Coward, Adrian V.; Papageorgiou, Demetrios T.; Smyrlis, Yiorgos S.
1994-01-01
In this paper the nonlinear stability of two-phase core-annular flow in a pipe is examined when the acting pressure gradient is modulated by time harmonic oscillations and viscosity stratification and interfacial tension is present. An exact solution of the Navier-Stokes equations is used as the background state to develop an asymptotic theory valid for thin annular layers, which leads to a novel nonlinear evolution describing the spatio-temporal evolution of the interface. The evolution equation is an extension of the equation found for constant pressure gradients and generalizes the Kuramoto-Sivashinsky equation with dispersive effects found by Papageorgiou, Maldarelli & Rumschitzki, Phys. Fluids A 2(3), 1990, pp. 340-352, to a similar system with time periodic coefficients. The distinct regimes of slow and moderate flow are considered and the corresponding evolution is derived. Certain solutions are described analytically in the neighborhood of the first bifurcation point by use of multiple scales asymptotics. Extensive numerical experiments, using dynamical systems ideas, are carried out in order to evaluate the effect of the oscillatory pressure gradient on the solutions in the presence of a constant pressure gradient.
Invariant Subspace Method and Fractional Modified Kuramoto-Sivashinsky Equation
Ouhadan, A.; Kinani, E. H. El
2015-01-01
In this paper, the invariant subspace method is applied to the time fractional modified Kuramoto-Sivashinsky partial differential equation. The obtained reduced system of nonlinear ordinary fractional equations is solved by the Laplace transform method and with using of some useful properties of Mittag-Leffler function. Then, some exact solutions of the time fractional nonlinear studied equation are found.
Controlling roughening processes in the stochastic Kuramoto-Sivashinsky equation
Kalliadasis, Serafim; Gomes, Susana; Papageorgiou, Demetrios; Pavliotis, Greg; Pradas, Marc
2017-11-01
We present a novel methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic Kuramoto-Sivashinsky equation. The original equation is split into a linear stochastic and a nonlinear deterministic equation so that we can apply linear feedback control methods. Our control strategy is then based on two steps: first, stabilize the zero solution of the deterministic part and, second, control the roughness of the stochastic linear equation. We consider both periodic controls and point actuated ones, observing in all cases that the second moment of the solution evolves in time according to a power-law until it saturates at the desired controlled value. Furthermore, our control framework allows us to force the interfaces to have a prescribed shape. We observe from our numerical experiments that our results are valid for different types of nonlinearity (in particular, the Burgers and KPZ ones) as well as white and coloured noise.
Travelling wave solutions to the Kuramoto-Sivashinsky equation
International Nuclear Information System (INIS)
Nickel, J.
2007-01-01
Combining the approaches given by Baldwin [Baldwin D et al. Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs. J Symbol Comput 2004;37:669-705], Peng [Peng YZ. A polynomial expansion method and new general solitary wave solutions to KS equation. Comm Theor Phys 2003;39:641-2] and by Schuermann [Schuermann HW, Serov VS. Weierstrass' solutions to certain nonlinear wave and evolution equations. Proc progress electromagnetics research symposium, 28-31 March 2004, Pisa. p. 651-4; Schuermann HW. Traveling-wave solutions to the cubic-quintic nonlinear Schroedinger equation. Phys Rev E 1996;54:4312-20] leads to a method for finding exact travelling wave solutions of nonlinear wave and evolution equations (NLWEE). The first idea is to generalize ansaetze given by Baldwin and Peng to find elliptic solutions of NLWEEs. Secondly, conditions used by Schuermann to find physical (real and bounded) solutions and to discriminate between periodic and solitary wave solutions are used. The method is shown in detail by evaluating new solutions of the Kuramoto-Sivashinsky equation
On blow-up of solutions of the Kuramoto-Sivashinsky equation
International Nuclear Information System (INIS)
Pokhozhaev, S I
2008-01-01
The problem of the absence of global solutions of initial-boundary value problems for the Kuramoto-Sivashinsky equation is considered. Sufficient conditions for the absence of global solutions of the problems under consideration are obtained both for bounded and unbounded domains. These conditions imply a priori the blow-up of the solution of the corresponding initial-boundary value problem. The proof uses a generalization of the method of non-linear capacity based on the choice of asymptotically optimal test functions. Bibliography: 20 titles.
Stability and symmetry of ion-induced surface patterning
Matthes, Christopher S. R.; Ghoniem, Nasr M.; Walgraef, Daniel
2017-12-01
We present a continuum model of ion-induced surface patterning. The model incorporates the atomic processes of sputtering, re-deposition and surface diffusion, and is shown to display the generic features of the damped Kuramoto-Sivashinsky (KS) equation of non-linear dynamics. Linear and non-linear stability analyses of the evolution equation give estimates of the emerging pattern wavelength and spatial symmetry. The analytical theory is confirmed by numerical simulations of the evolution equation with the Fast Fourier Transform method, where we show the influence of the incident ion angle, flux, and substrate surface temperature. It is shown that large local geometry variations resulting in quadratic non-linearities in the evolution equation dominate pattern selection and stability at long time scales.
The Kuramoto–Sivashinsky equation. A Local Attractor Filled with Unstable Periodic Solutions
Directory of Open Access Journals (Sweden)
Anatoli N. Kulikov
2018-01-01
Full Text Available A periodic boundary value problem is considered for one version of the KuramotoSivashinsky equation, which is widely known in mathematical physics. Local bifurcations in a neighborhood of the spatially homogeneous equilibrium points in the case when they change stability are studied. It is shown that the loss of stability of homogeneous equilibrium points leads to the appearance of a two-dimensional attractor on which all solutions are periodic functions of time, except one spatially inhomogeneous state. A spectrum of frequencies of the given family of periodic solutions fills the entire number line, and they are all unstable in a sense of Lyapunov definition in the metric of the phase space (space of initial conditions of the corresponding initial boundary value problem. It is chosen the Sobolev space as the phase space. For the periodic solutions which fill the two-dimensional attractor, the asymptotic formulas are given. In order to analyze the bifurcation problem it was used analysis methods for infinite-dimensional dynamical systems: the integral (invariant manifold method, the Poincare normal form theory, and asymptotic methods. The analysis of bifurcations for periodic boundary value problem was reduced to analysing the structure of the neighborhood of the zero solution of the homogeneous Dirichlet boundary value problem for the considered equation.
Dissipative Structures of the Kuramoto–Sivashinsky Equation
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N. A. Kudryashov
2015-01-01
Full Text Available In the present work, we study the features of dissipative structures formation described by the periodic boundary value problem for the Kuramoto-Sivashinsky equation. The numerical algorithm which is based on the pseudospectral method is presented. We prove the efficiency and accuracy of the proposed numerical method on the exact solution of the equation considered. Using this approach, we performed the numerical simulation of dissipative structure formations described by the Kuramoto–Sivashinsky equation. The influence of the problem parameters on these processes are studied. The quantitative and qualitative characteristics of dissipative structure formations are described. We have shown that there is a value of the control parameter at which the processes of dissipative structure formation are observed. In particular, using the cyclic convolution we define the average value of this parameter. Also, we find the dependence of the amplitude of the structures on the value of control parameter.
Parabolized stability equations
Herbert, Thorwald
1994-01-01
The parabolized stability equations (PSE) are a new approach to analyze the streamwise evolution of single or interacting Fourier modes in weakly nonparallel flows such as boundary layers. The concept rests on the decomposition of every mode into a slowly varying amplitude function and a wave function with slowly varying wave number. The neglect of the small second derivatives of the slowly varying functions with respect to the streamwise variable leads to an initial boundary-value problem that can be solved by numerical marching procedures. The PSE approach is valid in convectively unstable flows. The equations for a single mode are closely related to those of the traditional eigenvalue problems for linear stability analysis. However, the PSE approach does not exploit the homogeneity of the problem and, therefore, can be utilized to analyze forced modes and the nonlinear growth and interaction of an initial disturbance field. In contrast to the traditional patching of local solutions, the PSE provide the spatial evolution of modes with proper account for their history. The PSE approach allows studies of secondary instabilities without the constraints of the Floquet analysis and reproduces the established experimental, theoretical, and computational benchmark results on transition up to the breakdown stage. The method matches or exceeds the demonstrated capabilities of current spatial Navier-Stokes solvers at a small fraction of their computational cost. Recent applications include studies on localized or distributed receptivity and prediction of transition in model environments for realistic engineering problems. This report describes the basis, intricacies, and some applications of the PSE methodology.
Hidden physics models: Machine learning of nonlinear partial differential equations
Raissi, Maziar; Karniadakis, George Em
2018-03-01
While there is currently a lot of enthusiasm about "big data", useful data is usually "small" and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from small data. In particular, we introduce hidden physics models, which are essentially data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and nonlinear partial differential equations, to extract patterns from high-dimensional data generated from experiments. The proposed methodology may be applied to the problem of learning, system identification, or data-driven discovery of partial differential equations. Our framework relies on Gaussian processes, a powerful tool for probabilistic inference over functions, that enables us to strike a balance between model complexity and data fitting. The effectiveness of the proposed approach is demonstrated through a variety of canonical problems, spanning a number of scientific domains, including the Navier-Stokes, Schrödinger, Kuramoto-Sivashinsky, and time dependent linear fractional equations. The methodology provides a promising new direction for harnessing the long-standing developments of classical methods in applied mathematics and mathematical physics to design learning machines with the ability to operate in complex domains without requiring large quantities of data.
Stability of Functional Differential Equations
Lemm, Jeffrey M
1986-01-01
This book provides an introduction to the structure and stability properties of solutions of functional differential equations. Numerous examples of applications (such as feedback systrems with aftereffect, two-reflector antennae, nuclear reactors, mathematical models in immunology, viscoelastic bodies, aeroautoelastic phenomena and so on) are considered in detail. The development is illustrated by numerous figures and tables.
Stability theory of differential equations
Bellman, Richard Ernest
1953-01-01
Suitable for advanced undergraduates and graduate students, this was the first English-language text to offer detailed coverage of boundedness, stability, and asymptotic behavior of linear and nonlinear differential equations. It remains a classic guide, featuring material from original research papers, including the author's own studies.The linear equation with constant and almost-constant coefficients receives in-depth attention that includes aspects of matrix theory. No previous acquaintance with the theory is necessary, since author Richard Bellman derives the results in matrix theory from
Fuzzy Stability of Quadratic Functional Equations
Directory of Open Access Journals (Sweden)
Jang Sun-Young
2010-01-01
Full Text Available The fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated by Moslehian et al. In this paper, we prove the generalized Hyers-Ulam stability of the following quadratic functional equations and in fuzzy Banach spaces.
Fuzzy Stability of Quadratic Functional Equations
Dong Yun Shin; Choonkil Park; Sun-Young Jang; Jung Rye Lee
2010-01-01
The fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated by Moslehian et al. In this paper, we prove the generalized Hyers-Ulam stability of the following quadratic functional equations and in fuzzy Banach spaces.
Lattice Boltzmann model for high-order nonlinear partial differential equations
Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang
2018-01-01
In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂tϕ +∑k=1mαk∂xkΠk(ϕ ) =0 (1 ≤k ≤m ≤6 ), αk are constant coefficients, Πk(ϕ ) are some known differential functions of ϕ . As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K (n ,n ) -Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009), 10.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009), 10.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.
Feedback stabilization of semilinear heat equations
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V. Barbu
2003-01-01
Full Text Available This paper is concerned with the internal and boundary stabilization of the steady-state solutions to quasilinear heat equations via internal linear feedback controllers provided by an LQ control problem associated with the linearized equation.
Orthogonality preserving property, Wigner equation, and stability
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Chmieliński Jacek
2006-01-01
Full Text Available We deal with the stability of the orthogonality preserving property in the class of mappings phase-equivalent to linear or conjugate-linear ones. We give a characterization of approximately orthogonality preserving mappings in this class and we show some connections between the considered stability and the stability of the Wigner equation.
Controllability and stabilization of parabolic equations
Barbu, Viorel
2018-01-01
This monograph presents controllability and stabilization methods in control theory that solve parabolic boundary value problems. Starting from foundational questions on Carleman inequalities for linear parabolic equations, the author addresses the controllability of parabolic equations on a variety of domains and the spectral decomposition technique for representing them. This method is, in fact, designed for use in a wider class of parabolic systems that include the heat and diffusion equations. Later chapters develop another process that employs stabilizing feedback controllers with a finite number of unstable modes, with special attention given to its use in the boundary stabilization of Navier–Stokes equations for the motion of viscous fluid. In turn, these applied methods are used to explore related topics like the exact controllability of stochastic parabolic equations with linear multiplicative noise. Intended for graduate students and researchers working on control problems involving nonlinear diff...
Stability for retarded functional differential equations
Czech Academy of Sciences Publication Activity Database
Federson, M.; Schwabik, Štefan
2008-01-01
Roč. 60, č. 1 (2008), s. 121-140 ISSN 0041-5995 R&D Projects: GA AV ČR IAA100190702 Institutional research plan: CEZ:AV0Z10190503 Keywords : retarded functional differential equation * generalized differential equation * stability Subject RIV: BA - General Mathematics
Some Remarks on Stability of Generalized Equations
Czech Academy of Sciences Publication Activity Database
Outrata, Jiří; Henrion, R.; Kruger, A.Y.
2013-01-01
Roč. 159, č. 3 (2013), s. 681-697 ISSN 0022-3239 R&D Projects: GA AV ČR IAA100750802; GA ČR(CZ) GAP201/12/0671 Institutional support: RVO:67985556 Keywords : Parameterized generalized equation * Regular and limiting coderivative * Constant rank CQ * Mathematical program with equilibrium constraints Subject RIV: BA - General Mathematics Impact factor: 1.406, year: 2013 http://library.utia.cas.cz/separaty/2013/MTR/outrata-some remarks on stability of generalized equations.pdf
Compressible stability of growing boundary layers using parabolized stability equations
Chang, Chau-Lyan; Malik, Mujeeb R.; Erlebacher, Gordon; Hussaini, M. Y.
1991-01-01
The parabolized stability equation (PSE) approach is employed to study linear and nonlinear compressible stability with an eye to providing a capability for boundary-layer transition prediction in both 'quiet' and 'disturbed' environments. The governing compressible stability equations are solved by a rational parabolizing approximation in the streamwise direction. Nonparallel flow effects are studied for both the first- and second-mode disturbances. For oblique waves of the first-mode type, the departure from the parallel results is more pronounced as compared to that for the two-dimensional waves. Results for the Mach 4.5 case show that flow nonparallelism has more influence on the first mode than on the second. The disturbance growth rate is shown to be a strong function of the wall-normal distance due to either flow nonparallelism or nonlinear interactions. The subharmonic and fundamental types of breakdown are found to be similar to the ones in incompressible boundary layers.
Equations for studies of feedback stabilization
International Nuclear Information System (INIS)
Boozer, A.H.
1998-01-01
Important ideal magnetohydrodynamic (MHD) instabilities grow slowly when a conducting wall surrounds a toroidal plasma. Feedback stabilization of these instabilities may be required for tokamaks and other magnetic confinement concepts to achieve adequate plasma pressure and self-driven current for practical fusion power. Equations are derived for simulating feedback stabilization, which require the minimum information about an ideal plasma for an exact analysis. The equations are solved in the approximation of one unstable mode, one wall circuit, one feedback circuit, and one sensor circuit. The analysis based on a single unstable mode is shown to be mathematically equivalent to the standard analysis of feedback of the axisymmetric vertical instability of tokamaks. Unlike that analysis, the method presented here applies to multiple modes that are coupled by the wall and to arbitrary toroidal mode numbers. copyright 1998 American Institute of Physics
Fractal tracer distributions in turbulent field theories
DEFF Research Database (Denmark)
Hansen, J. Lundbek; Bohr, Tomas
1998-01-01
We study the motion of passive tracers in a two-dimensional turbulent velocity field generated by the Kuramoto-Sivashinsky equation. By varying the direction of the velocity-vector with respect to the field-gradient we can continuously vary the two Lyapunov exponents for the particle motion and t...
On the Stability of Quadratic Functional Equations
Lee, Jung Rye; An, Jong Su; Park, Choonkil
2008-01-01
Let $X,Y$ be vector spaces and $k$ a fixed positive integer. It is shown that a mapping $f(kx+y)+f(kx-y)=2{k}^{2}f(x)+2f(y)$ for all $x,y \\in X$ if and only if the mapping $f:X\\rightarrow Y$ satisfies $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y\\in X$ . Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.
On the Stability of Quadratic Functional Equations
Directory of Open Access Journals (Sweden)
Jung Rye Lee
2008-01-01
Full Text Available Let X,Y be vector spaces and k a fixed positive integer. It is shown that a mapping f(kx+y+f(kx-y=2k2f(x+2f(y for all x,y∈X if and only if the mapping f:X→Y satisfies f(x+y+f(x-y=2f(x+2f(y for all x,y∈X. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.
Lyapunov functionals and stability of stochastic functional differential equations
Shaikhet, Leonid
2013-01-01
Stability conditions for functional differential equations can be obtained using Lyapunov functionals. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations describes the general method of construction of Lyapunov functionals to investigate the stability of differential equations with delays. This work continues and complements the author’s previous book Lyapunov Functionals and Stability of Stochastic Difference Equations, where this method is described for discrete- and continuous-time difference equations. The text begins with a description of the peculiarities of deterministic and stochastic functional differential equations. There follow basic definitions for stability theory of stochastic hereditary systems, and a formal procedure of Lyapunov functionals construction is presented. Stability investigation is conducted for stochastic linear and nonlinear differential equations with constant and distributed delays. The proposed method is used for stability investigation of di...
Stability of quasi-linear differential equations with transition conditions
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Yubin Liu
2008-08-01
Full Text Available This paper investigates the stability of quasi-linear differential equations on certain time scales with transition condition (DETC. We establish Sufficient conditions for stability and illustrate our results with examples.
Stability of Vector Functional Differential Equations: A Survey | Gil ...
African Journals Online (AJOL)
This paper is a survey of the recent results of the author on the stability of linear and nonlinear vector differential equations with delay. Explicit conditions for the exponential and absolute stabilities are derived. Moreover, solution estimates for the considered equations are established. They provide the bounds for the regions ...
Internal stabilization of Maxwell's equations in heterogeneous media
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Cristina Pignotti
2005-09-01
Full Text Available We consider the internal stabilization of Maxwell's equations with Ohm's law with space variable coefficients in a bounded region with a smooth boundary. Our result is mainly based on an observability estimate, obtained in some particular cases by the multiplier method, a duality argument and a weakening of norm argument, and arguments used in internal stabilization of scalar wave equations.
Stability for Linear Volterra Difference Equations in Banach Spaces
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Rigoberto Medina
2018-01-01
Full Text Available This paper is devoted to studying the existence and stability of implicit Volterra difference equations in Banach spaces. The proofs of our results are carried out by using an appropriate extension of the freezing method to Volterra difference equations in Banach spaces. Besides, sharp explicit stability conditions are derived.
Stability of Difference Equations and Applications to Robustness Problems
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Sasu Bogdan
2010-01-01
Full Text Available The aim of this paper is to obtain new necessary and sufficient conditions for the uniform exponential stability of variational difference equations with applications to robustness problems. We prove characterizations for exponential stability of variational difference equations using translation invariant sequence spaces and emphasize the importance of each hypothesis. We introduce a new concept of stability radius for a variational system of difference equations with respect to a perturbation structure and deduce a very general estimate for the lower bound of . All the results are obtained without any restriction concerning the coefficients, being applicable for any system of variational difference equations.
On the stability of Jensen's functional equation on groups
Indian Academy of Sciences (India)
embedded into some group G such that the Jensen functional equation is stable on G. Keywords. Additive mapping ... direct product of groups; stability of functional equation, wreath product of groups. 1. Introduction ...... [3] Chung J K, Ebanks B R, Ng C T and Sahoo P K, On a quadratic-trigonometric functional equation and ...
Lyapunov functionals and stability of stochastic difference equations
Shaikhet, Leonid
2011-01-01
This book offers a general method of Lyapunov functional construction which lets researchers analyze the degree to which the stability properties of differential equations are preserved in their difference analogues. Includes examples from physical systems.
Stability of stationary and periodic solutions equations in Banach space
Directory of Open Access Journals (Sweden)
A. Ya. Dorogovtsev
1997-01-01
Full Text Available Linear difference and differential equations with operator coefficients and random stationary (periodic input are considered. Conditions are presented for the mean stability of stationary (periodic solutions under small perturbation of the coefficients.
Stability and instability of stationary solutions for sublinear parabolic equations
Kajikiya, Ryuji
2018-01-01
In the present paper, we study the initial boundary value problem of the sublinear parabolic equation. We prove the existence of solutions and investigate the stability and instability of stationary solutions. We show that a unique positive and a unique negative stationary solutions are exponentially stable and give the exact exponent. We prove that small stationary solutions are unstable. For one space dimensional autonomous equations, we elucidate the structure of stationary solutions and study the stability of all stationary solutions.
On the Fuzzy Stability of an Affine Functional Equation
Nasiruzzaman, Md.
2015-01-01
In this paper, we obtain the general solution of the following functional equation f(3x + y + z) + f(x + 3y + z) + f(x + y + 3z) + f(x) + f(y) + f(z) = 6f(x + y + z): We establish the Hyers-Ulam-Rassias stability of the above functional equation in the fuzzy normed spaces. Further we show the above functional equation is stable in the sense of Hyers and Ulam in fuzzy normed spaces.
Orbital stability of Gausson solutions to logarithmic Schrodinger equations
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Alex H. Ardila
2016-12-01
Full Text Available In this article we prove of the orbital stability of the ground state for logarithmic Schrodinger equation in any dimension and under nonradial perturbations. This general stability result was announced by Cazenave and Lions [9, Remark II.3], but no details were given there.
Stability analysis of a class of fractional delay differential equations
Indian Academy of Sciences (India)
In this paper we analyse stability of nonlinear fractional order delay differential equations of the form D y ( t ) = a f ( y ( t − ) ) − by ( t ) , where D is a Caputo fractional derivative of order 0 < ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic ...
On the stability of some systems of exponential difference equations
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N. Psarros
2018-01-01
Full Text Available In this paper we prove the stability of the zero equilibria of two systems of difference equations of exponential type, which are some extensions of an one-dimensional biological model. The stability of these systems is investigated in the special case when one of the eigenvalues is equal to -1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. In addition, we study the existence and uniqueness of positive equilibria, the attractivity and the global asymptotic stability of these equilibria of some related systems of difference equations.
Periodic feedback stabilization for linear periodic evolution equations
Wang, Gengsheng
2016-01-01
This book introduces a number of recent advances regarding periodic feedback stabilization for linear and time periodic evolution equations. First, it presents selected connections between linear quadratic optimal control theory and feedback stabilization theory for linear periodic evolution equations. Secondly, it identifies several criteria for the periodic feedback stabilization from the perspective of geometry, algebra and analyses respectively. Next, it describes several ways to design periodic feedback laws. Lastly, the book introduces readers to key methods for designing the control machines. Given its coverage and scope, it offers a helpful guide for graduate students and researchers in the areas of control theory and applied mathematics.
An Alternative Stability Equation For Rock Armoured Rubble Mound Breakwaters
DEFF Research Database (Denmark)
Hald, Tue; Burcharth, H. F.
2000-01-01
Rubble mound breakwaters are by far the most common type of breakwater, the importance of which is clearly reflected in the vast amount of published research. Especially, the hydraulic stability of the main armour layer has been studied in order to obtain reliable design equations. It should...... be straight forward to assume, that a relation between the exerted wave loads and the stability of the armour layer would be of major interest, but the normal approach to the problem has simply been to relate easy accessible wave parameters, e.g. wave height and period, directly to the single unit weight...... equations and model test results still exists. When turning toward prototype the situation is even worse. With the objective to reduce some of the variability an alternative approach based on force considerations is presented. The paper will describe a new stability equation for rock armoured slopes derived...
pth Moment Exponential Stability of Nonlinear Hybrid Stochastic Heat Equations
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Xuetao Yang
2014-01-01
Full Text Available We are concerned with the exponential stability problem of a class of nonlinear hybrid stochastic heat equations (known as stochastic heat equations with Markovian switching in an infinite state space. The fixed point theory is utilized to discuss the existence, uniqueness, and pth moment exponential stability of the mild solution. Moreover, we also acquire the Lyapunov exponents by combining the fixed point theory and the Gronwall inequality. At last, two examples are provided to verify the effectiveness of our obtained results.
Stability of the Exponential Functional Equation in Riesz Algebras
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Bogdan Batko
2014-01-01
Full Text Available We deal with the stability of the exponential Cauchy functional equation F(x+y=F(xF(y in the class of functions F:G→L mapping a group (G, + into a Riesz algebra L. The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem.
Classical and nonclassical symmetries analysis for initial value problems
International Nuclear Information System (INIS)
Zhang Zhiyong; Chen Yufu
2010-01-01
Classical and nonclassical symmetries are considered to reduce evolution equations with initial conditions in two independent variables. First of all, we rearrange the classical infinitesimal operators such that they leave the initial value problems invariant. Secondly, we give a sufficient condition for the nonclassical symmetry reductions of initial value problems. The generalized Kuramoto-Sivashinsky equation with dispersive effects is considered to examine the algorithms.
STABILITY OF A FUNCTIONAL EQUATION IN COMPLEX BANACH SPACES
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PRATAP MONDAL
2016-12-01
Full Text Available Using fixed point technique, in the present paper , we wish to examine gen- eralization of the Hyers-Ulam-Rassias stability theorem for the functional equations f ( 2 x + i y + f ( x + 2 i y = 4 f ( x + i y + f ( x + f ( y (0.1 and f ( 2 x + i y .
Fuzzy Stability of Jensen-Type Quadratic Functional Equations
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Sun-Young Jang
2009-01-01
Full Text Available We prove the generalized Hyers-Ulam stability of the following quadratic functional equations 2((+/2+2((−/2=(+( and (++(−=22(+22( in fuzzy Banach spaces for a nonzero real number with ≠±1/2.
The Local Stability of Solutions for a Nonlinear Equation
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Haibo Yan
2014-01-01
Full Text Available The approach of Kruzkov’s device of doubling the variables is applied to establish the local stability of strong solutions for a nonlinear partial differential equation in the space L1(R by assuming that the initial value only lies in the space L1(R∩L∞(R.
Exponential stability of dynamic equations on time scales
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Raffoul Youssef N
2005-01-01
Full Text Available We investigate the exponential stability of the zero solution to a system of dynamic equations on time scales. We do this by defining appropriate Lyapunov-type functions and then formulate certain inequalities on these functions. Several examples are given.
STABILITY OF A FUNCTIONAL EQUATION IN COMPLEX BANACH SPACES
PRATAP MONDAL; T.K. SAMANTA
2016-01-01
Using fixed point technique, in the present paper , we wish to examine gen- eralization of the Hyers-Ulam-Rassias stability theorem for the functional equations f ( 2 x + i y ) + f ( x + 2 i y ) = 4 f ( x + i y ) + f ( x ) + f ( y ) (0.1) and f ( 2 x + i y ) .
Stability of Solutions to Semilinear Stochastic Evolution Equations
Czech Academy of Sciences Publication Activity Database
Leha, G.; Maslowski, Bohdan; Ritter, G.
1999-01-01
Roč. 17, č. 6 (1999), s. 1009-1051 ISSN 0736-2994 R&D Projects: GA ČR GA201/95/0629 Institutional research plan: CEZ:AV0Z1019905; CEZ:AV0Z1019905 Keywords : stochastic evolution equations * Lyapunov stability * forward inequality Subject RIV: BA - General Mathematics Impact factor: 0.263, year: 1999
Exponential Stability of Stochastic Differential Equation with Mixed Delay
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Wenli Zhu
2014-01-01
Full Text Available This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.
Stability theory for dynamic equations on time scales
Martynyuk, Anatoly A
2016-01-01
This monograph is a first in the world to present three approaches for stability analysis of solutions of dynamic equations. The first approach is based on the application of dynamic integral inequalities and the fundamental matrix of solutions of linear approximation of dynamic equations. The second is based on the generalization of the direct Lyapunovs method for equations on time scales, using scalar, vector and matrix-valued auxiliary functions. The third approach is the application of auxiliary functions (scalar, vector, or matrix-valued ones) in combination with differential dynamic inequalities. This is an alternative comparison method, developed for time continuous and time discrete systems. In recent decades, automatic control theory in the study of air- and spacecraft dynamics and in other areas of modern applied mathematics has encountered problems in the analysis of the behavior of solutions of time continuous-discrete linear and/or nonlinear equations of perturbed motion. In the book “Men of Ma...
Stability analysis of a class of fractional delay differential equations
Indian Academy of Sciences (India)
equation (8) satisfy. Re(λi ) < 0, ∀i. (9). Let λ = u + ıv, u,v ∈ . A change in stability can occur only when the value of λ crosses the imaginary axis at λ = ıv and the characteristic equation becomes. (ıv)α + b = af (y. ∗) exp(−ıvτ). (10). Using ıv = v exp(ıπ/2), v ∈ and separating real and imaginary parts in (10) we get b + vα cos. (.
Stabilization and real world satellite problem. [transformations for stabilizing orbital equations
Velez, C. E.
1975-01-01
The use of transformations of orbital equations has been considered in connection with requirements for more accurate data. The reported investigation is concerned with an evaluation of the relative merits of such transformations. The formulations tested include the classical Cowell formulation, the time regularized formulation, stabilization by the use of integrals, and stabilization by the use of elements. It is found that irrespective of efficiency considerations, stabilizing transformation makes it possible to obtain precisions which are unattainable with the Cowell formulations.
Local asymptotic stability for nonlinear quadratic functional integral equations
Directory of Open Access Journals (Sweden)
Bapurao Dhage
2008-03-01
Full Text Available In the present study, using the characterizations of measures of noncompactness we prove a theorem on the existence and local asymptotic stability of solutions for a quadratic functional integral equation via a fixed point theorem of Darbo. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. An example is indicated to demonstrate the natural realizations of abstract result presented in the paper.
Generalized Stability of Euler-Lagrange Quadratic Functional Equation
Directory of Open Access Journals (Sweden)
Hark-Mahn Kim
2012-01-01
Full Text Available The main goal of this paper is the investigation of the general solution and the generalized Hyers-Ulam stability theorem of the following Euler-Lagrange type quadratic functional equation f(ax+by+af(x-by=(a+1b2f(y+a(a+1f(x, in (β,p-Banach space, where a,b are fixed rational numbers such that a≠-1,0 and b≠0.
Remarks on the stability of some quadratic functional equations
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Zygfryd Kominek
2008-01-01
Full Text Available Stability problems concerning the functional equations of the form \\[f(2x+y=4f(x+f(y+f(x+y-f(x-y,\\tag{1}\\] and \\[f(2x+y+f(2x-y=8f(x+2f(y\\tag{2}\\] are investigated. We prove that if the norm of the difference between the LHS and the RHS of one of equations \\((1\\ or \\((2\\, calculated for a function \\(g\\ is say, dominated by a function \\(\\varphi\\ in two variables having some standard properties then there exists a unique solution \\(f\\ of this equation and the norm of the difference between \\(g\\ and \\(f\\ is controlled by a function depending on \\(\\varphi\\.
On the Stability of a New Pexider-Type Functional Equation
Directory of Open Access Journals (Sweden)
Lee Juri
2008-01-01
Full Text Available Abstract We establish the generalized Hyers-Ulam stability of a Pexider-type functonal equation , which is mixed of a quadratic and an additive functional equations. Also, we obtain its general solution from the stability results.
Stabilization and asymptotic behavior of a generalized telegraph equation
Nicaise, Serge
2015-12-01
We analyze the stability of different models of the telegraph equation set in a real interval. They correspond to the coupling between a first-order hyperbolic system and a first-order differential equation of parabolic type. We show that some models have an exponential decay rate, while other ones are only polynomially stable. When the parameters are constant, we show that the obtained polynomial decay is optimal and in the case of an exponential decay that the decay rate is equal to the spectral abscissa. These optimality results are based on a careful spectral analysis of the operator. In particular, we characterize its full spectrum that is made of a discrete set of eigenvalues and an essential spectrum reduced to one point.
Stochastic modeling of mode interactions via linear parabolized stability equations
Ran, Wei; Zare, Armin; Hack, M. J. Philipp; Jovanovic, Mihailo
2017-11-01
Low-complexity approximations of the Navier-Stokes equations have been widely used in the analysis of wall-bounded shear flows. In particular, the parabolized stability equations (PSE) and Floquet theory have been employed to capture the evolution of primary and secondary instabilities in spatially-evolving flows. We augment linear PSE with Floquet analysis to formally treat modal interactions and the evolution of secondary instabilities in the transitional boundary layer via a linear progression. To this end, we leverage Floquet theory by incorporating the primary instability into the base flow and accounting for different harmonics in the flow state. A stochastic forcing is introduced into the resulting linear dynamics to model the effect of nonlinear interactions on the evolution of modes. We examine the H-type transition scenario to demonstrate how our approach can be used to model nonlinear effects and capture the growth of the fundamental and subharmonic modes observed in direct numerical simulations and experiments.
Ulam-Hyers Stability of Trigonometric Functional Equation with Involution
Directory of Open Access Journals (Sweden)
Jaeyoung Chung
2015-01-01
Full Text Available Let S and G be a commutative semigroup and a commutative group, respectively, C and R+ the sets of complex numbers and nonnegative real numbers, respectively, and σ:S→S or σ:G→G an involution. In this paper, we first investigate general solutions of the functional equation f(x+σy=f(xg(y-g(xf(y for all x,y∈S, where f,g:S→C. We then prove the Hyers-Ulam stability of the functional equation; that is, we study the functional inequality |f(x+σy-f(xg(y+g(xf(y|≤ψ(y for all x,y∈G, where f,g:G→C and ψ:G→R+.
Stability by fixed point theory for functional differential equations
Burton, T A
2006-01-01
This book is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques. It contains an extensive collection of new and classical examples worked in detail and presented in an elementary manner. Most of this text relies on three principles: a complete metric space, the contraction mapping principle, and an elementary variation of parameters formula. The material is highly accessible to upper-level undergraduate students in the mathematical sciences, as well as working biologists, chemists, economists, engineers, mathematicia
Stability theorems for stochastic differential equations driven by G-Brownian motion
Zhang, Defei
2011-01-01
In this paper, stability theorems for stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion are obtained. We show the existence and uniqueness of solutions to forward-backward stochastic differential equations driven by G-Brownian motion. Stability theorem for forward-backward stochastic differential equations driven by G-Brownian motion is also presented.
Directory of Open Access Journals (Sweden)
Samuel Castillo
2015-03-01
Full Text Available This work concerns the existence of almost periodic solutions for certain differential equations with piecewise constant arguments. The coefficients of these equations are almost periodic and the equation can be seen as perturbations of a linear equation satisfying an exponential dichotomy on a difference equation. The stability of that solution on a semi-axis is also studied.
Stability of the complex generalized Hartree-Fock equations.
Goings, Joshua J; Ding, Feizhi; Frisch, Michael J; Li, Xiaosong
2015-04-21
For molecules with complex and competing magnetic interactions, it is often the case that the lowest energy Hartree-Fock solution may only be obtained by removing the spin and time-reversal symmetry constraints of the exact non-relativistic Hamiltonian. To do so results in the complex generalized Hartree-Fock (GHF) method. However, with the loss of variational constraints comes the greater possibility of converging to higher energy minima. Here, we report the implementation of stability test of the complex GHF equations, along with an orbital update scheme should an instability be found. We apply the methodology to finding the local minima of several spin-frustrated hydrogen rings, as well as the non-collinear molecular magnet Cr3, illustrating the utility of the broken symmetry GHF method and some of its lesser-known nuances.
Directory of Open Access Journals (Sweden)
Coşkun Yakar
2010-01-01
Full Text Available The qualitative behavior of a perturbed fractional-order differential equation with Caputo's derivative that differs in initial position and initial time with respect to the unperturbed fractional-order differential equation with Caputo's derivative has been investigated. We compare the classical notion of stability to the notion of initial time difference stability for fractional-order differential equations in Caputo's sense. We present a comparison result which again gives the null solution a central role in the comparison fractional-order differential equation when establishing initial time difference stability of the perturbed fractional-order differential equation with respect to the unperturbed fractional-order differential equation.
Improved algorithm for solving nonlinear parabolized stability equations
International Nuclear Information System (INIS)
Zhao Lei; Zhang Cun-bo; Liu Jian-xin; Luo Ji-sheng
2016-01-01
Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. (paper)
Improved algorithm for solving nonlinear parabolized stability equations
Zhao, Lei; Zhang, Cun-bo; Liu, Jian-xin; Luo, Ji-sheng
2016-08-01
Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. Project supported by the National Natural Science Foundation of China (Grant Nos. 11332007 and 11402167).
Stability of Quadratic Functional Equations via the Fixed Point and Direct Method
Directory of Open Access Journals (Sweden)
Son Eunyoung
2010-01-01
Full Text Available Cădariu and Radu applied the fixed point theorem to prove the stability theorem of Cauchy and Jensen functional equations. In this paper, we prove the generalized Hyers-Ulam stability via the fixed point method and investigate new theorems via direct method concerning the stability of a general quadratic functional equation.
Stability analysis of a boundary layer over a hump using parabolized stability equations
Energy Technology Data Exchange (ETDEWEB)
Gao, B; Park, D H; Park, S O, E-mail: sopark@kaist.ac.kr [Division of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Gusong-dong, Yusong-gu, Daejeon 305-701 (Korea, Republic of)
2011-10-15
Parabolized stability equations (PSEs) were used to investigate the stability of boundary layer flows over a small hump. The applicability of PSEs to flows with a small separation bubble was examined by comparing the result with DNS data. It was found that PSEs can efficiently track the disturbance waves with an acceptable accuracy in spite of a small separation bubble. A typical evolution scenario of Tollmien-Schlichting (TS) wave is presented. The adverse pressure gradient and the flow separation due to the hump have a strong effect on the amplification of the disturbances. The effect of hump width and height is also examined. When the width of the hump is reduced, the amplification factor is increased. The height of the hump is found to obviously influence the stability only when it is greater than the critical layer thickness.
Stability analysis of a boundary layer over a hump using parabolized stability equations
International Nuclear Information System (INIS)
Gao, B; Park, D H; Park, S O
2011-01-01
Parabolized stability equations (PSEs) were used to investigate the stability of boundary layer flows over a small hump. The applicability of PSEs to flows with a small separation bubble was examined by comparing the result with DNS data. It was found that PSEs can efficiently track the disturbance waves with an acceptable accuracy in spite of a small separation bubble. A typical evolution scenario of Tollmien-Schlichting (TS) wave is presented. The adverse pressure gradient and the flow separation due to the hump have a strong effect on the amplification of the disturbances. The effect of hump width and height is also examined. When the width of the hump is reduced, the amplification factor is increased. The height of the hump is found to obviously influence the stability only when it is greater than the critical layer thickness.
Hyers-Ulam Stability of Iterative Equation in the Class of Lipschitz Functions
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Chao Xia
2014-01-01
Full Text Available Hyers-Ulam stability is a basic sense of stability for functional equations. In the present paper we discuss the Hyers-Ulam stability of a kind of iterative equations in the class of Lipschitz functions. By the construction of a uniformly convergent sequence of functions we prove that, for every approximate solution of such an equation, there exists an exact solution near it.
Directory of Open Access Journals (Sweden)
Diamandescu Aurel
2016-07-01
Full Text Available It is proved (necessary and sufficient conditions for Ψ– conditional exponential asymptotic stability of the trivial solution of nonlinear Lyapunov matrix differential equations
On the Ψ-Conditional Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations
Directory of Open Access Journals (Sweden)
Diamandescu Aurel
2015-12-01
Full Text Available It is proved (necessary and sufficient conditions for Ψ − conditional asymptotic stability of the trivial solution of linear or nonlinear Lyapunov matrix differential equations.
Refined stability of additive and quadratic functional equations in modular spaces
Directory of Open Access Journals (Sweden)
Hark-Mahn Kim
2017-06-01
Full Text Available Abstract The purpose of this paper is to obtain refined stability results and alternative stability results for additive and quadratic functional equations using direct method in modular spaces.
On the nonlinear interfacial instability of rotating core-annular flow
Coward, Aidrian V.; Hall, Philip
1993-01-01
The interfacial stability of rotating core-annular flows is investigated. The linear and nonlinear effects are considered for the case when the annular region is very thin. Both asymptotic and numerical methods are used to solve the flow in the core and film regions which are coupled by a difference in viscosity and density. The long-term behavior of the fluid-fluid interface is determined by deriving its nonlinear evolution in the form of a modified Kuramoto-Sivashinsky equation. We obtain a generalization of this equation to three dimensions. The flows considered are applicable to a wide array of physical problems where liquid films are used to lubricate higher or lower viscosity core fluids, for which a concentric arrangement is desired. Linearized solutions show that the effects of density and viscosity stratification are crucial to the stability of the interface. Rotation generally destabilizes non-axisymmetric disturbances to the interface, whereas the centripetal forces tend to stabilize flows in which the film contains the heavier fluid. Nonlinear affects allow finite amplitude helically travelling waves to exist when the fluids have different viscosities.
Difference equations in normed spaces stability and oscillations
Gil, Michael
2007-01-01
Difference equations appear as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete. They also appear in the applications of discretization methods for differential, integral and integro-differential equations. The application of the theory of difference equations is rapidly increasing to various fields, such as numerical analysis, control theory, finite mathematics, and computer sciences. This book is devoted to linear and nonlinear difference equations in a normed space. The main methodology presented in this book is based on a combined use of recent norm estimates for operator-valued functions with the following methods and results: The freezing methodThe Liapunov type equationThe method of majorantsThe multiplicative representation of solutionsDeals systematically with difference equations in normed spaces Considers new classes of equations that could not be studied in the frameworks of ordinary and partial difference equationsDevelops ...
Hyers-Ulam Stability and Existence of Solutions for Nigmatullin’s Fractional Diffusion Equation
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Zhuoyan Gao
2017-01-01
Full Text Available We discuss stability of time-fractional order heat conduction equations and prove the Hyers-Ulam and generalized Hyers-Ulam-Rassias stability of time-fractional order heat conduction equations via fractional Green function involving Wright function. In addition, an interesting existence result for solution is given.
Stability of Quartic Functional Equations in the Spaces of Generalized Functions
Directory of Open Access Journals (Sweden)
2009-03-01
Full Text Available We consider the general solution of quartic functional equations and prove the Hyers-Ulam-Rassias stability. Moreover, using the pullbacks and the heat kernels we reformulate and prove the stability results of quartic functional equations in the spaces of tempered distributions and Fourier hyperfunctions.
Hyers-Ulam Stability for a Class of Quadratic Functional Equations via a Typical Form
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Chang Il Kim
2013-01-01
Full Text Available We investigate the following typical form of a certain class of quadratic functional equations: . Furthermore, we provide a systematic program to prove the generalized Hyers-Ulam stability for the class of functional equations via the stability for the typical form.
Domoshnitsky, Alexander; Maghakyan, Abraham; Berezansky, Leonid
2017-01-01
In this paper a method for studying stability of the equation [Formula: see text] not including explicitly the first derivative is proposed. We demonstrate that although the corresponding ordinary differential equation [Formula: see text] is not exponentially stable, the delay equation can be exponentially stable.
Stability in terms of two measures for a class of semilinear impulsive parabolic equations
International Nuclear Information System (INIS)
Dvirnyj, Aleksandr I; Slyn'ko, Vitalij I
2013-01-01
The problem of stability in terms of two measures is considered for semilinear impulsive parabolic equations. A new version of the comparison method is proposed, and sufficient conditions for stability in terms of two measures are obtained on this basis. An example of a hybrid impulsive system formed by a system of ordinary differential equations coupled with a partial differential equation of parabolic type is given. The efficiency of the described approaches is demonstrated. Bibliography: 24 titles.
Stabilization analysis of Euler-Bernoulli beam equation with locally distributed disturbance
Directory of Open Access Journals (Sweden)
Pengcheng HAN
2017-12-01
Full Text Available In order to enrich the system stability theory of the control theories, taking Euler-Bernoulli beam equation as the research subject, the stability of Euler-Bernoulli beam equation with locally distributed disturbance is studied. A feedback controller based on output is designed to reduce the effects of the disturbances. The well-posedness of the nonlinear closed-loop system is investigated by the theory of maximal monotone operator, namely the existence and uniqueness of solutions for the closed-loop system. An appropriate state space is established, an appropriate inner product is defined, and a non-linear operator satisfying this state space is defined. Then, the system is transformed into the form of evolution equation. Based on this, the existence and uniqueness of solutions for the closed-loop system are proved. The asymptotic stability of the system is studied by constructing an appropriate Lyapunov function, which proves the asymptotic stability of the closed-loop system. The result shows that designing proper anti-interference controller is the foundation of investigating the system stability, and the research of the stability of Euler-bernoulli beam equation with locally distributed disturbance can prove the asymptotic stability of the system. This method can be extended to study the other equations such as wave equation, Timoshenko beam equation, Schrodinger equation, etc.
Stability with respect to initial time difference for generalized delay differential equations
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Ravi Agarwal
2015-02-01
Full Text Available Stability with initial data difference for nonlinear delay differential equations is introduced. This type of stability generalizes the known concept of stability in the literature. It gives us the opportunity to compare the behavior of two nonzero solutions when both initial values and initial intervals are different. Several sufficient conditions for stability and for asymptotic stability with initial time difference are obtained. Lyapunov functions as well as comparison results for scalar ordinary differential equations are employed. Several examples are given to illustrate the theory.
Stability of stationary inverse transport equation in diffusion scaling
Chen, Ke; Li, Qin; Wang, Li
2018-02-01
We consider the inverse problem of reconstructing the optical parameters for the stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales, characterized by the magnitude of a dimensionless parameter—the Knudsen number ( \
Stability analysis of a class of fractional delay differential equations
Indian Academy of Sciences (India)
Abstract. In this paper we analyse stability of nonlinear fractional order delay differential equa- tions of the form Dα y(t) = af (y(t − τ )) − by(t), where Dα is a Caputo fractional derivative of order 0 < α ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic ...
Stability Conditions of Second Order Integrodifferential Equations with Variable Delay
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Dingheng Pi
2014-01-01
Full Text Available We investigate integrodifferential functional differential equations ẍ+f(t,x,ẋẋ+∫t-r(tta(t,sg(x(sds=0 with variable delay. By using the fixed point theory, we obtain conditions which ensure that the zero solution of this equation is stable under an exponentially weighted metric. Then we establish necessary and sufficient conditions ensuring that the zero solution is asymptotically stable. We will give an example to apply our results.
Almost Surely Exponential Stability of Numerical Solutions for Stochastic Pantograph Equations
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Shaobo Zhou
2014-01-01
Full Text Available Our effort is to develop a criterion on almost surely exponential stability of numerical solution to stochastic pantograph differential equations, with the help of the discrete semimartingale convergence theorem and the technique used in stable analysis of the exact solution. We will prove that the Euler-Maruyama (EM method can preserve almost surely exponential stability of stochastic pantograph differential equations under the linear growth conditions. And the backward EM method can reproduce almost surely exponential stability for highly nonlinear stochastic pantograph differential equations. A highly nonlinear example is provided to illustrate the main theory.
Unconditional Stability of a Numerical Method for the Dual-Phase-Lag Equation
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M. A. Castro
2017-01-01
Full Text Available The stability properties of a numerical method for the dual-phase-lag (DPL equation are analyzed. The DPL equation has been increasingly used to model micro- and nanoscale heat conduction in engineering and bioheat transfer problems. A discretization method for the DPL equation that could yield efficient numerical solutions of 3D problems has been previously proposed, but its stability properties were only suggested by numerical experiments. In this work, the amplification matrix of the method is analyzed, and it is shown that its powers are uniformly bounded. As a result, the unconditional stability of the method is established.
Stability, bifurcation and a new chaos in the logistic differential equation with delay
International Nuclear Information System (INIS)
Jiang Minghui; Shen Yi; Jian Jigui; Liao Xiaoxin
2006-01-01
This Letter is concerned with bifurcation and chaos in the logistic delay differential equation with a parameter r. The linear stability of the logistic equation is investigated by analyzing the associated characteristic transcendental equation. Based on the normal form approach and the center manifold theory, the formula for determining the direction of Hopf bifurcation and the stability of bifurcation periodic solution in the first bifurcation values is obtained. By theoretical analysis and numerical simulation, we found a new chaos in the logistic delay differential equation
On the stability of evolution equations | Egwurube | Global Journal of ...
African Journals Online (AJOL)
accretive operator is considered and conditions which guarantee asymptotic stability of its solution in a dense subset of the space are given. Global Jouranl of Mathematical Sciences Vol. 6 (1) 2007: pp. 27-30 ...
Stability of generalized Runge-Kutta methods for stiff kinetics coupled differential equations
International Nuclear Information System (INIS)
Aboanber, A E
2006-01-01
A stability and efficiency improved class of generalized Runge-Kutta methods of order 4 are developed for the numerical solution of stiff system kinetics equations for linear and/or nonlinear coupled differential equations. The determination of the coefficients required by the method is precisely obtained from the so-called equations of condition which in turn are derived by an approach based on Butcher series. Since the equations of condition are fewer in number, free parameters can be chosen for optimizing any desired feature of the process. A further related coefficient set with different values of these parameters and the region of absolute stability of the method have been introduced. In addition, the A(α) stability properties of the method are investigated. Implementing the method in a personal computer estimated the accuracy and speed of calculations and verified the good performances of the proposed new schemes for several sample problems of the stiff system point kinetics equations with reactivity feedback
Stability and square integrability of solutions of nonlinear fourth order differential equations
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Moussadek Remili
2016-05-01
Full Text Available The aim of the present paper is to establish a new result, which guarantees the asymptotic stability of zero solution and square integrability of solutions and their derivatives to nonlinear differential equations of fourth order.
Stability of the Simplest Periodic Solutions in the Stuart–Landau Equation with Large Delay
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A. A. Kashchenko
2012-01-01
Full Text Available We study the local dynamics of the Stuart–Landau equation with large delay in the neibourhood of periodic solutions. We find sufficient conditions of instability of periodic solutions and sufficient conditions of their stability.
Stability Criteria for Differential Equations with Variable Time Delays
Schley, D.; Shail, R.; Gourley, S. A.
2002-01-01
Time delays are an important aspect of mathematical modelling, but often result in highly complicated equations which are difficult to treat analytically. In this paper it is shown how careful application of certain undergraduate tools such as the Method of Steps and the Principle of the Argument can yield significant results. Certain delay…
Exponential p-stability of impulsive stochastic differential equations with delays
International Nuclear Information System (INIS)
Yang Zhiguo; Xu Daoyi; Xiang Li
2006-01-01
In this Letter, we establish a method to study the exponential p-stability of the zero solution of impulsive stochastic differential equations with delays. By establishing an L-operator inequality and using the properties of M-cone and stochastic analysis technique, we obtain some new conditions ensuring the exponential p-stability of the zero solution of impulsive stochastic differential equations with delays. Two illustrative examples have been provided to show the effectiveness of our results
Chadha, Alka; Bora, Swaroop Nandan
2017-11-01
This paper studies the existence, uniqueness, and exponential stability in mean square for the mild solution of neutral second order stochastic partial differential equations with infinite delay and Poisson jumps. By utilizing the Banach fixed point theorem, first the existence and uniqueness of the mild solution of neutral second order stochastic differential equations is established. Then, the mean square exponential stability for the mild solution of the stochastic system with Poisson jumps is obtained with the help of an established integral inequality.
Stability of a simple Levi–Civitá functional equation on non-unital ...
Indian Academy of Sciences (India)
Additive function; exponential function; functional inequality; Hyers–. Ulam stability; Levi–Civitá equation; non-unital semigroup; 2-divisible group. 2010 Mathematics Subject Classification. Primary: 39B82. 1. Introduction. A certain formula or equation is applicable to model a physical process if a small change in the formula ...
On Polynomial Stability of Variational Nonautonomous Difference Equations in Banach Spaces
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Mihail Megan
2013-01-01
Full Text Available Our goal in this paper is to give characterizations for some concepts of polynomial stability for variational nonautonomous difference equations. The obtained results can be considered generalizations for the case of variational nonautonomous difference equations of some theorems proved by Barbashin (1967, Datko (1973, and Lyapunov (1992, for evolution operators.
Stabilization of solutions to higher-order nonlinear Schrodinger equation with localized damping
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Eleni Bisognin
2007-01-01
Full Text Available We study the stabilization of solutions to higher-order nonlinear Schrodinger equations in a bounded interval under the effect of a localized damping mechanism. We use multiplier techniques to obtain exponential decay in time of the solutions of the linear and nonlinear equations.
Directory of Open Access Journals (Sweden)
Dandan Guo
2017-08-01
Full Text Available In this article we consider the boundary stabilization of a wave equation with variable coefficients. This equation has an acceleration term and a delayed velocity term on the boundary. Under suitable geometric conditions, we obtain the exponential decay for the solutions. Our proof relies on the geometric multiplier method and the Lyapunov approach.
A Fixed Point Approach to the Stability of Quadratic Functional Equation with Involution
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Lee Zoon-Hee
2008-01-01
Full Text Available Abstract Cădariu and Radu applied the fixed point method to the investigation of Cauchy and Jensen functional equations. In this paper, we will adopt the idea of Cădariu and Radu to prove the Hyers-Ulam-Rassias stability of the quadratic functional equation with involution.
A Fixed Point Approach to the Stability of Quadratic Functional Equation with Involution
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Zoon-Hee Lee
2008-06-01
Full Text Available CÃ„Âƒdariu and Radu applied the fixed point method to the investigation of Cauchy and Jensen functional equations. In this paper, we will adopt the idea of CÃ„Âƒdariu and Radu to prove the Hyers-Ulam-Rassias stability of the quadratic functional equation with involution.
Stability of negative solitary waves for an integrable modified Camassa-Holm equation
International Nuclear Information System (INIS)
Yin Jiuli; Tian Lixin; Fan Xinghua
2010-01-01
In this paper, we prove that the modified Camassa-Holm equation is Painleve integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation.
Lyapunov stability and its application to systems of ordinary differential equations
Kennedy, E. W.
1979-01-01
An outline and a brief introduction to some of the concepts and implications of Lyapunov stability theory are presented. Various aspects of the theory are illustrated by the inclusion of eight examples, including the Cartesian coordinate equations of the two-body problem, linear and nonlinear (Van der Pol's equation) oscillatory systems, and the linearized Kustaanheimo-Stiefel element equations for the unperturbed two-body problem.
Two remarks on the stability of ordniary differential equations
International Nuclear Information System (INIS)
Vidossich, G.
1979-01-01
In this note the definition of Liapunov functions is extended, obtaining a generalization of the recent result by Lakshmikantham and Leela on the perturbation of Liapunov functions, and two sufficient conditions for the solvability of Olech's problem on global stability are pointed out. (author)
Stability of embedded solitons in higher-order NLS equations
Energy Technology Data Exchange (ETDEWEB)
Pal, Debabrata; Ali, Sk Golam; Talukdar, B, E-mail: binoy123@bsnl.in [Department of Physics, Visva-Bharati University, Santiniketan 731235 (India)
2011-02-15
We consider two models for femtosecond pulse propagation through optical fibers. The first one involves a nonlinear Schroedinger (NLS) equation with a perturbing term arising due to third-order dispersion in the medium, whereas the second one incorporates two additional effects-self-steepening (SS) and stimulated Raman scattering (SRS)-that have their physical origin in molecular vibration. We make use of the theory of Espinosa-Ceron et al (2003 Phys. Scr. 67 314) to analytically demonstrate that the third-order NLS equation involving terms due to SS and SRS obeys the radiation inhibition condition and thereby supports exponentially localized solitons. On the other hand, the purely third-order NLS equation invalidates the condition for radiation inhibition and its traveling wave solution shows oscillatory behavior due to the emission of radiation. We verify both these conclusions by numerical simulation and conclude that the effects of SS and SRS could be judiciously manipulated for the unattenuated propagation of femtosecond pulses through fibers.
Unique continuation for the Helmholtz equation using stabilized finite element methods
Burman, Erik; Nechita, Mihai; Oksanen, Lauri
2017-01-01
We solve a unique continuation problem for the Helmholtz equation using a stabilized finite element method. We first derive conditional stability estimates with constants independent of the wave number $k$, which we then use to obtain error estimates for the finite element method that are robust and explicit with respect to $k$. Some numerical illustrations are given.
Existence and Stability of Solutions for Hadamard-Stieltjes Fractional Integral Equations
Directory of Open Access Journals (Sweden)
Saïd Abbas
2015-01-01
Full Text Available We give some existence results and Ulam stability results for a class of Hadamard-Stieltjes integral equations. We present two results: the first one is an existence result based on Schauder’s fixed point theorem and the second one is about the generalized Ulam-Hyers-Rassias stability.
Ashyralyev, Allaberen; Cakir, Zafer
2016-08-01
In this work, we investigate initial-boundary value problems for fractional parabolic equations with the Neumann boundary condition. Stability estimates for the solution of this problem are established. Difference schemes for approximate solution of initial-boundary value problem are constructed. Furthermore, we give theorem on coercive stability estimates for the solution of the difference schemes.
Exponential stability of nonlinear time-varying differential equations and applications
Directory of Open Access Journals (Sweden)
N. M. Linh
2001-05-01
Full Text Available In this paper, we give sufficient conditions for the exponential stability of a class of nonlinear time-varying differential equations. We use the Lyapunov method with functions that are not necessarily differentiable; hence we extend previous results. We also provide an application to exponential stability for nonlinear time-varying control systems.
Gil', M. I.
2005-08-01
We consider a class of nonautonomous functional-differential equations in a Banach space with unbounded nonlinear history-responsive operators, which have the local Lipshitz property. Conditions for the boundedness of solutions, Lyapunov stability, absolute stability and input-output one are established. Our approach is based on a combined usage of properties of sectorial operators and spectral properties of commuting operators.
Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays
Directory of Open Access Journals (Sweden)
Chang-Wen Zhao
2008-07-01
Full Text Available We consider the mean square asymptotic stability of a generalized linear neutral stochastic differential equation with variable delays by using the fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton, Zhang and Luo. Two examples are also given to illustrate our results.
Directory of Open Access Journals (Sweden)
Zhanhua Yu
2011-01-01
Full Text Available We study the almost surely asymptotic stability of exact solutions to neutral stochastic pantograph equations (NSPEs, and sufficient conditions are obtained. Based on these sufficient conditions, we show that the backward Euler method (BEM with variable stepsize can preserve the almost surely asymptotic stability. Numerical examples are demonstrated for illustration.
Directory of Open Access Journals (Sweden)
Sherif Amirov
2017-08-01
Full Text Available The recent work on the solvability of the boundary value problem for the nonlinear analogue of the Boussinesq equation has been further extended to focus on the characteristics of the solution. Since this type of equation does not have a known analytical solution for arbitrary boundary conditions, the problem has been solved numerically. The stability of the solution and the effect of the input function on the stability have been investigated from the physics point of view. For the special case of a discontinuous function at the right hand side of the equation, the solution has been analyzed around the discontinuity points.
International Nuclear Information System (INIS)
Scannapieco, A.J.; Cranfill, C.W.
1978-11-01
There now exists an inertial confinement stability code called DOC, which runs as a postprocessor. DOC (a code that has evolved from a previous code, PANSY) is a spherical harmonic linear stability code that integrates, in time, a set of Lagrangian perturbation equations. Effects due to real equations of state, asymmetric energy deposition, thermal conduction, shock propagation, and a time-dependent zeroth-order state are handled in the code. We present here a detailed derivation of the physical equations that are solved in the code
The Lyapunov stabilization of satellite equations of motion using integrals
Nacozy, P. E.
1973-01-01
A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.
Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations
Abdulle, Assyr
2013-01-01
We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the step size reduction faced by standard explicit methods. The family is based on the standard second order orthogonal Runge-Kutta-Chebyshev (ROCK2) methods for deterministic problems. The convergence, meansquare, and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results. © 2013 Society for Industrial and Applied Mathematics.
Stability of the Filter Equation for a Time-Dependent Signal on Rd
International Nuclear Information System (INIS)
Stannat, Wilhelm
2005-01-01
Stability of the pathwise filter equation for a time-dependent signal process induced by a d-dimensional stochastic differential equation and a linear observation is studied, using a variational approach. A lower bound for the rate of stability is identified in terms of the mass-gap of a parabolic ground state transform associated with the generator of the signal process and the square of the observation. The lower bound can be easily calculated a priori and provides hints on how precisely to measure the signal in order to reach a certain rate of stability. Ergodicity of the signal process is not needed
Stability and bifurcation analysis of a generalized scalar delay differential equation.
Bhalekar, Sachin
2016-08-01
This paper deals with the stability and bifurcation analysis of a general form of equation D(α)x(t)=g(x(t),x(t-τ)) involving the derivative of order α ∈ (0, 1] and a constant delay τ ≥ 0. The stability of equilibrium points is presented in terms of the stability regions and critical surfaces. We provide a necessary condition to exist chaos in the system also. A wide range of delay differential equations involving a constant delay can be analyzed using the results proposed in this paper. The illustrative examples are provided to explain the theory.
Polynomial asymptotic stability of damped stochastic differential equations
Directory of Open Access Journals (Sweden)
John Appleby
2004-08-01
Full Text Available The paper studies the polynomial convergence of solutions of a scalar nonlinear It\\^{o} stochastic differential equation\\[dX(t = -f(X(t\\,dt + \\sigma(t\\,dB(t\\] where it is known, {\\it a priori}, that $\\lim_{t\\rightarrow\\infty} X(t=0$, a.s. The intensity of the stochastic perturbation $\\sigma$ is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function $f$ obeys $\\lim_{x\\rightarrow 0}\\mbox{sgn}(xf(x/|x|^\\beta = a$, for some $\\beta>1$, and $a>0$.We study two asymptotic regimes: when $\\sigma$ tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when $\\sigma\\equiv0$. When $\\sigma$ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for $\\sigma$ to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.
Razumikhin-type stability criteria for differential equations with delayed impulses
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Qing Wang
2013-01-01
Full Text Available This paper studies stability problems of general impulsive differential equations where time delays occur in both differential and difference equations. Based on the method of Lyapunov functions, Razumikhin technique and mathematical induction, several stability criteria are obtained for differential equations with delayed impulses. Our results show that some systems with delayed impulses may be exponentially stabilized by impulses even if the system matrices are unstable. Some less restrictive sufficient conditions are also given to keep the good stability property of systems subject to certain type of impulsive perturbations. Examples with numerical simulations are discussed to illustrate the theorems. Our results may be applied to complex problems where impulses depend on both current and past states.
Mathieu's Equation and its Generalizations: Overview of Stability Charts and their Features
DEFF Research Database (Denmark)
Kovacic, Ivana; Rand, Richard H.; Sah, Si Mohamed
2018-01-01
This work is concerned with Mathieu's equation - a classical differential equation, which has the form of a linear second-order ordinary differential equation with Cosine-type periodic forcing of the stiffness coefficient, and its different generalizations/extensions. These extensions include......: the effects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure...... and features, and how it differs from that of the classical Mathieu's equation....
A Stability Result for Stochastic Differential Equations Driven by Fractional Brownian Motions
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Bruno Saussereau
2012-01-01
Full Text Available We study the stability of the solutions of stochastic differential equations driven by fractional Brownian motions with Hurst parameter greater than half. We prove that when the initial conditions, the drift, and the diffusion coefficients as well as the fractional Brownian motions converge in a suitable sense, then the sequence of the solutions of the corresponding equations converge in Hölder norm to the solution of a stochastic differential equation. The limit equation is driven by the limit fractional Brownian motion and its coefficients are the limits of the sequence of the coefficients.
Tweten, Dennis J.; Lipp, Genevieve M.; Khasawneh, Firas A.; Mann, Brian P.
2012-08-01
This paper provides the first comparison of the semi-discretization, spectral element, and Legendre collocation methods. Each method is a technique for solving delay differential equations (DDEs) as well as determining regions of stability in the DDE parameter space. We present the necessary concepts, assumptions, and equations required to implement each method. To compare the relative performance between the methods, the convergence rate and computational time for each method is compared in three numerical studies consisting of a ship stability example, the delayed damped Mathieu equation, and a helicopter rotor control problem. For each study, we present one or more stability diagrams in the parameter space and one or more convergence plots. The spectral element method is demonstrated to have the quickest convergence rate while the Legendre collocation method requires the least computational time. The semi-discretization method on the other hand has both the slowest convergence rate and requires the most computational time.
International Nuclear Information System (INIS)
Grigoriu, Mircea; Samorodnitsky, Gennady
2004-01-01
Two methods are considered for assessing the asymptotic stability of the trivial solution of linear stochastic differential equations driven by Poisson white noise, interpreted as the formal derivative of a compound Poisson process. The first method attempts to extend a result for diffusion processes satisfying linear stochastic differential equations to the case of linear equations with Poisson white noise. The developments for the method are based on Ito's formula for semimartingales and Lyapunov exponents. The second method is based on a geometric ergodic theorem for Markov chains providing a criterion for the asymptotic stability of the solution of linear stochastic differential equations with Poisson white noise. Two examples are presented to illustrate the use and evaluate the potential of the two methods. The examples demonstrate limitations of the first method and the generality of the second method
A memory type boundary stabilization of a mildly damped wave equation
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Mokhtar Kirane
1999-01-01
Full Text Available We consider the wave equation with a mild internal dissipation. It is proved that any small dissipation inside the domain is sufficient to uniformly stabilize the solution of this equation by means of a nonlinear feedback of memory type acting on a part of the boundary. This is established without any restriction on the space dimension and without geometrical conditions on the domain or its boundary.
Stability of the equation of homomorphism and completeness of the underlying space
Directory of Open Access Journals (Sweden)
Zenon Moszner
2008-01-01
Full Text Available We prove that all assumptions of a Theorem of Forti and Schwaiger (cf. [G. L. Forti, J. Schwaiger, Stability of homomorphisms and completeness, C. R. Math. Rep. Acad. Sci. Canada 11 (1989, 215–220] on the coherence of stability of the equation of homomorphism with the completeness of the space of values of all these homomorphisms, are essential. We give some generalizations of this theorem and certain examples of applications.
Yu, Jie; Liu, Yikan; Yamamoto, Masahiro
2018-04-01
In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic operators and an auxiliary Carleman estimate, we establish local Hölder stability with either partial boundary or interior measurements under certain geometrical conditions. For numerical reconstruction, we minimize a Tikhonov functional which penalizes the gradient of the unknown function. Based on the resulting variational equation, we design an iteration method which is updated by solving a Poisson equation at each step. One-dimensional prototype examples illustrate the numerical performance of the proposed iteration.
Susuki, Yoshihiko; Hikihara Takashi; Chiang, HD
2004-01-01
This paper discusses stability boundaries in an electric power system with dc transmission based on a differential-algebraic equation (DAE) system. The DAE system is derived to analyze transient stability of the ac/dc power system: the differential equation represents the dynamics of the generator and the dc transmission, and the algebraic equation the active and reactive power relationship between the ac system and the dc transmission. In this paper complete characterization of stability bou...
Stabilizing local boundary conditions for two-dimensional shallow water equations
Dia, Ben Mansour
2018-03-27
In this article, we present a sub-critical two-dimensional shallow water flow regulation. From the energy estimate of a set of one-dimensional boundary stabilization problems, we obtain a set of polynomial equations with respect to the boundary values as a requirement for the energy decrease. Using the Riemann invariant analysis, we build stabilizing local boundary conditions that guarantee the stability of the hydrodynamical state around a given steady state. Numerical results for the controller applied to the nonlinear problem demonstrate the performance of the method.
Integral equation based stability analysis of short wavelength drift modes in tokamaks
International Nuclear Information System (INIS)
Hirose, A.; Elia, M.
2003-01-01
Linear stability of electron skin-size drift modes in collisionless tokamak discharges has been investigated in terms of electromagnetic, kinetic integral equations in which neither ions nor electrons are assumed to be adiabatic. A slab-like ion temperature gradient mode persists in such a short wavelength regime. However, toroidicity has a strong stabilizing influence on this mode. In the electron branch, the toroidicity induced skin-size drift mode previously predicted in terms of local kinetic analysis has been recovered. The mode is driven by positive magnetic shear and strongly stabilized for negative shear. The corresponding mixing length anomalous thermal diffusivity exhibits favourable isotope dependence. (author)
Stabilization of the Wave Equation with Boundary Time-Varying Delay
Directory of Open Access Journals (Sweden)
Hao Li
2014-01-01
Full Text Available We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying delay term in the time-varying, weakly nonlinear boundary feedbacks. By the Riemannian geometry methods and a suitable assumption of nonlinearity, we obtain the uniform decay of the energy of the closed loop system.
On the stability of pexider functional equation in non-archimedean spaces
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Vaezpour Seiyed
2011-01-01
Full Text Available Abstract In this paper, the Hyers-Ulam stability of the Pexider functional equation in a non-Archimedean space is investigated, where σ is an involution in the domain of the given mapping f. MSC 2010:26E30, 39B52, 39B72, 46S10
Bonito, Andrea
2013-10-03
We establish the L2-stability of an entropy viscosity technique applied to nonlinear scalar conservation equations. First-and second-order explicit time-stepping techniques using continuous finite elements in space are considered. The method is shown to be stable independently of the polynomial degree of the space approximation under the standard CFL condition. © 2013 American Mathematical Society.
Stability of finite difference schemes for generalized von Foerster equations with renewal
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Henryk Leszczyński
2014-01-01
Full Text Available We consider a von Foerster-type equation describing the dynamics of a population with the production of offsprings given by the renewal condition. We construct a finite difference scheme for this problem and give sufficient conditions for its stability with respect to \\(l^1\\ and \\(l^\\infty\\ norms.
On a Uniform Fuzzy Direct Method for Stability of Functional Equations
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N. Eghbali∗
2015-09-01
Full Text Available In this paper, we consider the Hyers-Ulam-Rassias stability of the following equation f(x = af(h(x + bf(−h(x in the fuzzy normed spaces with some conditions imposed on the constants a, b and the function h on a nonempty set X.
Stability of Pexiderized Quadratic Functional Equation in Random 2-Normed Spaces
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Mohammed A. Alghamdi
2015-01-01
Full Text Available The aim of this paper is to investigate the stability of Hyers-Ulam-Rassias type theorems by considering the pexiderized quadratic functional equation in the setting of random 2-normed spaces (RTNS, while the concept of random 2-normed space has been recently studied by Goleţ (2005.
Stability of a simple Levi–Civitá functional equation on non-unital ...
Indian Academy of Sciences (India)
g(x) h(y)+k(y) on non-unital commutative semigroups by investigating the functional inequalities |f (x +y)−f ... [3], [14] and [11]. In this paper, we investigate the Hyers–Ulam stability of the functional equation (1.2) ...... supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) ...
Stabilization of the higher order nonlinear Schrödinger equation with ...
Indian Academy of Sciences (India)
18
can describe the propagation of ultrashort light pulses which are shorter than ∼ 10−13s. In practical applications, sometimes we may want to prevent the propagation of some signal in optical fibre, the stabilization of the HNLS equation can be viewed as a good mathematical representation of this phenomenon. To state the ...
Approximation of the unsteady Brinkman-Forchheimer equations by the pressure stabilization method
Louaked, Mohammed
2017-07-20
In this work, we propose and analyze the pressure stabilization method for the unsteady incompressible Brinkman-Forchheimer equations. We present a time discretization scheme which can be used with any consistent finite element space approximation. Second-order error estimate is proven. Some numerical results are also given.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017
Analytical Approximation Methods for the Stabilizing Solution of the Hamilton-Jacobi Equation
Sakamoto, Noboru; van der Schaft, Arjan J.
2008-01-01
In this paper, two methods for approximating the stabilizing solution of the Hamilton-Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable
Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation
Sakamoto, Noboru; Schaft, Arjan J. van der
2008-01-01
In this paper, two methods for approximating the stabilizing solution of the Hamilton–Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable
DEFF Research Database (Denmark)
Backi, Christoph Josef; Bendtsen, Jan Dimon; Leth, John
2015-01-01
In this work the stability properties of a partial differential equation (PDE) with state-dependent parameters and asymmetric boundary conditions are investigated. The PDE describes the temperature distribution inside foodstuff, but can also hold for other applications and phenomena. We show...
Directory of Open Access Journals (Sweden)
Xu Liguang
2009-01-01
Full Text Available The exponential stability of singularly perturbed impulsive delay integrodifferential equations (SPIDIDEs is concerned. By establishing an impulsive delay integrodifferential inequality (IDIDI, some sufficient conditions ensuring the exponentially stable of any solution of SPIDIDEs for sufficiently small are obtained. A numerical example shows the effectiveness of our theoretical results.
Oscillation and asymptotic stability of a delay differential equation with Richard's nonlinearity
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Leonid Berezansky
2005-04-01
Full Text Available We obtain sufficient conditions for oscillation of solutions, and for asymptotical stability of the positive equilibrium, of the scalar nonlinear delay differential equation $$ frac{dN}{dt} = r(tN(tBig[a-Big(sum_{k=1}^m b_k N(g_k(tBig^{gamma}Big], $$ where $ g_k(tleq t$.
New stability and boundedness results to Volterra integro-differential equations with delay
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Cemil Tunç
2016-04-01
Full Text Available In this paper, we consider a certain non-linear Volterra integro-differential equations with delay. We study stability and boundedness of solutions. The technique of proof involves defining suitable Lyapunov functionals. Our results improve and extend the results obtained in literature.
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Erdal Korkmaz
2017-06-01
Full Text Available Abstract In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov’s second method. The results obtained essentially improve, include and complement the results in the literature.
Korkmaz, Erdal
2017-01-01
In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov's second method. The results obtained essentially improve, include and complement the results in the literature.
Generalized Hyers-Ulam Stability of Quadratic Functional Equations: A Fixed Point Approach
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Choonkil Park
2008-03-01
Full Text Available Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation f(2x+y=4f(x+f(y+f(x+yÃ¢ÂˆÂ’f(xÃ¢ÂˆÂ’y in Banach spaces.
Symbolic-numeric stability investigations of Jameson's scheme for this-layer Navier-Stokes equations
Ganzha, V.G.; Vorozhtsov, E.V.; Boers, J.; van Hulzen, J.A.; van Hulzen, J.A.
1994-01-01
The Navier-Stokes equations governing the three-dimensional flows of viscous, compressible, heat-conducting gas and augmented by turbulence modeling present the most realistic model for gas flows around the elements of aircraft configurations. We study the stability of one of the Jameson's schemes
Chen, Guiling
2013-01-01
This thesis studies asymptotic behavior and stability of determinsitic and stochastic delay differential equations. The approach used in this thesis is based on fixed point theory, which does not resort to any Liapunov function or Liapunov functional. The main contribution of this thesis is to study
Stabilization of the higher order nonlinear Schrödinger equation with ...
Indian Academy of Sciences (India)
18
Abstract. We study the internal stabilization of the higher order nonlinear Schrödinger equation with constant coefficients. ... cattering, respectively. (1.2) also arised in the optical communications (see [1, 8, 16]). They can be applied to the long distance communications and ultrafast signalrouting systems. The HNLS models.
A result concerning the stability of some difference equations and its ...
Indian Academy of Sciences (India)
M. Senthilkumar (Newgen Imaging) 1461 1996 Oct 15 13:05:22
Ulam stability problem for eq. (1.1). That is, the difference equation (1.1) is stable in the sense of Hyers and Ulam under the conditions subject to. (2.1) and (2.2). Theorem 2.1. Suppose that functions f, ψ : N × N → R and ϕ satisfy the inequality.
Role of secondary instability theory and parabolized stability equations in transition modeling
El-Hady, Nabil M.; Dinavahi, Surya P.; Chang, Chau-Lyan; Zang, Thomas A.
1993-01-01
In modeling the laminar-turbulent transition region, the designer depends largely on benchmark data from experiments and/or direct numerical simulations that are usually extremely expensive. An understanding of the evolution of the Reynolds stresses, turbulent kinetic energy, and quantifies in the transport equations like the dissipation and production is essential in the modeling process. The secondary instability theory and the parabolized stability equations method are used to calculate these quantities, which are then compared with corresponding quantities calculated from available direct numerical simulation data for the incompressible boundary-layer flow of laminar-turbulent transition conditions. The potential of the secondary instability theory and the parabolized stability equations approach in predicting these quantities is discussed; results indicate that inexpensive data that are useful for transition modeling in the early stages of the transition region can be provided by these tools.
Spectral stability of undercompressive shock profile solutions of a modified KdV-Burgers equation
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Jeff Dodd
2007-10-01
Full Text Available It is shown that certain undercompressive shock profile solutions of the modified Korteweg-de Vries-Burgers equation $$ partial_t u + partial_x(u^3 = partial_x^3 u + alpha partial_x^2 u, quad alpha geq 0 $$ are spectrally stable when $alpha$ is sufficiently small, in the sense that their linearized perturbation equations admit no eigenvalues having positive real part except a simple eigenvalue of zero (due to the translation invariance of the linearized perturbation equations. This spectral stability makes it possible to apply a theory of Howard and Zumbrun to immediately deduce the asymptotic orbital stability of these undercompressive shock profiles when $alpha$ is sufficiently small and positive.
Zhang, Ling
2017-01-01
The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs). It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order [Formula: see text] to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.
On the Gross–Pitaevskii Equation with Pumping and Decay: Stationary States and Their Stability
Sierra Nunez, Jesus Alfredo
2015-02-11
We investigate the behavior of solutions of the complex Gross–Pitaevskii equation, a model that describes the dynamics of pumped decaying Bose–Einstein condensates. The stationary radially symmetric solutions of the equation are studied, and their linear stability with respect to two-dimensional perturbations is analyzed. Using numerical continuation, we calculate not only the ground state of the system, but also a number of excited states. Accurate numerical integration is employed to study the general nonlinear evolution of the system from the unstable stationary solutions to the formation of stable vortex patterns.
Stability of line solitons for the KP-II equation in R2
Mizumachi, Tetsu
2015-01-01
The author proves nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as x\\to\\infty. He finds that the amplitude of the line soliton converges to that of the line soliton at initial time whereas jumps of the local phase shift of the crest propagate in a finite speed toward y=\\pm\\infty. The local amplitude and the phase shift of the crest of the line solitons are described by a system of 1D wave equations with diffraction terms.
Wu, Jie; Zhan, Xi-Sheng; Zhang, Xian-He; Gao, Hong-Liang
2012-05-01
A kind of discrete logistic model with distributed delays obtained by the Euler method is investigated, where the discrete delay τ is regarded as a parameter. By analyzing the associated characteristic equation, it is found that the stability of the positive equilibrium and Hopf occurs when τ crosses some critical value. Then the explicit formulae which determine the stability, direction and other properties of the bifurcating periodic solution are derived by using the theory of normal form and center manifold. Finally, numerical simulations are performed to verify and illustrate the analytical results.
Directory of Open Access Journals (Sweden)
Cemil Tunç
2017-10-01
Full Text Available In this article, the authors obtain some clear assumptions for the asymptotic stability (AS and boundedness (B of solutions of non-linear retarded Volterra integro-differential equations (VIDEs of first order by constructing a new Lyapunov functional (LF. The results obtained are new and differ from those found in the literature, and they also contain and improve a result found in the literature under more less restrictive conditions. We establish an example and give a discussion to indicate the applicability of the weaker conditions obtained. We also employ MATLAB-Simulink to display the behaviors of the orbits of the (VIDEs considered. Keywords: Nonlinear, Volterra integro-differential equations, First order, Asymptotic stability, Boundedness, Lyapunov functional, MSC: 34D05, 34K20, 45J05
On global asymptotic stability of solutions of a system of ordinary differential equations
Filimonov, M. Yu.
2017-12-01
To investigate global asymptotic stability in general of an equilibrium position of an autonomous system of ordinary differential equations, considered by V.A. Pliss, a function different from the Lyapunov functions is applied. V.A. Pliss proved that for this system it is impossible to construct the Lyapunov function as a sum of a quadratic form and an integral of some nonlinear function defined by the right-hand side of the system.
The stability of coupled renewal-differential equations with econometric applications
Rhoten, R. P.; Aggarwal, J. K.
1969-01-01
Concepts and results are presented in the fields of mathematical modeling, economics, and stability analysis. A coupled renewal-differential equation structure is presented as a modeling form for systems possessing hereditary characteristics, and this structure is applied to a model of the Austrian theory of business cycles. For realistic conditions, the system is shown to have an infinite number of poles, and conditions are presented which are both necessary and sufficient for all poles to lie strictly in the left half plane.
Directory of Open Access Journals (Sweden)
Xiaolin Zhu
2014-01-01
Full Text Available This paper studies the T-stability of the Heun method and balanced method for solving stochastic differential delay equations (SDDEs. Two T-stable conditions of the Heun method are obtained for two kinds of linear SDDEs. Moreover, two conditions under which the balanced method is T-stable are obtained for two kinds of linear SDDEs. Some numerical examples verify the theoretical results proposed.
The generalized hyers-ulam stability of sextic functional equation in ...
African Journals Online (AJOL)
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following generalized sextic functional equation. Df (x, y):= f (mx+ y) +f (mx− y) +f(x+ my) +f(x− my) − (m4+ m2) [f(x+ y+f(x− y)] −2(m6− m4− m2+ 1) [f(x) + f(y)]. in matrix fuzzy normed spaces. Furthermore, using the fixed point method, we also ...
Variational Approach to the Orbital Stability of Standing Waves of the Gross-Pitaevskii Equation
Hadj Selem, Fouad
2014-08-26
This paper is concerned with the mathematical analysis of a masssubcritical nonlinear Schrödinger equation arising from fiber optic applications. We show the existence and symmetry of minimizers of the associated constrained variational problem. We also prove the orbital stability of such solutions referred to as standing waves and characterize the associated orbit. In the last section, we illustrate our results with few numerical simulations. © 2014 Springer Basel.
Ulam stability for fractional differential equations in the sense of Caputo operator
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Rabha W. Ibrahim
2012-12-01
Full Text Available In this paper, we consider the Hyers-Ulam stability for the following fractional differential equations, in the sense ofcomplex Caputo fractional derivative defined, in the unit disk: cDßzf(z=G(f(z, cDázf(z,zf‘(z;z 0<á<1<ß<2 . Furthermore,a generalization of the admissible functions in complex Banach spaces is imposed and applications are illustrated.
Gerbi, Stéphane
2011-12-01
In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the KelvinVoigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained. © 2011 Elsevier Ltd. All rights reserved.
Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue
2018-01-01
This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0 . Furthermore, we prove the global existence and uniqueness of C^{α ,β } -solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1 -space. The exponential convergence rate is also derived.
One-way spatial integration of Navier-Stokes equations: stability of wall-bounded flows
Rigas, Georgios; Colonius, Tim; Towne, Aaron; Beyar, Michael
2016-11-01
For three-dimensional flows, questions of stability, receptivity, secondary flows, and coherent structures require the solution of large partial-derivative eigenvalue problems. Reduced-order approximations are thus required for engineering prediction since these problems are often computationally intractable or prohibitively expensive. For spatially slowly evolving flows, such as jets and boundary layers, a regularization of the equations of motion sometimes permits a fast spatial marching procedure that results in a huge reduction in computational cost. Recently, a novel one-way spatial marching algorithm has been developed by Towne & Colonius. The new method overcomes the principle flaw observed in Parabolized Stability Equations (PSE), namely the ad hoc regularization that removes upstream propagating modes. The one-way method correctly parabolizes the flow equations based on estimating, in a computationally efficient way, the local spectrum in each cross-stream plane and an efficient spectral filter eliminates modes with upstream group velocity. Results from the application of the method to wall-bounded flows will be presented and compared with predictions from the full linearized compressible Navier-Stokes equations and PSE.
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Park Choonkil
2007-01-01
Full Text Available We prove the Hyers-Ulam-Rassias stability of homomorphisms in real Banach algebras and of generalized derivations on real Banach algebras for the following Cauchy-Jensen functional equations: , , which were introduced and investigated by Baak (2006. The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias' stability theorem that appeared in his paper (1978.
Stability of stationary states of non-local equations with singular interaction potentials
Fellner, Klemens
2011-04-01
We study the large-time behaviour of a non-local evolution equation for the density of particles or individuals subject to an external and an interaction potential. In particular, we consider interaction potentials which are singular in the sense that their first derivative is discontinuous at the origin.For locally attractive singular interaction potentials we prove under a linear stability condition local non-linear stability of stationary states consisting of a finite sum of Dirac masses. For singular repulsive interaction potentials we show the stability of stationary states of uniformly bounded solutions under a convexity condition.Finally, we present numerical simulations to illustrate our results. © 2010 Elsevier Ltd.
Kuehl, Joseph
2016-11-01
The parabolized stability equations (PSE) have been developed as an efficient and powerful tool for studying the stability of advection-dominated laminar flows. In this work, a new "wavepacket" formulation of the PSE is presented. This method accounts for the influence of finite-bandwidth-frequency distributions on nonlinear stability calculations. The methodology is motivated by convolution integrals and is found to appropriately represent nonlinear energy transfer between primary modes and harmonics, in particular nonlinear feedback, via a "nonlinear coupling coefficient." It is found that traditional discrete mode formulations overestimate nonlinear feedback by approximately 70%. This results in smaller maximum disturbance amplitudes than those observed experimentally. The new formulation corrects this overestimation, accounts for the generation of side lobes responsible for spectral broadening and results in disturbance saturation amplitudes consistent with experiment. A Mach 6 flared-cone example is presented. Support from the AFOSR Young Investigator Program via Grant FA9550-15-1-0129 is gratefully acknowledges.
Principles and equations for measuring and interpreting protein stability: From monomer to tetramer.
Bedouelle, Hugues
2016-02-01
The ability to measure the thermodynamic stability of proteins with precision is important for both academic and applied research. Such measurements rely on mathematical models of the protein denaturation profile, i.e. the relation between a global protein signal, corresponding to the folding states in equilibrium, and the variable value of a denaturing agent, either heat or a chemical molecule, e.g. urea or guanidinium hydrochloride. In turn, such models rely on a handful of physical laws: the laws of mass action and conservation, the law that relates the protein signal and concentration, and the one that relates stability and denaturant value. So far, equations have been derived mainly for the denaturation profiles of homomeric proteins. Here, we review the underlying basic physical laws and show in detail how to derive model equations for the unfolding equilibria of homomeric or heteromeric proteins up to trimers and potentially tetramers, with or without folding intermediates, and give full demonstrations. We show that such equations cannot be derived for pentamers or higher oligomers except in special degenerate cases. We expand the method to signals that do not correspond to extensive protein properties. We review and expand methods for uncovering hidden intermediates of unfolding. Finally, we review methods for comparing and interpreting the thermodynamic parameters that derive from stability measurements for cognate wild-type and mutant proteins. This work should provide a robust theoretical basis for measuring the stability of complex proteins. Copyright © 2015 Elsevier B.V. and Société Française de Biochimie et Biologie Moléculaire (SFBBM). All rights reserved.
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Roberts Jason A
2006-01-01
Full Text Available We consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation , where α, β, σ, τ ≥ 0 are real constants, and W(t is a standard Wiener process. The areas of the regions of asymptotic stability for the class of methods considered, indicated by the sufficient conditions for the discrete system, are shown to be equal in size to each other and we show that an upper bound can be put on the time-step parameter for the numerical method for which the system is asymptotically mean-square stable. We illustrate our results by means of numerical experiments and various stability diagrams. We examine the extent to which the continuous system can tolerate stochastic perturbations before losing its stability properties and we illustrate how one may accurately choose a numerical method to preserve the stability properties of the original problem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high.
International Nuclear Information System (INIS)
Gorshkov, A V
2003-01-01
The problem of the stabilization of a semilinear equation in the exterior of a bounded domain is considered. In view of the impossibility of an exponential stabilization of the form e -σt of the solution of a parabolic equation in an unbounded domain no matter what the boundary control is, one poses the problem of power-like stabilization by means of a boundary control. For a fixed initial condition and parameter k>0 of the rate of stabilization the existence of a boundary control such that the solution approaches zero at the rate 1/t k is demonstrated
Fixed Points and Fuzzy Stability of Functional Equations Related to Inner Product
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Hassan Azadi Kenary
2012-04-01
Full Text Available In , Th.M. Rassias introduced the following equality sum_{i,j=1}^m |x_i - x_j |^2 = 2m sum_{i=1}^m|x_i|^2, qquad sum_{i=1}^m x_i =0 for a fixed integer $m ge 3$. Let $V, W$ be real vector spaces. It is shown that if a mapping $f : V ightarrow W$ satisfies sum_{i,j=1}^m f(x_i - x_j = 2m sum_{i=1}^m f(x_i for all $x_1, ldots, x_{m} in V$ with $sum_{i=1}^m x_i =0$, then the mapping $f : V ightarrow W$ is realized as the sum of an additive mapping and a quadratic mapping. From the above equality we can define the functional equation f(x-y +f(2x+y + f(x+2y= 3f(x+ 3f(y + 3f(x+y , which is called a {it quadratic functional equation}. Every solution of the quadratic functional equation is said to be a {it quadratic mapping}. Using fixed point theorem we prove the Hyers-Ulam stability of the functional equation ( in fuzzy Banach spaces.
Conditional stability in determination of initial data for stochastic parabolic equations
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Yuan, Ganghua
2017-01-01
In this paper, we solve two kinds of inverse problems in determination of the initial data for stochastic parabolic equations. One is determination of the initial data by lateral boundary observation on arbitrary portion of the boundary, the second one is determination of the initial data by internal observation in a subregion inside the domain. We obtain conditional stability for the two kinds of inverse problems. To prove the results, we estimate the initial data by a terminal observation near the initial time, then we estimate this terminal observation by lateral boundary observation on arbitrary portion of the boundary or internal observation in a subregion inside the domain. To achieve those goals, we derive several new Carleman estimates for stochastic parabolic equations in this paper. (paper)
Conditional stability in determination of initial data for stochastic parabolic equations
Yuan, Ganghua
2017-03-01
In this paper, we solve two kinds of inverse problems in determination of the initial data for stochastic parabolic equations. One is determination of the initial data by lateral boundary observation on arbitrary portion of the boundary, the second one is determination of the initial data by internal observation in a subregion inside the domain. We obtain conditional stability for the two kinds of inverse problems. To prove the results, we estimate the initial data by a terminal observation near the initial time, then we estimate this terminal observation by lateral boundary observation on arbitrary portion of the boundary or internal observation in a subregion inside the domain. To achieve those goals, we derive several new Carleman estimates for stochastic parabolic equations in this paper.
Stability study of pre-stack seismic inversion based on the full Zoeppritz equation
Liang, Lifeng; Zhang, Hongbing; Guo, Qiang; Saeed, Wasif; Shang, Zuoping; Huang, Guojiao
2017-10-01
Pre-stack seismic inversion is highly important and complicated. Its result is non-unique, and the process is unstable because pre-stack seismic inversion is an ill-posed problem that simultaneously obtains the results of multiple parameters. Combining the full Zoeppritz equation and additional assumptions with edge-preserving regularization (EPR) can help mitigate the problem. To achieve this combination, we developed an inversion method by constructing a new objective function, which includes the EPR and the Markov random field. The method directly gains reflectivity R PP by the full Zoeppritz equation instead of its approximations and effectively controls the stability of simultaneous inversion by two additional assumptions: the sectional constant V S/V P and the generalized Gardner equation. Thus, the simultaneous inversion of multiple parameters is directed toward to V P, ΔL S (the fitting deviation of V S) and density, and the generalized Gardner equation is regarded as a constraint from which the fitting relationship is derived. We applied the fast simulated annealing algorithm to solve the nonlinear optimization problem. The test results on 2D synthetic data indicated that the stability of simultaneous inversion for V P, ΔL S and density is better than these for V P, V S, and density. The inverted result of density gradually worsens as the deviation ΔL D (the fitting deviation of the density) increases. Moreover, the inverted results were acceptable when using the fitting relationships with error, although they showed varying degrees of influence. We constructed time-varying and space-varying fitting relationships using the logging data in pre-stack inversion of the field seismic data. This improved the inverted results of the simultaneous inversion for complex geological models. Finally, the inverted results of the field data distinctly revealed more detailed information about the layers and matched well with the logging data along the wells over most
Stability of the Cauchy Additive Functional Equation on Tangle Space and Applications
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Soo Hwan Kim
2016-01-01
Full Text Available We introduce real tangle and its operations, as a generalization of rational tangle and its operations, to enumerating tangles by using the calculus of continued fraction and moreover we study the analytical structure of tangles, knots, and links by using new operations between real tangles which need not have the topological structure. As applications of the analytical structure, we prove the generalized Hyers-Ulam stability of the Cauchy additive functional equation fx⊕y=fx⊕fy in tangle space which is a set of real tangles with analytic structure and describe the DNA recombination as the action of some enzymes on tangle space.
The Explicit Green's Approach with stability enhancement for solving the bioheat transfer equation
Loureiro, FS; Mansur, WJ; Wrobel, LC; Silva, JEA
2014-01-01
The aim of this paper is to propose a strategy for performing a stability enhancement into the Explicit Green’s Approach (ExGA) method applied to the bioheat transfer equation. The ExGA method is a time-stepping technique that uses numerical Green’s functions in the time domain; these functions are here computed by the FEM. Basically, a new two nonequal time substeps procedure is proposed to compute Green’s functions at the first time step. This is accomplished by adopting the standard explic...
Multidimensional stability of traveling fronts in combustion and non-KPP monostable equations
Bu, Zhen-Hui; Wang, Zhi-Cheng
2018-02-01
This paper is concerned with the multidimensional stability of traveling fronts for the combustion and non-KPP monostable equations. Our study contains two parts: in the first part, we first show that the two-dimensional V-shaped traveling fronts are asymptotically stable in R^{n+2} with n≥1 under any (possibly large) initial perturbations that decay at space infinity, and then, we prove that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which implies that even very small perturbations to the V-shaped traveling front can lead to permanent oscillation. In the second part, we establish the multidimensional stability of planar traveling front in R^{n+1} with n≥1.
Stability of linear delay differential equations a numerical approach with Matlab
Breda, Dimitri; Vermiglio, Rossana
2015-01-01
This book presents the authors' recent work on the numerical methods for the stability analysis of linear autonomous and periodic delay differential equations, which consist in applying pseudospectral techniques to discretize either the solution operator or the infinitesimal generator and in using the eigenvalues of the resulting matrices to approximate the exact spectra. The purpose of the book is to provide a complete and self-contained treatment, which includes the basic underlying mathematics and numerics, examples from population dynamics and engineering applications, and Matlab programs implementing the proposed numerical methods. A number of proofs is given to furnish a solid foundation, but the emphasis is on the (unifying) idea of the pseudospectral technique for the stability analysis of DDEs. It is aimed at advanced students and researchers in applied mathematics, in dynamical systems and in various fields of science and engineering, concerned with delay systems. A relevant feature of the book is t...
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Xue-Lian Jin
2017-01-01
Full Text Available The exponential stability of the monotubular heat exchanger equation with boundary observation possessing a time delay and inner control was investigated. Firstly, the close-loop system was translated into an abstract Cauchy problem in the suitable state space. A uniformly bounded C0-semigroup generated by the close-loop system, which implies that the unique solution of the system exists, was shown. Secondly, the spectrum configuration of the closed-loop system was analyzed and the eventual differentiability and the eventual compactness of the semigroup were shown by the resolvent estimates on some resolvent sets. This implies that the spectrum-determined growth assumption holds. Finally, a sufficient condition, which is related to the physical parameters in the system and is independent of the time delay, of the exponential stability of the closed-loop system was given.
Lyapunov stability and Poisson structure of the thermal TDHF and RPA equations
International Nuclear Information System (INIS)
Veneroni, M.; Balian, R.
1989-01-01
The thermal TDHF equation is analyzed in the Liouville representation of quantum mechanics, where the matrix elements of the single-particle (s.p.) density ρ behave as classical dynamical variables. By introducing the Lie-Poisson bracket associated with the unitary group of the s.p. Hilbert space, we show that TDHF has a hamiltonian, but non-canonical, classical form. Within this Poisson structure, either the s.p. energy or the s.p. grand potential Ω(ρ) act as a Hamilton function. The Lyapunov stability of both the TDHF and RPA equations around a HF state then follows, since the HF approximation for thermal equilibrium is determined by minimizing Ω(ρ). The RPA matrix in the Liouville space is expressed as the product of the Poisson tensor with the HF stability matrix, interpreted as a metric tensor generated by the entropy. This factorization displays the roles of the energy and entropy terms arising from Ω(ρ) in the RPA dynamics, and it helps to construct the RPA modes. Several extensions are considered
Stability and oscillation of two coupled Duffing equations with time delay state feedback
International Nuclear Information System (INIS)
El-Bassiouny, A F
2006-01-01
This paper presents an analytical study of the simultaneous principal parametric resonances of two coupled Duffing equations with time delay state feedback. The concept of an equivalent damping related to the delay feedback is proposed and the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. The method of multiple scales is used to determine a set of ordinary differential equations governing the modulation of the amplitudes and phases of the two modes. The first order approximation of the resonances are derived and the effect of time delay on the resonances is investigated. The fixed points correspond to a periodic motion for the starting system and we show the frequency-response curves. We analyse the effect of time delay and the other different parameters on these oscillations. The stability of the fixed points is examined by using the variational method. Numerical solutions are carried out and graphical representations of the results are presented and discussed. Increasing in the time delay τ given decreasing and increasing in the regions of definition and stability respectively and the first mode has decreased magnitudes. The multivalued solutions disappear when decreasing the coefficients of cubic nonlinearities of the second mode α 3 and the detuning parameter σ 2 respectively. Both modes shift to the left for increasing linear feedback gain v 1 and the coefficient of parametric excitation f 1 respectively
Energy Technology Data Exchange (ETDEWEB)
Zhou, Xiafeng, E-mail: zhou-xf11@mails.tsinghua.edu.cn; Guo, Jiong, E-mail: guojiong12@tsinghua.edu.cn; Li, Fu, E-mail: lifu@tsinghua.edu.cn
2015-12-15
Highlights: • NEMs are innovatively applied to solve convection diffusion equation. • Stability, accuracy and numerical diffusion for NEM are analyzed for the first time. • Stability and numerical diffusion depend on the NEM expansion order and its parity. • NEMs have higher accuracy than both second order upwind and QUICK scheme. • NEMs with different expansion orders are integrated into a unified discrete form. - Abstract: The traditional finite difference method or finite volume method (FDM or FVM) is used for HTGR thermal-hydraulic calculation at present. However, both FDM and FVM require the fine mesh sizes to achieve the desired precision and thus result in a limited efficiency. Therefore, a more efficient and accurate numerical method needs to be developed. Nodal expansion method (NEM) can achieve high accuracy even on the coarse meshes in the reactor physics analysis so that the number of spatial meshes and computational cost can be largely decreased. Because of higher efficiency and accuracy, NEM can be innovatively applied to thermal-hydraulic calculation. In the paper, NEMs with different orders of basis functions are successfully developed and applied to multi-dimensional steady convection diffusion equation. Numerical results show that NEMs with three or higher order basis functions can track the reference solutions very well and are superior to second order upwind scheme and QUICK scheme. However, the false diffusion and unphysical oscillation behavior are discovered for NEMs. To explain the reasons for the above-mentioned behaviors, the stability, accuracy and numerical diffusion properties of NEM are analyzed by the Fourier analysis, and by comparing with exact solutions of difference and differential equation. The theoretical analysis results show that the accuracy of NEM increases with the expansion order. However, the stability and numerical diffusion properties depend not only on the order of basis functions but also on the parity of
Stability analysis of Eulerian-Lagrangian methods for the one-dimensional shallow-water equations
Casulli, V.; Cheng, R.T.
1990-01-01
In this paper stability and error analyses are discussed for some finite difference methods when applied to the one-dimensional shallow-water equations. Two finite difference formulations, which are based on a combined Eulerian-Lagrangian approach, are discussed. In the first part of this paper the results of numerical analyses for an explicit Eulerian-Lagrangian method (ELM) have shown that the method is unconditionally stable. This method, which is a generalized fixed grid method of characteristics, covers the Courant-Isaacson-Rees method as a special case. Some artificial viscosity is introduced by this scheme. However, because the method is unconditionally stable, the artificial viscosity can be brought under control either by reducing the spatial increment or by increasing the size of time step. The second part of the paper discusses a class of semi-implicit finite difference methods for the one-dimensional shallow-water equations. This method, when the Eulerian-Lagrangian approach is used for the convective terms, is also unconditionally stable and highly accurate for small space increments or large time steps. The semi-implicit methods seem to be more computationally efficient than the explicit ELM; at each time step a single tridiagonal system of linear equations is solved. The combined explicit and implicit ELM is best used in formulating a solution strategy for solving a network of interconnected channels. The explicit ELM is used at channel junctions for each time step. The semi-implicit method is then applied to the interior points in each channel segment. Following this solution strategy, the channel network problem can be reduced to a set of independent one-dimensional open-channel flow problems. Numerical results support properties given by the stability and error analyses. ?? 1990.
International Nuclear Information System (INIS)
Ignat'yev, A O
2003-01-01
A system of ordinary differential equations with impulse action at fixed moments of time is considered. The system is assumed to have the zero solution. It is shown that the existence of a corresponding Lyapunov function is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution. Restrictions on perturbations of the right-hand sides of differential equations and impulse actions are obtained under which the uniform asymptotic stability of the zero solution of the 'unperturbed' system implies the uniform asymptotic stability of the zero solution of the 'perturbed' system
Energy Technology Data Exchange (ETDEWEB)
Tokuda, Shinji [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment; Watanabe, Tomoko
1996-08-01
The matching problem in resistive MagnetoHydroDynamic stability analysis by the asymptotic matching method has been reformulated as an initial-boundary value problem for the inner-layer equations describing the plasma dynamics in the thin layer around a rational surface. The third boundary conditions at boundaries of a finite interval are imposed on the inner layer equations in the formulation instead of asymptotic conditions at infinities. The finite difference method for this problem has been applied to model equations whose solutions are known in a closed form. It has been shown that the initial value problem and the associated eigenvalue problem for the model equations can be solved by the finite difference method with numerical stability. The formulation presented here enables the asymptotic matching method to be a practical method for the resistive MHD stability analysis. (author)
Soliton stability criterion for generalized nonlinear Schrödinger equations.
Quintero, Niurka R; Mertens, Franz G; Bishop, A R
2015-01-01
A stability criterion for solitons of the driven nonlinear Schrödinger equation (NLSE) has been conjectured. The criterion states that p'(v)0 is a necessary condition for stability; here, v is the soliton velocity and p=P/N, where P and N are the soliton momentum and norm, respectively. To date, the curve p(v) was calculated approximately by a collective coordinate theory, and the criterion was confirmed by simulations. The goal of this paper is to calculate p(v) exactly for several classes and cases of the generalized NLSE: a soliton moving in a real potential, in particular a time-dependent ramp potential, and a time-dependent confining quadratic potential, where the nonlinearity in the NLSE also has a time-dependent coefficient. Moreover, we investigate a logarithmic and a cubic NLSE with a time-independent quadratic potential well. In the latter case, there is a bisoliton solution that consists of two solitons with asymmetric shapes, forming a bound state in which the shapes and the separation distance oscillate. Finally, we consider a cubic NLSE with parametric driving. In all cases, the p(v) curve is calculated either analytically or numerically, and the stability criterion is confirmed.
Variational approach to stability boundary for the Taylor-Goldstein equation
Hirota, Makoto; Morrison, Philip J.
2015-11-01
Linear stability of inviscid stratified shear flow is studied by developing an efficient method for finding neutral (i.e., marginally stable) solutions of the Taylor-Goldstein equation. The classical Miles-Howard criterion states that stratified shear flow is stable if the local Richardson number JR is greater than 1/4 everywhere. In this work, the case of JR > 0 everywhere is considered by assuming strictly monotonic and smooth profiles of the ambient shear flow and density. It is shown that singular neutral modes that are embedded in the continuous spectrum can be found by solving one-parameter families of self-adjoint eigenvalue problems. The unstable ranges of wavenumber are searched for accurately and efficiently by adopting this method in a numerical algorithm. Because the problems are self-adjoint, the variational method can be applied to ascertain the existence of singular neutral modes. For certain shear flow and density profiles, linear stability can be proven by showing the non-existence of a singular neutral mode. New sufficient conditions, extensions of the Rayleigh-Fjortoft stability criterion for unstratified shear flows, are derived in this manner. This work was supported by JSPS Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation # 55053270.
International Nuclear Information System (INIS)
Fujimura, Kaoru
1980-11-01
The numerical treatment of Orr-Sommerfeld equation which is the fundamental equation of linear hydrodynamic stability theory is described. Present calculation procedure is applied to the two-dimensional quasi-parallel flow for which linearized disturbance equation (Orr-Sommerfeld equation) contains one simple turning point and αR >> 1. The numerical procedure for this problem and one numerical example for Jeffery-Hamel flow (J-H III 1 ) are presented. These treatment can be extended to the other velocity profiles by slight midifications. (author)
Li, Bing; Li, Yongkun; Zhang, Xuemei
2016-01-01
In this paper, by using the existence of the exponential dichotomy of linear dynamic equations on time scales and the theory of calculus on time scales, we study the existence and global exponential stability of periodic solutions for a class of n-dimensional neutral dynamic equations on time scales. We also present an example to illustrate the feasibility of our results. The results of this paper are completely new and complementary to the previously known results even in both the case of differential equations (time scale [Formula: see text]) and the case of difference equations (time scale [Formula: see text]).
International Nuclear Information System (INIS)
Fewo, S I; Kenfack-Jiotsa, A; Kofane, T C
2006-01-01
With the help of the one-dimensional quintic complex Ginzburg-Landau equation (CGLE) as perturbations of the nonlinear Schroedinger equation (NLSE), we derive the equations of motion of pulse parameters called collective variables (CVs), of a pulse propagating in dispersion-managed (DM) fibre optic links. The equations obtained are investigated numerically in order to view the evolution of pulse parameters along the propagation distance, and also to analyse effects of initial amplitude and width on the propagating pulse. Nonlinear gain is shown to be beneficial in stabilizing DM solitons. A fully numerical simulation of the one-dimensional quintic CGLE as perturbations of NLSE finally tests the results of the CV theory. A good agreement is observed between both methods
Nonlinear Stability of ρ-Functional Equations in Latticetic Random Banach Lattice Spaces
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Mohammad Maleki V.
2018-02-01
Full Text Available In this paper, we prove the generalized nonlinear stability of the first and second of the following ρ -functional equations, G ( | a | Δ A * | b | Δ B * G ( | a | Δ A * * | b | − G ( | a | Δ B * * G ( | b | = ρ ( 2 G | a | Δ A * | b | 2 Δ B * G | a | Δ A * * | b | 2 − G ( | a | Δ B * * G ( | b | , and 2 G | a | Δ A * | b | 2 Δ B * G | a | Δ A * * | b | 2 − G ( | a | Δ B * * G ( | b | = ρ G ( | a | Δ A * | b | Δ B * G ( | a | Δ A * * | b | − G ( | a | Δ B * * G ( | b | in latticetic random Banach lattice spaces, where ρ is a fixed real or complex number with ρ ≠ 1 .
On the Generalized Hyers-Ulam-Rassias Stability of Quadratic Functional Equations
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M. Eshaghi Gordji
2009-01-01
Full Text Available We achieve the general solution and the generalized Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities for quadratic functional equations f(ax+by+f(ax−by=(b(a+b/2f(x+y+(b(a+b/2f(x−y+(2a2−ab−b2f(x+(b2−abf(y where a, b are nonzero fixed integers with b≠±a,−3a, and f(ax+by+f(ax−by=2a2f(x+2b2f(y for fixed integers a, b with a,b≠0 and a±b≠0.
International Nuclear Information System (INIS)
Pelinovsky, Dmitry E.; Yang Jianke
2005-01-01
We study the generalized third-order nonlinear Schroedinger (NLS) equation which admits a one-parameter family of single-hump embedded solitons. Analyzing the spectrum of the linearization operator near the embedded soliton, we show that there exists a resonance pole in the left half-plane of the spectral parameter, which explains linear stability, rather than nonlinear semistability, of embedded solitons. Using exponentially weighted spaces, we approximate the resonance pole both analytically and numerically. We confirm in a near-integrable asymptotic limit that the resonance pole gives precisely the linear decay rate of parameters of the embedded soliton. Using conserved quantities, we qualitatively characterize the stable dynamics of embedded solitons
Bellavia, Stefania; Berrone, Stefano
2007-10-01
In this work we study the numerical solution of nonlinear systems arising from stabilized FEM discretizations of Navier-Stokes equations. This is a very challenging problem and often inexact Newton solvers present severe difficulties to converge. Then, they must be wrapped into a globalization strategy. We consider the classical backtracking procedure, a subspace trust-region strategy and an hybrid approach. This latter strategy is proposed with the aim of improve the robustness of backtracking and it is obtained combining the backtracking procedure and the elliptical subspace trust-region strategy. Under standard assumptions, we prove global and fast convergence of the inexact Newton methods embedded in this new strategy as well as in the subspace trust-region procedure. Computational results on classical CFD benchmarks are performed. Comparisons among the classical backtracking strategy, the elliptical subspace trust-region approach and the hybrid procedure are presented. Our numerical experiments show the effectiveness of the proposed hybrid technique.
State equations and stability of shock wave fronts in homogeneous and heterogeneous metallic medium
International Nuclear Information System (INIS)
Romain, Jean-Pierre
1977-01-01
This research thesis in physical sciences reports a theoretical and experimental study of some mechanical and thermodynamic aspects related to a shock wave propagation in homogeneous and heterogeneous metallic media: state equations, stability and instability of shock wave fronts. In the first part, the author reports the study of the Grueneisen coefficient for some metallic elements with known static and dynamic compression properties. The second part reports the experimental investigation of dynamic compressibility of some materials (lamellar Al-Cu compounds). The front shock wave propagation has been visualised, and experimental Hugoniot curves are compared with those deduced from a developed numeric model and other models. The bismuth Hugoniot curve is also determined, and the author compares the existence and nature of phase transitions obtained by static and dynamic compression
International Nuclear Information System (INIS)
Wong, Pring; Pang, Li-Hui; Huang, Long-Gang; Li, Yan-Qing; Lei, Ming; Liu, Wen-Jun
2015-01-01
The study of the complex Ginzburg–Landau equation, which can describe the fiber laser system, is of significance for ultra-fast laser. In this paper, dromion-like structures for the complex Ginzburg–Landau equation are considered due to their abundant nonlinear dynamics. Via the modified Hirota method and simplified assumption, the analytic dromion-like solution is obtained. The partial asymmetry of structure is particularly discussed, which arises from asymmetry of nonlinear and dispersion terms. Furthermore, the stability of dromion-like structures is analyzed. Oscillation structure emerges to exhibit strong interference when the dispersion loss is perturbed. Through the appropriate modulation of modified exponent parameter, the oscillation structure is transformed into two dromion-like structures. It indicates that the dromion-like structure is unstable, and the coherence intensity is affected by the modified exponent parameter. Results in this paper may be useful in accounting for some nonlinear phenomena in fiber laser systems, and understanding the essential role of modified Hirota method
Modeling boundary-layer transition in DNS and LES using Parabolized Stability Equations
Lozano-Duran, Adrian; Hack, M. J. Philipp; Moin, Parviz
2016-11-01
The modeling of the laminar region and the prediction of the point of transition remain key challenges in the numerical simulation of boundary layers. The issue is of particular relevance for wall-modeled large eddy simulations which require 10 to 100 times higher grid resolution in the thin laminar region than in the turbulent regime. Our study examines the potential of the nonlinear parabolized stability equations (PSE) to provide an accurate, yet computationally efficient treatment of the growth of disturbances in the pre-transitional flow regime. The PSE captures the nonlinear interactions that eventually induce breakdown to turbulence, and can as such identify the onset of transition without relying on empirical correlations. Since the local PSE solution at the point of transition is the solution of the Navier-Stokes equations, it provides a natural inflow condition for large eddy and direct simulations by avoiding unphysical transients. We show that in a classical H-type transition scenario, a combined PSE/DNS approach can reproduce the skin-friction distribution obtained in reference direct numerical simulations. The computational cost in the laminar region is reduced by several orders of magnitude. Funded by the Air Force Office of Scientific Research.
Lozano-Durán, A.; Hack, M. J. P.; Moin, P.
2018-02-01
We examine the potential of the nonlinear parabolized stability equations (PSE) to provide an accurate yet computationally efficient treatment of the growth of disturbances in H-type transition to turbulence. The PSE capture the nonlinear interactions that eventually induce breakdown to turbulence and can as such identify the onset of transition without relying on empirical correlations. Since the local PSE solution at the onset of transition is a close approximation of the Navier-Stokes equations, it provides a natural inflow condition for direct numerical simulations (DNS) and large-eddy simulations (LES) by avoiding nonphysical transients. We show that a combined PSE-DNS approach, where the pretransitional region is modeled by the PSE, can reproduce the skin-friction distribution and downstream turbulent statistics from a DNS of the full domain. When the PSE are used in conjunction with wall-resolved and wall-modeled LES, the computational cost in both the laminar and turbulent regions is reduced by several orders of magnitude compared to DNS.
Transient Growth Analysis of Compressible Boundary Layers with Parabolized Stability Equations
Paredes, Pedro; Choudhari, Meelan M.; Li, Fei; Chang, Chau-Lyan
2016-01-01
The linear form of parabolized linear stability equations (PSE) is used in a variational approach to extend the previous body of results for the optimal, non-modal disturbance growth in boundary layer flows. This methodology includes the non-parallel effects associated with the spatial development of boundary layer flows. As noted in literature, the optimal initial disturbances correspond to steady counter-rotating stream-wise vortices, which subsequently lead to the formation of stream-wise-elongated structures, i.e., streaks, via a lift-up effect. The parameter space for optimal growth is extended to the hypersonic Mach number regime without any high enthalpy effects, and the effect of wall cooling is studied with particular emphasis on the role of the initial disturbance location and the value of the span-wise wavenumber that leads to the maximum energy growth up to a specified location. Unlike previous predictions that used a basic state obtained from a self-similar solution to the boundary layer equations, mean flow solutions based on the full Navier-Stokes (NS) equations are used in select cases to help account for the viscous-inviscid interaction near the leading edge of the plate and also for the weak shock wave emanating from that region. These differences in the base flow lead to an increasing reduction with Mach number in the magnitude of optimal growth relative to the predictions based on self-similar mean-flow approximation. Finally, the maximum optimal energy gain for the favorable pressure gradient boundary layer near a planar stagnation point is found to be substantially weaker than that in a zero pressure gradient Blasius boundary layer.
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Elfelsoufi, Z.; Azrar, L.
2016-01-01
In this paper, a mathematical modeling of flutter and divergence analyses of fluid conveying pipes based on integral equation formulations is presented. Dynamic stability problems related to fluid pressure, velocity, tension, topography slope and viscoelastic supports and foundations are formulated. A methodological approach is presented and the required matrices, associated to the influencing fluid and pipe parameters, are explicitly given. Internal discretizations are used allowing to investigate the deformation, the bending moment, slope and shear force at internal points. Velocity–frequency, pressure-frequency and tension-frequency curves are analyzed for various fluid parameters and internal elastic supports. Critical values of divergence and flutter behaviors with respect to various fluid parameters are investigated. This model is general and allows the study of dynamic stability of tubes crossed by stationary and instationary fluid on various types of supports. Accurate predictions can be obtained and are of particular interest for a better performance and for an optimal safety of piping system installations. - Highlights: • Modeling the flutter and divergence of fluid conveying pipes based on RBF. • Dynamic analysis of a fluid conveying pipe with generalized boundary conditions. • Considered parameters fluid are the pressure, tension, slopes topography, velocity. • Internal support increase the critical velocity value. • This methodologies determine the fluid parameters effects.
Chen, Huangxin
2017-09-01
In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i.e., the linear implicit scheme for time discretization with the finite difference method (FDM) on staggered grids for spatial discretization, pressure-correction schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations, and pressure-stabilization schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations. The energy stability estimates are obtained for the above each fully discrete scheme. The upwind scheme is used in the discretization of the convection term which plays an important role in the design of unconditionally stable discrete schemes. Numerical results are given to verify the theoretical analysis.
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Ding Xiaohua; Su Huan; Liu Mingzhu
2008-01-01
The paper analyzes a discrete second-order, nonlinear delay differential equation with negative feedback. The characteristic equation of linear stability is solved, as a function of two parameters describing the strength of the feedback and the damping in the autonomous system. The existence of local Hopf bifurcations is investigated, and the direction and stability of periodic solutions bifurcating from the Hopf bifurcation of the discrete model are determined by the Hopf bifurcation theory of discrete system. Finally, some numerical simulations are performed to illustrate the analytical results found
Rouz, Omid Farkhondeh; Ahmadian, Davood; Milev, Mariyan
2017-12-01
This paper establishes exponential mean square stability of two classes of theta Milstein methods, namely split-step theta Milstein (SSTM) method and stochastic theta Milstein (STM) method, for stochastic differential delay equations (SDDEs). We consider the SDDEs problem under a coupled monotone condition on drift and diffusion coefficients, as well as a necessary linear growth condition on the last term of theta Milstein method. It is proved that the SSTM method with θ ∈ [0, ½] can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize, but for θ ∈ (½, 1], we proved that the stability results hold for any stepsize. Then, based on the stability results of SSTM method, we examine the exponential mean square stability of the STM method and obtain the similar stability results to that of the SSTM method. In the numerical section the figures show thevalidity of our claims.
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Sinha S. C.
1996-01-01
Full Text Available Chebyshev polynomials are utilized to obtain solutions of a set of p th order linear differential equations with periodic coefficients. For this purpose, the operational matrix of differentiation associated with the shifted Chebyshev polynomials of the first kind is derived. Utilizing the properties of this matrix, the solution of a system of differential equations can be found by solving a set of linear algebraic equations without constructing the equivalent integral equations. The Floquet Transition Matrix (FTM can then be computed and its eigenvalues (Floquet multipliers subsequently analyzed for stability. Two straightforward methods, the ‘differential state space formulation’ and the ‘differential direct formulation’, are presented and the results are compared with those obtained from other available techniques. The well-known Mathieu equation and a higher order system are used as illustrative examples.
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S. C. Sinha
1996-01-01
Full Text Available Chebyshev polynomials are utilized to obtain solutions of a set of pth order linear differential equations with periodic coefficients. For this purpose, the operational matrix of differentiation associated with the shifted Chebyshev polynomials of the first kind is derived. Utilizing the properties of this matrix, the solution of a system of differential equations can be found by solving a set of linear algebraic equations without constructing the equivalent integral equations. The Floquet Transition Matrix (FTM can then be computed and its eigenvalues (Floquet multipliers subsequently analyzed for stability. Two straightforward methods, the ‘differential state space formulation’ and the ‘differential direct formulation’, are presented and the results are compared with those obtained from other available techniques. The well-known Mathieu equation and a higher order system are used as illustrative examples.
Yue, Chen; Seadawy, Aly; Lu, Dianchen
The propagation of hydrodynamic wave packets and media with negative refractive index is studied in a quintic derivative nonlinear Schrödinger (DNLS) equation. The quintic DNLS equation describe the wave propagation on a discrete electrical transmission line. We obtain a Lagrangian and the invariant variational principle for quintic DNLS equation. By using a class of ordinary differential equation, we found four types of exact solutions of the quintic DNLS equation, which are kink-type solitary wave solution, antikink-type solitary wave solution, sinusoidal solitary wave solution, bell-type solitary wave solution. By applying the modulation instability to discuss stability analysis of the obtained solutions. Modulation instabilities of continuous waves and localized solutions on a zero background have been investigated.
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Chen Yue
2016-01-01
Full Text Available The propagation of hydrodynamic wave packets and media with negative refractive index is studied in a quintic derivative nonlinear Schrödinger (DNLS equation. The quintic DNLS equation describe the wave propagation on a discrete electrical transmission line. We obtain a Lagrangian and the invariant variational principle for quintic DNLS equation. By using a class of ordinary differential equation, we found four types of exact solutions of the quintic DNLS equation, which are kink-type solitary wave solution, antikink-type solitary wave solution, sinusoidal solitary wave solution, bell-type solitary wave solution. By applying the modulation instability to discuss stability analysis of the obtained solutions. Modulation instabilities of continuous waves and localized solutions on a zero background have been investigated.
A note on stability of the integral-differential equation of the hyperbolic type in a Hilbert space
M. Ashyraliyev (Maksat)
2008-01-01
htmlabstractIn this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme
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Agnieszka Chlebowicz
2014-01-01
Full Text Available Using the technique of measures of noncompactness and, in particular, a consequence of Sadovskii’s fixed point theorem, we prove a theorem about the existence and asymptotic stability of solutions of a functional integral equation. Moreover, in order to illustrate our results, we include one example and compare our results with those obtained in other papers appearing in the literature.
El-Tom, M E A
1974-01-01
A procedure, using spine functions of degree m, deficiency k-1, for obtaining approximate solutions to nonlinear Volterra integral equations of the second kind is presented. The paper is an investigation of the numerical stability of the procedure for various values of m and k. (5 refs).
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Tian Zhou Xu
2012-01-01
Full Text Available The objective of the present paper is to determine the generalized Hyers-Ulam stability of the mixed additive-cubic functional equation in n-Banach spaces by the direct method. In addition, we show under some suitable conditions that an approximately mixed additive-cubic function can be approximated by a mixed additive and cubic mapping.
Integral φ0-Stability in terms of Two Measures for Impulsive Differential Equations with “Supremum”
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Peiguang Wang
2014-01-01
Full Text Available This paper establishes a criterion on integral φ0-stability in terms of two measures for impulsive differential equations with “supremum” by using the cone-valued piecewise continuous Lyapunov functions, Razumikhin method, and comparative method. Meantime, an example is given to illustrate our result.
International Nuclear Information System (INIS)
Xu Tianzhou; Rassias, John Michael; Xu Wanxin
2010-01-01
We establish some stability results concerning the general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces. In addition, we establish some results of approximately general mixed additive-cubic mappings in non-Archimedean fuzzy normed spaces. The results improve and extend some recent results.
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Giai Giang Vo
2015-01-01
Full Text Available This paper is devoted to the study of a wave equation with a boundary condition of many-point type. The existence of weak solutions is proved by using the Galerkin method. Also, the uniqueness and the stability of solutions are established.
International Nuclear Information System (INIS)
Itasse, Maxime; Brazier, Jean-Philippe; Léon, Olivier; Casalis, Grégoire
2015-01-01
Nonlinear evolution of disturbances in an axisymmetric, high subsonic, high Reynolds number hot jet with forced eigenmodes is studied using the Parabolized Stability Equations (PSE) approach to understand how modes interact with one another. Both frequency and azimuthal harmonic interactions are analyzed by setting up one or two modes at higher initial amplitudes and various phases. While single mode excitation leads to harmonic growth and jet noise amplification, controlling the evolution of a specific mode has been made possible by forcing two modes (m 1 , n 1 ), (m 2 , n 2 ), such that the difference in azimuth and in frequency matches the desired “target” mode (m 1 − m 2 , n 1 − n 2 ). A careful setup of the initial amplitudes and phases of the forced modes, defined as the “killer” modes, has allowed the minimizing of the initially dominant instability in the near pressure field, as well as its estimated radiated noise with a 15 dB loss. Although an increase of the overall sound pressure has been found in the range of azimuth and frequency analyzed, the present paper reveals the possibility to make the initially dominant instability ineffective acoustically using nonlinear interactions with forced eigenmodes
Energy Technology Data Exchange (ETDEWEB)
Itasse, Maxime, E-mail: Maxime.Itasse@onera.fr; Brazier, Jean-Philippe, E-mail: Jean-Philippe.Brazier@onera.fr; Léon, Olivier, E-mail: Olivier.Leon@onera.fr; Casalis, Grégoire, E-mail: Gregoire.Casalis@onera.fr [Onera - The French Aerospace Lab, F-31055 Toulouse (France)
2015-08-15
Nonlinear evolution of disturbances in an axisymmetric, high subsonic, high Reynolds number hot jet with forced eigenmodes is studied using the Parabolized Stability Equations (PSE) approach to understand how modes interact with one another. Both frequency and azimuthal harmonic interactions are analyzed by setting up one or two modes at higher initial amplitudes and various phases. While single mode excitation leads to harmonic growth and jet noise amplification, controlling the evolution of a specific mode has been made possible by forcing two modes (m{sub 1}, n{sub 1}), (m{sub 2}, n{sub 2}), such that the difference in azimuth and in frequency matches the desired “target” mode (m{sub 1} − m{sub 2}, n{sub 1} − n{sub 2}). A careful setup of the initial amplitudes and phases of the forced modes, defined as the “killer” modes, has allowed the minimizing of the initially dominant instability in the near pressure field, as well as its estimated radiated noise with a 15 dB loss. Although an increase of the overall sound pressure has been found in the range of azimuth and frequency analyzed, the present paper reveals the possibility to make the initially dominant instability ineffective acoustically using nonlinear interactions with forced eigenmodes.
Controllability and stability of 3D heat conduction equation in a submicroscale thin film
Heidari, H.; Zwart, Heiko J.; Malek, Alaeddin
We obtain a closed form analytic solution for the Dual Phase Lagging equation. This equation is a linear, time-independent partial differential equation modeling the heat distribution in a thin film. The spatial domain is of micrometer and nanometer geometries. We show that the solution is described
International Nuclear Information System (INIS)
Zhang Zaiyun; Miao Xiujin; Chen Yuezhong; Liu Zhenhai
2011-01-01
In this paper, we prove the existence, uniqueness, and uniform stability of strong and weak solutions of the nonlinear generalized Klein-Gordon equation (1.1) 1 (see Sec. I) in bounded domains with nonlinear damped boundary conditions given by (1.1) 3 (see Sec. I) with some restrictions on function f(u), h(∇u), g(u t ), and b(x), we prove the existence and uniqueness by means of nonlinear semigroup method and obtain the uniform stabilization by using the multiplier technique.
Shen, Wenxian
2017-09-01
This paper is concerned with the stability of transition waves and strictly positive entire solutions of random and nonlocal dispersal evolution equations of Fisher-KPP type with general time and space dependence, including time and space periodic or almost periodic dependence as special cases. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability of uniformly continuous transition waves connecting the unique strictly positive entire solution and the trivial solution zero and satisfying certain decay property at the end close to the trivial solution zero (if it exists). The existence of transition waves has been studied in Liang and Zhao (2010 J. Funct. Anal. 259 857-903), Nadin (2009 J. Math. Pures Appl. 92 232-62), Nolen et al (2005 Dyn. PDE 2 1-24), Nolen and Xin (2005 Discrete Contin. Dyn. Syst. 13 1217-34) and Weinberger (2002 J. Math. Biol. 45 511-48) for random dispersal Fisher-KPP equations with time and space periodic dependence, in Nadin and Rossi (2012 J. Math. Pures Appl. 98 633-53), Nadin and Rossi (2015 Anal. PDE 8 1351-77), Nadin and Rossi (2017 Arch. Ration. Mech. Anal. 223 1239-67), Shen (2010 Trans. Am. Math. Soc. 362 5125-68), Shen (2011 J. Dynam. Differ. Equ. 23 1-44), Shen (2011 J. Appl. Anal. Comput. 1 69-93), Tao et al (2014 Nonlinearity 27 2409-16) and Zlatoš (2012 J. Math. Pures Appl. 98 89-102) for random dispersal Fisher-KPP equations with quite general time and/or space dependence, and in Coville et al (2013 Ann. Inst. Henri Poincare 30 179-223), Rawal et al (2015 Discrete Contin. Dyn. Syst. 35 1609-40) and Shen and Zhang (2012 Comm. Appl. Nonlinear Anal. 19 73-101) for nonlocal dispersal Fisher-KPP equations with time and/or space periodic dependence. The stability result established in this paper implies that the transition waves obtained in many of the above mentioned papers are asymptotically stable for well-fitted perturbation. Up to the author
Cho, Yonggeun
2016-05-04
This paper is devoted to the mathematical analysis of a class of nonlinear fractional Schrödinger equations with a general Hartree-type integrand. We show the well-posedness of the associated Cauchy problem and prove the existence and stability of standing waves under suitable assumptions on the nonlinearity. Our proofs rely on a contraction argument in mixed functional spaces and the concentration-compactness method. © 2015 World Scientific Publishing Company
Nefedov, N. N.; Nikulin, E. I.
2018-01-01
A singularly perturbed periodic in time problem for a parabolic reaction-diffusion equation in a two-dimensional domain is studied. The case of existence of an internal transition layer under the conditions of balanced and unbalanced rapid reaction is considered. An asymptotic expansion of a solution is constructed. To justify the asymptotic expansion thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is investigated.
Asymptotic behavior and stability of second order neutral delay differential equations
Chen, G.L.; van Gaans, O.W.; Verduyn Lunel, Sjoerd
2014-01-01
We study the asymptotic behavior of a class of second order neutral delay differential equations by both a spectral projection method and an ordinary differential equation method approach. We discuss the relation of these two methods and illustrate some features using examples. Furthermore, a fixed
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WU Renchao
2016-06-01
Full Text Available In this paper, we consider three dimensional compressible viscous magnetohydro dynamic equations(MHD with external potentialforce. We first derive the corresponding non-constantstationary solutions. Then we show global well-posedness of the initial value problem for the three dimensional compressible viscous magnetohydrodynamic equations, provided that rescribed initial data is close to the stationary solution.
International Nuclear Information System (INIS)
Zheng Youqi; Wu Hongchun; Cao Liangzhi
2013-01-01
This paper describes the stability analysis for the coarse mesh finite difference (CMFD) acceleration used in the wavelet expansion method. The nonlinear CMFD acceleration scheme is transformed by linearization and the Fourier ansatz is introduced into the linearized formulae. The spectral radius is defined as the stability criterion, which is the least upper bound (LUB) of the largest eigenvalue of Fourier analysis matrix. The stability analysis considers the effect of mesh size (spectral length), coarse mesh division and scattering ratio. The results show that for the wavelet expansion method, the CMFD acceleration is conditionally stable. The small size of fine mesh brings stability and fast convergent. With the increase of the mesh size, the stability becomes worse. The scattering ratio does not impact the stability obviously. It makes the CMFD acceleration highly efficient in the strong scattering case. The results of Fourier analysis are verified by the numerical tests based on a homogeneous slab problem.
Peng, Wei-Qi; Tian, Shou-Fu; Zou, Li; Zhang, Tian-Tian
2018-01-01
In this paper, the extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms is investigated, whose particular cases are the Hirota equation, the Sasa-Satsuma equation and Lakshmanan-Porsezian-Daniel equation by selecting some specific values on the parameters of higher-order terms. We first study the stability analysis of the equation. Then, using the ansatz method, we derive its bright, dark solitons and some constraint conditions which can guarantee the existence of solitons. Moreover, the Ricatti equation extension method is employed to derive some exact singular solutions. The outstanding characteristics of these solitons are analyzed via several diverting graphics.
Alagoz, Baris Baykant
2017-09-01
With power mapping (conformal mapping), stability analyses of fractional order linear time invariant (LTI) systems are carried out by consideration of the root locus of expanded degree integer order polynomials in the principal Riemann sheet. However, it is essential to show the left half plane (LHP) stability analysis of fractional order characteristic polynomials in the s plane in order to close the gap emerging in stability analyses of fractional order and integer order systems. In this study, after briefly discussing the relation between the characteristic root orientations and the system stability, the author presents a methodology to establish principal characteristic polynomials to perform the LHP stability analysis of fractional order systems. The principal characteristic polynomials are formed by factorizing principal characteristic roots. Then, the LHP stability analysis of fractional order systems can be carried out by using the root equivalency of fractional order principal characteristic polynomials. Illustrative examples are presented to explain how to find equivalent roots of fractional order principal characteristic polynomials in order to carry out the LHP stability analyses of fractional order nominal and interval systems. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Global stability of periodic orbits of non-autonomous difference equations and population biology
Elaydi, Saber; Sacker, Robert J.
Elaydi and Yakubu showed that a globally asymptotically stable(GAS) periodic orbit in an autonomous difference equation must in fact be a fixed point whenever the phase space is connected. In this paper we extend this result to periodic nonautonomous difference equations via the concept of skew-product dynamical systems. We show that for a k-periodic difference equation, if a periodic orbit of period r is GAS, then r must be a divisor of k. In particular sub-harmonic, or long periodic, oscillations cannot occur. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our methods are then applied to prove a conjecture by J. Cushing and S. Henson concerning a non-autonomous Beverton-Holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates.
Stability Properties of the Regular Set for the Navier-Stokes Equation
D'Ancona, Piero; Lucà, Renato
2017-11-01
We investigate the size of the regular set for small perturbations of some classes of strong large solutions to the Navier-Stokes equation. We consider perturbations of the data that are small in suitable weighted L2 spaces but can be arbitrarily large in any translation invariant Banach space. We give similar results in the small data setting.
Height fluctuations and intermittency of V sub 2 O sub 5 films by atomic force microscopy
Iraji-Zad, A; Tabar, M R R; Allaei, S M V
2003-01-01
The spatial scaling law and intermittency of the V sub 2 O sub 5 surface roughness has been investigated by atomic force microscopy. The intermittency of the height fluctuations has been checked by two different methods, first, by measuring the scaling exponent of the q-th moment of height-difference fluctuations i.e. C sub q = (|h(x sub 1) - h(x sub 2)|), and second, by defining the generating function Z(q, N) and generalized multi-fractal dimension D sub q. These methods predict that there is no intermittency in the height fluctuations. The observed roughness and dynamical exponents can be explained by numerical simulation on the basis of the forced Kuramoto-Sivashinsky equation.
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John Michael Rassias
2017-07-01
Full Text Available The aim of this paper is to investigate generalized Ulam-Hyers stabilities of the following Euler-Lagrange-Jensen-$(a,b$-sextic functional equation $$ f(ax+by+f(bx+ay+(a-b^6\\left[f\\left(\\frac{ax-by}{a-b}\\right+f\\left(\\frac{bx-ay}{b-a}\\right\\right]\\\\ = 64(ab^2\\left(a^2+b^2\\right\\left[f\\left(\\frac{x+y}{2}\\right+f\\left(\\frac{x-y}{2}\\right\\right]\\\\ +2\\left(a^2-b^2\\right\\left(a^4-b^4\\right[f(x+f(y] $$ where $a\
International Nuclear Information System (INIS)
Tang, X H; Zou, Xingfu
2009-01-01
By employing Schauder's fixed point theorem and a non-Liapunov method (matrix theory, inequality analysis), we obtain some new criteria that ensure existence and global exponential stability of a periodic solution to a class of functional differential equations. Applying these criteria to a cellular neural network with time delays (delayed cellular neural network, DCNN) under a periodic environment leads to some new results that improve and generalize many existing ones we know on this topic. These results are of great significance in designs and applications of globally stable periodic DCNNs
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Vasile Dr ̆agan
2017-06-01
Full Text Available We investigate the problem for solving a discrete-time periodic gen- eralized Riccati equation with an indefinite sign of the quadratic term. A necessary condition for the existence of bounded and stabilizing solution of the discrete-time Riccati equation with an indefinite quadratic term is derived. The stabilizing solution is positive semidefinite and satisfies the introduced sign conditions. The proposed condition is illustrated via a numerical example.
Araújo, Adérito; Barbeiro, Sílvia; Ghalati, Maryam Khaksar
2017-05-01
In this work we discuss the numerical discretization of the time-dependent Maxwell's equations using a fully explicit leap-frog type discontinuous Galerkin method. We present a sufficient condition for the stability, for cases of typical boundary conditions, either perfect electric, perfect magnetic or first order Silver-M\\"uller. The bounds of the stability region point out the influence of not only the mesh size but also the dependence on the choice of the numerical flux and the degree of the polynomials used in the construction of the finite element space, making possible to balance accuracy and computational efficiency. In the model we consider heterogeneous anisotropic permittivity tensors which arise naturally in many applications of interest. Numerical results supporting the analysis are provided.
Balsara, Dinshaw S.; Käppeli, Roger
2017-05-01
In this paper we focus on the numerical solution of the induction equation using Runge-Kutta Discontinuous Galerkin (RKDG)-like schemes that are globally divergence-free. The induction equation plays a role in numerical MHD and other systems like it. It ensures that the magnetic field evolves in a divergence-free fashion; and that same property is shared by the numerical schemes presented here. The algorithms presented here are based on a novel DG-like method as it applies to the magnetic field components in the faces of a mesh. (I.e., this is not a conventional DG algorithm for conservation laws.) The other two novel building blocks of the method include divergence-free reconstruction of the magnetic field and multidimensional Riemann solvers; both of which have been developed in recent years by the first author. Since the method is linear, a von Neumann stability analysis is carried out in two-dimensions to understand its stability properties. The von Neumann stability analysis that we develop in this paper relies on transcribing from a modal to a nodal DG formulation in order to develop discrete evolutionary equations for the nodal values. These are then coupled to a suitable Runge-Kutta timestepping strategy so that one can analyze the stability of the entire scheme which is suitably high order in space and time. We show that our scheme permits CFL numbers that are comparable to those of traditional RKDG schemes. We also analyze the wave propagation characteristics of the method and show that with increasing order of accuracy the wave propagation becomes more isotropic and free of dissipation for a larger range of long wavelength modes. This makes a strong case for investing in higher order methods. We also use the von Neumann stability analysis to show that the divergence-free reconstruction and multidimensional Riemann solvers are essential algorithmic ingredients of a globally divergence-free RKDG-like scheme. Numerical accuracy analyses of the RKDG
Stability and uniqueness of slowly oscillating periodic solutions to Wright's equation
Jaquette, Jonathan; Lessard, Jean-Philippe; Mischaikow, Konstantin
2017-12-01
In this paper, we prove that Wright's equation y‧ (t) = - αy (t - 1) { 1 + y (t) } has a unique slowly oscillating periodic solution (SOPS) for all parameter values α ∈ [ 1.9 , 6.0 ], up to time translation. Our proof is based on the same strategy employed earlier by Xie [27]; show that every SOPS is asymptotically stable. We first introduce a branch and bound algorithm to control all SOPS using bounding functions at all parameter values α ∈ [ 1.9 , 6.0 ]. Once the bounding functions are constructed, we then control the Floquet multipliers of all possible SOPS by solving rigorously an eigenvalue problem, again using a formulation introduced by Xie. Using these two main steps, we prove that all SOPS of Wright's equation are asymptotically stable for α ∈ [ 1.9 , 6.0 ], and the proof follows. This result is a step toward the proof of the Jones' Conjecture formulated in 1962.
DEFF Research Database (Denmark)
Fuhrmann, David R.; Bingham, Harry B.; Madsen, Per A.
2004-01-01
This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly nonlinear and extremely dispersive water waves. The analysis demonstrates the near-equivalence of classical linear Fourier (von Neumann...
A result concerning the stability of some difference equations and its ...
Indian Academy of Sciences (India)
M. Senthilkumar (Newgen Imaging) 1461 1996 Oct 15 13:05:22
, Ulam [18] raised a question concerning the stability of group homomorphisms: Let G1 be a group and let G2 be a metric group with the metric d(·, ·). Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality.
Directory of Open Access Journals (Sweden)
Chung Jae-Young
2010-01-01
Full Text Available Let be the set of positive real numbers, a Banach space, and , with . We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality in restricted domains of the form for fixed with or and . As consequences of the results we obtain asymptotic behaviors of the inequality as .
Directory of Open Access Journals (Sweden)
Zhanhua Yu
2011-01-01
convergence theorem. It is shown that the Euler method and the backward Euler method can reproduce the almost surely asymptotic stability of exact solutions to NSDDEs under additional conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results.
Stability of plane wave solutions in complex Ginzburg-Landau equation with delayed feedback
Puzyrev, D.; Yanchuk, S.; Vladimirov, A. G.; Gurevich, S. V.
2013-01-01
We perform bifurcation analysis of plane wave solutions in one-dimensional cubic-quintic Ginzburg-Landau equation with delayed feedback. Our study reveals how multistability and snaking behavior of plane waves emerge as time delay is introduced. For intermediate values of the delay, bifurcation diagrams are obtained by a combination of analytical and numerical methods. For large delays, using an asymptotic approach we classify plane wave solutions into strongly unstable, weakly unstable, and ...
International Nuclear Information System (INIS)
Karimov, Ruslan Kh; Kozhevnikova, Larisa M
2010-01-01
The first mixed problem with homogeneous Dirichlet boundary condition and initial function with compact support is considered for quasilinear second order parabolic equations in a cylindrical domain D=(0,∞)xΩ. Upper bounds are obtained, which give the rate of decay of the solutions as t→∞ as a function of the geometry of the unbounded domain Ω subset of R n , n≥2. Bibliography: 18 titles.
Stability of contact discontinuities to 1-D piston problem for the compressible Euler equations
Ding, Min
2018-03-01
We consider 1-D piston problem for the compressible Euler equations when the piston is static relatively to the gas in the tube. By a modified wave front tracking method, we prove that a contact discontinuity is structurally stable under the assumptions that the total variation of the initial data and the perturbation of the piston velocity are both sufficiently small. Meanwhile, we study the asymptotic behavior of the solutions by the generalized characteristic method and approximate conservation law theory as t → + ∞.
Stabilization of the solution of a doubly nonlinear parabolic equation
International Nuclear Information System (INIS)
Andriyanova, È R; Mukminov, F Kh
2013-01-01
The method of Galerkin approximations is employed to prove the existence of a strong global (in time) solution of a doubly nonlinear parabolic equation in an unbounded domain. The second integral identity is established for Galerkin approximations, and passing to the limit in it an estimate for the decay rate of the norm of the solution from below is obtained. The estimates characterizing the decay rate of the solution as x→∞ obtained here are used to derive an upper bound for the decay rate of the solution with respect to time; the resulting estimate is pretty close to the lower one. Bibliography: 17 titles
New exponential stability conditions for linear delayed systems of differential equations
Directory of Open Access Journals (Sweden)
Leonid Berezansky
2016-08-01
where $t\\ge 0$, $m$ and $r_{ij}$, $i,j=1,\\dots,m$ are natural numbers, $a_{ij}^{k}\\colon [0,\\infty\\to\\mathbb{R}$ are measurable coefficients, and $h_{ij}^{k}\\colon [0,\\infty\\to\\mathbb{R}$ are measurable delays. The progress was achieved by using a new technique making it possible to replace the constant $1$ by the constant $1+{1}/{\\mathrm{e}}$ on the right-hand sides of crucial inequalities ensuring exponential stability.
Numerical study of solitary wave stability in cubic nonlinear Dirac equations in 1D
Lakoba, T. I.
2018-02-01
Recently there has occurred a controversy between the semi-analytical prediction of linear stability of the soliton of the massive Gross-Neveu model and direct numerical observations of its instability for small values of the frequency. We revisit the problem of numerical computation of this soliton, find a mechanism behind the numerical instability observed in earlier studies, and propose methods to stably compute the soliton over long times. Thus, we confirm the semi-analytical prediction of the soliton's being linearly stable.
Hannah, S. R.; Palazotto, A. N.
1978-01-01
A new trigonometric approach to the finite difference calculus was applied to the problem of beam buckling as represented by virtual work and equilibrium equations. The trigonometric functions were varied by adjusting a wavelength parameter in the approximating Fourier series. Values of the critical force obtained from the modified approach for beams with a variety of boundary conditions were compared to results using the conventional finite difference method. The trigonometric approach produced significantly more accurate approximations for the critical force than the conventional approach for a relatively wide range in values of the wavelength parameter; and the optimizing value of the wavelength parameter corresponded to the half-wavelength of the buckled mode shape. It was found from a modal analysis that the most accurate solutions are obtained when the approximating function closely represents the actual displacement function and matches the actual boundary conditions.
Feshchenko, R. M.
Recently a new exact transparent boundary condition (TBC) for the 3D parabolic wave equation (PWE) in rectangular computational domain was derived. However in the obtained form it does not appear to be unconditionally stable when used with, for instance, the Crank-Nicolson finite-difference scheme. In this paper two new formulations of the TBC for the 3D PWE in rectangular computational domain are reported, which are likely to be unconditionally stable. They are based on an unconditionally stable fully discrete TBC for the Crank-Nicolson scheme for the 2D PWE. These new forms of the TBC can be used for numerical solution of the 3D PWE when a higher precision is required.
Existence and asymptotic stability of a viscoelastic wave equation with a delay
Kirane, Mokhtar
2011-07-07
In this paper, we consider the viscoelastic wave equation with a delay term in internal feedbacks; namely, we investigate the following problem, together with initial conditions and boundary conditions of Dirichlet type. Here (x, t) ∈ Ω × (0, ∞), g is a positive real valued decreasing function and μ1, μ2 are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the weight of the term without delay, using the Faedo-Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Under the same assumptions, general decay results of the energy are established via suitable Lyapunov functionals. © 2011 Springer Basel AG.
International Nuclear Information System (INIS)
Souza, Altivo Monteiro de
2008-12-01
The world energy consumption has been increasing strongly in recent years. Nuclear energy has been regarded as a suitable option to supply this growing energy demand in industrial scale. In view of the need of improving the understanding and capacity of analysis of nuclear power plants, modern simulation techniques for flow and heat transfer problems are gaining greater importance. A large number of problems found in nuclear reactor engineering can be dealt assuming axial symmetry. Thus, in this work a stabilized finite element formulation for the solution of the Navier-Stokes and energy equations for axisymmetric problems have been developed and tested. The formulation has been implemented in the NS S OLVER M PI 2 D A program developed at the Parallel Computation Laboratory of the Instituto de Engenharia Nuclear (LCP/IEN) and is now available either for safety analysis or design of nuclear systems. (author)
Stumpf, Eugen
2016-01-01
This work is concerned with the study of a scalar delay differential equation \\begin{equation*} z^{\\prime\\prime}(t)=h^2\\,V(z(t-1)-z(t))+h\\,z^\\prime(t) \\end{equation*} motivated by a simple car-following model on an unbounded straight line. Here, the positive real $h$ denotes some parameter, and $V$ is a so-called \\textit{optimal velocity function} of the traffic model involved. We analyze the existence and local stability properties of solutions $z(t)=c\\,t+d$, $t\\in\\mathbb{R}$, with $c,d\\in\\m...
International Nuclear Information System (INIS)
Meyer, Chad D.; Balsara, Dinshaw S.; Aslam, Tariq D.
2014-01-01
Parabolic partial differential equations appear in several physical problems, including problems that have a dominant hyperbolic part coupled to a sub-dominant parabolic component. Explicit methods for their solution are easy to implement but have very restrictive time step constraints. Implicit solution methods can be unconditionally stable but have the disadvantage of being computationally costly or difficult to implement. Super-time-stepping methods for treating parabolic terms in mixed type partial differential equations occupy an intermediate position. In such methods each superstep takes “s” explicit Runge–Kutta-like time-steps to advance the parabolic terms by a time-step that is s 2 times larger than a single explicit time-step. The expanded stability is usually obtained by mapping the short recursion relation of the explicit Runge–Kutta scheme to the recursion relation of some well-known, stable polynomial. Prior work has built temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Chebyshev polynomials. Since their stability is based on the boundedness of the Chebyshev polynomials, these methods have been called RKC1 and RKC2. In this work we build temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Legendre polynomials. We call these methods RKL1 and RKL2. The RKL1 method is first-order accurate in time; the RKL2 method is second-order accurate in time. We verify that the newly-designed RKL1 and RKL2 schemes have a very desirable monotonicity preserving property for one-dimensional problems – a solution that is monotone at the beginning of a time step retains that property at the end of that time step. It is shown that RKL1 and RKL2 methods are stable for all values of the diffusion coefficient up to the maximum value. We call this a convex monotonicity preserving property and show by examples that it is very useful
Kamynin, V. L.; Bukharova, T. I.
2017-01-01
We prove the estimates of stability with respect to perturbations of input data for the solutions of inverse problems for degenerate parabolic equations with unbounded coefficients. An important feature of these estimates is that the constants in these estimates are written out explicitly by the input data of the problem.
Reddy, T. S. R.
1986-01-01
The process of performing an automated stability analysis for an elastic-bladed helicopter rotor is discussed. A symbolic manipulation program, written in FORTRAN, is used to aid in the derivation of the governing equations of motion for the rotor. The blades undergo coupled bending and torsional deformations. Two-dimensional quasi-steady aerodynamics below stall are used. Although reversed flow effects are neglected, unsteady effects, modeled as dynamic inflow are included. Using a Lagrangian approach, the governing equations are derived in generalized coordinates using the symbolic program. The program generates the steady and perturbed equations and writes into subroutines to be called by numerical routines. The symbolic program can operate on both expressions and matrices. For the case of hovering flight, the blade and dynamic inflow equations are converted to equations in a multiblade coordinate system by rearranging the coefficients of the equations. For the case of forward flight, the multiblade equations are obtained through the symbolic program. The final multiblade equations are capable of accommodating any number of elastic blade modes. The computer implementation of this procedure consists of three stages: (1) the symbolic derivation of equations; (2) the coding of the equations into subroutines; and (3) the numerical study after identifying mass, damping, and stiffness coefficients. Damping results are presented in hover and in forward flight with and without dynamic inflow effects for various rotor blade models, including rigid blade lag-flap, elastic flap-lag, flap-lag-torsion, and quasi-static torsion. Results from dynamic inflow effects which are obtained from a lift deficiency function for a quasi-static inflow model in hover are also presented.
International Nuclear Information System (INIS)
Ben Joud, Hajer
2009-01-01
In this paper, we study the inverse boundary value problem of determining the magnetic field and the electric potential of the magnetic Schrödinger equation from the Dirichlet to Neumann map when the boundary measurement is taken on any open subset Γ 0 of the boundary. Assuming that in a neighborhood of the boundary, the potentials coefficients are known, we prove the logarithmic stability estimate. The proofs involve the use of the complex geometrical optics solutions of the magnetic Schrödinger equation
International Nuclear Information System (INIS)
Mukminov, F Kh; Bikkulov, I M
2004-01-01
The behaviour as t→∞ of the solution of a mixed problem for parabolic equations in an unbounded domain with two exits to infinity is studied. A certain class of domains is distinguished, in which an estimate characterizing the stabilization of solutions and determined by the geometry of the domain is established. This estimate is proved to be sharp in a certain sense for a broad class of domains with two exits to infinity.
DEFF Research Database (Denmark)
leMesurier, B.J.; Christiansen, Peter Leth; Gaididei, Yuri Borisovich
2004-01-01
The effect of attractive linear potentials on self-focusing in-waves modeled by a nonlinear Schrodinger equation is considered. It is shown that the attractive potential can prevent both singular collapse and dispersion that are generic in the cubic Schrodinger equation in the critical dimension 2...
International Nuclear Information System (INIS)
Lyczkowski, R.W.; Gidaspow, D.; Solbrig, C.W.; Hughes, E.D.
1975-01-01
Equation systems describing one-dimensional, transient, two-phase flow with separate continuity, momentum, and energy equations for each phase are classified by use of the method of characteristics. Little attempt is made to justify the physics of these equations. Many of the equation systems possess complex-valued characteristics and hence, according to well-known mathematical theorems, are not well-posed as initial-value problems (IVPs). Real-valued characteristics are necessary but not sufficient to insure well-posedness. In the absence of lower order source or sink terms (potential type flows), which can affect the well-posedness of IVPs, the complex characteristics associated with these two-phase flow equations imply unbounded exponential growth for disturbances of all wavelengths. Analytical and numerical examples show that the ill-posedness of IVPs for the two-phase flow partial differential equations which possess complex characteristics produce unstable numerical schemes. These unstable numerical schemes can produce apparently stable and even accurate results if the growth rate resulting from the complex characteristics remains small throughout the time span of the numerical experiment or if sufficient numerical damping is present for the increment size used. Other examples show that clearly nonphysical numerical instabilities resulting from the complex characteristics can be produced. These latter types of numerical instabilities are shown to be removed by the addition of physically motivated differential terms which eliminate the complex characteristics. (auth)
Energy Technology Data Exchange (ETDEWEB)
Valat, J. [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires
1960-12-15
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) [French] Pour les equations du genre de Hill-Meissner a coefficients creneles, on a calcule des diagrammes universels de stabilite et ceux-ci ont ete verifies experimentalement. L'etude de ces equations dans le plan de phase a permis ensuite d'etendre le calcul des solutions periodiques au cas des equations differentielles non lineaires a coefficients periodiques creneles. Cette theorie a ete verifiee experimentalement. Pour Jes systemes couples non lineaires a coefficients constants, on a d'abord cherche les solutions menant a des mouvements algebriques. Les fonctions elliptiques et fuchsiennes uniformisent de tels mouvements. L'etude de mouvements non algebriques est plus delicate, a part l'etude des mouvements de Lissajous non lineaires. Une analyse fonctionnelle montre qu'il est toutefois possible dans certains cas de decoupler le systeme et de
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
Directory of Open Access Journals (Sweden)
Tian Zhou Xu
2010-01-01
Full Text Available Using the fixed point methods, we prove the generalized Hyers-Ulam stability of the general mixed additive-quadratic-cubic-quartic functional equation f(x+ky+f(x−ky=k2f(x+y+k2f(x−y+2(1−k2f(x+((k4−k2/12[f(2y+f(−2y−4f(y−4f(−y] for a fixed integer k with k≠0,±1 in non-Archimedean normed spaces.
Directory of Open Access Journals (Sweden)
Qianqian Yang
2010-01-01
fractional Fokker-Planck equations with a nonlinear source term (TSFFPENST, which involve the Caputo time fractional derivative (CTFD of order ∈ (0, 1 and the symmetric Riesz space fractional derivative (RSFD of order ∈ (1, 2]. Approximating the CTFD and RSFD using the L1-algorithm and shifted Grünwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.
Makarov, D. V.; Uleysky, M. Yu.
2017-02-01
Binary discrete nonlinear Schrödinger equation is used to describe dynamics of two-species Bose-Einstein condensate loaded into an optical lattice. Linear inter-species coupling leads to Rabi transitions between the species. In the regime of strong nonlinearity, a wavepacket corresponding to condensate separates into localized and ballistic fractions. Localized fraction is predominantly formed by immiscible solitons consisted of only one species. Immiscible solitons are formed from initially non-separated states after transient chaotic regime. We calculate the finite-time Lyapunov exponent as a rate of wavepacket divergence in the Hilbert space. Appearance of immiscible solitons to spontaneous self-stabilization of the wavepacket. It is found that onset of chaos is accompanied by fast variations of interaction energy and energy of inter-site tunneling. Crossover to self-stabilization is accompanied by reduction of condensate density due to emittance of ballistically propagating waves.
Singer, Bart A.; Choudhari, Meelan; Li, Fei
1995-01-01
A multiple-scales approach is used to approximate the effects of nonparallelism and streamwise surface curvature on the growth of stationary crossflow vortices in incompressible, three-dimesional boundary layers. The results agree with results predicted by solving the parabolized stability equations in regions where the nonparallelism is sufficiently weak. As the nonparallelism increases, the agreement between the two approaches worsens. An attempt has been made to quantify the nonparallelism on flow stability in terms of a nondimensional number that describes the rate of change of the mean flow relative to the disturbance wavelength. We find that the above nondimensional number provides useful information about the adequacy of the multiple-scales approximation for different disturbances for a given flow geometry, but the number does not collapse data for different flow geometries onto a single curve.
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Muhammad H. Al-Malack
2016-07-01
Full Text Available Fuel oil flyash (FFA produced in power and water desalination plants firing crude oils in the Kingdom of Saudi Arabia is being disposed in landfills, which increases the burden on the environment, therefore, FFA utilization must be encouraged. In the current research, the effect of adding FFA on the engineering properties of two indigenous soils, namely sand and marl, was investigated. FFA was added at concentrations of 5%, 10% and 15% to both soils with and without the addition of Portland cement. Mixtures of the stabilized soils were thoroughly evaluated using compaction, California Bearing Ratio (CBR, unconfined compressive strength (USC and durability tests. Results of these tests indicated that stabilized sand mixtures could not attain the ACI strength requirements. However, marl was found to satisfy the ACI strength requirement when only 5% of FFA was added together with 5% of cement. When the FFA was increased to 10% and 15%, the mixture’s strength was found to decrease to values below the ACI requirements. Results of the Toxicity Characteristics Leaching Procedure (TCLP, which was performed on samples that passed the ACI requirements, indicated that FFA must be cautiously used in soil stabilization.
International Nuclear Information System (INIS)
Song, T.; Ma, Q.; Sun, X.W.; Liu, Z.J.; Fu, Z.J.; Wei, X.P.; Wang, T.; Tian, J.H.
2016-01-01
The phase transition, electronic band structure, and equation of state (EOS) of cubic TcN are investigated by first-principles pseudopotential method based on density-functional theory. The calculated enthalpies show that TcN has a transformation between zincblende and rocksalt phases and the pressure determined by the relative enthalpy is 32 GPa. The calculated band structure indicates the metallic feature and it might make cubic TcN a better candidate for hard materials. Particular attention is paid to the predictions of volume, bulk modulus and its pressure derivative which play a central role in the formulation of approximate EOSs using the quasi-harmonic Debye model. - Highlights: • The phase transition pressure and electronic band structure for cubic TcN are determined. • Particular attention is paid to investigate the equation of state parameters for cubic TcN. • The thermodynamic properties up to 80 GPa and 3000 K are successfully predicted.
DEFF Research Database (Denmark)
Fuhrman, David R.; Bingham, Harry B.; Madsen, Per A.
2004-01-01
This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly non-linear and extremely dispersive water waves. The analysis demonstrates the near-equivalence of classical linear Fourier (von Neumann...
DEFF Research Database (Denmark)
Fuhrman, David R.; Bingham, Harry B.; Madsen, Per A.
2004-01-01
This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly non-linear and extremely dispersive water waves. The analysis demonstrates the near-equivalence of classical linear Fourier (von Neuman...
Functional equations with causal operators
Corduneanu, C
2003-01-01
Functional equations encompass most of the equations used in applied science and engineering: ordinary differential equations, integral equations of the Volterra type, equations with delayed argument, and integro-differential equations of the Volterra type. The basic theory of functional equations includes functional differential equations with causal operators. Functional Equations with Causal Operators explains the connection between equations with causal operators and the classical types of functional equations encountered by mathematicians and engineers. It details the fundamentals of linear equations and stability theory and provides several applications and examples.
International Nuclear Information System (INIS)
Rieiro, I.; Fernandez, A.; Martinez, A.; Casi, M.
1998-01-01
Has been developed a fast and easy method for to evaluate the efficiency of the process and some or possible stabilities situations in the hot working process for the polycrystalline metallic materials (p.m.m.), by the obtained dates in the Garofalo's equation resolution for the steady state creep and for wide ranges of the work variables, stress, strain rate and temperature. These method use the fitting parameters obtained for the equation mentioned and of their physical meaning. Has been developed the numerical treatment from our previously developed software for the analysis of creep and we can obtained the efficiency energetic maps for the creep and the more generally advises areas for the hot working. Further more has been obtained maps for parameters of great physical significance; f.e. the effective activation energy for different areas of the materials hot working, and in addition has been developed a method for obtained the values of {n P L} for the different ranges of power-law application, when has been obtained the value of { n G } in the Garofalo's equation. (Author) 13 refs
Butuzov, V. F.
2017-06-01
We construct and justify asymptotic expansions of solutions of a singularly perturbed elliptic problem with Dirichlet boundary conditions in the case when the corresponding degenerate equation has a triple root. In contrast to the case of a simple root, the expansion is with respect to fractional (non-integral) powers of the small parameter, the boundary-layer variables have another scaling, and the boundary layer has three zones. This gives rise to essential modifications in the algorithm for constructing the boundary functions. Solutions of the elliptic problem are stationary solutions of the corresponding parabolic problem. We prove that such a stationary solution is asymptotically stable and find its global domain of attraction.
Joslin, Ronald D.; Streett, Craig L.; Chang, Chau-Lyan
1992-01-01
Spatially evolving instabilities in a boundary layer on a flat plate are computed by direct numerical simulation (DNS) of the incompressible Navier-Stokes equations. In a truncated physical domain, a nonstaggered mesh is used for the grid. A Chebyshev-collocation method is used normal to the wall; finite difference and compact difference methods are used in the streamwise direction; and a Fourier series is used in the spanwise direction. For time stepping, implicit Crank-Nicolson and explicit Runge-Kutta schemes are used to the time-splitting method. The influence-matrix technique is used to solve the pressure equation. At the outflow boundary, the buffer-domain technique is used to prevent convective wave reflection or upstream propagation of information from the boundary. Results of the DNS are compared with those from both linear stability theory (LST) and parabolized stability equation (PSE) theory. Computed disturbance amplitudes and phases are in very good agreement with those of LST (for small inflow disturbance amplitudes). A measure of the sensitivity of the inflow condition is demonstrated with both LST and PSE theory used to approximate inflows. Although the DNS numerics are very different than those of PSE theory, the results are in good agreement. A small discrepancy in the results that does occur is likely a result of the variation in PSE boundary condition treatment in the far field. Finally, a small-amplitude wave triad is forced at the inflow, and simulation results are compared with those of LST. Again, very good agreement is found between DNS and LST results for the 3-D simulations, the implication being that the disturbance amplitudes are sufficiently small that nonlinear interactions are negligible.
Arshad, Muhammad; Seadawy, Aly R.; Lu, Dianchen
2017-12-01
In optical fibers, the higher order non-linear Schrödinger equation (NLSE) with cubic quintic nonlinearity describes the propagation of extremely short pulses. We constructed bright and dark solitons, solitary wave and periodic solitary wave solutions of generalized higher order NLSE in cubic quintic non Kerr medium by applying proposed modified extended mapping method. These obtained solutions have key applications in physics and mathematics. Moreover, we have also presented the formation conditions on solitary wave parameters in which dark and bright solitons can exist for this media. We also gave graphically the movement of constructed solitary wave and soliton solutions, that helps to realize the physical phenomena's of this model. The stability of the model in normal dispersion and anomalous regime is discussed by using the modulation instability analysis, which confirms that all constructed solutions are exact and stable. Many other such types of models arising in applied sciences can also be solved by this reliable, powerful and effective method.
Arshad, Muhammad; Seadawy, Aly R.; Lu, Dianchen
2018-01-01
In mono-mode optical fibers, the higher order non-linear Schrödinger equation (NLSE) describes the propagation of enormously short light pulses. We constructed optical solitons and, solitary wave solutions of higher order NLSE mono-mode optical fibers via employing modified extended mapping method which has important applications in Mathematics and physics. Furthermore, the formation conditions are also given on parameters in which optical bright and dark solitons can exist for this media. The moment of the obtained solutions are also given graphically, that helps to realize the physical phenomena's of this model. The modulation instability analysis is utilized to discuss the model stability, which verifies that all obtained solutions are exact and stable. Many other such types of models arising in applied sciences can also be solved by this reliable, powerful and effective method. The method can also be functional to other sorts of higher order nonlinear problems in contemporary areas of research.
International Nuclear Information System (INIS)
Mathews, G.J.; Wilson, J.R.; Evans, C.R.; Detweiler, S.L.
1987-12-01
The dynamics of the final stages of the coalescence of two neturon stars (such as the binary pulsar PSR 1913+16) is an unsolved problem in astrophysics. Such systems are probably efficient generators of gravitational radiation, and may be significant contributors to heavy-element nucleosynthesis. The input physics for the study of such systems is similar to that required for the strudy of heavy-ion collision hydrodynamics; e.g., a finite temperature nuclear equation of state, properties of nuclei away from stability, etc. We discuss the development of a relativistic hydrodynamics code in three spatial dimensions for the purpose of studying such neutron-star systems. The properties of the mass-radius relation (determined by the nuclear equation of state) may lead to a proposed mechanism by which hot, highly neutronized matter is ejected from the coalescing stars. This material is photodisintegrated into a free (mostly) neutron gas which may subsequently experience rapid-neutron capture (r-process) nucleosynthesis. 15 refs., 4 figs
Fehn, Niklas; Wall, Wolfgang A.; Kronbichler, Martin
2017-12-01
The present paper deals with the numerical solution of the incompressible Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods for discretization in space. For DG methods applied to the dual splitting projection method, instabilities have recently been reported that occur for small time step sizes. Since the critical time step size depends on the viscosity and the spatial resolution, these instabilities limit the robustness of the Navier-Stokes solver in case of complex engineering applications characterized by coarse spatial resolutions and small viscosities. By means of numerical investigation we give evidence that these instabilities are related to the discontinuous Galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure. Integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method. Since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity, a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high-order accuracy with respect to the temporal discretization. This new formulation is stable in the limit of small time steps for both equal-order and mixed-order polynomial approximations. Although the dual splitting scheme itself includes inf-sup stabilizing contributions, we demonstrate that spurious pressure oscillations appear for equal-order polynomials and small time steps highlighting the necessity to consider inf-sup stability explicitly.
Formation of Suncups on Snowfields Exposed to Solar Radiation
Tiedje, T.
2005-03-01
A mathematical model is proposed to explain the ablation hollows (suncups) that are observed on snowfields exposed to intense solar radiation. The model is derived by first expressing the distribution of scattered sunlight in the snow in terms of the local slope and curvature of the surface. From this expression, we use a perturbation method valid in the limit of weak surface topography to obtain a differential equation for the snow surface morphology. The resulting non-linear equation is the Kuramoto Sivashinsky equation except with the addition of a conservative non-linear term. In simplified form the equation is: th=cF( ∇^2h-∇^4h+( ∇h )^2+∇^2( ∇h )^2 ) where is the spectral average of the n^th power of the photon diffusion length. Multiple scattering from one concave part of the surface to another is treated self consistently. Numerical solutions of this equation with published values for the optical properties of snow yield spontaneous ordered patterns with a characteristic length of 25-50 cm and characteristic formation time under full solar illumination of 5-15 days, depending on the microstructure of the snow. The spontaneous pattern consists of a hexagonal array of parabolic valleys separated by sharp ridges that closely resemble suncups in size, shape and growth time.
Radiation transport model for ablation hollows on snowfields
Tiedje, T.; Mitchell, Kevin A.; Lau, Bayo; Ballestad, A.; Nodwell, E.
2006-06-01
The ablation hollows or "suncups" that form on the surface of snowfields in summer are a wonderful example of pattern formation in nature. Suncups reduce the albedo of the snow and set a characteristic length for interaction of wind with the snowpack. They also contain information about the properties of the snow and its ablation rate, which could be extracted if we had a more quantitative understanding of how suncups form. A mathematical model is proposed that explains the shape, size, and dynamical behavior of suncups in terms of the interaction of solar radiation with the snowpack. Using a perturbation method, we derive a nonlinear partial differential equation for the time-dependent shape of the snow surface from an approximate physical model for the interaction of solar radiation with snow. The resulting equation, which is similar to the Kuramoto-Sivashinsky equation in fluid mechanics, has solutions with characteristic length and amplitude. We find expressions for the characteristic size of suncups in terms of the spectrally averaged diffusion length of solar radiation in snow. The model correctly describes the shape of suncups, with their spatially ordered patterns of parabolic valleys and V-shaped ridges. It is also in remarkably good agreement with the observed length scales and growth rates. Depending on the relative values of the coefficients of the nonlinear terms in the differential equation, the suncup patterns can be either stationary in time or chaotic.
Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data
Pathak, Jaideep; Lu, Zhixin; Hunt, Brian R.; Girvan, Michelle; Ott, Edward
2017-12-01
We use recent advances in the machine learning area known as "reservoir computing" to formulate a method for model-free estimation from data of the Lyapunov exponents of a chaotic process. The technique uses a limited time series of measurements as input to a high-dimensional dynamical system called a "reservoir." After the reservoir's response to the data is recorded, linear regression is used to learn a large set of parameters, called the "output weights." The learned output weights are then used to form a modified autonomous reservoir designed to be capable of producing an arbitrarily long time series whose ergodic properties approximate those of the input signal. When successful, we say that the autonomous reservoir reproduces the attractor's "climate." Since the reservoir equations and output weights are known, we can compute the derivatives needed to determine the Lyapunov exponents of the autonomous reservoir, which we then use as estimates of the Lyapunov exponents for the original input generating system. We illustrate the effectiveness of our technique with two examples, the Lorenz system and the Kuramoto-Sivashinsky (KS) equation. In the case of the KS equation, we note that the high dimensional nature of the system and the large number of Lyapunov exponents yield a challenging test of our method, which we find the method successfully passes.
International Nuclear Information System (INIS)
Sung, Chang Kyung; Chun, Moon Hyun
1996-01-01
A two-step approach has been used to obtain a new criterion for the onset of slug formation : (1) In the first step, a more general expression than the existing models for the onset of slug flow criterion has been derived from the analysis of singular points and neutral stability conditions of the transient one-dimensional two-phase flow equations of two-fluid model. (2) In the second step, introducing simplifications and incorporating a parameter into the general expression obtained in the first step to satisfy a number of physical conditions a priori specified, a new simple criterion for the onset of slug flow has been derived. Comparisons of the present model with existing models and experimental data show that the present model agree very closely with Taitel and Dukler's model and experimental data in horizontal pipes. In an inclined pipe (θ=50 deg ), however, the difference between the predictions of the present model and those of existing models is appreciably large and the present model gives the best agreement with Ohnuki et al.'s data. 17 refs., 5 figs., 1 tab. (author)
Betchov, R
2012-01-01
Stability of Parallel Flows provides information pertinent to hydrodynamical stability. This book explores the stability problems that occur in various fields, including electronics, mechanics, oceanography, administration, economics, as well as naval and aeronautical engineering. Organized into two parts encompassing 10 chapters, this book starts with an overview of the general equations of a two-dimensional incompressible flow. This text then explores the stability of a laminar boundary layer and presents the equation of the inviscid approximation. Other chapters present the general equation
Li, Qianxiao; Dietrich, Felix; Bollt, Erik M; Kevrekidis, Ioannis G
2017-10-01
Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) 51 and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables which spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky partical differential equation as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy, we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.
Ensemble/Variational Estimation (EnVE) and its application to canonical turbulent flow realizations
Colburn, Christopher; Cessna, Joseph; Bewley, Thomas
2008-11-01
The recently-developed hybrid EnVE method for data assimilation incorporates successive adjoint optimizations to update the initial conditions of a flow model, over various horizons of interest, in order to reconcile this model with recent measurements. Such adjoint optimizations typically require the trajectory to be saved over the entire interval over which the optimization is performed; in high-dimensional systems, this can lead to significant storage problems, which can be partially alleviated via checkpointing. In the EnVE framework, this requirement is eliminated, and supplanted by a requirement to march the state of the system backward in time simultaneously with the adjoint. If the system is derived from a PDE with a diffusive component, this backward-in-time state march is ill conditioned, and requires regularization/smoothing to prevent errors from accumulating rapidly at the small scales. The present talk focuses on this peculiar requirement of the EnVE algorithm. As the forecasting problem may itself be considered as a smoothing problem, it is, in fact, expected to find a ``smoothing'' ingredient at the heart of an algorithm of this sort. Various strategies are proposed and tested for accomplishing the required smoothing in the EnVE setting, and are tested on both a chaotic 1D PDE (the Kuramoto-Sivashinsky equation) as well as our in-house spectral 3D DNS/LES code, diablo.
Li, Qianxiao; Dietrich, Felix; Bollt, Erik M.; Kevrekidis, Ioannis G.
2017-10-01
Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD)51 and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables which spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky partical differential equation as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy, we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.
Peng, Qiujin
2017-09-18
In this paper, we present two second-order numerical schemes to solve the fourth order parabolic equation derived from a diffuse interface model with Peng-Robinson Equation of state (EOS) for pure substance. The mass conservation, energy decay property, unique solvability and L-infinity convergence of these two schemes are proved. Numerical results demonstrate the good approximation of the fourth order equation and confirm reliability of these two schemes.
Ordinary differential equations
Pontryagin, Lev Semenovich
1962-01-01
Ordinary Differential Equations presents the study of the system of ordinary differential equations and its applications to engineering. The book is designed to serve as a first course in differential equations. Importance is given to the linear equation with constant coefficients; stability theory; use of matrices and linear algebra; and the introduction to the Lyapunov theory. Engineering problems such as the Watt regulator for a steam engine and the vacuum-tube circuit are also presented. Engineers, mathematicians, and engineering students will find the book invaluable.
Arshad, M.; Seadawy, Aly R.; Lu, Dianchen
2017-08-01
The higher-order nonlinear Schrödinger equation (NLSE) with fourth-order dispersion, cubic-quintic terms, self-steepening and nonlinear dispersive terms describes the propagation of extremely short pulses in optical fibers. In this paper, the elliptic function, bright and dark solitons and solitary wave solutions of higher-order NLSE are constructed by employing a modified extended direct algebraic method, which has important applications in applied mathematics and physics. Furthermore, we also present the formation conditions of the bright and dark solitons for this equation. The modulation instability is utilized to discuss the stability of these solutions, which shows that all solutions are exact and stable. Many other higher-order nonlinear evolution equations arising in applied sciences can also be solved by this powerful, effective and reliable method.
Elliptic partial differential equations
Volpert, Vitaly
If we had to formulate in one sentence what this book is about it might be "How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species". These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Mathematical anaylsis of reaction-diffusion equations will be based on the theory of Fredholm operators presented in the first volume. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equ...
Iterative Splitting Methods for Differential Equations
Geiser, Juergen
2011-01-01
Iterative Splitting Methods for Differential Equations explains how to solve evolution equations via novel iterative-based splitting methods that efficiently use computational and memory resources. It focuses on systems of parabolic and hyperbolic equations, including convection-diffusion-reaction equations, heat equations, and wave equations. In the theoretical part of the book, the author discusses the main theorems and results of the stability and consistency analysis for ordinary differential equations. He then presents extensions of the iterative splitting methods to partial differential
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.
Tiofack, C. G. L.; Ndzana, F., II; Mohamadou, A.; Kofane, T. C.
2018-03-01
We investigate the existence and stability of solitons in parity-time (PT )-symmetric optical media characterized by a generic complex hyperbolic refractive index distribution and fourth-order diffraction (FOD). For the linear case, we demonstrate numerically that the FOD parameter can alter the PT -breaking points. For nonlinear cases, the exact analytical expressions of the localized modes are obtained both in one- and two-dimensional nonlinear Schrödinger equations with self-focusing and self-defocusing Kerr nonlinearity. The effect of FOD on the stability structure of these localized modes is discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. Examples of stable and unstable solutions are given. The transverse power flow density associated with these localized modes is also discussed. It is found that the relative strength of the FOD coefficient can utterly change the direction of the power flow, which may be used to control the energy exchange among gain or loss regions.
International Nuclear Information System (INIS)
Khisamutdinova, N A
2003-01-01
The behaviour as t→∞ of the solution of the mixed problem for the system of Navier-Stokes equations with a Dirichlet condition at the boundary is studied in an unbounded two-dimensional domain with several exits to infinity. A class of domains is distinguished in which an estimate characterizing the decay of solutions in terms of the geometry of the domain is proved for exponentially decreasing initial velocities. A similar estimate of the solution of the first mixed problem for the heat equation is sharp in a broad class of domains with several exits to infinity
International Nuclear Information System (INIS)
Kozhevnikova, L M; Mukminov, F Kh
2000-01-01
A quasilinear system of parabolic equations with energy inequality is considered in a cylindrical domain {t>0}xΩ. In a broad class of unbounded domains Ω two geometric characteristics of a domain are identified which determine the rate of convergence to zero as t→∞ of the L 2 -norm of a solution. Under additional assumptions on the coefficients of the quasilinear system estimates of the derivatives and uniform estimates of the solution are obtained; they are proved to be best possible in the order of convergence to zero in the case of one semilinear equation
Some functional equations originating from number theory
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations. Keywords. Functional equation; stability; multiplicative function. 1. Introduction. In 1940, Ulam gave a wide ranging talk before the Mathematics ...
Czech Academy of Sciences Publication Activity Database
Penel, P.; Straškraba, Ivan
2007-01-01
Roč. 345, č. 1 (2007), s. 67-72 ISSN 0249-6291 R&D Projects: GA ČR GA201/05/0005 Institutional research plan: CEZ:AV0Z10190503 Keywords : 1D-Navier-Stokes equations * compressible flow * global behavior Subject RIV: BA - General Mathematics
Computational partial differential equations using Matlab
Li, Jichun
2008-01-01
Brief Overview of Partial Differential Equations The parabolic equations The wave equations The elliptic equations Differential equations in broader areasA quick review of numerical methods for PDEsFinite Difference Methods for Parabolic Equations Introduction Theoretical issues: stability, consistence, and convergence 1-D parabolic equations2-D and 3-D parabolic equationsNumerical examples with MATLAB codesFinite Difference Methods for Hyperbolic Equations IntroductionSome basic difference schemes Dissipation and dispersion errors Extensions to conservation lawsThe second-order hyperbolic PDE
Stochastic nonlinear beam equations
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Maslowski, Bohdan; Seidler, Jan
2005-01-01
Roč. 132, č. 1 (2005), s. 119-149 ISSN 0178-8051 R&D Projects: GA ČR(CZ) GA201/01/1197 Institutional research plan: CEZ:AV0Z10190503 Keywords : stochastic beam equation * stability Subject RIV: BA - General Mathematics Impact factor: 0.896, year: 2005
Fay, Temple H.
2010-01-01
Through numerical investigations, we study examples of the forced quadratic spring equation [image omitted]. By performing trial-and-error numerical experiments, we demonstrate the existence of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions, investigate the resonance boundary in the [omega]…
1991-03-01
sections. Stated in English . these stability criteria sound almost trivial, since they immediately result from fundamental thermodynamic statements, which...Armament R&D Establishment ATTN: Ian Cullis Fort Halstead Severnoaks, Kent TN14 7BJ ENGLAND 1 Centre d’Etudes de Gramat ATTN: SOLVE Gerard 46500
Functional equations in matrix normed spaces
Indian Academy of Sciences (India)
Cauchy additive functional equation and the quadratic functional equation in matrix normed spaces. Keywords. Operator space; fixed point; Hyers–Ulam stability; Cauchy additive functional equation; quadratic functional equation. 2000 Mathematics Subject Classification. 47L25, 47H10, 39B82, 46L07, 39B52. 1.
Breda, D.; Diekmann, O.; Gyllenberg, M.; Scarabel, F.; Vermiglio, R.
2016-01-01
We apply the pseudospectral discretization approach to nonlinear delay models described by delay differential equations, renewal equations, or systems of coupled renewal equations and delay differential equations. The aim is to derive ordinary differential equations and to investigate the stability
Nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Developments in functional equations and related topics
Ciepliński, Krzysztof; Rassias, Themistocles
2017-01-01
This book presents current research on Ulam stability for functional equations and inequalities. Contributions from renowned scientists emphasize fundamental and new results, methods and techniques. Detailed examples are given to theories to further understanding at the graduate level for students in mathematics, physics, and engineering. Key topics covered in this book include: Quasi means Approximate isometries Functional equations in hypergroups Stability of functional equations Fischer-Muszély equation Haar meager sets and Haar null sets Dynamical systems Functional equations in probability theory Stochastic convex ordering Dhombres functional equation Nonstandard analysis and Ulam stability This book is dedicated in memory of Staniłsaw Marcin Ulam, who posed the fundamental problem concerning approximate homomorphisms of groups in 1940; which has provided the stimulus for studies in the stability of functional equations and inequalities.
Energy Technology Data Exchange (ETDEWEB)
Young, C.W. [Applied Research Associates, Inc., Albuquerque, NM (United States)
1997-10-01
In 1967, Sandia National Laboratories published empirical equations to predict penetration into natural earth materials and concrete. Since that time there have been several small changes to the basic equations, and several more additions to the overall technique for predicting penetration into soil, rock, concrete, ice, and frozen soil. The most recent update to the equations was published in 1988, and since that time there have been changes in the equations to better match the expanding data base, especially in concrete penetration. This is a standalone report documenting the latest version of the Young/Sandia penetration equations and related analytical techniques to predict penetration into natural earth materials and concrete. 11 refs., 6 tabs.
Khalili, Mey; Liwo, Adam; Rakowski, Franciszek; Grochowski, Paweł; Scheraga, Harold A.
2008-01-01
The Lagrange formalism was implemented to derive the equations of motion for the physics-based united-residue (UNRES) force field developed in our laboratory. The Cα…Cα and Cα…SC (SC denoting a side-chain center) virtual-bond vectors were chosen as variables. The velocity Verlet algorithm was adopted to integrate the equations of motion. Tests on the unblocked Ala10 polypeptide showed that the algorithm is stable in short periods of time up to the time step of 1.467 fs; however, even with the shorter time step of 0.489 fs, some drift of the total energy occurs because of momentary jumps of the acceleration. These jumps are caused by numerical instability of the forces arising from the Urot component of UNRES that describes the energetics of side-chain-rotameric states. Test runs on the Gly10 sequence (in which Urot is not present) and on the Ala10 sequence with Urot replaced by a simple numerically stable harmonic potential confirmed this observation; oscillations of the total energy were observed only up to the time step of 7.335 fs, and some drift in the total energy or instability of the trajectories started to appear in long-time (2 ns and longer) trajectories only for the time step of 9.78 fs. These results demonstrate that the present Urot components (which are statistical potentials derived from the Protein Data Bank) must be replaced with more numerically stable functions; this work is under way in our laboratory. For the purpose of our present work, a nonsymplectic variable-time-step algorithm was introduced to reduce the energy drift for regular polypeptide sequences. The algorithm scales down the time step at a given point of a trajectory if the maximum change of acceleration exceeds a selected cutoff value. With this algorithm, the total energy is reasonably conserved up to a time step of 2.445 fs, as tested on the unblocked Ala10 polypeptide. We also tried a symplectic multiple-time-step reversible RESPA algorithm and achieved satisfactory
Energy Technology Data Exchange (ETDEWEB)
Souza, Altivo Monteiro de
2008-12-15
The world energy consumption has been increasing strongly in recent years. Nuclear energy has been regarded as a suitable option to supply this growing energy demand in industrial scale. In view of the need of improving the understanding and capacity of analysis of nuclear power plants, modern simulation techniques for flow and heat transfer problems are gaining greater importance. A large number of problems found in nuclear reactor engineering can be dealt assuming axial symmetry. Thus, in this work a stabilized finite element formulation for the solution of the Navier-Stokes and energy equations for axisymmetric problems have been developed and tested. The formulation has been implemented in the NS{sub S}OLVER{sub M}PI{sub 2}D{sub A} program developed at the Parallel Computation Laboratory of the Instituto de Engenharia Nuclear (LCP/IEN) and is now available either for safety analysis or design of nuclear systems. (author)
Tricomi, FG
2013-01-01
Based on his extensive experience as an educator, F. G. Tricomi wrote this practical and concise teaching text to offer a clear idea of the problems and methods of the theory of differential equations. The treatment is geared toward advanced undergraduates and graduate students and addresses only questions that can be resolved with rigor and simplicity.Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and diff
Macroscopic plasma properties and stability theory
International Nuclear Information System (INIS)
Sakanaka, P.H.
1981-01-01
1. Two-fluid equations: (a) Boltzmann equation: complete set of equations; collision models - Vlasov, BGK, Fokker-Planck-Landau, Boltzmann. (b) Moments of the Boltzmann equation: problem of closure. (c) Two-fluid equations. 2. One-fluid equation: (a) One-fluid variables. (b) One-fluid equations: quasi-neutrality. (c) Resistive MHD equations. (d) Ideal MHD equations: one-adiabatic approximation; double-adiabatic approximation - CGL. 3. MHD stability problem - energy principle: (a) Linearized ideal MHD equations: force-operator equation. (b) Boundary conditions. (c) Self-adjointness of force operator. (d) The energy principle. 4. Stability problems: application of the energy principle; stability of sharp-boundary plasmas. 5. Thermodynamic approach for stability of plasmas: Newcomb and Rosenbluth's stability criteria. (author)
Some Functional Equations Originating from Number Theory
Indian Academy of Sciences (India)
We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.
Some functional equations originating from number theory
Indian Academy of Sciences (India)
We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.
Differential equations methods and applications
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. .
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.)
Thermoviscous Model Equations in Nonlinear Acoustics
DEFF Research Database (Denmark)
Rasmussen, Anders Rønne
Four nonlinear acoustical wave equations that apply to both perfect gasses and arbitrary fluids with a quadratic equation of state are studied. Shock and rarefaction wave solutions to the equations are studied. In order to assess the accuracy of the wave equations, their solutions are compared...... to solutions of the basic equations from which the wave equations are derived. A straightforward weakly nonlinear equation is the most accurate for shock modeling. A higher order wave equation is the most accurate for modeling of smooth disturbances. Investigations of the linear stability properties...... of solutions to the wave equations, reveal that the solutions may become unstable. Such instabilities are not found in the basic equations. Interacting shocks and standing shocks are investigated....
Numerical methods for stochastic differential equations.
Wilkie, Joshua
2004-01-01
Stochastic differential equations (SDE's) play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability. A general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here. High-order numerical methods are developed for the integration of stochastic differential equations with strong solutions. We demonstrate the accuracy of the resulting integration schemes by computing the errors in approximate solutions for SDE's which have known exact solutions.
Full-order optimal compensators for flow control: the multiple inputs case
Semeraro, Onofrio; Pralits, Jan O.
2018-03-01
Flow control has been the subject of numerous experimental and theoretical works. We analyze full-order, optimal controllers for large dynamical systems in the presence of multiple actuators and sensors. The full-order controllers do not require any preliminary model reduction or low-order approximation: this feature allows us to assess the optimal performance of an actuated flow without relying on any estimation process or further hypothesis on the disturbances. We start from the original technique proposed by Bewley et al. (Meccanica 51(12):2997-3014, 2016. https://doi.org/10.1007/s11012-016-0547-3), the adjoint of the direct-adjoint (ADA) algorithm. The algorithm is iterative and allows bypassing the solution of the algebraic Riccati equation associated with the optimal control problem, typically infeasible for large systems. In this numerical work, we extend the ADA iteration into a more general framework that includes the design of controllers with multiple, coupled inputs and robust controllers (H_{∞} methods). First, we demonstrate our results by showing the analytical equivalence between the full Riccati solutions and the ADA approximations in the multiple inputs case. In the second part of the article, we analyze the performance of the algorithm in terms of convergence of the solution, by comparing it with analogous techniques. We find an excellent scalability with the number of inputs (actuators), making the method a viable way for full-order control design in complex settings. Finally, the applicability of the algorithm to fluid mechanics problems is shown using the linearized Kuramoto-Sivashinsky equation and the Kármán vortex street past a two-dimensional cylinder.
Stability of Linear Equations--Algebraic Approach
Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.
2012-01-01
This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…
Stochastic partial differential equations
Chow, Pao-Liu
2014-01-01
Preliminaries Introduction Some Examples Brownian Motions and Martingales Stochastic Integrals Stochastic Differential Equations of Itô Type Lévy Processes and Stochastic IntegralsStochastic Differential Equations of Lévy Type Comments Scalar Equations of First Order Introduction Generalized Itô's Formula Linear Stochastic Equations Quasilinear Equations General Remarks Stochastic Parabolic Equations Introduction Preliminaries Solution of Stochastic Heat EquationLinear Equations with Additive Noise Some Regularity Properties Stochastic Reaction-Diffusion Equations Parabolic Equations with Grad
Tam, Christopher; Krothapalli, A
1993-01-01
The research program for the first year of this project (see the original research proposal) consists of developing an explicit marching scheme for solving the parabolized stability equations (PSE). Performing mathematical analysis of the computational algorithm including numerical stability analysis and the determination of the proper boundary conditions needed at the boundary of the computation domain are implicit in the task. Before one can solve the parabolized stability equations for high-speed mixing layers, the mean flow must first be found. In the past, instability analysis of high-speed mixing layer has mostly been performed on mean flow profiles calculated by the boundary layer equations. In carrying out this project, it is believed that the boundary layer equations might not give an accurate enough nonparallel, nonlinear mean flow needed for parabolized stability analysis. A more accurate mean flow can, however, be found by solving the parabolized Navier-Stokes equations. The advantage of the parabolized Navier-Stokes equations is that its accuracy is consistent with the PSE method. Furthermore, the method of solution is similar. Hence, the major part of the effort of the work of this year has been devoted to the development of an explicit numerical marching scheme for the solution of the Parabolized Navier-Stokes equation as applied to the high-seed mixing layer problem.
Concepts of stability and control
International Nuclear Information System (INIS)
Parks, P.C.
1976-01-01
The first part of the paper deals with the role transfer functions play in control problems (closed loop, Nyquist stability criterion, sampled data systems and z-transforms; the ''hog cycle''; spring oscillations; Lyapunov functions; the Zubov method; positive-real functions and the Popov criterion; the circle criterion; linear time-delay systems; equations with periodic coefficients; stability of repeated processes). In the second part the author considers the control of systems which are described by partial differential equations (heat-conduction equation; wave equation; control of the heat and the wave equations; parasitic oscillations; noise in linear systems; discrete noise processes). Many examples are given and briefly discussed. (author)
On two functional equations originating from number theory
Indian Academy of Sciences (India)
Reducing the functional equations introduced in Proc. Indian Acad. Sci. (Math. Sci.) 113(2) (2003) 91–98 and in Appl. Math. Lett. 21 (2008) 974–977 to equations in complex variables and quaternions, we find general solutions of the equations. We also obtain the stability of the equations.
Blowup in the complex Ginzburg-Landau equation
Schans, Martin van der
2013-01-01
In this thesis, we study the stability of a finite-time blowup solution of a partial di erential equation (PDE). Partial di erential equations can be used to model phenomena in a wide range of applications. Examples of well known partial di erential equations are: the heat equation which models heat
Generalized reduced MHD equations
International Nuclear Information System (INIS)
Kruger, S.E.; Hegna, C.C.; Callen, J.D.
1998-07-01
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general toroidal configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson
Elements of magnetohydrodynamic stability theory
International Nuclear Information System (INIS)
Spies, G.O.
1976-11-01
The nonlinear equations of ideal magnetohydrodynamics are discussed along with the following topics: (1) static equilibrium, (2) strict linear theory, (3) stability of a system with one degree of freedom, (4) spectrum and variational principles in magnetohydrodynamics, (5) elementary proof of the modified energy principle, (6) sufficient stability criteria, (7) local stability, and (8) normal modes
International Nuclear Information System (INIS)
Shore, B.W.
1981-01-01
The equations of motion are discussed which describe time dependent population flows in an N-level system, reviewing the relationship between incoherent (rate) equations, coherent (Schrodinger) equations, and more general partially coherent (Bloch) equations. Approximations are discussed which replace the elaborate Bloch equations by simpler rate equations whose coefficients incorporate long-time consequences of coherence
Applied analysis and differential equations
Cârj, Ovidiu
2007-01-01
This volume contains refereed research articles written by experts in the field of applied analysis, differential equations and related topics. Well-known leading mathematicians worldwide and prominent young scientists cover a diverse range of topics, including the most exciting recent developments. A broad range of topics of recent interest are treated: existence, uniqueness, viability, asymptotic stability, viscosity solutions, controllability and numerical analysis for ODE, PDE and stochastic equations. The scope of the book is wide, ranging from pure mathematics to various applied fields such as classical mechanics, biomedicine, and population dynamics.
Discrete nonlinear Schrodinger equations with arbitrarily high-order nonlinearities
DEFF Research Database (Denmark)
Khare, A.; Rasmussen, Kim Ø; Salerno, M.
2006-01-01
A class of discrete nonlinear Schrodinger equations with arbitrarily high-order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrodinger equation and the Ablowitz......-Ladik equation. As a common property, these equations possess three kinds of exact analytical stationary solutions for which the Peierls-Nabarro barrier is zero. Several properties of these solutions, including stability, discrete breathers, and moving solutions, are investigated....
Analytical Solution of Pantograph Equation with Incommensurate Delay
Patade, Jayvant; Bhalekar, Sachin
2017-08-01
Pantograph equation is a delay differential equation (DDE) arising in electrodynamics. This paper studies the pantograph equation with two delays. The existence, uniqueness, stability and convergence results for DDEs are presented. The series solution of the proposed equation is obtained by using Daftardar-Gejji and Jafari method and given in terms of a special function. This new special function has several properties and relations with other functions. Further, we generalize the proposed equation to fractional-order case and obtain its solution.
Analyticity estimates for the Navier-Stokes equations
DEFF Research Database (Denmark)
Herbst, I.; Skibsted, Erik
We study spatial analyticity properties of solutions of the Navier-Stokes equation and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier-Stokes equation with data in and prove a stability result...
Dynamics of partial differential equations
Wayne, C Eugene
2015-01-01
This book contains two review articles on the dynamics of partial differential equations that deal with closely related topics but can be read independently. Wayne reviews recent results on the global dynamics of the two-dimensional Navier-Stokes equations. This system exhibits stable vortex solutions: the topic of Wayne's contribution is how solutions that start from arbitrary initial conditions evolve towards stable vortices. Weinstein considers the dynamics of localized states in nonlinear Schrodinger and Gross-Pitaevskii equations that describe many optical and quantum systems. In this contribution, Weinstein reviews recent bifurcations results of solitary waves, their linear and nonlinear stability properties, and results about radiation damping where waves lose energy through radiation. The articles, written independently, are combined into one volume to showcase the tools of dynamical systems theory at work in explaining qualitative phenomena associated with two classes of partial differential equ...
Partial Differential Equations
1988-01-01
The volume contains a selection of papers presented at the 7th Symposium on differential geometry and differential equations (DD7) held at the Nankai Institute of Mathematics, Tianjin, China, in 1986. Most of the contributions are original research papers on topics including elliptic equations, hyperbolic equations, evolution equations, non-linear equations from differential geometry and mechanics, micro-local analysis.
Magnetic equations with FreeFem++: The Grad-Shafranov equation & the current hole
Directory of Open Access Journals (Sweden)
Sadaka Georges
2011-11-01
Full Text Available FreeFem++ [11] is a software for the numerical solution of partial differential equations. It is based on finite element method. The FreeFem++ platform aims at facilitating teaching and basic research through prototyping. For the moment this platform is restricted to the numerical simulations of problems which admit a variational formulation. Our goal in this work is to evaluate the FreeFem++ tool on basic magnetic equations arising in Fusion Plasma in the context of the ITER project. First we consider the Grad-Shafranov equation, which is derived from the static ideal MHD equations assuming axisymetry. Some of the properties of the equation and its analytical solutions are discussed. Second we discretize a reduced resistive MHD model which admits solutions of the Grad-Shafranov equation as stationary solutions. Then the physical stability of these stationary solutions is investigated through numerical experiments and the numerical stability of the algorithm is discussed.
The concept of stability in numerical mathematics
Hackbusch, Wolfgang
2014-01-01
In this book, the author compares the meaning of stability in different subfields of numerical mathematics. Concept of Stability in numerical mathematics opens by examining the stability of finite algorithms. A more precise definition of stability holds for quadrature and interpolation methods, which the following chapters focus on. The discussion then progresses to the numerical treatment of ordinary differential equations (ODEs). While one-step methods for ODEs are always stable, this is not the case for hyperbolic or parabolic differential equations, which are investigated next. The final chapters discuss stability for discretisations of elliptic differential equations and integral equations. In comparison among the subfields we discuss the practical importance of stability and the possible conflict between higher consistency order and stability.
Stability of Approximate Quadratic Mappings
Directory of Open Access Journals (Sweden)
Kim Hark-Mahn
2010-01-01
Full Text Available We investigate the general solution of the quadratic functional equation , in the class of all functions between quasi- -normed spaces, and then we prove the generalized Hyers-Ulam stability of the equation by using direct method and fixed point method.
Difference equations by differential equation methods
Hydon, Peter E
2014-01-01
Most well-known solution techniques for differential equations exploit symmetry in some form. Systematic methods have been developed for finding and using symmetries, first integrals and conservation laws of a given differential equation. Here the author explains how to extend these powerful methods to difference equations, greatly increasing the range of solvable problems. Beginning with an introduction to elementary solution methods, the book gives readers a clear explanation of exact techniques for ordinary and partial difference equations. The informal presentation is suitable for anyone who is familiar with standard differential equation methods. No prior knowledge of difference equations or symmetry is assumed. The author uses worked examples to help readers grasp new concepts easily. There are 120 exercises of varying difficulty and suggestions for further reading. The book goes to the cutting edge of research; its many new ideas and methods make it a valuable reference for researchers in the field.
Ordinary differential equations a graduate text
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...
Stability of charged strange quark stars
Energy Technology Data Exchange (ETDEWEB)
Arbañil, José D. V.; Malheiro, Manuel [Departamento de Física, Instituto Tecnológico de Aeronáutica, Centro Técnico Aeroespacial, 12228-900 São José dos Campos, SP (Brazil)
2015-12-17
We investigate the hydrostatic equilibrium and the stability of charged stars made of a charged perfect fluid. The matter contained in the star follows the MIT bag model equation of state and the charge distribution to a power-law of the radial coordinate. The hydrostatic equilibrium and the stability of charged strange stars are analyzed using the Tolman-Oppenheimer-Volkoff equation and the Chandrasekhar’s equation pulsation, respectively. These two equation are modified from their original form to the inclusion of the electric charge. We found that the stability of the star decreases with the increment of the central energy density and with the increment of the amount of charge.
General Solutions of Two Quadratic Functional Equations of Pexider Type on Orthogonal Vectors
Fochi, Margherita
2012-01-01
Based on the studies on the Hyers-Ulam stability and the orthogonal stability of some Pexider-quadratic functional equations, in this paper we find the general solutions of two quadratic functional equations of Pexider type. Both equations are studied in restricted domains: the first equation is studied on the restricted domain of the orthogonal vectors in the sense of Rätz, and the second equation is considered on the orthogonal vectors in the inner product spaces with the usual orthogonality.
Handbook of integral equations
Polyanin, Andrei D
2008-01-01
This handbook contains over 2,500 integral equations with solutions as well as analytical and numerical methods for solving linear and nonlinear equations. It explores Volterra, Fredholm, WienerHopf, Hammerstein, Uryson, and other equations that arise in mathematics, physics, engineering, the sciences, and economics. This second edition includes new chapters on mixed multidimensional equations and methods of integral equations for ODEs and PDEs, along with over 400 new equations with exact solutions. With many examples added for illustrative purposes, it presents new material on Volterra, Fredholm, singular, hypersingular, dual, and nonlinear integral equations, integral transforms, and special functions.
Lie theoretic aspects of the Riccati equation
Hermann, R.; Martin, C.
1977-01-01
Various features of the application of Lie theory to matrix Riccati equations, of basic importance in control and system theories, are discussed. Particular consideration is given to centralizer foliation, the Cartan decomposition, matrix Riccati equations as Lie systems on Grassmanians, local analysis near a zero point of a vector field, linearization in homogeneous space, the tangent bundle in terms of partitioned matrices, and stability properties of fixed points of Riccati vector fields.
Investigation of the Stability via Shadowing Property
Directory of Open Access Journals (Sweden)
Koh Heejeong
2009-01-01
Full Text Available The shadowing property is to find an exact solution to an iterated map that remains close to an approximate solution. In this article, using shadowing property, we show the stability of the following equation in normed group: , where , and is a mapping. And we prove that the even mapping which satisfies the above equation is quadratic and also the Hyers-Ulam stability of the functional equation in Banach spaces.
Uraltseva, N N
1995-01-01
This collection focuses on nonlinear problems in partial differential equations. Most of the papers are based on lectures presented at the seminar on partial differential equations and mathematical physics at St. Petersburg University. Among the topics explored are the existence and properties of solutions of various classes of nonlinear evolution equations, nonlinear imbedding theorems, bifurcations of solutions, and equations of mathematical physics (Navier-Stokes type equations and the nonlinear Schrödinger equation). The book will be useful to researchers and graduate students working in p
Introduction to differential equations
Taylor, Michael E
2011-01-01
The mathematical formulations of problems in physics, economics, biology, and other sciences are usually embodied in differential equations. The analysis of the resulting equations then provides new insight into the original problems. This book describes the tools for performing that analysis. The first chapter treats single differential equations, emphasizing linear and nonlinear first order equations, linear second order equations, and a class of nonlinear second order equations arising from Newton's laws. The first order linear theory starts with a self-contained presentation of the exponen
General particle transport equation. Final report
International Nuclear Information System (INIS)
Lafi, A.Y.; Reyes, J.N. Jr.
1994-12-01
The general objectives of this research are as follows: (1) To develop fundamental models for fluid particle coalescence and breakage rates for incorporation into statistically based (Population Balance Approach or Monte Carlo Approach) two-phase thermal hydraulics codes. (2) To develop fundamental models for flow structure transitions based on stability theory and fluid particle interaction rates. This report details the derivation of the mass, momentum and energy conservation equations for a distribution of spherical, chemically non-reacting fluid particles of variable size and velocity. To study the effects of fluid particle interactions on interfacial transfer and flow structure requires detailed particulate flow conservation equations. The equations are derived using a particle continuity equation analogous to Boltzmann's transport equation. When coupled with the appropriate closure equations, the conservation equations can be used to model nonequilibrium, two-phase, dispersed, fluid flow behavior. Unlike the Eulerian volume and time averaged conservation equations, the statistically averaged conservation equations contain additional terms that take into account the change due to fluid particle interfacial acceleration and fluid particle dynamics. Two types of particle dynamics are considered; coalescence and breakage. Therefore, the rate of change due to particle dynamics will consider the gain and loss involved in these processes and implement phenomenological models for fluid particle breakage and coalescence
Nonlinear physical systems spectral analysis, stability and bifurcations
Kirillov, Oleg N
2013-01-01
Bringing together 18 chapters written by leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics, this book presents state-of-the-art approaches to a wide spectrum of new and challenging stability problems.Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations focuses on problems of spectral analysis, stability and bifurcations arising in the nonlinear partial differential equations of modern physics. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynam
Particle methods for Boltzmann equation
International Nuclear Information System (INIS)
Hermeline, F.
1985-05-01
This work is aimed at showing how to discretize an equation such as Boltzmann equation in its most general form, by particle methods. Then method is applied to some equations of plasma physics which appear as peculiar cases of Boltzmann equation, such as Vlasov equation, Bhatnager-Gross-Krook equation, Fokker-Planck equation and neutron transport equation [fr
Differential Equations and Computational Simulations
1999-06-18
given in (6),(7) in Taylor series of e. Equating coefficients of same power of e in both side of equity , we obtain a sequence of linear boundary value...fields, 3). structural instability and block stability of divergence-free vector fields on 2D compact manifolds with nonzero genus , and 4). structural...circle bands. Definition 3.1 Let N be a compact manifold without boundary and with genus k > 0. A closed domain fi C N is called a pseudo-manifold
Stability and response bounds of non-conservative linear systems
DEFF Research Database (Denmark)
Pommer, Christian
2003-01-01
For a linear system of second order differential equations the stability is studied by Lyapunov's direct method. The Lyapunov matrix equation is solved and a sufficient condition for stability is expressed by the system matrices. For a system which satisfies the condition for stability the Lyapunov...
A remark on Dickey's stabilizing chain
International Nuclear Information System (INIS)
Svinin, Andrei K.
2011-01-01
We observe that Dickey's stabilizing chain can be naturally included into two-dimensional chain of infinitely many copies of equations of KP hierarchy. -- Highlights: → In this study we consider Dickey's stabilizing chain. → We construct two-dimensional chain of dressing truncated operators. → We show that Dickey's stabilizing chain can be included into two-dimensional chain of KP hierarchies.
Beginning partial differential equations
O'Neil, Peter V
2014-01-01
A broad introduction to PDEs with an emphasis on specialized topics and applications occurring in a variety of fields Featuring a thoroughly revised presentation of topics, Beginning Partial Differential Equations, Third Edition provides a challenging, yet accessible,combination of techniques, applications, and introductory theory on the subjectof partial differential equations. The new edition offers nonstandard coverageon material including Burger's equation, the telegraph equation, damped wavemotion, and the use of characteristics to solve nonhomogeneous problems. The Third Edition is or
Ordinary differential equations
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
Bottura, L
2014-01-01
Superconductor stability is at the core of the design of any successful cable and magnet application. This chapter reviews the initial understanding of the stability mechanism, and reviews matters of importance for stability such as the nature and magnitude of the perturbation spectrum and the cooling mechanisms. Various stability strategies are studied, providing criteria that depend on the desired design and operating conditions.
Fay, Temple H.
2002-01-01
We investigate the pendulum equation [theta] + [lambda][squared] sin [theta] = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin [theta] do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are…
Functional Equations in Fuzzy Banach Spaces
Directory of Open Access Journals (Sweden)
M. Eshaghi Gordji
2012-01-01
generalized Hyers-Ulam stability of the following additive-quadratic functional equation f(x+ky+f(x−ky=f(x+y+f(x−y+(2(k+1/kf(ky−2(k+1f(y for fixed integers k with k≠0,±1 in fuzzy Banach spaces.
A Comparison of IRT Equating and Beta 4 Equating
Kim, Dong-In; Brennan, Robert; Kolen, Michael
2005-01-01
Four equating methods (3PL true score equating, 3PL observed score equating, beta 4 true score equating, and beta 4 observed score equating) were compared using four equating criteria: first-order equity (FOE), second-order equity (SOE), conditional-mean-squared-error (CMSE) difference, and the equi-percentile equating property. True score…
On the Stability of Cubic Mappings and Quadratic Mappings in Random Normed Spaces
Directory of Open Access Journals (Sweden)
Cho YJ
2008-01-01
Full Text Available Abstract Recently, the stability of the cubic functional equation in fuzzy normed spaces was proved in earlier work; and the stability of the additive functional equations , in random normed spaces was proved as well. In this paper, we prove the stability of the cubic functional equation in random normed spaces by an alternative proof which provides a better estimation. Finally, we prove the stability of the quartic functional equation in random normed spaces.
Wu, Guo-Cheng; Baleanu, Dumitru; Zeng, Sheng-Da
2018-04-01
This study investigates finite-time stability of Caputo delta fractional difference equations. A generalized Gronwall inequality is given on a finite time domain. A finite-time stability criterion is proposed for fractional differential equations. Then the idea is extended to the discrete fractional case. A linear fractional difference equation with constant delays is considered and finite-time stable conditions are provided. One example is numerically illustrated to support the theoretical result.
International Nuclear Information System (INIS)
Zhalij, Alexander
2002-01-01
We classify (1+3)-dimensional Pauli equations for a spin-(1/2) particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the 11 classes of vector-potentials of the electro-magnetic field A(t,x(vector sign))=(A 0 (t,x(vector sign)), A(vector sign)(t,x(vector sign))) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is its equivalence to the system of two uncoupled Schroedinger equations. In addition, the magnetic field has to be independent of spatial variables. We prove that coordinate systems and the vector-potentials of the electro-magnetic field providing the separability of the corresponding Pauli equations coincide with those for the Schroedinger equations. Furthermore, an efficient algorithm for constructing all coordinate systems providing the separability of Pauli equation with a fixed vector-potential of the electro-magnetic field is developed. Finally, we describe all vector-potentials A(t,x(vector sign)) that (a) provide the separability of Pauli equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe non-zero magnetic field
An introduction to geometric theory of fully nonlinear parabolic equations
International Nuclear Information System (INIS)
Lunardi, A.
1991-01-01
We study a class of nonlinear evolution equations in general Banach space being an abstract version of fully nonlinear parabolic equations. In addition to results of existence, uniqueness and continuous dependence on the data, we give some qualitative results about stability of the stationary solutions, existence and stability of the periodic orbits. We apply such results to some parabolic problems arising from combustion theory. (author). 24 refs
A textbook on ordinary differential equations
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...
Advances in iterative methods for nonlinear equations
Busquier, Sonia
2016-01-01
This book focuses on the approximation of nonlinear equations using iterative methods. Nine contributions are presented on the construction and analysis of these methods, the coverage encompassing convergence, efficiency, robustness, dynamics, and applications. Many problems are stated in the form of nonlinear equations, using mathematical modeling. In particular, a wide range of problems in Applied Mathematics and in Engineering can be solved by finding the solutions to these equations. The book reveals the importance of studying convergence aspects in iterative methods and shows that selection of the most efficient and robust iterative method for a given problem is crucial to guaranteeing a good approximation. A number of sample criteria for selecting the optimal method are presented, including those regarding the order of convergence, the computational cost, and the stability, including the dynamics. This book will appeal to researchers whose field of interest is related to nonlinear problems and equations...
Stability of Approximate Quadratic Mappings
Directory of Open Access Journals (Sweden)
Juri Lee
2010-01-01
Full Text Available We investigate the general solution of the quadratic functional equation f(2x+y+3f(2x−y=4f(x−y+12f(x, in the class of all functions between quasi-β-normed spaces, and then we prove the generalized Hyers-Ulam stability of the equation by using direct method and fixed point method.
The Price Equation, Gradient Dynamics, and Continuous Trait Game Theory.
Lehtonen, Jussi
2018-01-01
A recent article convincingly nominated the Price equation as the fundamental theorem of evolution and used it as a foundation to derive several other theorems. A major section of evolutionary theory that was not addressed is that of game theory and gradient dynamics of continuous traits with frequency-dependent fitness. Deriving fundamental results in these fields under the unifying framework of the Price equation illuminates similarities and differences between approaches and allows a simple, unified view of game-theoretical and dynamic concepts. Using Taylor polynomials and the Price equation, I derive a dynamic measure of evolutionary change, a condition for singular points, the convergence stability criterion, and an alternative interpretation of evolutionary stability. Furthermore, by applying the Price equation to a multivariable Taylor polynomial, the direct fitness approach to kin selection emerges. Finally, I compare these results to the mean gradient equation of quantitative genetics and the canonical equation of adaptive dynamics.
Stability Limits in Resonant Planetary Systems
Barnes, Rory; Greenberg, Richard
2007-01-01
The relationship between the boundaries for Hill and Lagrange stability in orbital element space is modified in the case of resonantly interacting planets. Hill stability requires the ordering of the planets to remain constant while Lagrange stability also requires all planets to remain bound to the central star. The Hill stability boundary is defined analytically, but no equations exist to define the Lagrange boundary, so we perform numerical experiments to estimate the location of this boun...
Directory of Open Access Journals (Sweden)
Wei Khim Ng
2009-02-01
Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.
Partial differential equations
Evans, Lawrence C
2010-01-01
This text gives a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDEs) with particular emphasis on nonlinear equations. The exposition is divided into three parts: representation formulas for solutions; theory for linear partial differential equations; and theory for nonlinear partial differential equations. Included are complete treatments of the method of characteristics; energy methods within Sobolev spaces; regularity for second-order elliptic, parabolic, and hyperbolic equations; maximum principles; the multidimensional calculus of variations; viscosity solutions of Hamilton-Jacobi equations; shock waves and entropy criteria for conservation laws; and, much more.The author summarizes the relevant mathematics required to understand current research in PDEs, especially nonlinear PDEs. While he has reworked and simplified much of the classical theory (particularly the method of characteristics), he primarily emphasizes the modern interplay between funct...
Differential equations for dummies
Holzner, Steven
2008-01-01
The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
Degenerate nonlinear diffusion equations
Favini, Angelo
2012-01-01
The aim of these notes is to include in a uniform presentation style several topics related to the theory of degenerate nonlinear diffusion equations, treated in the mathematical framework of evolution equations with multivalued m-accretive operators in Hilbert spaces. The problems concern nonlinear parabolic equations involving two cases of degeneracy. More precisely, one case is due to the vanishing of the time derivative coefficient and the other is provided by the vanishing of the diffusion coefficient on subsets of positive measure of the domain. From the mathematical point of view the results presented in these notes can be considered as general results in the theory of degenerate nonlinear diffusion equations. However, this work does not seek to present an exhaustive study of degenerate diffusion equations, but rather to emphasize some rigorous and efficient techniques for approaching various problems involving degenerate nonlinear diffusion equations, such as well-posedness, periodic solutions, asympt...
Reactimeter dispersion equation
A.G. Yuferov
2016-01-01
The aim of this work is to derive and analyze a reactimeter metrological model in the form of the dispersion equation which connects reactimeter input/output signal dispersions with superimposed random noise at the inlet. It is proposed to standardize the reactimeter equation form, presenting the main reactimeter computing unit by a convolution equation. Hence, the reactimeter metrological characteristics are completely determined by this unit hardware function which represents a transient re...
Differential equations I essentials
REA, Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Differential Equations I covers first- and second-order equations, series solutions, higher-order linear equations, and the Laplace transform.
Travelling waves in a singularly perturbed sine-Gordon equation
Derks, Gianne; Doelman, Arjen; van Gils, Stephanus A.; Visser, T.P.P.
2003-01-01
We determine the linearised stability of travelling front solutions of a perturbed sine-Gordon equation. This equation models the long Josephson junction using the RCSJ model for currents across the junction and includes surface resistance for currents along the junction. The travelling waves
Travelling waves in a singularly perturbed sine-Gordon equations
Derks, G.L.A.; Derks, Gianne; Doelman, Arjen; van Gils, Stephanus A.; Visser, T.P.P.
2003-01-01
We determine the linearised stability of travelling front solutions of a perturbed sine-Gordon equation. This equation models the long Josephson junction using the RCSJ model for currents across the junction and includes surface resistance for currents along the junction. The travelling waves
A heat equation for freezing processes with phase change
DEFF Research Database (Denmark)
Backi, Christoph Josef; Bendtsen, Jan Dimon; Leth, John-Josef
2016-01-01
In this work, the stability properties as well as possible applications of a partial differential equation (PDE) with state-dependent parameters are investigated. Among other things, the PDE describes freezing of foodstuff, and is closely related to the (potential) Burgers’ equation. We show...
Modulation equations for spatially periodic systems: derivation and solutions
Schielen, R.; Doelman, A.
1996-01-01
We study a class of partial dierential equations in one spatial dimension, which can be seen as model equations for the analysis of pattern formation in physical systems dened on unbounded, weakly oscillating domains. We perform a linear and weakly nonlinear stability analysis for solutions that
The 'golden' algebraic equations
International Nuclear Information System (INIS)
Stakhov, A.; Rozin, B.
2006-01-01
The special case of the (p + 1)th degree algebraic equations of the kind x p+1 = x p + 1 (p = 1, 2, 3, ?) is researched in the present article. For the case p = 1, the given equation is reduced to the well-known Golden Proportion equation x 2 = x + 1. These equations are called the golden algebraic equations because the golden p-proportions τ p , special irrational numbers that follow from Pascal's triangle, are their roots. A research on the general properties of the roots of the golden algebraic equations is carried out in this article. In particular, formulas are derived for the golden algebraic equations that have degree greater than p + 1. There is reason to suppose that algebraic equations derived by the authors in the present article will interest theoretical physicists. For example, these algebraic equations could be found in the research of the energy relationships within the structures of many compounds and physical particles. For the case of butadiene (C 4 H 6 ), this fact is proved by the famous physicist Richard Feynman
Element diameter free stability parameters for stabilized methods applied to fluids
International Nuclear Information System (INIS)
Franca, L.P.; Madureira, A.L.
1992-08-01
Stability parameters for stabilized methods in fluids are suggested. The computation of the largest eigenvalue of a generalized eigenvalue problem replaces controversial definitions of element diameters and inverse estimate constants, used heretofore to compute these stability parameters. The design is employed in the advective-diffusive model, incompressible Navier-Stokes equations and the Stokes problem. (author)
DEFF Research Database (Denmark)
Backi, Christoph Josef; Bendtsen, Jan Dimon; Leth, John-Josef
2014-01-01
In this work the stability properties of a nonlinear partial differential equation (PDE) with state–dependent parameters is investigated. Among other things, the PDE describes freezing of foodstuff, and is closely related to the (Potential) Burgers’ Equation. We show that for certain forms of coe...
fractional differential equations
Indian Academy of Sciences (India)
We apply this method for solving space–time fractional Cahn--Allen equation and space--time fractional Klein–Gordon equation. The fractional derivatives are described in the sense of modified Riemann--Lioville. As a result of some exact solution in the form of hyperbolic, trigonometric and rational solutions are deduced.
Directory of Open Access Journals (Sweden)
Hannelore Breckner
2000-01-01
Full Text Available We consider a stochastic equation of Navier-Stokes type containing a noise part given by a stochastic integral with respect to a Wiener process. The purpose of this paper is to approximate the solution of this nonlinear equation by the Galerkin method. We prove the convergence in mean square.
Indian Academy of Sciences (India)
role in converting the Fokas equation into Hirota's bilinear form. Keywords. Bilinearization; multisoliton solution; Fokas equation; Hirota's bilinear method. PACS Nos 05.45.Yv; 04.20.Jb; 02.30.Jr. 1. Introduction. As pointed out by Drazin and Johnson [1], it is not easy to give a comprehensive and precise definition of a soliton.
M. Hazewinkel (Michiel)
1995-01-01
textabstractDedication: I dedicate this paper to Prof. P.C. Baayen, at the occasion of his retirement on 20 December 1994. The beautiful equation which forms the subject matter of this paper was invented by Wouthuysen after he retired. The four complex variable Wouthuysen equation arises from an
Elliptic Quadratic Operator Equations
Ganikhodjaev, Rasul; Mukhamedov, Farrukh; Saburov, Mansoor
2017-01-01
In the present paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator equations. The iterative Newton-Kantorovich method is also presented for stable solutions.
Differential Equation of Equilibrium
African Journals Online (AJOL)
user
differential equation of equilibrium, comparable to that of beam on elastic foundation, was derived from static principles on the ... tedious and more time saving than the classical method in the solution of the aforementioned differential equation. ... silos, pipelines, bridge arches or wind turbine towers [3]. The objective of this ...
Partial differential equations
Indian Academy of Sciences (India)
been a regular stream of high quality work done in these areas. Talking of elliptic partial differen- tial equations, important contributions have been made in the ...... [6] Evans L C 1992 Periodic homogenisation of certain fully nonlinear partial differential equations; Proc. Roy. Soc. Edinburgh Sect. A 120 No. 3–4, 245–265.
Energy Technology Data Exchange (ETDEWEB)
Yagi, M. [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment; Horton, W. [Texas Univ., Austin, TX (United States). Inst. for Fusion Studies
1993-11-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite {beta} that we solve the perpendicular component of Ohm`s law to conserve the physical energy while ensuring the relation {del} {center_dot} j = 0.
International Nuclear Information System (INIS)
Yagi, M.; Horton, W.
1993-11-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite β that we solve the perpendicular component of Ohm's law to conserve the physical energy while ensuring the relation ∇ · j = 0
Alternative equations of gravitation
International Nuclear Information System (INIS)
Pinto Neto, N.
1983-01-01
It is shown, trough a new formalism, that the quantum fluctuation effects of the gravitational field in Einstein's equations are analogs to the effects of a continuum medium in Maxwell's Electrodynamics. Following, a real example of the applications of these equations is studied. Qunatum fluctuations effects as perturbation sources in Minkowski and Friedmann Universes are examined. (L.C.) [pt
Nearly Radical Quadratic Functional Equations in p-2-Normed Spaces
Directory of Open Access Journals (Sweden)
M. Eshaghi Gordji
2012-01-01
Full Text Available We establish some stability results in 2-normed spaces for the radical quadratic functional equation (∑=1(+2+(∑=1(−2=2∑=1((+( and then use subadditive functions to prove its stability in -2-normed spaces.
ANALYTIC SOLUTIONS OF MATRIX RICCATI EQUATIONS WITH ANALYTIC COEFFICIENTS
Curtain, Ruth; Rodman, Leiba
2010-01-01
For matrix Riccati equations of platoon-type systems and of systems arising from PDEs, assuming the coefficients are analytic or rational functions in a suitable domain, analyticity of the stabilizing solution is proved under various hypotheses. General results on analytic behavior of stabilizing
On Reynolds stress and neutral azimuthal modes in the stability ...
Indian Academy of Sciences (India)
We consider the linear stability problem of inviscid, incompressible swirling flows with radius-dependent density with respect to two-dimensional disturbances. Some results of Miles on the parallel flow stability theory are extended to the swirling flow stability theory. In particular, series solutions for the stability equation for ...
Applied partial differential equations
Logan, J David
2004-01-01
This primer on elementary partial differential equations presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. What makes this book unique is that it is a brief treatment, yet it covers all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. Mathematical ideas are motivated from physical problems, and the exposition is presented in a concise style accessible to science and engineering students; emphasis is on motivation, concepts, methods, and interpretation, rather than formal theory. This second edition contains new and additional exercises, and it includes a new chapter on the applications of PDEs to biology: age structured models, pattern formation; epidemic wave fronts, and advection-diffusion processes. The student who reads through this book and solves many of t...
Numerical study of fractional nonlinear Schrodinger equations
Klein, Christian
2014-10-08
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.
Vector domain decomposition schemes for parabolic equations
Vabishchevich, P. N.
2017-09-01
A new class of domain decomposition schemes for finding approximate solutions of timedependent problems for partial differential equations is proposed and studied. A boundary value problem for a second-order parabolic equation is used as a model problem. The general approach to the construction of domain decomposition schemes is based on partition of unity. Specifically, a vector problem is set up for solving problems in individual subdomains. Stability conditions for vector regionally additive schemes of first- and second-order accuracy are obtained.
Convergence of method of lines approximations to partial differential equations
International Nuclear Information System (INIS)
Verwer, J.G.; Sanz-Serna, J.M.
1984-01-01
Many existing numerical schemes for evolutionary problems in partial differential equations (PDEs) can be viewed as method of lines (MOL) schemes. This paper treats the convergence of one-step MOL schemes. The main purpose is to set up a general framework for a convergence analysis applicable to nonlinear problems. The stability materials for this framework are taken from the field of nonlinear stiff ODEs. In this connection, important concepts are the logarithmic matrix norm and C-stability. A nonlinear parabolic equation and the cubic Schroedinger equation are used for illustrating the ideas. (Auth.)
Numerical simulation of fractional Cable equation of spiny neuronal dendrites
Directory of Open Access Journals (Sweden)
N.H. Sweilam
2014-03-01
Full Text Available In this article, numerical study for the fractional Cable equation which is fundamental equations for modeling neuronal dynamics is introduced by using weighted average of finite difference methods. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. A simple and an accurate stability criterion valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced and checked numerically. Numerical results, figures, and comparisons have been presented to confirm the theoretical results and efficiency of the proposed method.
Ordinary differential equations
Ince, Edward Lindsay
1956-01-01
The theory of ordinary differential equations in real and complex domains is here clearly explained and analyzed. Not only classical theory, but also the main developments of modern times are covered. Exhaustive sections on the existence and nature of solutions, continuous transformation groups, the algebraic theory of linear differential systems, and the solution of differential equations by contour integration are as valuable to the pure mathematician as the fine treatment of the equations of Legendre, Bessel, and Mathieu, the conditions for the oscillatory character of solutions of a diffe
Uncertain differential equations
Yao, Kai
2016-01-01
This book introduces readers to the basic concepts of and latest findings in the area of differential equations with uncertain factors. It covers the analytic method and numerical method for solving uncertain differential equations, as well as their applications in the field of finance. Furthermore, the book provides a number of new potential research directions for uncertain differential equation. It will be of interest to researchers, engineers and students in the fields of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, automation, economics, and management science.
Hyperbolic partial differential equations
Witten, Matthew
1986-01-01
Hyperbolic Partial Differential Equations III is a refereed journal issue that explores the applications, theory, and/or applied methods related to hyperbolic partial differential equations, or problems arising out of hyperbolic partial differential equations, in any area of research. This journal issue is interested in all types of articles in terms of review, mini-monograph, standard study, or short communication. Some studies presented in this journal include discretization of ideal fluid dynamics in the Eulerian representation; a Riemann problem in gas dynamics with bifurcation; periodic M
Wu Zhuo Qun; Li Hui Lai; Zhao Jun Ning
2001-01-01
Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which
Ordinary differential equations
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,
Differential equations problem solver
Arterburn, David R
2012-01-01
REA's Problem Solvers is a series of useful, practical, and informative study guides. Each title in the series is complete step-by-step solution guide. The Differential Equations Problem Solver enables students to solve difficult problems by showing them step-by-step solutions to Differential Equations problems. The Problem Solvers cover material ranging from the elementary to the advanced and make excellent review books and textbook companions. They're perfect for undergraduate and graduate studies.The Differential Equations Problem Solver is the perfect resource for any class, any exam, and
Fully nonlinear elliptic equations
Caffarelli, Luis A
1995-01-01
The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equa
Partial differential equations
Friedman, Avner
2008-01-01
This three-part treatment of partial differential equations focuses on elliptic and evolution equations. Largely self-contained, it concludes with a series of independent topics directly related to the methods and results of the preceding sections that helps introduce readers to advanced topics for further study. Geared toward graduate and postgraduate students of mathematics, this volume also constitutes a valuable reference for mathematicians and mathematical theorists.Starting with the theory of elliptic equations and the solution of the Dirichlet problem, the text develops the theory of we
Beginning partial differential equations
O'Neil, Peter V
2011-01-01
A rigorous, yet accessible, introduction to partial differential equations-updated in a valuable new edition Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addres
Analysis of spectral methods for the homogeneous Boltzmann equation
Filbet, Francis
2011-04-01
The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound). © 2010 American Mathematical Society.
Determination of source terms in a degenerate parabolic equation
International Nuclear Information System (INIS)
Cannarsa, P; Tort, J; Yamamoto, M
2010-01-01
In this paper, we prove Lipschitz stability results for inverse source problems relative to parabolic equations. We use the method introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates. What is new here is that we study a class of one-dimensional degenerate parabolic equations. In our model, the diffusion coefficient vanishes at one extreme point of the domain. Instead of the classical Carleman estimates obtained by Fursikov and Imanuvilov for non degenerate equations, we use and extend some recent Carleman estimates for degenerate equations obtained by Cannarsa, Martinez and Vancostenoble. Finally, we obtain Lipschitz stability results in inverse source problems for our class of degenerate parabolic equations both in the case of a boundary observation and in the case of a locally distributed observation
Saaty, Thomas L
1981-01-01
Covers major types of classical equations: operator, functional, difference, integro-differential, and more. Suitable for graduate students as well as scientists, technologists, and mathematicians. "A welcome contribution." - Math Reviews. 1964 edition.
Equations For Rotary Transformers
Salomon, Phil M.; Wiktor, Peter J.; Marchetto, Carl A.
1988-01-01
Equations derived for input impedance, input power, and ratio of secondary current to primary current of rotary transformer. Used for quick analysis of transformer designs. Circuit model commonly used in textbooks on theory of ac circuits.
Applied partial differential equations
Logan, J David
2015-01-01
This text presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. Emphasis is placed on motivation, concepts, methods, and interpretation, rather than on formal theory. The concise treatment of the subject is maintained in this third edition covering all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. In this third edition, text remains intimately tied to applications in heat transfer, wave motion, biological systems, and a variety other topics in pure and applied science. The text offers flexibility to instructors who, for example, may wish to insert topics from biology or numerical methods at any time in the course. The exposition is presented in a friendly, easy-to-read, style, with mathematical ideas motivated from physical problems. Many exercises and worked e...
Applied partial differential equations
DuChateau, Paul
2012-01-01
Book focuses mainly on boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data, first-order equations, and more. Numerous exercises included.
dimensional nonlinear evolution equations
Indian Academy of Sciences (India)
–. (4)) by applying the exp-function method. The computer symbolic systems such as. Maple and Mathematica allow us to perform complicated and tedious calculations. 2. Solutions of (N + 1)-dimensional generalized Boussinesq equation.
Problems in differential equations
Brenner, J L
2013-01-01
More than 900 problems and answers explore applications of differential equations to vibrations, electrical engineering, mechanics, and physics. Problem types include both routine and nonroutine, and stars indicate advanced problems. 1963 edition.
Regularized Structural Equation Modeling
Jacobucci, Ross; Grimm, Kevin J.; McArdle, John J.
2016-01-01
A new method is proposed that extends the use of regularization in both lasso and ridge regression to structural equation models. The method is termed regularized structural equation modeling (RegSEM). RegSEM penalizes specific parameters in structural equation models, with the goal of creating easier to understand and simpler models. Although regularization has gained wide adoption in regression, very little has transferred to models with latent variables. By adding penalties to specific parameters in a structural equation model, researchers have a high level of flexibility in reducing model complexity, overcoming poor fitting models, and the creation of models that are more likely to generalize to new samples. The proposed method was evaluated through a simulation study, two illustrative examples involving a measurement model, and one empirical example involving the structural part of the model to demonstrate RegSEM’s utility. PMID:27398019
On Fuzzy Solutions for Diffusion Equation
Directory of Open Access Journals (Sweden)
Jefferson Leite
2015-01-01
Full Text Available Our main goal is to define a fuzzy solution for problems involving diffusion. To this end, the solution of fuzzy diffusion-reaction-advection equation will be defined as Zadeh’s extension of deterministic solution of the associated problem. Important aspects such as unity and stability of these solutions will also be studied. Graphical representations of these solutions will be presented.
Differential equations and applications recent advances
2014-01-01
Differential Equations and Applications : Recent Advances focus on the latest developments in Nonlinear Dynamical Systems, Neural Networks, Fluid Dynamics, Fractional Differential Systems, Mathematical Modelling and Qualitative Theory. Different aspects such as Existence, Stability, Controllability, Viscosity and Numerical Analysis for different systems have been discussed in this book. This book will be of great interest and use to researchers in Applied Mathematics, Engineering and Mathematical Physics.
Ordinary differential equations.
Lebl, Jiří
2013-01-01
In this chapter we provide an overview of the basic theory of ordinary differential equations (ODE). We give the basics of analytical methods for their solutions and also review numerical methods. The chapter should serve as a primer for the basic application of ODEs and systems of ODEs in practice. As an example, we work out the equations arising in Michaelis-Menten kinetics and give a short introduction to using Matlab for their numerical solution.
Indian Academy of Sciences (India)
continuous medium is μgrav ≡ μ + 3p/c2. Particular versions of this equation had been obtained earlier, at centers of symmetry by Tolman and Synge (see Raychaud- .... (8) by l ˙l and integrate to find. 3( ˙l)2 − κμl2 − Λl2 = const. (10). This is just the Friedmann equation which governs the time evolution of FLRW universe ...
Equations of radiation hydrodynamics
International Nuclear Information System (INIS)
Mihalas, D.
1982-01-01
The purpose of this paper is to give an overview of the role of radiation in the transport of energy and momentum in a combined matter-radiation fluid. The transport equation for a moving radiating fluid is presented in both a fully Eulerian and a fully Lagrangian formulation, along with conservation equations describing the dynamics of the fluid. Special attention is paid to the problem of deriving equations that are mutually consistent in each frame, and between frames, to 0(v/c). A detailed analysis is made to show that in situations of broad interest, terms that are formally of 0(v/c) actually dominate the solution, demonstrating that it is esential (1) to pay scrupulous attention to the question of the frame dependence in formulating the equations; and (2) to solve the equations to 0(v/c) in quite general circumstances. These points are illustrated in the context of the nonequilibrium radiation diffusion limit, and a sketch of how the Lagrangian equations are to be solved will be presented
Nonlinear stability of supersonic jets
Tiwari, S. N. (Principal Investigator); Bhat, T. R. S. (Principal Investigator)
1996-01-01
The stability calculations made for a shock-free supersonic jet using the model based on parabolized stability equations are presented. In this analysis the large scale structures, which play a dominant role in the mixing as well as the noise radiated, are modeled as instability waves. This model takes into consideration non-parallel flow effects and also nonlinear interaction of the instability waves. The stability calculations have been performed for different frequencies and mode numbers over a range of jet operating temperatures. Comparisons are made, where appropriate, with the solutions to Rayleigh's equation (linear, inviscid analysis with the assumption of parallel flow). The comparison of the solutions obtained using the two approaches show very good agreement.
INVESTIGATION OF DYNAMIC STABILITY OF FLEXIBLE WAVEGUIDES FOR ULTRASONIC THROMBECTOMY
Directory of Open Access Journals (Sweden)
D. A. Stepanenko
2011-01-01
Full Text Available The paper presents results of mathematical modelling of dynamic stability of flexible ultrasonic waveguides applied in engineering and medicine, particularly, in minimally-invasive surgery. By means of the Bubnov-Galerkin method an equation of parametric flexural vibrations of a waveguide is reduced to the Mathieu equation. Stability of the equation solutions is determined by values of its coefficients and may be represented graphically by means of the Ince-Strutt diagram. Parameters of the Mathieu equation determining the stability of its solution are determined by means of finite element method using ANSYS software and APDL programming language.
MATLAB Stability and Control Toolbox Trim and Static Stability Module
Kenny, Sean P.; Crespo, Luis
2012-01-01
MATLAB Stability and Control Toolbox (MASCOT) utilizes geometric, aerodynamic, and inertial inputs to calculate air vehicle stability in a variety of critical flight conditions. The code is based on fundamental, non-linear equations of motion and is able to translate results into a qualitative, graphical scale useful to the non-expert. MASCOT was created to provide the conceptual aircraft designer accurate predictions of air vehicle stability and control characteristics. The code takes as input mass property data in the form of an inertia tensor, aerodynamic loading data, and propulsion (i.e. thrust) loading data. Using fundamental nonlinear equations of motion, MASCOT then calculates vehicle trim and static stability data for the desired flight condition(s). Available flight conditions include six horizontal and six landing rotation conditions with varying options for engine out, crosswind, and sideslip, plus three take-off rotation conditions. Results are displayed through a unique graphical interface developed to provide the non-stability and control expert conceptual design engineer a qualitative scale indicating whether the vehicle has acceptable, marginal, or unacceptable static stability characteristics. If desired, the user can also examine the detailed, quantitative results.
The Bernoulli-Poiseuille Equation.
Badeer, Henry S.; Synolakis, Costas E.
1989-01-01
Describes Bernoulli's equation and Poiseuille's equation for fluid dynamics. Discusses the application of the combined Bernoulli-Poiseuille equation in real flows, such as viscous flows under gravity and acceleration. (YP)
Derivation of stable Burnett equations for rarefied gas flows.
Singh, Narendra; Jadhav, Ravi Sudam; Agrawal, Amit
2017-07-01
A set of constitutive relations for the stress tensor and heat flux vector for the hydrodynamic description of rarefied gas flows is derived in this work. A phase density function consistent with Onsager's reciprocity principle and H theorem is utilized to capture nonequilibrium thermodynamics effects. The phase density function satisfies the linearized Boltzmann equation and the collision invariance property. Our formulation provides the correct value of the Prandtl number as it involves two different relaxation times for momentum and energy transport by diffusion. Generalized three-dimensional constitutive equations for different kinds of molecules are derived using the phase density function. The derived constitutive equations involve cross single derivatives of field variables such as temperature and velocity, with no higher-order derivative in higher-order terms. This is remarkable feature of the equations as the number of boundary conditions required is the same as needed for conventional Navier-Stokes equations. Linear stability analysis of the equations is performed, which shows that the derived equations are unconditionally stable. A comparison of the derived equations with existing Burnett-type equations is presented and salient features of our equations are outlined. The classic internal flow problem, force-driven compressible plane Poiseuille flow, is chosen to verify the stable Burnett equations and the results for equilibrium variables are presented.
Stability investigation of quadratic systems with delay
Directory of Open Access Journals (Sweden)
Vladimir Davydov
2000-01-01
Full Text Available Systems of differential equations with quadratic right-hand sides with delay are considered in the paper. Compact matrix notation form is proposed for the systems of such type. Stability investigations are performed by Lyapunov's second method with functions of quadratic form. Stability conditions of quadratic systems with delay, uniformly by argument deviation, and with delay depending on the system's parameters are derived. A guaranteed radius of the ball of asymptotic stability region for zero solution is obtained.
General Solutions of Two Quadratic Functional Equations of Pexider Type on Orthogonal Vectors
Directory of Open Access Journals (Sweden)
Margherita Fochi
2012-01-01
Full Text Available Based on the studies on the Hyers-Ulam stability and the orthogonal stability of some Pexider-quadratic functional equations, in this paper we find the general solutions of two quadratic functional equations of Pexider type. Both equations are studied in restricted domains: the first equation is studied on the restricted domain of the orthogonal vectors in the sense of Rätz, and the second equation is considered on the orthogonal vectors in the inner product spaces with the usual orthogonality.
Soliton multidimensional equations and integrable evolutions preserving Laplace's equation
International Nuclear Information System (INIS)
Fokas, A.S.
2008-01-01
The KP equation, which is an integrable nonlinear evolution equation in 2+1, i.e., two spatial and one temporal dimensions, is a physically significant generalization of the KdV equation. The question of constructing an integrable generalization of the KP equation in 3+1, has been one of the central open problems in the field of integrability. By complexifying the independent variables of the KP equation, I obtain an integrable nonlinear evolution equation in 4+2. The requirement that real initial conditions remain real under this evolution, implies that the dependent variable satisfies a nonlinear evolution equation in 3+1 coupled with Laplace's equation. A reduction of this system of equations to a single equation in 2+1 contains as particular cases certain singular integro-differential equations which appear in the theory of water waves
Introduction to partial differential equations
Greenspan, Donald
2000-01-01
Designed for use in a one-semester course by seniors and beginning graduate students, this rigorous presentation explores practical methods of solving differential equations, plus the unifying theory underlying the mathematical superstructure. Topics include basic concepts, Fourier series, second-order partial differential equations, wave equation, potential equation, heat equation, approximate solution of partial differential equations, and more. Exercises appear at the ends of most chapters. 1961 edition.
Control Design of a Nonlinear Controller to Stabilize the Nonlinear ...
African Journals Online (AJOL)
inyangs
is used to formularize the delay differential equation in equation. (13), which accounts for the delayed terms in the system. By applying the Leibniz integration rule, the derivative. •. V shows that the system is asymptotically stable with a negative definite solution in equation (14). The conditions for the stability are: •. The delay ...
On the behaviour of solutions of parabolic equations for large values of time
International Nuclear Information System (INIS)
Denisov, V N
2005-01-01
This paper is a survey of classical and new results on stabilization of solutions of the Cauchy problem and mixed problems for second-order linear parabolic equations. Proofs are given for some new results about exact sufficient conditions on the behaviour of lower-order coefficients of the parabolic equation; these conditions ensure stabilization of a solution of the Cauchy problem for the parabolic equation in the class of bounded or increasing initial functions
Stabilities of Cubic Mappings in Fuzzy Normed Spaces
Directory of Open Access Journals (Sweden)
Ghaffari Ali
2010-01-01
Full Text Available Rassias(2001 introduced the pioneering cubic functional equation in the history of mathematical analysis: and solved the pertinent famous Ulam stability problem for this inspiring equation. This Rassias cubic functional equation was the historic transition from the following famous Euler-Lagrange-Rassias quadratic functional equation: to the cubic functional equations. In this paper, we prove the Ulam-Hyers stability of the cubic functional equation: in fuzzy normed linear spaces. We use the definition of fuzzy normed linear spaces to establish a fuzzy version of a generalized Hyers-Ulam-Rassias stability for above equation in the fuzzy normed linear space setting. The fuzzy sequentially continuity of the cubic mappings is discussed.
Nonlinear differential equations
Struble, Raimond A
2017-01-01
Detailed treatment covers existence and uniqueness of a solution of the initial value problem, properties of solutions, properties of linear systems, stability of nonlinear systems, and two-dimensional systems. 1962 edition.
CIME course on Control of Partial Differential Equations
Alabau-Boussouira, Fatiha; Glass, Olivier; Le Rousseau, Jérôme; Zuazua, Enrique
2012-01-01
The term “control theory” refers to the body of results - theoretical, numerical and algorithmic - which have been developed to influence the evolution of the state of a given system in order to meet a prescribed performance criterion. Systems of interest to control theory may be of very different natures. This monograph is concerned with models that can be described by partial differential equations of evolution. It contains five major contributions and is connected to the CIME Course on Control of Partial Differential Equations that took place in Cetraro (CS, Italy), July 19 - 23, 2010. Specifically, it covers the stabilization of evolution equations, control of the Liouville equation, control in fluid mechanics, control and numerics for the wave equation, and Carleman estimates for elliptic and parabolic equations with application to control. We are confident this work will provide an authoritative reference work for all scientists who are interested in this field, representing at the same time a fri...
Optimal control of stochastic difference Volterra equations an introduction
Shaikhet, Leonid
2015-01-01
This book showcases a subclass of hereditary systems, that is, systems with behaviour depending not only on their current state but also on their past history; it is an introduction to the mathematical theory of optimal control for stochastic difference Volterra equations of neutral type. As such, it will be of much interest to researchers interested in modelling processes in physics, mechanics, automatic regulation, economics and finance, biology, sociology and medicine for all of which such equations are very popular tools. The text deals with problems of optimal control such as meeting given performance criteria, and stabilization, extending them to neutral stochastic difference Volterra equations. In particular, it contrasts the difference analogues of solutions to optimal control and optimal estimation problems for stochastic integral Volterra equations with optimal solutions for corresponding problems in stochastic difference Volterra equations. Optimal Control of Stochastic Difference Volterra Equation...
Boussinesq evolution equations
DEFF Research Database (Denmark)
Bredmose, Henrik; Schaffer, H.; Madsen, Per A.
2004-01-01
This paper deals with the possibility of using methods and ideas from time domain Boussinesq formulations in the corresponding frequency domain formulations. We term such frequency domain models "evolution equations". First, we demonstrate that the numerical efficiency of the deterministic...... Boussinesq evolution equations of Madsen and Sorensen [Madsen, P.A., Sorensen, O.R., 1993. Bound waves and triad interactions in shallow water. Ocean Eng. 20 359-388] can be improved by using Fast Fourier Transforms to evaluate the nonlinear terms. For a practical example of irregular waves propagating over...... a submerged bar, it is demonstrated that evolution equations utilising FFT can be solved around 100 times faster than the corresponding time domain model. Use of FFT provides an efficient bridge between the frequency domain and the time domain. We utilise this by adapting the surface roller model for wave...
Quadratic Diophantine equations
Andreescu, Titu
2015-01-01
This monograph treats the classical theory of quadratic Diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. These new techniques combined with the latest increases in computational power shed new light on important open problems. The authors motivate the study of quadratic Diophantine equations with excellent examples, open problems, and applications. Moreover, the exposition aptly demonstrates many applications of results and techniques from the study of Pell-type equations to other problems in number theory. The book is intended for advanced undergraduate and graduate students as well as researchers. It challenges the reader to apply not only specific techniques and strategies, but also to employ methods and tools from other areas of mathematics, such as algebra and analysis.
Stochastic porous media equations
Barbu, Viorel; Röckner, Michael
2016-01-01
Focusing on stochastic porous media equations, this book places an emphasis on existence theorems, asymptotic behavior and ergodic properties of the associated transition semigroup. Stochastic perturbations of the porous media equation have reviously been considered by physicists, but rigorous mathematical existence results have only recently been found. The porous media equation models a number of different physical phenomena, including the flow of an ideal gas and the diffusion of a compressible fluid through porous media, and also thermal propagation in plasma and plasma radiation. Another important application is to a model of the standard self-organized criticality process, called the "sand-pile model" or the "Bak-Tang-Wiesenfeld model". The book will be of interest to PhD students and researchers in mathematics, physics and biology.
Linear stability analysis of heated parallel channels
International Nuclear Information System (INIS)
Nourbakhsh, H.P.; Isbin, H.S.
1982-01-01
An analyis is presented of thermal hydraulic stability of flow in parallel channels covering the range from inlet subcooling to exit superheat. The model is based on a one-dimensional drift velocity formulation of the two phase flow conservation equations. The system of equations is linearized by assuming small disturbances about the steady state. The dynamic response of the system to an inlet flow perturbation is derived yielding the characteristic equation which predicts the onset of instabilities. A specific application is carried out for homogeneous and regional uniformly heated systems. The particular case of equal characteristic frequencies of two-phase and single phase vapor region is studied in detail. The D-partition method and the Mikhailov stability criterion are used for determining the marginal stability boundary. Stability predictions from the present analysis are compared with the experimental data from the solar test facility. 8 references
Equations of mathematical physics
Tikhonov, A N
2011-01-01
Mathematical physics plays an important role in the study of many physical processes - hydrodynamics, elasticity, and electrodynamics, to name just a few. Because of the enormous range and variety of problems dealt with by mathematical physics, this thorough advanced-undergraduate or graduate-level text considers only those problems leading to partial differential equations. The authors - two well-known Russian mathematicians - have focused on typical physical processes and the principal types of equations deailing with them. Special attention is paid throughout to mathematical formulation, ri
Partial differential equations
Agranovich, M S
2002-01-01
Mark Vishik's Partial Differential Equations seminar held at Moscow State University was one of the world's leading seminars in PDEs for over 40 years. This book celebrates Vishik's eightieth birthday. It comprises new results and survey papers written by many renowned specialists who actively participated over the years in Vishik's seminars. Contributions include original developments and methods in PDEs and related fields, such as mathematical physics, tomography, and symplectic geometry. Papers discuss linear and nonlinear equations, particularly linear elliptic problems in angles and gener
Generalized estimating equations
Hardin, James W
2002-01-01
Although powerful and flexible, the method of generalized linear models (GLM) is limited in its ability to accurately deal with longitudinal and clustered data. Developed specifically to accommodate these data types, the method of Generalized Estimating Equations (GEE) extends the GLM algorithm to accommodate the correlated data encountered in health research, social science, biology, and other related fields.Generalized Estimating Equations provides the first complete treatment of GEE methodology in all of its variations. After introducing the subject and reviewing GLM, the authors examine th
Li, Tatsien
2017-01-01
This book focuses on nonlinear wave equations, which are of considerable significance from both physical and theoretical perspectives. It also presents complete results on the lower bound estimates of lifespan (including the global existence), which are established for classical solutions to the Cauchy problem of nonlinear wave equations with small initial data in all possible space dimensions and with all possible integer powers of nonlinear terms. Further, the book proposes the global iteration method, which offers a unified and straightforward approach for treating these kinds of problems. Purely based on the properties of solut ions to the corresponding linear problems, the method simply applies the contraction mapping principle.
Systematic Equation Formulation
DEFF Research Database (Denmark)
Lindberg, Erik
2007-01-01
A tutorial giving a very simple introduction to the set-up of the equations used as a model for an electrical/electronic circuit. The aim is to find a method which is as simple and general as possible with respect to implementation in a computer program. The “Modified Nodal Approach”, MNA, and th......, and the “Controlled Source Approach”, CSA, for systematic equation formulation are investigated. It is suggested that the kernel of the P Spice program based on MNA is reprogrammed....
Test equating methods and practices
Kolen, Michael J
1995-01-01
In recent years, many researchers in the psychology and statistical communities have paid increasing attention to test equating as issues of using multiple test forms have arisen and in response to criticisms of traditional testing techniques This book provides a practically oriented introduction to test equating which both discusses the most frequently used equating methodologies and covers many of the practical issues involved The main themes are - the purpose of equating - distinguishing between equating and related methodologies - the importance of test equating to test development and quality control - the differences between equating properties, equating designs, and equating methods - equating error, and the underlying statistical assumptions for equating The authors are acknowledged experts in the field, and the book is based on numerous courses and seminars they have presented As a result, educators, psychometricians, professionals in measurement, statisticians, and students coming to the subject for...
Conservation laws for equations related to soil water equations
C. M. Khalique; F. M. Mahomed
2005-01-01
We obtain all nontrivial conservation laws for a class of ( 2+1 ) nonlinear evolution partial differential equations which are related to the soil water equations. It is also pointed out that nontrivial conservation laws exist for certain classes of equations which admit point symmetries. Moreover, we associate symmetries with conservation laws for special classes of these equations.
Conservation laws for equations related to soil water equations
Directory of Open Access Journals (Sweden)
Khalique C. M.
2005-01-01
Full Text Available We obtain all nontrivial conservation laws for a class of ( 2+1 nonlinear evolution partial differential equations which are related to the soil water equations. It is also pointed out that nontrivial conservation laws exist for certain classes of equations which admit point symmetries. Moreover, we associate symmetries with conservation laws for special classes of these equations.
Coupled Higgs field equation and Hamiltonian amplitude equation ...
Indian Academy of Sciences (India)
Abstract. In this paper, coupled Higgs field equation and Hamiltonian amplitude equation are studied using the Lie classical method. Symmetry reductions and exact solutions are reported for Higgs equation and Hamiltonian amplitude equation. We also establish the travelling wave solutions involving parameters of the ...
Coupled Higgs field equation and Hamiltonian amplitude equation ...
Indian Academy of Sciences (India)
In this paper, coupled Higgs field equation are studied using the Lie classical method. Symmetry reductions and exact solutions are reported for Higgs equation and Hamiltonian amplitude equation. We also establish the travelling wave solutions involving parameters of the coupled Higgs equation and Hamiltonian ...
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
International Nuclear Information System (INIS)
Kanki, Takashi; Uyama, Tadao; Tokuda, Shinji.
1995-07-01
In the numerical method to compute the matching data which are necessary for resistive MHD stability analyses, it is required to solve the eigenvalue problem and the associated singular equation. An iterative method is developed to solve the eigenvalue problem and the singular equation. In this method, the eigenvalue problem is replaced with an equivalent nonlinear equation and a singular equation is derived from Newton's method for the nonlinear equation. The multi-grid method (MGM), a high speed iterative method, can be applied to this method. The convergence of the eigenvalue and the eigenvector, and the CPU time in this method are investigated for a model equation. It is confirmed from the numerical results that this method is effective for solving the eigenvalue problem and the singular equation with numerical stability and high accuracy. It is shown by improving the MGM that the CPU time for this method is 50 times shorter than that of the direct method. (author)
Stability and periodic solutions of ordinary and functional differential equations
Burton, T A
1985-01-01
In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank mat
Lipschitz stability of the K-quadratic functional equation | Chahbi ...
African Journals Online (AJOL)
Let N be the set of all positive integers, G an Abelian group with a metric d and E a normed space. For any f : G → E we define the k-quadratic difference of the function f by the formula Qk ƒ(x; y) := 2ƒ(x) + 2k2ƒ(y) - f(x + ky) - f(x - ky) for x; y ∈ G and k ∈ N. Under some assumptions about f and Qkƒ we prove that if Qkƒ is ...
Implicit fractional differential and integral equations existence and stability
Abbas, Saïd; Graef, John R; Henderson, Johnny
2018-01-01
The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Please submit book proposals toJürgen Appell.
Stability result for Navier-Stokes equations with entropy transport
Czech Academy of Sciences Publication Activity Database
Michálek, Martin
2015-01-01
Roč. 17, č. 2 (2015), s. 279-285 ISSN 1422-6928 R&D Projects: GA ČR GA13-00522S Institutional support: RVO:67985840 Keywords : compressible Navier-Stokes system * entropy transport * effective viscous flux Subject RIV: BA - General Mathematics Impact factor: 1.023, year: 2015 http://link.springer.com/article/10.1007%2Fs00021-015-0205-x
Equation of state and stability of hot asymmetric nuclear matter
International Nuclear Information System (INIS)
Samaddar, S.K.
1989-01-01
The nuclear incompressibility as obtained from different sources, from nuclei, high energy nuclear collisions, supernova and neutron stars are briefly reviewed. All these data in general favour a compression modulus, K α ∼ 300 Mev with a minimum uncertainty ∼ 50 MeV. Using a finite rang e density and momentum dependent two-body effective interaction, variation of nucl ear incompressibility with temperature, asymmetry and density is discussed in a non-relativistic mean field approach. The same formalism has also been used to study the limiting temperatures of infinite as well as finite nuclear systems in the astrophysical context as well as in high energy heavy ion collisions. (autho r). 16 refs., 6 figs., 1 tab
Weak stability of Lagrangian solutions to the semigeostrophic equations
International Nuclear Information System (INIS)
Faria, Josiane C O; Lopes Filho, Milton C; Nussenzveig Lopes, Helena J
2009-01-01
In (Cullen and Feldman 2006 SIAM J. Math. Anal. 37 137–95), Cullen and Feldman proved the existence of Lagrangian solutions for the semigeostrophic system in physical variables with initial potential vorticity in L p , p > 1. Here, we show that a subsequence of the Lagrangian solutions corresponding to a strongly convergent sequence of initial potential vorticities in L 1 converges strongly in L q , q < ∞, to a Lagrangian solution, in particular extending the existence result of Cullen and Feldman to the case p = 1. We also present a counterexample for Lagrangian solutions corresponding to a sequence of initial potential vorticities converging in BM. The analytical tools used include techniques from optimal transportation, Ambrosio's results on transport by BV vector fields and Orlicz spaces
Mixed, Nonsplit, Extended Stability, Stiff Integration of Reaction Diffusion Equations
Alzahrani, Hasnaa H.
2016-07-26
A tailored integration scheme is developed to treat stiff reaction-diffusion prob- lems. The construction adapts a stiff solver, namely VODE, to treat reaction im- plicitly together with explicit treatment of diffusion. The second-order Runge-Kutta- Chebyshev (RKC) scheme is adjusted to integrate diffusion. Spatial operator is de- scretised by second-order finite differences on a uniform grid. The overall solution is advanced over S fractional stiff integrations, where S corresponds to the number of RKC stages. The behavior of the scheme is analyzed by applying it to three simple problems. The results show that it achieves second-order accuracy, thus, preserving the formal accuracy of the original RKC. The presented development sets the stage for future extensions, particularly, to multidimensional reacting flows with detailed chemistry.
African Journals Online (AJOL)
Petrophysical, Decompaction and Linear Regression techniques were used to investigate overpressure, degree of compaction and to derive a model compaction equation for. -1. -1 hydrostatic sandstones. Compaction coefficients obtained range from 0.0003 - 0.0005 m (averaging 0.0004 m ) and percentage compaction ...
Structural Equation Model Trees
Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman
2013-01-01
In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree…
Indian Academy of Sciences (India)
Design of Four-Link Mechanisms. Ashitava Ghosal. Ashitava Ghosal is a. Professor in the Depart- ment of Mechanical. Engineering and Centre for Product Design, IISc,. Bangalore. His research interests are in the areas of analysis and design of mechanisms and ... Freudenstein's thesis and the equation named af- ter him.
Indian Academy of Sciences (India)
Amalkumar Raychaudhuri's remarkable paper [1] for the first time gave a general derivation of the fundamental equation of gravitational attraction for pressure- free matter, showing the repulsive nature of a positive cosmological constant, and underlying the basic singularity theorem (see below). He used special coordinates.
Solving Equations Applet Project
Thatcher, Kimberly
2011-01-01
The purpose of this paper is to summarize a Masters Project for the MMath Degree. The purpose of the project was to create and evaluate an applet that maintains the advantages of the existent manipulatives (Hands-On Equations® and the NLVM applet) while also overcoming the limitations of each. Another product of this project is accompanying lesson plans for teachers.
Differential Equations as Actions
DEFF Research Database (Denmark)
Ronkko, Mauno; Ravn, Anders P.
1997-01-01
We extend a conventional action system with a primitive action consisting of a differential equation and an evolution invariant. The semantics is given by a predicate transformer. The weakest liberal precondition is chosen, because it is not always desirable that steps corresponding to differential...
Differential Equation of Equilibrium
African Journals Online (AJOL)
user
Department of Civil Engineering. University of Nigeria Nsukka. ABSTRACT. Analysis of underground circular cylindrical shell is carried out in this work. The forth order differential equation of equilibrium, comparable to that of beam on elastic foundation, was derived from static principles on the assumptions of P. L Pasternak.
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 15; Issue 8. The Freudenstein Equation - Design of Four-Link Mechanisms. Ashitava Ghosal. General Article Volume 15 Issue 8 August 2010 pp 699-710. Fulltext. Click here to view fulltext PDF. Permanent link:
Equational binary decision diagrams
J.F. Groote (Jan Friso); J.C. van de Pol (Jaco)
2000-01-01
textabstractWe incorporate equations in binary decision diagrams (BDD). The resulting objects are called EQ-BDDs. A straightforward notion of ordered EQ-BDDs (EQ-OBDD) is defined, and it is proved that each EQ-BDD is logically equivalent to an EQ-OBDD. Moreover, on EQ-OBDDs satisfiability and
Nonlinear Fokker-Planck Equations Fundamentals and Applications
Frank, Till Daniel
2005-01-01
Providing an introduction to the theory of nonlinear Fokker-Planck equations, this book discusses fundamental properties of transient and stationary solutions, emphasizing the stability analysis of stationary solutions by means of self-consistency equations, linear stability analysis, and Lyapunov's direct method. Also treated are Langevin equations and correlation functions. Nonlinear Fokker-Planck Equations addresses various phenomena such as phase transitions, multistability of systems, synchronization, anomalous diffusion, cut-off solutions, travelling-wave solutions and the emergence of power law solutions. A nonlinear Fokker-Planck perspective to quantum statistics, generalized thermodynamics, and linear nonequilibrium thermodynamics is given. Theoretical concepts are illustrated where possible by simple examples. The book also reviews several applications in the fields of condensed matter physics, the physics of porous media and liquid crystals, accelerator physics, neurophysics, social sciences, popul...
International Nuclear Information System (INIS)
Tian Chou.
1991-05-01
It is important but difficult to find the invariant groups for the differential equations. We found a new invariant group for the MKdV equation. In this paper, we present a new invariance for the CDF equation. By using this invariance, we obtain some new solutions of CDF equation. (author). 5 refs
Lectures on partial differential equations
Petrovsky, I G
1992-01-01
Graduate-level exposition by noted Russian mathematician offers rigorous, transparent, highly readable coverage of classification of equations, hyperbolic equations, elliptic equations and parabolic equations. Wealth of commentary and insight invaluable for deepening understanding of problems considered in text. Translated from the Russian by A. Shenitzer.
Quantum equations from Brownian motions
International Nuclear Information System (INIS)
Rajput, B.S.
2011-01-01
Classical Schrodinger and Dirac equations have been derived from Brownian motions of a particle, it has been shown that the classical Schrodinger equation can be transformed to usual Schrodinger Quantum equation on applying Heisenberg uncertainty principle between position and momentum while Dirac Quantum equation follows it's classical counter part on applying Heisenberg uncertainly principle between energy and time without applying any analytical continuation. (author)
Anticipated backward stochastic differential equations
Peng, Shige; Yang, Zhe
2007-01-01
In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.
Linear stability of tearing modes
International Nuclear Information System (INIS)
Cowley, S.C.; Kulsrud, R.M.; Hahm, T.S.
1986-05-01
This paper examines the stability of tearing modes in a sheared slab when the width of the tearing layer is much smaller than the ion Larmor radius. The ion response is nonlocal, and the quasineutrality retains its full integal form. An expansion procedure is introduced to solve the quasineutrality equation in powers of the width of the tearing layer over the ion Larmor radius. The expansion procedure is applied to the collisionless and semi-collisional tearing modes. The first order terms in the expansion we find to be strongly stabilizing. The physics of the mode and of the stabilization is discussed. Tearing modes are observed in experiments even though the slab theory predicts stability. It is proposed that these modes grow from an equilibrium with islands at the rational surfaces. If the equilibrium islands are wider than the ion Larmor radius, the mode is unstable when Δ' is positive
Elements of partial differential equations
Sneddon, Ian Naismith
1957-01-01
Geared toward students of applied rather than pure mathematics, this volume introduces elements of partial differential equations. Its focus is primarily upon finding solutions to particular equations rather than general theory.Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent st
Investigation of the stability of collision-less spherically symmetric systems
Energy Technology Data Exchange (ETDEWEB)
Polyachenko, V.L.; Shukhman, I.G. (AN SSSR, Leningrad. Inst. Teoreticheskoj Astronomii)
The matrix equation star is suggested for investigating the large- scale stability of general spherical collisionless star systems. The equation derived is used for a stability analysis of Idlis one-parametric series of distribution functions. It is shown that the stability condition restricts strongly the permissible degree of anisotropy of the stellar velocity distribution.
Witten, Matthew
1983-01-01
Hyperbolic Partial Differential Equations, Volume 1: Population, Reactors, Tides and Waves: Theory and Applications covers three general areas of hyperbolic partial differential equation applications. These areas include problems related to the McKendrick/Von Foerster population equations, other hyperbolic form equations, and the numerical solution.This text is composed of 15 chapters and begins with surveys of age specific population interactions, populations models of diffusion, nonlinear age dependent population growth with harvesting, local and global stability for the nonlinear renewal eq
Stable perturbations of linear differential equations generating a uniformly bounded group
Skazka, V. V.
2017-08-01
Stability problems for solutions of the differential equation u'(t)=Au+\\varepsilon B(t,u) in a Banach space are considered. It is assumed that for \\varepsilon=0 this equation generates a uniformly bounded group of class C_0. Sufficient conditions on B and A are found under which the solutions of this equation are bounded for small \\varepsilon. A linearization principle is proved for this equation under certain conditions on the operator B. Bibliography: 9 titles.
Superstability of the generalized orthogonality equation on restricted ...
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
(n ≥ 2) satisfies the functional inequality. ||〈f (x), f (y)〉| − |〈x,y〉|| ≤ ε for some ε ≥ 0 and for all x,y ∈ Rn. , then f is a solution of the generalized orthogonality equation (1). We will refer the reader to [3,6,8,12] for detailed definitions of stability and superstability of functional equations. By using ideas of Skof and Rassias [8,11], ...
Fixed Points, Inner Product Spaces, and Functional Equations
Directory of Open Access Journals (Sweden)
Park Choonkil
2010-01-01
Full Text Available Rassias introduced the following equality , , for a fixed integer . Let be real vector spaces. It is shown that, if a mapping satisfies the following functional equation for all with , which is defined by the above equality, then the mapping is realized as the sum of an additive mapping and a quadratic mapping. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.
Analytical approach for the Floquet theory of delay differential equations.
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.
Simple Numerical Schemes for the Korteweg-deVries Equation
International Nuclear Information System (INIS)
McKinstrie, C. J.; Kozlov, M.V.
2000-01-01
Two numerical schemes, which simulate the propagation of dispersive non-linear waves, are described. The first is a split-step Fourier scheme for the Korteweg-de Vries (KdV) equation. The second is a finite-difference scheme for the modified KdV equation. The stability and accuracy of both schemes are discussed. These simple schemes can be used to study a wide variety of physical processes that involve dispersive nonlinear waves
Functional differential equations with unbounded delay in extrapolation spaces
Directory of Open Access Journals (Sweden)
Mostafa Adimy
2014-08-01
Full Text Available We study the existence, regularity and stability of solutions for nonlinear partial neutral functional differential equations with unbounded delay and a Hille-Yosida operator on a Banach space X. We consider two nonlinear perturbations: the first one is a function taking its values in X and the second one is a function belonging to a space larger than X, an extrapolated space. We use the extrapolation techniques to prove the existence and regularity of solutions and we establish a linearization principle for the stability of the equilibria of our equation.
Stability analysis of nonlinear systems
Lakshmikantham, Vangipuram; Martynyuk, Anatoly A
2015-01-01
The book investigates stability theory in terms of two different measure, exhibiting the advantage of employing families of Lyapunov functions and treats the theory of a variety of inequalities, clearly bringing out the underlying theme. It also demonstrates manifestations of the general Lyapunov method, showing how this technique can be adapted to various apparently diverse nonlinear problems. Furthermore it discusses the application of theoretical results to several different models chosen from real world phenomena, furnishing data that is particularly relevant for practitioners. Stability Analysis of Nonlinear Systems is an invaluable single-sourse reference for industrial and applied mathematicians, statisticians, engineers, researchers in the applied sciences, and graduate students studying differential equations.
Resonance tongues in the linear Sitnikov equation
Misquero, Mauricio
2018-04-01
In this paper, we deal with a Hill's equation, depending on two parameters e\\in [0,1) and Λ >0, that has applications to some problems in Celestial Mechanics of the Sitnikov type. Due to the nonlinearity of the eccentricity parameter e and the coexistence problem, the stability diagram in the (e,Λ )-plane presents unusual resonance tongues emerging from points (0,(n/2)^2), n=1,2,\\ldots The tongues bounded by curves of eigenvalues corresponding to 2π -periodic solutions collapse into a single curve of coexistence (for which there exist two independent 2π -periodic eigenfunctions), whereas the remaining tongues have no pockets and are very thin. Unlike most of the literature related to resonance tongues and Sitnikov-type problems, the study of the tongues is made from a global point of view in the whole range of e\\in [0,1). Indeed, an interesting behavior of the tongues is found: almost all of them concentrate in a small Λ -interval [1, 9 / 8] as e→ 1^-. We apply the stability diagram of our equation to determine the regions for which the equilibrium of a Sitnikov (N+1)-body problem is stable in the sense of Lyapunov and the regions having symmetric periodic solutions with a given number of zeros. We also study the Lyapunov stability of the equilibrium in the center of mass of a curved Sitnikov problem.
Classical Diophantine equations
1993-01-01
The author had initiated a revision and translation of "Classical Diophantine Equations" prior to his death. Given the rapid advances in transcendence theory and diophantine approximation over recent years, one might fear that the present work, originally published in Russian in 1982, is mostly superseded. That is not so. A certain amount of updating had been prepared by the author himself before his untimely death. Some further revision was prepared by close colleagues. The first seven chapters provide a detailed, virtually exhaustive, discussion of the theory of lower bounds for linear forms in the logarithms of algebraic numbers and its applications to obtaining upper bounds for solutions to the eponymous classical diophantine equations. The detail may seem stark--- the author fears that the reader may react much as does the tourist on first seeing the centre Pompidou; notwithstanding that, Sprind zuk maintainsa pleasant and chatty approach, full of wise and interesting remarks. His emphases well warrant, ...
DEFF Research Database (Denmark)
Dyre, Jeppe
1995-01-01
energies chosen randomly according to a Gaussian. The random-walk model is here derived from Newton's laws by making a number of simplifying assumptions. In the second part of the paper an approximate low-temperature description of energy fluctuations in the random-walk modelthe energy master equation...... (EME)is arrived at. The EME is one dimensional and involves only energy; it is derived by arguing that percolation dominates the relaxational properties of the random-walk model at low temperatures. The approximate EME description of the random-walk model is expected to be valid at low temperatures...... of the random-walk model. The EME allows a calculation of the energy probability distribution at realistic laboratory time scales for an arbitrarily varying temperature as function of time. The EME is probably the only realistic equation available today with this property that is also explicitly consistent...
Damped nonlinear Schrodinger equation
International Nuclear Information System (INIS)
Nicholson, D.R.; Goldman, M.V.
1976-01-01
High frequency electrostatic plasma oscillations described by the nonlinear Schrodinger equation in the presence of damping, collisional or Landau, are considered. At early times, Landau damping of an initial soliton profile results in a broader, but smaller amplitude soliton, while collisional damping reduces the soliton size everywhere; soliton speeds at early times are unchanged by either kind of damping. For collisional damping, soliton speeds are unchanged for all time
Differential equations with Mathematica
Abell, Martha L
2004-01-01
The Third Edition of the Differential Equations with Mathematica integrates new applications from a variety of fields,especially biology, physics, and engineering. The new handbook is also completely compatible with recent versions of Mathematica and is a perfect introduction for Mathematica beginners.* Focuses on the most often used features of Mathematica for the beginning Mathematica user* New applications from a variety of fields, including engineering, biology, and physics* All applications were completed using recent versions of Mathematica
Fun with Differential Equations
Indian Academy of Sciences (India)
IAS Admin
tion of ® with ¼=2. One can use the uniqueness of solutions of differential equations to prove the addition formulae for sin(t1 +t2), etc. But instead of continuing with this thought process, let us do something more interesting. Now we shall consider another system. Fix 0 < < 1. I am looking for three real-valued functions x(t), ...
Mathematics and Maxwell's equations
International Nuclear Information System (INIS)
Boozer, Allen H
2010-01-01
The universality of mathematics and Maxwell's equations is not shared by specific plasma models. Computations become more reliable, efficient and transparent if specific plasma models are used to obtain only the information that would otherwise be missing. Constraints of high universality, such as those from mathematics and Maxwell's equations, can be obscured or lost by integrated computations. Recognition of subtle constraints of high universality is important for (1) focusing the design of control systems for magnetic field errors in tokamaks from perturbations that have little effect on the plasma to those that do, (2) clarifying the limits of applicability to astrophysics of computations of magnetic reconnection in fields that have a double periodicity or have B-vector =0 on a surface, as in a Harris sheet. Both require a degree of symmetry not expected in natural systems. Mathematics and Maxwell's equations imply that neighboring magnetic field lines characteristically separate exponentially with distance along a line. This remarkably universal phenomenon has been largely ignored, though it defines a trigger for reconnection through a critical magnitude of exponentiation. These and other examples of the importance of making distinctions and understanding constraints of high universality are explained.
DEFF Research Database (Denmark)
Jespersen, Jesper
2004-01-01
It is demonstrated that full employment and sustainable development not necessarily are conflicting goals. On the other hand macroeconomic stability cannot be obtained without a deliberate labour sharing policy and a shift in the composition of private consumption away from traditional material...
Propagating reaction fronts in moving fluids
Vilela Proaño, Pablo Martin
2015-01-01
La presente tesis tuvo como objetivo estudiar frentes de reacción modelados mediante la ecuación de Kuramoto-Sivashinsky sujetos a diferentes tipos de movimiento de fluido: flujo externo de Poiseuille, el cual es contrastado con el flujo de Couette, y flujo convectivo debido a la inestabilidad de Rayleigh-Taylor. En el primer caso, los frentes se propagan a favor o en contra de un flujo estacionario bidimensional entre dos placas paralelas que se conoce como flujo de Poiseuille. Para pequeñas...
Two stability problems related to resistive magnetohydrodynamics
International Nuclear Information System (INIS)
Tasso, H.
1994-01-01
Two general problems related to resistive magnetohydrodynamic stability are addressed in this paper. First, a general stability condition previously derived by the author for a class of real systems, occurring especially in plasma physics, is proved to persist to second order, despite the addition of several antisymmetric operators of first order in the linearized stability equation. Second, for a special but representative choice of the stability operators, a nonperturbative analysis demonstrates the existence of a critical density for the appearance of an overstability and the connected Hopf bifurcation, as suggested in a previous paper [Phys. Lett. A 180, 257 (1993)
Numerical approximations of difference functional equations and applications
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Zdzisław Kamont
2005-01-01
Full Text Available We give a theorem on the error estimate of approximate solutions for difference functional equations of the Volterra type. We apply this general result in the investigation of the stability of difference schemes generated by nonlinear first order partial differential functional equations and by parabolic problems. We show that all known results on difference methods for initial or initial boundary value problems can be obtained as particular cases of this general and simple result. We assume that the right hand sides of equations satisfy nonlinear estimates of the Perron type with respect to functional variables.
Almost Automorphic Solutions to Nonautonomous Stochastic Functional Integrodifferential Equations
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Li Xi-liang
2013-01-01
Full Text Available This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear nonautonomous functional integrodifferential stochastic evolution equations in real separable Hilbert spaces. Using the so-called “Acquistapace-Terreni” conditions and Banach contraction principle, the existence, uniqueness, and asymptotical stability results of square-mean almost automorphic mild solutions to such stochastic equations are established. As an application, square-mean almost automorphic solution to a concrete nonautonomous integro-differential stochastic evolution equation is analyzed to illustrate our abstract results.
On the Stability of Three-Dimensional Boundary Layers. Part 1; Linear and Nonlinear Stability
Janke, Erik; Balakumar, Ponnampalam
1999-01-01
The primary stability of incompressible three-dimensional boundary layers is investigated using the Parabolized Stability Equations (PSE). We compute the evolution of stationary and traveling disturbances in the linear and nonlinear region prior to transition. As model problems, we choose Swept Hiemenz Flow and the DLR Transition Experiment. The primary stability results for Swept Hiemenz Flow agree very well with computations by Malik et al. For the DLR Experiment, the mean flow profiles are obtained by solving the boundary layer equations for the measured pressure distribution. Both linear and nonlinear results show very good agreement with the experiment.
Stability and stabilization of nonlinear systems and Takagi-Sugeno's fuzzy models
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Blanco Yann
2001-01-01
Full Text Available This paper outlines a methodology to study the stability of Takagi-Sugeno's (TS fuzzy models. The stability analysis of the TS model is performed using a quadratic Liapunov candidate function. This paper proposes a relaxation of Tanaka's stability condition: unlike related works, the equations to be solved are not Liapunov equations for each rule matrix, but a convex combination of them. The coefficients of this sums depend on the membership functions. This method is applied to the design of continuous controllers for the TS model. Three different control structures are investigated, among which the Parallel Distributed Compensation (PDC. An application to the inverted pendulum is proposed here.
Asymptotic behavior of the plasma equation
International Nuclear Information System (INIS)
Kwong, Y.C.
1984-01-01
This paper is concerned with the plasma equation on a bounded smooth domain the N-dimensional Euclidean Space, with non-negative initial data and a homogenous Dirichlet boundary condition. It is known that there exists a finite extinction time T such that the solution decays to zero at T. Berryman and Holland investigated the stability of the profile of the solution as t is approaching T. However, they obtained their results at the expense of some very strong regularity assumptions. By invoking both the nonlinear semi-group theory and a standard regularizing scheme for the equation, the same results are proved without those assumptions by measuring the rate of decay of the solution and estimates are obtained on the time derivative as t is approaching T. As motivated by the regularity assumptions, both the interior and boundary regularities of the solution are studied. Finally, the nonlinearity of the plasma equation is perturbed and the same aspects for the perturbed equation are studied