memory requirement of approximately 120K of 8 bit bytes. This program was developed in 1983 and last updated in 1987.

A linear evolution for non-linear dynamics and correlations in realistic nuclei

International Nuclear Information System (INIS)

Levin, E.; Lublinsky, M.

2004-01-01

A new approach to high energy evolution based on a linear equation for QCD generating functional is developed. This approach opens a possibility for systematic study of correlations inside targets, and, in particular, inside realistic nuclei. Our results are presented as three new equations. The first one is a linear equation for QCD generating functional (and for scattering amplitude) that sums the 'fan' diagrams. For the amplitude this equation is equivalent to the non-linear Balitsky-Kovchegov equation. The second equation is a generalization of the Balitsky-Kovchegov non-linear equation to interactions with realistic nuclei. It includes a new correlation parameter which incorporates, in a model-dependent way, correlations inside the nuclei. The third equation is a non-linear equation for QCD generating functional (and for scattering amplitude) that in addition to the 'fan' diagrams sums the Glauber-Mueller multiple rescatterings

LIE GROUPS AND NUMERICAL SOLUTIONS OF DIFFERENTIAL EQUATIONS: INVARIANT DISCRETIZATION VERSUS DIFFERENTIAL APPROXIMATION

Directory of Open Access Journals (Sweden)

Decio Levi

2013-10-01

Full Text Available We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which the “first differential approximation” is invariant under the same Lie group as the original ordinary differential equation.

An algorithm for solving initial value problems of third order ordinary ...

African Journals Online (AJOL)

Abstract. We propose an implicit multi-step method for the solution of initial value problems (IVPs) of third order ordinary differential equations (ODE) which does not require reducing the ODE to first order before solving. The development of the method is based on collocation of the differential system and interpolation of the ...

Non-linear realizations and bosonic branes

International Nuclear Information System (INIS)

West, P.

2001-01-01

In this very short note, following hep-th/0001216, we express the well known bosonic brane as a non-linear realization. The reader may also consult hep-th/9912226, 0001216 and 0005270 where the branes of M theory are constructed as a non-linear realisation. The automorphisms of the supersymmetry algebra play an essential role. (author)

Modelling female fertility traits in beef cattle using linear and non-linear models.

Science.gov (United States)

Naya, H; Peñagaricano, F; Urioste, J I

2017-06-01

Female fertility traits are key components of the profitability of beef cattle production. However, these traits are difficult and expensive to measure, particularly under extensive pastoral conditions, and consequently, fertility records are in general scarce and somehow incomplete. Moreover, fertility traits are usually dominated by the effects of herd-year environment, and it is generally assumed that relatively small margins are kept for genetic improvement. New ways of modelling genetic variation in these traits are needed. Inspired in the methodological developments made by Prof. Daniel Gianola and co-workers, we assayed linear (Gaussian), Poisson, probit (threshold), censored Poisson and censored Gaussian models to three different kinds of endpoints, namely calving success (CS), number of days from first calving (CD) and number of failed oestrus (FE). For models involving FE and CS, non-linear models overperformed their linear counterparts. For models derived from CD, linear versions displayed better adjustment than the non-linear counterparts. Non-linear models showed consistently higher estimates of heritability and repeatability in all cases (h 2  linear models; h 2  > 0.23 and r > 0.24, for non-linear models). While additive and permanent environment effects showed highly favourable correlations between all models (>0.789), consistency in selecting the 10% best sires showed important differences, mainly amongst the considered endpoints (FE, CS and CD). In consequence, endpoints should be considered as modelling different underlying genetic effects, with linear models more appropriate to describe CD and non-linear models better for FE and CS. © 2017 Blackwell Verlag GmbH.

Linear algebra a first course with applications to differential equations

CERN Document Server

Apostol, Tom M

2014-01-01

Developed from the author's successful two-volume Calculus text this book presents Linear Algebra without emphasis on abstraction or formalization. To accommodate a variety of backgrounds, the text begins with a review of prerequisites divided into precalculus and calculus prerequisites. It continues to cover vector algebra, analytic geometry, linear spaces, determinants, linear differential equations and more.

Dynamics of unsymmetric piecewise-linear/non-linear systems using finite elements in time

Science.gov (United States)

Wang, Yu

1995-08-01

The dynamic response and stability of a single-degree-of-freedom system with unsymmetric piecewise-linear/non-linear stiffness are analyzed using the finite element method in the time domain. Based on a Hamilton's weak principle, this method provides a simple and efficient approach for predicting all possible fundamental and sub-periodic responses. The stability of the steady state response is determined by using Floquet's theory without any special effort for calculating transition matrices. This method is applied to a number of examples, demonstrating its effectiveness even for a strongly non-linear problem involving both clearance and continuous stiffness non-linearities. Close agreement is found between available published findings and the predictions of the finite element in time approach, which appears to be an efficient and reliable alternative technique for non-linear dynamic response and stability analysis of periodic systems.

A Bayesian approach to estimating hidden variables as well as missing and wrong molecular interactions in ordinary differential equation-based mathematical models.

Science.gov (United States)

Engelhardt, Benjamin; Kschischo, Maik; Fröhlich, Holger

2017-06-01

Ordinary differential equations (ODEs) are a popular approach to quantitatively model molecular networks based on biological knowledge. However, such knowledge is typically restricted. Wrongly modelled biological mechanisms as well as relevant external influence factors that are not included into the model are likely to manifest in major discrepancies between model predictions and experimental data. Finding the exact reasons for such observed discrepancies can be quite challenging in practice. In order to address this issue, we suggest a Bayesian approach to estimate hidden influences in ODE-based models. The method can distinguish between exogenous and endogenous hidden influences. Thus, we can detect wrongly specified as well as missed molecular interactions in the model. We demonstrate the performance of our Bayesian dynamic elastic-net with several ordinary differential equation models from the literature, such as human JAK-STAT signalling, information processing at the erythropoietin receptor, isomerization of liquid α -Pinene, G protein cycling in yeast and UV-B triggered signalling in plants. Moreover, we investigate a set of commonly known network motifs and a gene-regulatory network. Altogether our method supports the modeller in an algorithmic manner to identify possible sources of errors in ODE-based models on the basis of experimental data. © 2017 The Author(s).

Evidence from the Semarkona ordinary chondrite for /sup 26/Al heating of small planets

Energy Technology Data Exchange (ETDEWEB)

Hutcheon, I D; Hutchison, R

1989-01-19

We report the first observation of radiogenic /sup 26/Mg in non-refractory meteoritic material, a plagio-clase-bearing, olivine-pyroxene clast chondrule in the Semarkona ordinary chondrite. The inferred initial abundance of /sup 26/Al is sufficient to produce incipient melting in well insulated bodies of chondritic composition. We conclude that planetary accretion and differentiation must have begun on a timescale comparable to the half life of /sup 26/Al and that, even if widespread melting did not occur, /sup 26/Al heating played a significant role in thermal metamorphism on small planets.

Least median of squares and iteratively re-weighted least squares as robust linear regression methods for fluorimetric determination of α-lipoic acid in capsules in ideal and non-ideal cases of linearity.

Science.gov (United States)

Korany, Mohamed A; Gazy, Azza A; Khamis, Essam F; Ragab, Marwa A A; Kamal, Miranda F

2018-03-26

This study outlines two robust regression approaches, namely least median of squares (LMS) and iteratively re-weighted least squares (IRLS) to investigate their application in instrument analysis of nutraceuticals (that is, fluorescence quenching of merbromin reagent upon lipoic acid addition). These robust regression methods were used to calculate calibration data from the fluorescence quenching reaction (∆F and F-ratio) under ideal or non-ideal linearity conditions. For each condition, data were treated using three regression fittings: Ordinary Least Squares (OLS), LMS and IRLS. Assessment of linearity, limits of detection (LOD) and quantitation (LOQ), accuracy and precision were carefully studied for each condition. LMS and IRLS regression line fittings showed significant improvement in correlation coefficients and all regression parameters for both methods and both conditions. In the ideal linearity condition, the intercept and slope changed insignificantly, but a dramatic change was observed for the non-ideal condition and linearity intercept. Under both linearity conditions, LOD and LOQ values after the robust regression line fitting of data were lower than those obtained before data treatment. The results obtained after statistical treatment indicated that the linearity ranges for drug determination could be expanded to lower limits of quantitation by enhancing the regression equation parameters after data treatment. Analysis results for lipoic acid in capsules, using both fluorimetric methods, treated by parametric OLS and after treatment by robust LMS and IRLS were compared for both linearity conditions. Copyright © 2018 John Wiley & Sons, Ltd.

Linear and non-linear amplification of high-mode perturbations at the ablation front in HiPER targets

Energy Technology Data Exchange (ETDEWEB)

Olazabal-Loume, M; Breil, J; Hallo, L; Ribeyre, X [CELIA, UMR 5107 Universite Bordeaux 1-CNRS-CEA, 351 cours de la Liberation, 33405 Talence (France); Sanz, J, E-mail: olazabal@celia.u-bordeaux1.f [ETSI Aeronauticos, Universidad Politecnica de Madrid, Madrid 28040 (Spain)

2011-01-15

The linear and non-linear sensitivity of the 180 kJ baseline HiPER target to high-mode perturbations, i.e. surface roughness, is addressed using two-dimensional simulations and a complementary analysis by linear and non-linear ablative Rayleigh-Taylor models. Simulations provide an assessment of an early non-linear stage leading to a significant deformation of the ablation surface for modes of maximum linear growth factor. A design using a picket prepulse evidences an improvement in the target stability inducing a delay of the non-linear behavior. Perturbation evolution and shape, evidenced by simulations of the non-linear stage, are analyzed with existing self-consistent non-linear theory.

Non-linear aeroelastic prediction for aircraft applications

Science.gov (United States)

de C. Henshaw, M. J.; Badcock, K. J.; Vio, G. A.; Allen, C. B.; Chamberlain, J.; Kaynes, I.; Dimitriadis, G.; Cooper, J. E.; Woodgate, M. A.; Rampurawala, A. M.; Jones, D.; Fenwick, C.; Gaitonde, A. L.; Taylor, N. V.; Amor, D. S.; Eccles, T. A.; Denley, C. J.

2007-05-01

Current industrial practice for the prediction and analysis of flutter relies heavily on linear methods and this has led to overly conservative design and envelope restrictions for aircraft. Although the methods have served the industry well, it is clear that for a number of reasons the inclusion of non-linearity in the mathematical and computational aeroelastic prediction tools is highly desirable. The increase in available and affordable computational resources, together with major advances in algorithms, mean that non-linear aeroelastic tools are now viable within the aircraft design and qualification environment. The Partnership for Unsteady Methods in Aerodynamics (PUMA) Defence and Aerospace Research Partnership (DARP) was sponsored in 2002 to conduct research into non-linear aeroelastic prediction methods and an academic, industry, and government consortium collaborated to address the following objectives: To develop useable methodologies to model and predict non-linear aeroelastic behaviour of complete aircraft. To evaluate the methodologies on real aircraft problems. To investigate the effect of non-linearities on aeroelastic behaviour and to determine which have the greatest effect on the flutter qualification process. These aims have been very effectively met during the course of the programme and the research outputs include: New methods available to industry for use in the flutter prediction process, together with the appropriate coaching of industry engineers. Interesting results in both linear and non-linear aeroelastics, with comprehensive comparison of methods and approaches for challenging problems. Additional embryonic techniques that, with further research, will further improve aeroelastics capability. This paper describes the methods that have been developed and how they are deployable within the industrial environment. We present a thorough review of the PUMA aeroelastics programme together with a comprehensive review of the relevant research

Soliton-like solutions to the ordinary Schroedinger equation

International Nuclear Information System (INIS)

Zamboni-Rached, Michel; Recami, Erasmo

2011-01-01

In recent times it has been paid attention to the fact that (linear) wave equations admit of soliton-like solutions, known as Localized Waves or Non-diffracting Waves, which propagate without distortion in one direction. Such Localized Solutions (existing also for K-G or Dirac equations) are a priori suitable, more than Gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, Localized Solutions exist even for the ordinary Schroedinger equation within standard Quantum Mechanics; and we obtain both approximate and exact solutions, also setting forth for them particular examples. In the ideal case such solutions bear infinite energy, as well as plane or spherical waves: we show therefore how to obtain nite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential. (author)

Soliton-like solutions to the ordinary Schroedinger equation

Energy Technology Data Exchange (ETDEWEB)

Zamboni-Rached, Michel [Universidade Estadual de Campinas (DMO/FEEC/UNICAMP), Campinas, SP (Brazil). Fac. de Engenharia Eletrica e de Computacao. Dept. de Microondas e Optica; Recami, Erasmo, E-mail: recami@mi.infn.i [Universita Statale di Bergamo, Bergamo (Italy). Facolta di Ingegneria

2011-07-01

In recent times it has been paid attention to the fact that (linear) wave equations admit of soliton-like solutions, known as Localized Waves or Non-diffracting Waves, which propagate without distortion in one direction. Such Localized Solutions (existing also for K-G or Dirac equations) are a priori suitable, more than Gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, Localized Solutions exist even for the ordinary Schroedinger equation within standard Quantum Mechanics; and we obtain both approximate and exact solutions, also setting forth for them particular examples. In the ideal case such solutions bear infinite energy, as well as plane or spherical waves: we show therefore how to obtain nite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential. (author)

Sobolev gradients and differential equations

CERN Document Server

Neuberger, J W

2010-01-01

A Sobolev gradient of a real-valued functional on a Hilbert space is a gradient of that functional taken relative to an underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. For discrete versions of partial differential equations, corresponding Sobolev gradients are seen to be vastly more efficient than ordinary gradients. In fact, descent methods with these gradients generally scale linearly with the number of grid points, in sharp contrast with the use of ordinary gradients. Aside from the first edition of this work, this is the only known account of Sobolev gradients in book form. Most of the applications in this book have emerged since the first edition was published some twelve years ago. What remains of the first edition has been extensively revised. There are a number of plots of results from calculations and a sample MatLab code is included for a simple problem. Those working through a fair p...

Mathematical modelling of tissue formation on the basis of ordinary differential equations

Directory of Open Access Journals (Sweden)

Maxim N. Nazarov

2017-10-01

Full Text Available A mathematical model is proposed for describing the population dynamics of cellular clusters on the basis of systems of the first-order ordinary differential equations. The main requirement for the construction of model equations was to obtain a formal biological justification for their derivation, as well as proof of their correctness. In addition, for all the parameters involved in the model equations, the presence of biological meaning was guaranteed, as well as the possibility of evaluating them either during the experiment or by using models of intracellular biochemistry. In the desired model the intercellular exchange of a special signal molecules was chosen as the main mechanism for coordination of the tissue growth and new types selection during cell division. For simplicity, all signalling molecules that can create cells of the same type were not considered separately in the model, but were instead combined in a single complex of molecules: a ‘generalized signal’. Such an approach allows us to eventually assign signals as a functions of cell types and introduce their effects in the form of matrices in the models, where the rows are responsible for the types of cells receiving the signals, and the columns for the types of cells emitting signals.

Non-linear finite element analysis in structural mechanics

CERN Document Server

Rust, Wilhelm

2015-01-01

This monograph describes the numerical analysis of non-linearities in structural mechanics, i.e. large rotations, large strain (geometric non-linearities), non-linear material behaviour, in particular elasto-plasticity as well as time-dependent behaviour, and contact. Based on that, the book treats stability problems and limit-load analyses, as well as non-linear equations of a large number of variables. Moreover, the author presents a wide range of problem sets and their solutions. The target audience primarily comprises advanced undergraduate and graduate students of mechanical and civil engineering, but the book may also be beneficial for practising engineers in industry.

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

International Nuclear Information System (INIS)

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

Stability of numerical method for semi-linear stochastic pantograph differential equations

Directory of Open Access Journals (Sweden)

Yu Zhang

2016-01-01

Full Text Available Abstract As a particular expression of stochastic delay differential equations, stochastic pantograph differential equations have been widely used in nonlinear dynamics, quantum mechanics, and electrodynamics. In this paper, we mainly study the stability of analytical solutions and numerical solutions of semi-linear stochastic pantograph differential equations. Some suitable conditions for the mean-square stability of an analytical solution are obtained. Then we proved the general mean-square stability of the exponential Euler method for a numerical solution of semi-linear stochastic pantograph differential equations, that is, if an analytical solution is stable, then the exponential Euler method applied to the system is mean-square stable for arbitrary step-size h > 0 $h>0$ . Numerical examples further illustrate the obtained theoretical results.

Vapor-deposited non-crystalline phase vs ordinary glasses and supercooled liquids: Subtle thermodynamic and kinetic differences

International Nuclear Information System (INIS)

Bhattacharya, Deepanjan; Sadtchenko, Vlad

2015-01-01

Vapor deposition of molecules on a substrate often results in glassy materials of high kinetic stability and low enthalpy. The extraordinary properties of such glasses are attributed to high rates of surface diffusion during sample deposition, which makes it possible for constituents to find a configuration of much lower energy on a typical laboratory time scale. However, the exact nature of the resulting phase and the mechanism of its formation are not completely understood. Using fast scanning calorimetry technique, we show that out-of-equilibrium relaxation kinetics and possibly the enthalpy of vapor-deposited films of toluene and ethylbenzene, archetypical fragile glass formers, are distinct from those of ordinary supercooled phase even when the deposition takes place at temperatures above the ordinary glass softening transition temperatures. These observations along with the absolute enthalpy dependences on deposition temperatures support the conjecture that the vapor-deposition may result in formation of non-crystalline phase of unique structural, thermodynamic, and kinetic properties

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

Science.gov (United States)

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.

Analysis of the linear induction motor in transient operation

Energy Technology Data Exchange (ETDEWEB)

Gentile, G; Rotondale, N; Scarano, M

1987-05-01

The paper deals with the analysis of a bilateral linear induction motor in transient operation. We have considered an impressed voltage one-dimensional model which takes into account end effects. The real winding distribution of the armature has been represented as a lumped parameters system. By using the space vectors methodology, the partial differential equation of the sheet is solved bythe variable separation method. Therefore it's possible to arrange a system of ordinary differential equations where the unknown quantities are the space vectors of the air-gap flux density and sheet currents. Finally, we have analyzed the characteristic quantities for a no-load starting of small power motors.

Analytical approach for the Floquet theory of delay differential equations.

Science.gov (United States)

Simmendinger, C; Wunderlin, A; Pelster, A

1999-05-01

We present an analytical approach to deal with nonlinear delay differential equations close to instabilities of time periodic reference states. To this end we start with approximately determining such reference states by extending the Poincaré-Lindstedt and the Shohat expansions, which were originally developed for ordinary differential equations. Then we systematically elaborate a linear stability analysis around a time periodic reference state. This allows us to approximately calculate the Floquet eigenvalues and their corresponding eigensolutions by using matrix valued continued fractions.

Hamiltonian structures of some non-linear evolution equations

International Nuclear Information System (INIS)

Tu, G.Z.

1983-06-01

The Hamiltonian structure of the O(2,1) non-linear sigma model, generalized AKNS equations, are discussed. By reducing the O(2,1) non-linear sigma model to its Hamiltonian form some new conservation laws are derived. A new hierarchy of non-linear evolution equations is proposed and shown to be generalized Hamiltonian equations with an infinite number of conservation laws. (author)

The non-linear ion trap. Part 5. Nature of non-linear resonances and resonant ion ejection

Science.gov (United States)

Franzen, J.

1994-01-01

The superposition of higher order multipole fields on the basic quadrupole field in ion traps generates a non-harmonic oscillator system for the ions. Fourier analyses of simulated secular oscillations in non-linear ion traps, therefore, not only reveal the sideband frequencies, well-known from the Mathieu theory, but additionally a commonwealth of multipole-specific overtones (or higher harmonics), and corresponding sidebands of overtones. Non-linear resonances occur when the overtone frequencies match sideband frequencies. It can be shown that in each of the resonance conditions, not just one overtone matches one sideband, instead, groups of overtones match groups of sidebands. The generation of overtones is studied by Fourier analysis of computed ion oscillations in the direction of thez axis. Even multipoles (octopole, dodecapole, etc.) generate only odd orders of higher harmonics (3, 5, etc.) of the secular frequency, explainable by the symmetry with regard to the planez = 0. In contrast, odd multipoles (hexapole, decapole, etc.) generate all orders of higher harmonics. For all multipoles, the lowest higher harmonics are found to be strongest. With multipoles of higher orders, the strength of the overtones decreases weaker with the order of the harmonics. Forz direction resonances in stationary trapping fields, the function governing the amplitude growth is investigated by computer simulations. The ejection in thez direction, as a function of timet, follows, at least in good approximation, the equation wheren is the order of multipole, andC is a constant. This equation is strictly valid for the electrically applied dipole field (n = 1), matching the secular frequency or one of its sidebands, resulting in a linear increase of the amplitude. It is valid also for the basic quadrupole field (n = 2) outside the stability area, giving an exponential increase. It is at least approximately valid for the non-linear resonances by weak superpositions of all higher odd

A non-linear kinematic hardening function

International Nuclear Information System (INIS)

Ottosen, N.S.

1977-05-01

Based on the classical theory of plasticity, and accepting the von Mises criterion as the initial yield criterion, a non-linear kinematic hardening function applicable both to Melan-Prager's and to Ziegler's hardening rule is proposed. This non-linear hardening function is determined by means of the uniaxial stress-strain curve, and any such curve is applicable. The proposed hardening function considers the problem of general reversed loading, and a smooth change in the behaviour from one plastic state to another nearlying plastic state is obtained. A review of both the kinematic hardening theory and the corresponding non-linear hardening assumptions is given, and it is shown that material behaviour is identical whether Melan-Prager's or Ziegler's hardening rule is applied, provided that the von Mises yield criterion is adopted. (author)

Dual exponential polynomials and linear differential equations

Science.gov (United States)

Wen, Zhi-Tao; Gundersen, Gary G.; Heittokangas, Janne

2018-01-01

We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.

Investigation of heat and mass transfer of rotating MHD viscous flow between a stretching sheet and a porous surface

DEFF Research Database (Denmark)

Sheikholeslami, R; Ashorynejad, H.R; Barari, Amin

2013-01-01

the viscous dissipation is negligible. Originality/value – The equations of conservation of mass, momentum and energy are reduced to a non-linear ordinary differential equations system. Differential Transformation Method is utilized to approximate the solution for velocity and temperature profiles....

Asymptotic behaviour of a non-commutative rational series with a nonnegative linear representation

Directory of Open Access Journals (Sweden)

Philippe Dumas

2007-01-01

Full Text Available We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which originates from differential cryptanalysis, using tools from probability theory, and from analytic number theory. We derive a Fourier representation of a first-order summation function obtained by interpreting this rational series as a non-classical rational sequence via the octal numeration system. The method is applicable to a wide class of sequences rational with respect to a numeration system essentially under the condition that they admit a linear representation with nonnegative coefficients.

Non-linear dynamics of wind turbine wings

DEFF Research Database (Denmark)

Larsen, Jesper Winther; Nielsen, Søren R.K.

2006-01-01

The paper deals with the formulation of non-linear vibrations of a wind turbine wing described in a wing fixed moving coordinate system. The considered structural model is a Bernoulli-Euler beam with due consideration to axial twist. The theory includes geometrical non-linearities induced...

Non-linear seismic analysis of structures coupled with fluid

International Nuclear Information System (INIS)

Descleve, P.; Derom, P.; Dubois, J.

1983-01-01

This paper presents a method to calculate non-linear structure behaviour under horizontal and vertical seismic excitation, making possible the full non-linear seismic analysis of a reactor vessel. A pseudo forces method is used to introduce non linear effects and the problem is solved by superposition. Two steps are used in the method: - Linear calculation of the complete model. - Non linear analysis of thin shell elements and calculation of seismic induced pressure originating from linear and non linear effects, including permanent loads and thermal stresses. Basic aspects of the mathematical formulation are developed. It has been applied to axi-symmetric shell element using a Fourier series solution. For the fluid interaction effect, a comparison is made with a dynamic test. In an example of application, the displacement and pressure time history are given. (orig./GL)

On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order.

Science.gov (United States)

Tunç, Cemil; Tunç, Osman

2016-01-01

In this paper, certain system of linear homogeneous differential equations of second-order is considered. By using integral inequalities, some new criteria for bounded and [Formula: see text]-solutions, upper bounds for values of improper integrals of the solutions and their derivatives are established to the considered system. The obtained results in this paper are considered as extension to the results obtained by Kroopnick (2014) [1]. An example is given to illustrate the obtained results.

Non-Linear Rheological Properties and Neutron Scattering Investigation on Dilute Ring-Linear Blends

DEFF Research Database (Denmark)

Pyckhout-Hintzen, W.; Bras, A.R.; Wischnewski, A.

in a filament stretching rheometer, followed by quenching, strong anisotropic scattering patterns were obtained which were described by affinely deformed rings which function as giant, polymeric chemical crosslinks or sliplinks and more or less isotropic topological contributions from the entangling...... with interpenetrating linear chains. At the same time the non-linear rheological and mechanical data fit to a non-affine slip-tube model as for moderately crosslinked networks and to interchain pressure models or a modified non-linear Doi-Edwards description for the observed strain hardening during the extensional...

An extension of PPLS-DA for classification and comparison to ordinary PLS-DA.

Directory of Open Access Journals (Sweden)

Anna Telaar

Full Text Available Classification studies are widely applied, e.g. in biomedical research to classify objects/patients into predefined groups. The goal is to find a classification function/rule which assigns each object/patient to a unique group with the greatest possible accuracy (classification error. Especially in gene expression experiments often a lot of variables (genes are measured for only few objects/patients. A suitable approach is the well-known method PLS-DA, which searches for a transformation to a lower dimensional space. Resulting new components are linear combinations of the original variables. An advancement of PLS-DA leads to PPLS-DA, introducing a so called 'power parameter', which is maximized towards the correlation between the components and the group-membership. We introduce an extension of PPLS-DA for optimizing this power parameter towards the final aim, namely towards a minimal classification error. We compare this new extension with the original PPLS-DA and also with the ordinary PLS-DA using simulated and experimental datasets. For the investigated data sets with weak linear dependency between features/variables, no improvement is shown for PPLS-DA and for the extensions compared to PLS-DA. A very weak linear dependency, a low proportion of differentially expressed genes for simulated data, does not lead to an improvement of PPLS-DA over PLS-DA, but our extension shows a lower prediction error. On the contrary, for the data set with strong between-feature collinearity and a low proportion of differentially expressed genes and a large total number of genes, the prediction error of PPLS-DA and the extensions is clearly lower than for PLS-DA. Moreover we compare these prediction results with results of support vector machines with linear kernel and linear discriminant analysis.

The Importance of Non-Linearity on Turbulent Fluxes

DEFF Research Database (Denmark)

Rokni, Masoud

2007-01-01

Two new non-linear models for the turbulent heat fluxes are derived and developed from the transport equation of the scalar passive flux. These models are called as non-linear eddy diffusivity and non-linear scalar flux. The structure of these models is compared with the exact solution which...... is derived from the Cayley-Hamilton theorem and contains a three term-basis plus a non-linear term due to scalar fluxes. In order to study the performance of the model itself, all other turbulent quantities are taken from a DNS channel flow data-base and thus the error source has been minimized. The results...... are compared with the DNS channel flow and good agreement is achieved. It has been shown that the non-linearity parts of the models are important to capture the true path of the streamwise scalar fluxes. It has also been shown that one of model constant should have negative sign rather than positive, which had...

Entropy analysis of convective MHD flow of third grade non-Newtonian fluid over a stretching sheet

Directory of Open Access Journals (Sweden)

M.M. Rashidi

2017-03-01

Full Text Available The purpose of this article is to study and analyze the convective flow of a third grade non-Newtonian fluid due to a linearly stretching sheet subject to a magnetic field. The dimensionless entropy generation equation is obtained by solving the reduced momentum and energy equations. The momentum and energy equations are reduced to a system of ordinary differential equations by a similarity method. The optimal homotopy analysis method (OHAM is used to solve the resulting system of ordinary differential equations. The effects of the magnetic field, Biot number and Prandtl number on the velocity component and temperature are studied. The results show that the thermal boundary-layer thickness gets decreased with increasing the Prandtl number. In addition, Brownian motion plays an important role to improve thermal conductivity of the fluid. The main purpose of the paper is to study the effects of Reynolds number, dimensionless temperature difference, Brinkman number, Hartmann number and other physical parameters on the entropy generation. These results are analyzed and discussed.

On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

Directory of Open Access Journals (Sweden)

Tian-Xiao He

2009-01-01

Full Text Available Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

On a representation of linear differential equations

Czech Academy of Sciences Publication Activity Database

Neuman, František

2010-01-01

Roč. 52, 1-2 (2010), s. 355-360 ISSN 0895-7177 Grant - others:GA ČR(CZ) GA201/08/0469 Institutional research plan: CEZ:AV0Z10190503 Keywords : Brandt and Ehresmann groupoinds * transformations * canonical forms * linear differential equations Subject RIV: BA - General Mathematics Impact factor: 1.066, year: 2010 http://www.sciencedirect.com/science/article/pii/S0895717710001184

Surveys in differential-algebraic equations III

CERN Document Server

Reis, Timo

2015-01-01

The present volume comprises survey articles on various fields of Differential-Algebraic Equations (DAEs), which have widespread applications in controlled dynamical systems, especially in mechanical and electrical engineering and a strong relation to (ordinary) differential equations. The individual chapters provide reviews, presentations of the current state of research and new concepts in - Flexibility of DAE formulations - Reachability analysis and deterministic global optimization - Numerical linear algebra methods - Boundary value problems The results are presented in an accessible style, making this book suitable not only for active researchers but also for graduate students (with a good knowledge of the basic principles of DAEs) for self-study.

Applicability of linear and non-linear potential flow models on a Wavestar float

DEFF Research Database (Denmark)

Bozonnet, Pauline; Dupin, Victor; Tona, Paolino

2017-01-01

as a model based on non-linear potential flow theory and weakscatterer hypothesis are successively considered. Simple tests, such as dip tests, decay tests and captive tests enable to highlight the improvements obtained with the introduction of nonlinearities. Float motion under wave actions and without...... control action, limited to small amplitude motion with a single float, is well predicted by the numerical models, including the linear one. Still, float velocity is better predicted by accounting for non-linear hydrostatic and Froude-Krylov forces.......Numerical models based on potential flow theory, including different types of nonlinearities are compared and validated against experimental data for the Wavestar wave energy converter technology. Exact resolution of the rotational motion, non-linear hydrostatic and Froude-Krylov forces as well...

On the correspondence between the dynamics with odd and even brackets and generalized Nambu's mechanics

International Nuclear Information System (INIS)

Makhaldiani, N.; Voskresenskaya, O.

1998-01-01

We consider selected problems of Hamiltonization and integrability of a general dynamical system described by a system of ordinary differential equations; odd and even symplectic structures; a new approach to the linearization of non-linear systems by the extended Hamiltonian and Nambu's mechanics

On the Liouvillian solution of second-order linear differential equations and algebraic invariant curves

International Nuclear Information System (INIS)

Man, Yiu-Kwong

2010-01-01

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. (fast track communication)

The mixed BVP for second order nonlinear ordinary differential equation at resonance

Czech Academy of Sciences Publication Activity Database

Mukhigulashvili, Sulkhan

2017-01-01

Roč. 290, 2-3 (2017), s. 393-400 ISSN 0025-584X Institutional support: RVO:67985840 Keywords : mixed problem at resonance * nonlinear ordinary differencial equation Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.742, year: 2016

ADM For Solving Linear Second-Order Fredholm Integro-Differential Equations

Science.gov (United States)

Karim, Mohd F.; Mohamad, Mahathir; Saifullah Rusiman, Mohd; Che-Him, Norziha; Roslan, Rozaini; Khalid, Kamil

2018-04-01

In this paper, we apply Adomian Decomposition Method (ADM) as numerically analyse linear second-order Fredholm Integro-differential Equations. The approximate solutions of the problems are calculated by Maple package. Some numerical examples have been considered to illustrate the ADM for solving this equation. The results are compared with the existing exact solution. Thus, the Adomian decomposition method can be the best alternative method for solving linear second-order Fredholm Integro-Differential equation. It converges to the exact solution quickly and in the same time reduces computational work for solving the equation. The result obtained by ADM shows the ability and efficiency for solving these equations.

Non linear structures seismic analysis by modal synthesis

International Nuclear Information System (INIS)

Aita, S.; Brochard, D.; Guilbaud, D.; Gibert, R.J.

1987-01-01

The structures submitted to a seismic excitation, may present a great amplitude response which induces a non linear behaviour. These non linearities have an important influence on the response of the structure. Even in this case (local shocks) the modal synthesis method remains attractive. In this paper we will present the way of taking into account, a local non linearity (shock between structures) in the seismic response of structures, by using the modal synthesis method [fr

Numerical solution of mixed convection flow of an MHD Jeffery fluid over an exponentially stretching sheet in the presence of thermal radiation and chemical reaction

Science.gov (United States)

Shateyi, Stanford; Marewo, Gerald T.

2018-05-01

We numerically investigate a mixed convection model for a magnetohydrodynamic (MHD) Jeffery fluid flowing over an exponentially stretching sheet. The influence of thermal radiation and chemical reaction is also considered in this study. The governing non-linear coupled partial differential equations are reduced to a set of coupled non-linear ordinary differential equations by using similarity functions. This new set of ordinary differential equations are solved numerically using the Spectral Quasi-Linearization Method. A parametric study of physical parameters involved in this study is carried out and displayed in tabular and graphical forms. It is observed that the velocity is enhanced with increasing values of the Deborah number, buoyancy and thermal radiation parameters. Furthermore, the temperature and species concentration are decreasing functions of the Deborah number. The skin friction coefficient increases with increasing values of the magnetic parameter and relaxation time. Heat and mass transfer rates increase with increasing values of the Deborah number and buoyancy parameters.

Passive impedance-based second-order sliding mode control for non-linear teleoperators

Directory of Open Access Journals (Sweden)

Luis G García-Valdovinos

2017-02-01

Full Text Available Bilateral teleoperation systems have attracted significant attention in the last decade mainly because of technological advancements in both the communication channel and computers performance. In addition, non-linear multi-degree-of-freedom bilateral teleoperators along with state observers have become an open research area. In this article, a model-free exact differentiator is used to estimate the full state along with a chattering-free second-order sliding mode controller to guarantee a robust impedance tracking under both constant and an unknown time delay of non-linear multi-degree-of-freedom robots. The robustness of the proposed controller is improved by introducing a change of coordinates in terms of a new nominal reference similar to that used in adaptive control theory. Experimental results that validate the predicted behaviour are presented and discussed using a Phantom Premium 1.0 as the master robot and a Catalyst-5 virtual model as the slave robot. The dynamics of the Catalyst-5 system is solved online.

Construction of local and non-local conservation laws for non-linear field equations

International Nuclear Information System (INIS)

Vladimirov, V.S.; Volovich, I.V.

1984-08-01

A method of constructing conserved currents for non-linear field equations is presented. More explicitly for non-linear equations, which can be derived from compatibility conditions of some linear system with a parameter, a procedure of obtaining explicit expressions for local and non-local currents is developed. Some examples such as the classical Heisenberg spin chain and supersymmetric Yang-Mills theory are considered. (author)

Linear and non-linear autoregressive models for short-term wind speed forecasting

International Nuclear Information System (INIS)

Lydia, M.; Suresh Kumar, S.; Immanuel Selvakumar, A.; Edwin Prem Kumar, G.

2016-01-01

Highlights: • Models for wind speed prediction at 10-min intervals up to 1 h built on time-series wind speed data. • Four different multivariate models for wind speed built based on exogenous variables. • Non-linear models built using three data mining algorithms outperform the linear models. • Autoregressive models based on wind direction perform better than other models. - Abstract: Wind speed forecasting aids in estimating the energy produced from wind farms. The soaring energy demands of the world and minimal availability of conventional energy sources have significantly increased the role of non-conventional sources of energy like solar, wind, etc. Development of models for wind speed forecasting with higher reliability and greater accuracy is the need of the hour. In this paper, models for predicting wind speed at 10-min intervals up to 1 h have been built based on linear and non-linear autoregressive moving average models with and without external variables. The autoregressive moving average models based on wind direction and annual trends have been built using data obtained from Sotavento Galicia Plc. and autoregressive moving average models based on wind direction, wind shear and temperature have been built on data obtained from Centre for Wind Energy Technology, Chennai, India. While the parameters of the linear models are obtained using the Gauss–Newton algorithm, the non-linear autoregressive models are developed using three different data mining algorithms. The accuracy of the models has been measured using three performance metrics namely, the Mean Absolute Error, Root Mean Squared Error and Mean Absolute Percentage Error.

Estimating the Probabilities of Low-Weight Differential and Linear Approximations on PRESENT-like Ciphers

DEFF Research Database (Denmark)

Abdelraheem, Mohamed Ahmed

2012-01-01

We use large but sparse correlation and transition-difference-probability submatrices to find the best linear and differential approximations respectively on PRESENT-like ciphers. This outperforms the branch and bound algorithm when the number of low-weight differential and linear characteristics...

Non-linear numerical studies of the tearing mode

International Nuclear Information System (INIS)

Schnack, D.D. Jr.

1978-01-01

A non-linear, time dependent, hydromagnetic model is developed and applied to the tearing mode, one of a class of instabilities which can occur in a magnetically confined plasma when the constraint of infinite conductivity is relaxed. The model is based on the eight partial differential equations of resistive magnetohydrodynamics (MHD). The equations are expressed as a set of conservation laws which conserves magnetic flux, momentum, mass, and total energy. These equations are then written in general, orthogonal, curvilinear coordinates in two space dimensions, so that the model can readily be applied to a variety of geometries. No assumption about the ordering of terms is made. The resulting equations are then solved by the method of finite differences on an Eulerian mesh. The model is applied to several geometries

Noise and non-linearities in high-throughput data

International Nuclear Information System (INIS)

Nguyen, Viet-Anh; Lió, Pietro; Koukolíková-Nicola, Zdena; Bagnoli, Franco

2009-01-01

High-throughput data analyses are becoming common in biology, communications, economics and sociology. The vast amounts of data are usually represented in the form of matrices and can be considered as knowledge networks. Spectra-based approaches have proved useful in extracting hidden information within such networks and for estimating missing data, but these methods are based essentially on linear assumptions. The physical models of matching, when applicable, often suggest non-linear mechanisms, that may sometimes be identified as noise. The use of non-linear models in data analysis, however, may require the introduction of many parameters, which lowers the statistical weight of the model. According to the quality of data, a simpler linear analysis may be more convenient than more complex approaches. In this paper, we show how a simple non-parametric Bayesian model may be used to explore the role of non-linearities and noise in synthetic and experimental data sets

A differential-geometric approach to generalized linear models with grouped predictors

NARCIS (Netherlands)

Augugliaro, Luigi; Mineo, Angelo M.; Wit, Ernst C.

We propose an extension of the differential-geometric least angle regression method to perform sparse group inference in a generalized linear model. An efficient algorithm is proposed to compute the solution curve. The proposed group differential-geometric least angle regression method has important

Return-Volatility Relationship: Insights from Linear and Non-Linear Quantile Regression

NARCIS (Netherlands)

D.E. Allen (David); A.K. Singh (Abhay); R.J. Powell (Robert); M.J. McAleer (Michael); J. Taylor (James); L. Thomas (Lyn)

2013-01-01

textabstractThe purpose of this paper is to examine the asymmetric relationship between price and implied volatility and the associated extreme quantile dependence using linear and non linear quantile regression approach. Our goal in this paper is to demonstrate that the relationship between the

Seismic analysis of equipment system with non-linearities such as gap and friction using equivalent linearization method

International Nuclear Information System (INIS)

Murakami, H.; Hirai, T.; Nakata, M.; Kobori, T.; Mizukoshi, K.; Takenaka, Y.; Miyagawa, N.

1989-01-01

Many of the equipment systems of nuclear power plants contain a number of non-linearities, such as gap and friction, due to their mechanical functions. It is desirable to take such non-linearities into account appropriately for the evaluation of the aseismic soundness. However, in usual design works, linear analysis method with rough assumptions is applied from engineering point of view. An equivalent linearization method is considered to be one of the effective analytical techniques to evaluate non-linear responses, provided that errors to a certain extent are tolerated, because it has greater simplicity in analysis and economization in computing time than non-linear analysis. The objective of this paper is to investigate the applicability of the equivalent linearization method to evaluate the maximum earthquake response of equipment systems such as the CANDU Fuelling Machine which has multiple non- linearities

Some mathematical problems in non-linear Physics; Algunos problemas matematicos en fisica no-lineal

Energy Technology Data Exchange (ETDEWEB)

NONE

1983-07-01

The main results contained in this report are the following: I) A general analysis of non-autonomous conserved densities for simple linear evolution systems. II) Partial differential systems within a wide class are converted into Lagrange an form. III) Rigorous criteria for existence of integrating factor matrices. IV) Isolation of all third-order evolution equations with high order symmetries and conservation laws. (Author) 3 refs.

Non-linear dynamics in Parkinsonism

Directory of Open Access Journals (Sweden)

Olivier eDarbin

2013-12-01

Full Text Available Over the last 30 years, the functions (and dysfunctions of the sensory-motor circuitry have been mostly conceptualized using linear modelizations which have resulted in two main models: the "rate hypothesis" and the "oscillatory hypothesis". In these two models, the basal ganglia data stream is envisaged as a random temporal combination of independent simple patterns issued from its probability distribution of interval interspikes or its spectrum of frequencies respectively.More recently, non-linear analyses have been introduced in the modelization of motor circuitry activities, and they have provided evidences that complex temporal organizations exist in basal ganglia neuronal activities. Regarding movement disorders, these complex temporal organizations in the basal ganglia data stream differ between conditions (i.e. parkinsonism, dyskinesia, healthy control and are responsive to treatments (i.e. L-DOPA,DBS. A body of evidence has reported that basal ganglia neuronal entropy (a marker for complexity/irregularity in time series is higher in hypokinetic state. In line with these findings, an entropy-based model has been recently formulated to introduce basal ganglia entropy as a marker for the alteration of motor processing and a factor of motor inhibition. Importantly, non-linear features have also been identified as a marker of condition and/or treatment effects in brain global signals (EEG, muscular activities (EMG or kinetic of motor symptoms (tremor, gait of patients with movement disorders. It is therefore warranted that the non-linear dynamics of motor circuitry will contribute to a better understanding of the neuronal dysfunctions underlying the spectrum of parkinsonian motor symptoms including tremor, rigidity and hypokinesia.

Trends in differential equations and applications

CERN Document Server

Neble, María; Galván, José

2016-01-01

This work collects the most important results presented at the Congress on Differential Equations and Applications/Congress on Applied Mathematics (CEDYA/CMA) in Cádiz (Spain) in 2015. It supports further research in differential equations, numerical analysis, mechanics, control and optimization. In particular, it helps readers gain an overview of specific problems of interest in the current mathematical research related to different branches of applied mathematics. This includes the analysis of nonlinear partial differential equations, exact solutions techniques for ordinary differential equations, numerical analysis and numerical simulation of some models arising in experimental sciences and engineering, control and optimization, and also trending topics on numerical linear Algebra, dynamical systems, and applied mathematics for Industry. This volume is mainly addressed to any researcher interested in the applications of mathematics, especially in any subject mentioned above. It may be also useful to PhD s...

Vibration suppression in ultrasonic machining described by non-linear differential equations

International Nuclear Information System (INIS)

Kamel, M. M.; El-Ganaini, W. A. A.; Hamed, Y. S.

2009-01-01

Vibrations are usually undesired phenomena as they may cause damage or destruction of the system. However, sometimes they are desirable, as in ultrasonic machining (USM). In such case, the problem is a complicated one, as it is required to reduce the vibration of the machine head and have reasonable amplitude for the tool. In the present work, the coupling of two non-linear oscillators of the tool holder and tool representing ultrasonic cutting process is investigated. This leads to a two-degree-of-freedom system subjected to multi-external excitation force. The aim of this work is to control the tool holder behavior at simultaneous primary and internal resonance condition and have high amplitude for the tool. Multiple scale perturbation method is applied to obtain a solution up to the second order approximations. Other different resonance cases are reported and studied numerically. The stability of the system is investigated applying both phase-plane and frequency response techniques. The effects of the different parameters of the tool on the system behavior are studied numerically. Comparison with the available published work is reported

LSODKR, Stiff Ordinary Differential Equations (ODE) System Solver with Krylov Iteration and Root-finding

International Nuclear Information System (INIS)

Hindmarsh, A.D.; Brown, P.N.

1996-01-01

1 - Description of program or function: LSODKR is a new initial value ODE solver for stiff and non-stiff systems. It is a variant of the LSODPK and LSODE solvers, intended mainly for large stiff systems. The main differences between LSODKR and LSODE are the following: a) for stiff systems, LSODKR uses a corrector iteration composed of Newton iteration and one of four preconditioned Krylov subspace iteration methods. The user must supply routines for the preconditioning operations, b) within the corrector iteration, LSODKR does automatic switching between functional (fix point) iteration and modified Newton iteration, c) LSODKR includes the ability to find roots of given functions of the solution during the integration. 2 - Method of solution: Integration is by Adams or BDF (Backward Differentiation Formula) methods, at user option. Corrector iteration is by Newton or fix point iteration, determined dynamically. Linear system solution is by a preconditioned Krylov iteration, selected by user from Incomplete Orthogonalization Method, Generalized Minimum Residual Method, and two variants of Preconditioned Conjugate Gradient Method. Preconditioning is to be supplied by the user. 3 - Restrictions on the complexity of the problem: None

Oscillatory behaviour of solutions of linear neutral differential ...

African Journals Online (AJOL)

The paper considers the contribution of space-time noise to the oscillatory behaviour of solutions of a linear neutral stochastic delay differential equation. It was established that under certain conditions on the time lags and their speed of adjustments, the presence of noise generates oscillation in the solution of the equation ...

The importance of non-linearities in modern proton synchrotrons

International Nuclear Information System (INIS)

Wilson, E.J.N.

1977-01-01

The paper outlines the physics and mathematics of non-linear field errors in the quide fields of accelerators, with particular reference to large accelerators such as the SPS. These non-linearities give rise to closed orbital distortions and non-linear resonances or stopbands. Both of these effects are briefly discussed and the use of resonances for slow beam extraction is also described. Another problem considered is that of chromaticity of the particle beam. The use of sextupoles to correct chromaticity and the Landau damping of beam instabilities using octupoles are also discussed. In the final section the application of Hamiltonian mechanics to non-linearities is demonstrated. The author concludes that the effect of non-linearities on particle dynamics may place a more severe limit on intensity and storage time in large rings than any other effect in transverse phase space. (B.D.)

Numerical methods for differential equations and applications

International Nuclear Information System (INIS)

Ixaru, L.G.

1984-01-01

This book is addressed to persons who, without being professionals in applied mathematics, are often faced with the problem of numerically solving differential equations. In each of the first three chapters a definite class of methods is discussed for the solution of the initial value problem for ordinary differential equations: multistep methods; one-step methods; and piecewise perturbation methods. The fourth chapter is mainly focussed on the boundary value problems for linear second-order equations, with a section devoted to the Schroedinger equation. In the fifth chapter the eigenvalue problem for the radial Schroedinger equation is solved in several ways, with computer programs included. (Auth.)

An implicit meshless scheme for the solution of transient non-linear Poisson-type equations

KAUST Repository

Bourantas, Georgios

2013-07-01

A meshfree point collocation method is used for the numerical simulation of both transient and steady state non-linear Poisson-type partial differential equations. Particular emphasis is placed on the application of the linearization method with special attention to the lagging of coefficients method and the Newton linearization method. The localized form of the Moving Least Squares (MLS) approximation is employed for the construction of the shape functions, in conjunction with the general framework of the point collocation method. Computations are performed for regular nodal distributions, stressing the positivity conditions that make the resulting system stable and convergent. The accuracy and the stability of the proposed scheme are demonstrated through representative and well-established benchmark problems. © 2013 Elsevier Ltd.

An implicit meshless scheme for the solution of transient non-linear Poisson-type equations

KAUST Repository

Bourantas, Georgios; Burganos, Vasilis N.

2013-01-01

A meshfree point collocation method is used for the numerical simulation of both transient and steady state non-linear Poisson-type partial differential equations. Particular emphasis is placed on the application of the linearization method with special attention to the lagging of coefficients method and the Newton linearization method. The localized form of the Moving Least Squares (MLS) approximation is employed for the construction of the shape functions, in conjunction with the general framework of the point collocation method. Computations are performed for regular nodal distributions, stressing the positivity conditions that make the resulting system stable and convergent. The accuracy and the stability of the proposed scheme are demonstrated through representative and well-established benchmark problems. © 2013 Elsevier Ltd.

Multi-point boundary value problems for linear functional-differential equations

Czech Academy of Sciences Publication Activity Database

Domoshnitsky, A.; Hakl, Robert; Půža, Bedřich

2017-01-01

Roč. 24, č. 2 (2017), s. 193-206 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : boundary value problems * linear functional-differential equations * functional-differential inequalities Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2017.24.issue-2/gmj-2016-0076/gmj-2016-0076. xml

Multi-point boundary value problems for linear functional-differential equations

Czech Academy of Sciences Publication Activity Database

Domoshnitsky, A.; Hakl, Robert; Půža, Bedřich

2017-01-01

Roč. 24, č. 2 (2017), s. 193-206 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : boundary value problems * linear functional- differential equations * functional- differential inequalities Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2017.24.issue-2/gmj-2016-0076/gmj-2016-0076.xml

Non-linear wave packet dynamics of coherent states

Indian Academy of Sciences (India)

In recent years, the non-linear quantum dynamics of these states have revealed some striking features. It was found that under the action of a Hamil- tonian which is a non-linear function of the photon operator(s) only, an initial coherent state loses its coherent structure quickly due to quantum dephasing induced by the non-.

A canonical eight-dimensional formalism for linear and non-linear classical spin-orbit motion in storage rings

International Nuclear Information System (INIS)

Barber, D.P.; Heinemann, K.; Ripken, G.

1991-05-01

In the following report we begin to reformulate work by Derbenev on the behaviour of coupled quantized spin-orbit motion. To this end we present a classical symplectic treatment of linear and non-linear spin-orbit motion for storage rings using a fully coupled eight-dimensional formalism which generalizes earlier investigations of coupled synchro-betatron oscillations by introducing two additional canonical spin variables which behave, in a small-angle limit, like those already used in linearised spin theory. Thus in addition to the usual x-z-s couplings, both the spin to orbit and orbit to spin coupling are described canonically. Since the spin Hamiltonian can be expanded in a Taylor series in canonical variables, the formalism is convenient for use in 8-dimensional symplectic tracking calculations with the help, for example, of Lie algebra or differential algebra for the study of chaotic spin motion, for construction of spin normal forms and for the study of the effect of Stern-Gerlach forces. (orig.)

Pareto optimality in infinite horizon linear quadratic differential games

NARCIS (Netherlands)

Reddy, P.V.; Engwerda, J.C.

2013-01-01

In this article we derive conditions for the existence of Pareto optimal solutions for linear quadratic infinite horizon cooperative differential games. First, we present a necessary and sufficient characterization for Pareto optimality which translates to solving a set of constrained optimal

Non-linear thermal convection in a

Directory of Open Access Journals (Sweden)

Sachin Shaw

2016-06-01

Full Text Available Casson fluid flow has many practical applications such as food processing, metallurgy, drilling operations and bio-engineering operations. In this paper, we study Casson fluid flow through a plate with a convective boundary condition at the surface and quantify the effects of suction/injection, velocity ratio, and Soret and Dufour effects. Firstly we used a similarity transformation to change the governing equations to ordinary differential equations which were then solved numerically. The effect of the rheological parameters on the velocity, temperature, and concentration with skin friction, and heat and mass transfer are shown graphically and discussed briefly. It is observed that the velocity of the fluid at the surface decreases with increase of the velocity ratio while the nature of the flow is in opposite characteristics. The local Nusselt number decreases with increase in the velocity ratio. Skin friction at the surface is enhanced by buoyancy ratio and Casson number. Due to injection of the fluid in the system, the mass transfer rate at the surface increases while it decreases with the velocity ratio parameter.

Non-Linear Dynamics and Fundamental Interactions

CERN Document Server

Khanna, Faqir

2006-01-01

The book is directed to researchers and graduate students pursuing an advanced degree. It provides details of techniques directed towards solving problems in non-linear dynamics and chos that are, in general, not amenable to a perturbative treatment. The consideration of fundamental interactions is a prime example where non-perturbative techniques are needed. Extension of these techniques to finite temperature problems is considered. At present these ideas are primarily used in a perturbative context. However, non-perturbative techniques have been considered in some specific cases. Experts in the field on non-linear dynamics and chaos and fundamental interactions elaborate the techniques and provide a critical look at the present status and explore future directions that may be fruitful. The text of the main talks will be very useful to young graduate students who are starting their studies in these areas.

Heterotic sigma models and non-linear strings

International Nuclear Information System (INIS)

Hull, C.M.

1986-01-01

The two-dimensional supersymmetric non-linear sigma models are examined with respect to the heterotic string. The paper was presented at the workshop on :Supersymmetry and its applications', Cambridge, United Kingdom, 1985. The non-linear sigma model with Wess-Zumino-type term, the coupling of the fermionic superfields to the sigma model, super-conformal invariance, and the supersymmetric string, are all discussed. (U.K.)

Assessing ozone and nitrogen impact on net primary productivity with a Generalised non-Linear Model

International Nuclear Information System (INIS)

De Marco, Alessandra; Screpanti, Augusto; Attorre, Fabio; Proietti, Chiara; Vitale, Marcello

2013-01-01

Some studies suggest that in Europe the majority of forest growth increment can be accounted for N deposition and very little by elevated CO 2 . High ozone (O 3 ) concentrations cause reductions in carbon fixation in native plants by offsetting the effects of elevated CO 2 or N deposition. The cause-effect relationships between primary productivity (NPP) of Quercus cerris, Q. ilex and Fagus sylvatica plant species and climate and pollutants (O 3 and N deposition) in Italy have been investigated by application of Generalised Linear/non-Linear regression model (GLZ model). The GLZ model highlighted: i) cumulative O 3 concentration-based indicator (AOT40F) did not significantly affect NPP; ii) a differential action of oxidised and reduced nitrogen depositions to NPP was linked to the geographical location; iii) the species-specific variation of NPP caused by combination of pollutants and climatic variables could be a potentially important drive-factor for the plant species' shift as response to the future climate change. - Highlights: ► GLZ Models emphasized the role of combination of variables affecting NPP. ► A differential action of ox-N and red-N deposition to NPP was observed for plants. ► Different responses to climate and pollutants could affect the plant species' shift. - Ozone and nitrogen depositions have non-linear effects on primary productivity of tree species differently distributed in Italy.

Model Selection and Risk Estimation with Applications to Nonlinear Ordinary Differential Equation Systems

DEFF Research Database (Denmark)

Mikkelsen, Frederik Vissing

eective computational tools for estimating unknown structures in dynamical systems, such as gene regulatory networks, which may be used to predict downstream eects of interventions in the system. A recommended algorithm based on the computational tools is presented and thoroughly tested in various......Broadly speaking, this thesis is devoted to model selection applied to ordinary dierential equations and risk estimation under model selection. A model selection framework was developed for modelling time course data by ordinary dierential equations. The framework is accompanied by the R software...... package, episode. This package incorporates a collection of sparsity inducing penalties into two types of loss functions: a squared loss function relying on numerically solving the equations and an approximate loss function based on inverse collocation methods. The goal of this framework is to provide...

On the selection of ordinary differential equation models with application to predator-prey dynamical models.

Science.gov (United States)

Zhang, Xinyu; Cao, Jiguo; Carroll, Raymond J

2015-03-01

We consider model selection and estimation in a context where there are competing ordinary differential equation (ODE) models, and all the models are special cases of a "full" model. We propose a computationally inexpensive approach that employs statistical estimation of the full model, followed by a combination of a least squares approximation (LSA) and the adaptive Lasso. We show the resulting method, here called the LSA method, to be an (asymptotically) oracle model selection method. The finite sample performance of the proposed LSA method is investigated with Monte Carlo simulations, in which we examine the percentage of selecting true ODE models, the efficiency of the parameter estimation compared to simply using the full and true models, and coverage probabilities of the estimated confidence intervals for ODE parameters, all of which have satisfactory performances. Our method is also demonstrated by selecting the best predator-prey ODE to model a lynx and hare population dynamical system among some well-known and biologically interpretable ODE models. © 2014, The International Biometric Society.

Linear and non-linear Modified Gravity forecasts with future surveys

Science.gov (United States)

Casas, Santiago; Kunz, Martin; Martinelli, Matteo; Pettorino, Valeria

2017-12-01

Modified Gravity theories generally affect the Poisson equation and the gravitational slip in an observable way, that can be parameterized by two generic functions (η and μ) of time and space. We bin their time dependence in redshift and present forecasts on each bin for future surveys like Euclid. We consider both Galaxy Clustering and Weak Lensing surveys, showing the impact of the non-linear regime, with two different semi-analytical approximations. In addition to these future observables, we use a prior covariance matrix derived from the Planck observations of the Cosmic Microwave Background. In this work we neglect the information from the cross correlation of these observables, and treat them as independent. Our results show that η and μ in different redshift bins are significantly correlated, but including non-linear scales reduces or even eliminates the correlation, breaking the degeneracy between Modified Gravity parameters and the overall amplitude of the matter power spectrum. We further apply a Zero-phase Component Analysis and identify which combinations of the Modified Gravity parameter amplitudes, in different redshift bins, are best constrained by future surveys. We extend the analysis to two particular parameterizations of μ and η and consider, in addition to Euclid, also SKA1, SKA2, DESI: we find in this case that future surveys will be able to constrain the current values of η and μ at the 2-5% level when using only linear scales (wavevector k < 0 . 15 h/Mpc), depending on the specific time parameterization; sensitivity improves to about 1% when non-linearities are included.

An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation

KAUST Repository

Liu, Da-Yan; Tian, Yang; Boutat, Driss; Laleg-Kirati, Taous-Meriem

2015-01-01

This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann-Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.

An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation

KAUST Repository

Liu, Da-Yan

2015-04-30