Some blow-up problems for a semilinear parabolic equation with a potential
Cheng, Ting; Zheng, Gao-Feng
The blow-up rate estimate for the solution to a semilinear parabolic equation u=Δu+V(x)|u in Ω×(0,T) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=Mφ(x) as M goes to infinity, which have been found in [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006], is improved under some reasonable and weaker conditions compared with [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006].
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
The present paper deals with the long-time behavior of a class of nonautonomous retarded semilinear parabolic differential equations. When the time delays are small enough and the spectral gap conditions hold, the inertial manifolds of the nonautonomous retard parabolic equations are constructed by using the Lyapunov-Perron method.
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.
Dynamical symmetries of semi-linear Schrodinger and diffusion equations
International Nuclear Information System (INIS)
Stoimenov, Stoimen; Henkel, Malte
2005-01-01
Conditional and Lie symmetries of semi-linear 1D Schrodinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrodinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra (conf 3 ) C . We consider non-hermitian representations and also include a dimensionful coupling constant of the non-linearity. The corresponding representations of the parabolic and almost-parabolic subalgebras of (conf 3 ) C are classified and the complete list of conditionally invariant semi-linear Schrodinger equations is obtained. Possible applications to the dynamical scaling behaviour of phase-ordering kinetics are discussed
Directory of Open Access Journals (Sweden)
Weifeng Wang
2014-01-01
Full Text Available We study an optimal control problem governed by a semilinear parabolic equation, whose control variable is contained only in the boundary condition. An existence theorem for the optimal control is obtained.
Classical and weak solutions for semilinear parabolic equations with Preisach hysteresis
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Mathias Jais
2008-01-01
Full Text Available We consider the solvability of the semilinear parabolic differential equation \\[\\frac{\\partial u}{\\partial t}(x,t- \\Delta u(x,t + c(x,tu(x,t = \\mathcal{P}(u + \\gamma (x,t\\] in a cylinder \\(D=\\Omega \\times (0,T\\, where \\(\\mathcal{P}\\ is a hysteresis operator of Preisach type. We show that the corresponding initial boundary value problems have unique classical solutions. We further show that using this existence and uniqueness result, one can determine the properties of the Preisach operator \\(\\mathcal{P}\\ from overdetermined boundary data.
On the controllability of the semilinear heat equation with hysteresis
International Nuclear Information System (INIS)
Bagagiolo, Fabio
2012-01-01
We study the null controllability problem for a semilinear parabolic equation, with hysteresis entering in the semilinearity. Under suitable hypotheses, we prove the controllability result and explicitly treat the cases where the hysteresis relationship is given by a Play or a Preisach operator.
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Rubio Gerardo
2011-03-01
Full Text Available We consider the Cauchy problem in ℝd for a class of semilinear parabolic partial differential equations that arises in some stochastic control problems. We assume that the coefficients are unbounded and locally Lipschitz, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution by approximation with linear parabolic equations. The linear equations involved can not be solved with the traditional results. Therefore, we construct a classical solution to the linear Cauchy problem under the same hypotheses on the coefficients for the semilinear equation. Our approach is using stochastic differential equations and parabolic differential equations in bounded domains. Finally, we apply the results to a stochastic optimal consumption problem. Nous considérons le problème de Cauchy dans ℝd pour une classe d’équations aux dérivées partielles paraboliques semi linéaires qui se pose dans certains problèmes de contrôle stochastique. Nous supposons que les coefficients ne sont pas bornés et sont localement Lipschitziennes, pas nécessairement différentiables, avec des données continues et ellipticité local uniforme. Nous construisons une solution classique par approximation avec les équations paraboliques linéaires. Les équations linéaires impliquées ne peuvent être résolues avec les résultats traditionnels. Par conséquent, nous construisons une solution classique au problème de Cauchy linéaire sous les mêmes hypothèses sur les coefficients pour l’équation semi-linéaire. Notre approche utilise les équations différentielles stochastiques et les équations différentielles paraboliques dans les domaines bornés. Enfin, nous appliquons les résultats à un problème stochastique de consommation optimale.
Semilinear Kolmogorov Equations and Applications to Stochastic Optimal Control
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Masiero, Federica
2005-01-01
Semilinear parabolic differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. Applications to stochastic optimal control problems are studied by solving the associated Hamilton-Jacobi-Bellman equation. These results are applied to some controlled stochastic partial differential equations
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
Partial differential equations of parabolic type
Friedman, Avner
2008-01-01
This accessible and self-contained treatment provides even readers previously unacquainted with parabolic and elliptic equations with sufficient background to understand research literature. Author Avner Friedman - Director of the Mathematical Biosciences Institute at The Ohio State University - offers a systematic and thorough approach that begins with the main facts of the general theory of second order linear parabolic equations. Subsequent chapters explore asymptotic behavior of solutions, semi-linear equations and free boundary problems, and the extension of results concerning fundamenta
Energy Technology Data Exchange (ETDEWEB)
Addona, Davide, E-mail: d.addona@campus.unimib.it [Università degli Studi di Milano Bicocca, (MILANO BICOCCA) Dipartimento di Matematica (Italy)
2015-08-15
We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton–Jacobi–Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified.
Blowing-up Semilinear Wave Equation with Exponential ...
Indian Academy of Sciences (India)
Blowing-up Semilinear Wave Equation with Exponential Nonlinearity in Two Space ... We investigate the initial value problem for some semi-linear wave equation in two space dimensions with exponential nonlinearity growth. ... Current Issue
A geometric theory for semilinear almost-periodic parabolic partial differential equations on RN
International Nuclear Information System (INIS)
Vuillermot, P.A.
1991-01-01
In this short expository article we review various applications of some geometric methods which have been recently devised to investigate the long time behaviour of classical solutions to certain semilinear almost-periodic reaction-diffusion equations on R N . As a consequence, we also show how to construct almost-periodic attractors for such equations and how to investigate their stability properties. The class of problems which we analyse here contains in particular well known equations of population genetics. (author). 17 refs
Optimal Wentzell Boundary Control of Parabolic Equations
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Luo, Yousong
2017-01-01
This paper deals with a class of optimal control problems governed by an initial-boundary value problem of a parabolic equation. The case of semi-linear boundary control is studied where the control is applied to the system via the Wentzell boundary condition. The differentiability of the state variable with respect to the control is established and hence a necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained convex problems. A second order condition is also derived.
Optimal Wentzell Boundary Control of Parabolic Equations
Energy Technology Data Exchange (ETDEWEB)
Luo, Yousong, E-mail: yousong.luo@rmit.edu.au [RMIT University, School of Mathematical and Geospatial Sciences (Australia)
2017-04-15
This paper deals with a class of optimal control problems governed by an initial-boundary value problem of a parabolic equation. The case of semi-linear boundary control is studied where the control is applied to the system via the Wentzell boundary condition. The differentiability of the state variable with respect to the control is established and hence a necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained convex problems. A second order condition is also derived.
Lp Theory for Super-Parabolic Backward Stochastic Partial Differential Equations in the Whole Space
International Nuclear Information System (INIS)
Du Kai; Qiu, Jinniao; Tang Shanjian
2012-01-01
This paper is concerned with semi-linear backward stochastic partial differential equations (BSPDEs for short) of super-parabolic type. An L p -theory is given for the Cauchy problem of BSPDEs, separately for the case of p∈(1,2] and for the case of p∈(2,∞). A comparison theorem is also addressed.
Talaei, Behzad; Jagannathan, Sarangapani; Singler, John
2018-04-01
In this paper, neurodynamic programming-based output feedback boundary control of distributed parameter systems governed by uncertain coupled semilinear parabolic partial differential equations (PDEs) under Neumann or Dirichlet boundary control conditions is introduced. First, Hamilton-Jacobi-Bellman (HJB) equation is formulated in the original PDE domain and the optimal control policy is derived using the value functional as the solution of the HJB equation. Subsequently, a novel observer is developed to estimate the system states given the uncertain nonlinearity in PDE dynamics and measured outputs. Consequently, the suboptimal boundary control policy is obtained by forward-in-time estimation of the value functional using a neural network (NN)-based online approximator and estimated state vector obtained from the NN observer. Novel adaptive tuning laws in continuous time are proposed for learning the value functional online to satisfy the HJB equation along system trajectories while ensuring the closed-loop stability. Local uniformly ultimate boundedness of the closed-loop system is verified by using Lyapunov theory. The performance of the proposed controller is verified via simulation on an unstable coupled diffusion reaction process.
POSITIVE SOLUTIONS TO SEMI-LINEAR SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS IN BANACH SPACE
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
In this paper,we study the existence of positive periodic solution to some second- order semi-linear differential equation in Banach space.By the fixed point index theory, we prove that the semi-linear differential equation has two positive periodic solutions.
Parametric Borel summability for some semilinear system of partial differential equations
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Hiroshi Yamazawa
2015-01-01
Full Text Available In this paper we study the Borel summability of formal solutions with a parameter of first order semilinear system of partial differential equations with \\(n\\ independent variables. In [Singular perturbation of linear systems with a regular singularity, J. Dynam. Control. Syst. 8 (2002, 313-322], Balser and Kostov proved the Borel summability of formal solutions with respect to a singular perturbation parameter for a linear equation with one independent variable. We shall extend their results to a semilinear system of equations with general independent variables.
Global existence of small solutions to semilinear Schroedinger equations
International Nuclear Information System (INIS)
Chihara, Hiroyuki
1996-01-01
We present global existence theorem for semilinear Schrodinger equations. In general, Schrodinger-type equations do not admit the classical energy estimates. To avoid this difficulty, we use S. Doi's method for linear Schrodinger-type equations. Combining his method and L p -L q estimates, we prove the global existence of solutions with small initial data
Stability of numerical method for semi-linear stochastic pantograph differential equations
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Yu Zhang
2016-01-01
Full Text Available Abstract As a particular expression of stochastic delay differential equations, stochastic pantograph differential equations have been widely used in nonlinear dynamics, quantum mechanics, and electrodynamics. In this paper, we mainly study the stability of analytical solutions and numerical solutions of semi-linear stochastic pantograph differential equations. Some suitable conditions for the mean-square stability of an analytical solution are obtained. Then we proved the general mean-square stability of the exponential Euler method for a numerical solution of semi-linear stochastic pantograph differential equations, that is, if an analytical solution is stable, then the exponential Euler method applied to the system is mean-square stable for arbitrary step-size h > 0 $h>0$ . Numerical examples further illustrate the obtained theoretical results.
Exponential decay for solutions to semilinear damped wave equation
Gerbi, Sté phane; Said-Houari, Belkacem
2011-01-01
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Intro- ducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data
Semilinear hyperbolic systems and equations with singular initial data
International Nuclear Information System (INIS)
Gramchev, T.
1991-07-01
We study the weak limits of solutions u ε (t, ·) for ε→0 to semilinear strictly hyperbolic systems and wave equations with initial data u ε (0, ·) approximating a distribution κ, 0 ε (t, ·) for ε→0 exists. 13 refs
Existence of solutions of abstract fractional impulsive semilinear evolution equations
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K. Balachandran
2010-01-01
Full Text Available In this paper we prove the existence of solutions of fractional impulsive semilinear evolution equations in Banach spaces. A nonlocal Cauchy problem is discussed for the evolution equations. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the theory.
Differential equations inverse and direct problems
Favini, Angelo
2006-01-01
DEGENERATE FIRST ORDER IDENTIFICATION PROBLEMS IN BANACH SPACES A NONISOTHERMAL DYNAMICAL GINZBURG-LANDAU MODEL OF SUPERCONDUCTIVITY. EXISTENCE AND UNIQUENESS THEOREMSSOME GLOBAL IN TIME RESULTS FOR INTEGRODIFFERENTIAL PARABOLIC INVERSE PROBLEMSFOURTH ORDER ORDINARY DIFFERENTIAL OPERATORS WITH GENERAL WENTZELL BOUNDARY CONDITIONSTUDY OF ELLIPTIC DIFFERENTIAL EQUATIONS IN UMD SPACESDEGENERATE INTEGRODIFFERENTIAL EQUATIONS OF PARABOLIC TYPE EXPONENTIAL ATTRACTORS FOR SEMICONDUCTOR EQUATIONSCONVERGENCE TO STATIONARY STATES OF SOLUTIONS TO THE SEMILINEAR EQUATION OF VISCOELASTICITY ASYMPTOTIC BEHA
Error Analysis of Galerkin's Method for Semilinear Equations
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Tadashi Kawanago
2012-01-01
Full Text Available We establish a general existence result for Galerkin's approximate solutions of abstract semilinear equations and conduct an error analysis. Our results may be regarded as some extension of a precedent work (Schultz 1969. The derivation of our results is, however, different from the discussion in his paper and is essentially based on the convergence theorem of Newton’s method and some techniques for deriving it. Some of our results may be applicable for investigating the quality of numerical verification methods for solutions of ordinary and partial differential equations.
Exponential decay for solutions to semilinear damped wave equation
Gerbi, Stéphane
2011-10-01
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Intro- ducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4].
Radial solutions to semilinear elliptic equations via linearized operators
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Phuong Le
2017-04-01
Full Text Available Let $u$ be a classical solution of semilinear elliptic equations in a ball or an annulus in $\\mathbb{R}^N$ with zero Dirichlet boundary condition where the nonlinearity has a convex first derivative. In this note, we prove that if the $N$-th eigenvalue of the linearized operator at $u$ is positive, then $u$ must be radially symmetric.
Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay
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Hassane Bouzahir
2007-02-01
Full Text Available We describe a semigroup of abstract semilinear functional differential equations with infinite delay by the use of the Crandall Liggett theorem. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We clarify the properties of the phase space ensuring equivalence between the equation under investigation and the nonlinear semigroup.
THE EXPONENTIAL STABILIZATION FOR A SEMILINEAR WAVE EQUATION WITH LOCALLY DISTRIBUTED FEEDBACK
Institute of Scientific and Technical Information of China (English)
JIA CHAOHUA; FENG DEXING
2005-01-01
This paper considers the exponential decay of the solution to a damped semilinear wave equation with variable coefficients in the principal part by Riemannian multiplier method. A differential geometric condition that ensures the exponential decay is obtained.
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
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...
Liu, Jinghuai; Zhang, Litao
2016-01-01
In this paper, we investigate the existence of anti-periodic (or anti-periodic differentiable) mild solutions to the semilinear differential equation [Formula: see text] with nondense domain. Furthermore, an example is given to illustrate our results.
Estimates for mild solutions to semilinear Cauchy problems
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Kresimir Burazin
2014-09-01
Full Text Available The existence (and uniqueness results on mild solutions of the abstract semilinear Cauchy problems in Banach spaces are well known. Following the results of Tartar (2008 and Burazin (2008 in the case of decoupled hyperbolic systems, we give an alternative proof, which enables us to derive an estimate on the mild solution and its time of existence. The nonlinear term in the equation is allowed to be time-dependent. We discuss the optimality of the derived estimate by testing it on three examples: the linear heat equation, the semilinear heat equation that models dynamic deflection of an elastic membrane, and the semilinear Schrodinger equation with time-dependent nonlinearity, that appear in the modelling of numerous physical phenomena.
Non-local quasi-linear parabolic equations
International Nuclear Information System (INIS)
Amann, H
2005-01-01
This is a survey of the most common approaches to quasi-linear parabolic evolution equations, a discussion of their advantages and drawbacks, and a presentation of an entirely new approach based on maximal L p regularity. The general results here apply, above all, to parabolic initial-boundary value problems that are non-local in time. This is illustrated by indicating their relevance for quasi-linear parabolic equations with memory and, in particular, for time-regularized versions of the Perona-Malik equation of image processing
International Workshop on Elliptic and Parabolic Equations
Schrohe, Elmar; Seiler, Jörg; Walker, Christoph
2015-01-01
This volume covers the latest research on elliptic and parabolic equations and originates from the international Workshop on Elliptic and Parabolic Equations, held September 10-12, 2013 at the Leibniz Universität Hannover. It represents a collection of refereed research papers and survey articles written by eminent scientist on advances in different fields of elliptic and parabolic partial differential equations, including singular Riemannian manifolds, spectral analysis on manifolds, nonlinear dispersive equations, Brownian motion and kernel estimates, Euler equations, porous medium type equations, pseudodifferential calculus, free boundary problems, and bifurcation analysis.
Directory of Open Access Journals (Sweden)
Yan-Tao Bian
2014-04-01
Full Text Available In this article, we study weighted asymptotic behavior of solutions to the semilinear integro-differential equation $$ u'(t=Au(t+\\alpha\\int_{-\\infty}^{t}e^{-\\beta(t-s}Au(sds+f(t,u(t, \\quad t\\in \\mathbb{R}, $$ where $\\alpha, \\beta \\in \\mathbb{R}$, with $\\beta > 0, \\alpha \
The parabolic equation method for outdoor sound propagation
DEFF Research Database (Denmark)
Arranz, Marta Galindo
The parabolic equation method is a versatile tool for outdoor sound propagation. The present study has focused on the Cranck-Nicolson type Parabolic Equation method (CNPE). Three different applications of the CNPE method have been investigated. The first two applications study variations of the g......The parabolic equation method is a versatile tool for outdoor sound propagation. The present study has focused on the Cranck-Nicolson type Parabolic Equation method (CNPE). Three different applications of the CNPE method have been investigated. The first two applications study variations...
Solving Variable Coefficient Fourth-Order Parabolic Equation by ...
African Journals Online (AJOL)
Solving Variable Coefficient Fourth-Order Parabolic Equation by Modified initial guess Variational ... variable coefficient fourth order parabolic partial differential equations. The new method shows rapid convergence to the exact solution.
International Nuclear Information System (INIS)
Casas, E.; Troeltzsch, F.
1999-01-01
In this paper we are concerned with some optimal control problems governed by semilinear elliptic equations. The case of a boundary control is studied. We consider pointwise constraints on the control and a finite number of equality and inequality constraints on the state. The goal is to derive first- and second-order optimality conditions satisfied by locally optimal solutions of the problem
International Nuclear Information System (INIS)
Agarwal, Ravi P.; Baghli, Selma; Benchohra, Mouffak
2009-01-01
The controllability of mild solutions defined on the semi-infinite positive real interval for two classes of first order semilinear functional and neutral functional differential evolution equations with infinite delay is studied in this paper. Our results are obtained using a recent nonlinear alternative due to Avramescu for sum of compact and contraction operators in Frechet spaces, combined with the semigroup theory
Linear and quasi-linear equations of parabolic type
Ladyženskaja, O A; Ural′ceva, N N; Uralceva, N N
1968-01-01
Equations of parabolic type are encountered in many areas of mathematics and mathematical physics, and those encountered most frequently are linear and quasi-linear parabolic equations of the second order. In this volume, boundary value problems for such equations are studied from two points of view: solvability, unique or otherwise, and the effect of smoothness properties of the functions entering the initial and boundary conditions on the smoothness of the solutions.
Notes on spectrum and exponential decay in nonautonomous evolutionary equations
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Christian Pötzsche
2016-08-01
Full Text Available We first determine the dichotomy (Sacker-Sell spectrum for certain nonautonomous linear evolutionary equations induced by a class of parabolic PDE systems. Having this information at hand, we underline the applicability of our second result: If the widths of the gaps in the dichotomy spectrum are bounded away from $0$, then one can rule out the existence of super-exponentially decaying (i.e. slow solutions of semi-linear evolutionary equations.
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
Critical spaces for quasilinear parabolic evolution equations and applications
Prüss, Jan; Simonett, Gieri; Wilke, Mathias
2018-02-01
We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal Lp-regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier-Stokes problem, convection-diffusion equations, the Nernst-Planck-Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and some other well-known parabolic equations are given.
A Priori Regularity of Parabolic Partial Differential Equations
Berkemeier, Francisco
2018-01-01
In this thesis, we consider parabolic partial differential equations such as the heat equation, the Fokker-Planck equation, and the porous media equation. Our aim is to develop methods that provide a priori estimates for solutions with singular
Moving interfaces and quasilinear parabolic evolution equations
Prüss, Jan
2016-01-01
In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasilinear parabolic evolution equations, elliptic and parabolic boundary value problems, transmission problems, one- and two-phase Stokes problems, and the equations of incompressible viscous one- and two-phase fluid flows. The theory of maximal regularity, an essential element, is also fully developed. The authors present a modern approach based on powerful tools in classical analysis, functional analysis, and vector-valued harmonic analysis. The theory is applied to problems in two-phase fluid dynamics and phase transitions, one-phase generalized Newtonian fluids, nematic liquid crystal flows, Maxwell-Stefan diffusion, and a variety of geometric evolution equations. The book also includes a discussion of the underlying physical and thermodynamic principles governing the equations of fluid flows and phase transitions...
An inverse problem in a parabolic equation
Directory of Open Access Journals (Sweden)
Zhilin Li
1998-11-01
Full Text Available In this paper, an inverse problem in a parabolic equation is studied. An unknown function in the equation is related to two integral equations in terms of heat kernel. One of the integral equations is well-posed while another is ill-posed. A regularization approach for constructing an approximate solution to the ill-posed integral equation is proposed. Theoretical analysis and numerical experiment are provided to support the method.
A Note on a Semilinear Fractional Differential Equation of Neutral Type with Infinite Delay
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Gisle M. Mophou
2010-01-01
Full Text Available We deal in this paper with the mild solution for the semilinear fractional differential equation of neutral type with infinite delay: Dαx(t+Ax(t=f(t,xt, t∈[0,T], x(t=ϕ(t, t∈]−∞,0], with T>0 and 0<α<1. We prove the existence (and uniqueness of solutions, assuming that −A is a linear closed operator which generates an analytic semigroup (T(tt≥0 on a Banach space 𝕏 by means of the Banach's fixed point theorem. This generalizes some recent results.
Regularity of solutions in semilinear elliptic theory
Indrei, Emanuel
2016-07-08
We study the semilinear Poisson equation Δu=f(x,u)inB1. (1) Our main results provide conditions on f which ensure that weak solutions of (1) belong to C1,1(B1/2). In some configurations, the conditions are sharp.
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
Integration of equations of parabolic type by the method of nets
Saul'Yev, V K; Stark, M; Ulam, S
1964-01-01
International Series of Monographs in Pure and Applied Mathematics, Volume 54: Integration of Equations of Parabolic Type by the Method of Nets deals with solving parabolic partial differential equations using the method of nets. The first part of this volume focuses on the construction of net equations, with emphasis on the stability and accuracy of the approximating net equations. The method of nets or method of finite differences (used to define the corresponding numerical method in ordinary differential equations) is one of many different approximate methods of integration of partial diff
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
Nonlinear anisotropic parabolic equations in Lm
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Fares Mokhtari
2014-01-01
Full Text Available In this paper, we give a result of regularity of weak solutions for a class of nonlinear anisotropic parabolic equations with lower-order term when the right-hand side is an Lm function, with m being ”small”. This work generalizes some results given in [2] and [3].
Identifying the principal coefficient of parabolic equations with non-divergent form
International Nuclear Information System (INIS)
Jiang, L S; Bian, B J
2005-01-01
We deal with an inverse problem of determining a coefficient a(x, t) of principal part for second order parabolic equations with non-divergent form when the solution is known. Such a problem has important applications in a large fields of applied science. We propose a well-posed approximate algorithm to identify the coefficient. The existence, uniqueness and stability of such solutions a(x, t) are proved. A necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced. Our numerical simulations show that the coefficient is recovered very well
Identifying the principal coefficient of parabolic equations with non-divergent form
Jiang, L. S.; Bian, B. J.
2005-01-01
We deal with an inverse problem of determining a coefficient a(x, t) of principal part for second order parabolic equations with non-divergent form when the solution is known. Such a problem has important applications in a large fields of applied science. We propose a well-posed approximate algorithm to identify the coefficient. The existence, uniqueness and stability of such solutions a(x, t) are proved. A necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced. Our numerical simulations show that the coefficient is recovered very well.
Degenerate parabolic stochastic partial differential equations
Czech Academy of Sciences Publication Activity Database
span class="emphasis">Hofmanová, Martinaspan>
2013-01-01
Roč. 123, č. 12 (2013), s. 4294-4336 ISSN 0304-4149 R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : kinetic solutions * degenerate stochastic parabolic equations Subject RIV: BA - General Mathematics Impact factor: 1.046, year: 2013 http://library.utia.cas.cz/separaty/2013/SI/hofmanova-0397241.pdf
Interior Gradient Estimates for Nonuniformly Parabolic Equations II
Directory of Open Access Journals (Sweden)
Lieberman Gary M
2007-01-01
Full Text Available We prove interior gradient estimates for a large class of parabolic equations in divergence form. Using some simple ideas, we prove these estimates for several types of equations that are not amenable to previous methods. In particular, we have no restrictions on the maximum eigenvalue of the coefficient matrix and we obtain interior gradient estimates for so-called false mean curvature equation.
On some perturbation techniques for quasi-linear parabolic equations
Directory of Open Access Journals (Sweden)
Igor Malyshev
1990-01-01
Full Text Available We study a nonhomogeneous quasi-linear parabolic equation and introduce a method that allows us to find the solution of a nonlinear boundary value problem in explicit form. This task is accomplished by perturbing the original equation with a source function, which is then found as a solution of some nonlinear operator equation.
A gradient estimate for solutions to parabolic equations with discontinuous coefficients
Directory of Open Access Journals (Sweden)
Jishan Fan
2013-04-01
Full Text Available Li-Vogelius and Li-Nirenberg gave a gradient estimate for solutions of strongly elliptic equations and systems of divergence forms with piecewise smooth coefficients, respectively. The discontinuities of the coefficients are assumed to be given by manifolds of codimension 1, which we called them emph{manifolds of discontinuities}. Their gradient estimate is independent of the distances between manifolds of discontinuities. In this paper, we gave a parabolic version of their results. That is, we gave a gradient estimate for parabolic equations of divergence forms with piecewise smooth coefficients. The coefficients are assumed to be independent of time and their discontinuities are likewise the previous elliptic equations. As an application of this estimate, we also gave a pointwise gradient estimate for the fundamental solution of a parabolic operator with piecewise smooth coefficients. Both gradient estimates are independent of the distances between manifolds of discontinuities.
Elliptic and parabolic equations for measures
Energy Technology Data Exchange (ETDEWEB)
Bogachev, Vladimir I [M. V. Lomonosov Moscow State University, Moscow (Russian Federation); Krylov, Nikolai V [University of Minnesota, Minneapolis, MN (United States); Roeckner, Michael [Universitat Bielefeld, Bielefeld (Germany)
2009-12-31
This article gives a detailed account of recent investigations of weak elliptic and parabolic equations for measures with unbounded and possibly singular coefficients. The existence and differentiability of densities are studied, and lower and upper bounds for them are discussed. Semigroups associated with second-order elliptic operators acting in L{sup p}-spaces with respect to infinitesimally invariant measures are investigated. Bibliography: 181 titles.
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.
ε-neighbourhoods of orbits of parabolic diffeomorphisms and cohomological equations
International Nuclear Information System (INIS)
Resman, Maja
2014-01-01
In this article, we study the analyticity of (directed) areas of ε-neighbourhoods of orbits of parabolic germs. The article is motivated by the question of analytic classification using ε-neighbourhoods of orbits in the simplest formal class. We show that the coefficient in front of the ε 2 term in the asymptotic expansion in ε, which we call the principal part of the area, is a sectorially analytic function in the initial point of the orbit. It satisfies a cohomological equation similar to the standard trivialization equation for parabolic diffeomorphisms. We give necessary and sufficient conditions on a diffeomorphism f for the existence of a globally analytic solution of this equation. Furthermore, we introduce a new classification type for diffeomorphisms implied by this new equation and investigate the relative position of its classes with respect to the analytic classes. (paper)
A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations
Directory of Open Access Journals (Sweden)
Jiebao Sun
2011-01-01
parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time-periodic solution for this system.
Parabolic Equation Modeling of Propagation over Terrain Using Digital Elevation Model
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Xiao-Wei Guan
2018-01-01
Full Text Available The parabolic equation method based on digital elevation model (DEM is applied on propagation predictions over irregular terrains. Starting from a parabolic approximation to the Helmholtz equation, a wide-angle parabolic equation is deduced under the assumption of forward propagation and the split-step Fourier transform algorithm is used to solve it. The application of DEM is extended to the Cartesian coordinate system and expected to provide a precise representation of a three-dimensional surface with high efficiency. In order to validate the accuracy, a perfectly conducting Gaussian terrain profile is simulated and the results are compared with the shift map. As a consequence, a good agreement is observed. Besides, another example is given to provide a theoretical basis and reference for DEM selection. The simulation results demonstrate that the prediction errors will be obvious only when the resolution of the DEM used is much larger than the range step in the PE method.
Stability test for a parabolic partial differential equation
Vajta, Miklos
2001-01-01
The paper describes a stability test applied to coupled parabolic partial differential equations. The PDE's describe the temperature distribution of composite structures with linear inner heat sources. The distributed transfer functions are developed based on the transmission matrix of each layer.
Numerical Solution of Parabolic Equations
DEFF Research Database (Denmark)
Østerby, Ole
These lecture notes are designed for a one-semester course on finite-difference methods for parabolic equations. These equations which traditionally are used for describing diffusion and heat-conduction problems in Geology, Physics, and Chemistry have recently found applications in Finance Theory...... ? and how do boundary value approximations affect the overall order of the method. Knowledge of a reliable order and error estimate enables us to determine (near-)optimal step sizes to meet a prescribed error tolerance, and possibly to extrapolate to get (higher order and) better accuracy at a minimal...... expense. Problems in two space dimensions are effectively handled using the Alternating Direction Implicit (ADI) technique. We present a systematic way of incorporating inhomogeneous terms and derivative boundary conditions in ADI methods as well as mixed derivative terms....
The fundamental solutions for fractional evolution equations of parabolic type
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Mahmoud M. El-Borai
2004-01-01
Full Text Available The fundamental solutions for linear fractional evolution equations are obtained. The coefficients of these equations are a family of linear closed operators in the Banach space. Also, the continuous dependence of solutions on the initial conditions is studied. A mixed problem of general parabolic partial differential equations with fractional order is given as an application.
Harnack's Inequality for Degenerate and Singular Parabolic Equations
DiBenedetto, Emmanuele; Vespri, Vincenzo
2012-01-01
Degenerate and singular parabolic equations have been the subject of extensive research for the last 25 years. Despite important achievements, the issue of the Harnack inequality for non-negative solutions to these equations, both of p-Laplacian and porous medium type, while raised by several authors, has remained basically open. Recently considerable progress has been made on this issue, to the point that, except for the singular sub-critical range, both for the p-laplacian and the porous medium equations, the theory is reasonably complete. It seemed therefore timely to trace a comprehensive
Real-time optical laboratory solution of parabolic differential equations
Casasent, David; Jackson, James
1988-01-01
An optical laboratory matrix-vector processor is used to solve parabolic differential equations (the transient diffusion equation with two space variables and time) by an explicit algorithm. This includes optical matrix-vector nonbase-2 encoded laboratory data, the combination of nonbase-2 and frequency-multiplexed data on such processors, a high-accuracy optical laboratory solution of a partial differential equation, new data partitioning techniques, and a discussion of a multiprocessor optical matrix-vector architecture.
Iterated Crank-Nicolson method for hyperbolic and parabolic equations in numerical relativity
International Nuclear Information System (INIS)
Leiler, Gregor; Rezzolla, Luciano
2006-01-01
The iterated Crank-Nicolson is a predictor-corrector algorithm commonly used in numerical relativity for the solution of both hyperbolic and parabolic partial differential equations. We here extend the recent work on the stability of this scheme for hyperbolic equations by investigating the properties when the average between the predicted and corrected values is made with unequal weights and when the scheme is applied to a parabolic equation. We also propose a variant of the scheme in which the coefficients in the averages are swapped between two corrections leading to systematically larger amplification factors and to a smaller numerical dispersion
A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations
Sun, Jiebao; Zhang, Dazhi; Wu, Boying
2011-01-01
We consider a cooperating two-species Lotka-Volterra model of degenerate parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time-periodic solution for this system.
Darboux transformations and linear parabolic partial differential equations
International Nuclear Information System (INIS)
Arrigo, Daniel J.; Hickling, Fred
2002-01-01
Solutions for a class of linear parabolic partial differential equation are provided. These solutions are obtained by first solving a system of (n+1) nonlinear partial differential equations. This system arises as the coefficients of a Darboux transformation and is equivalent to a matrix Burgers' equation. This matrix equation is solved using a generalized Hopf-Cole transformation. The solutions for the original equation are given in terms of solutions of the heat equation. These results are applied to the (1+1)-dimensional Schroedinger equation where all bound state solutions are obtained for a 2n-parameter family of potentials. As a special case, the solutions for integral members of the regular and modified Poeschl-Teller potentials are recovered. (author). Letter-to-the-editor
Rothe's method for parabolic equations on non-cylindrical domains
Czech Academy of Sciences Publication Activity Database
Dasht, J.; Engström, J.; Kufner, Alois; Persson, L.E.
2006-01-01
Roč. 1, č. 1 (2006), s. 59-80 ISSN 0973-2306 Institutional research plan: CEZ:AV0Z10190503 Keywords : parabolic equations * non-cylindrical domains * Rothe's method * time-discretization Subject RIV: BA - General Mathematics
An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology
Beretta, Elena; Cavaterra, Cecilia; Cerutti, M. Cristina; Manzoni, Andrea; Ratti, Luca
2017-10-01
In this paper we develop theoretical analysis and numerical reconstruction techniques for the solution of an inverse boundary value problem dealing with the nonlinear, time-dependent monodomain equation, which models the evolution of the electric potential in the myocardial tissue. The goal is the detection of an inhomogeneity \
Parametrices and exact paralinearization of semi-linear boundary problems
DEFF Research Database (Denmark)
Johnsen, Jon
2008-01-01
The subject is parametrices for semi-linear problems, based on parametrices for linear boundary problems and on non-linearities that decompose into solution-dependent linear operators acting on the solutions. Non-linearities of product type are shown to admit this via exact paralinearization...... of homogeneous distributions, tensor products and halfspace extensions have been revised. Examples include the von Karman equation....
A Priori Regularity of Parabolic Partial Differential Equations
Berkemeier, Francisco
2018-05-13
In this thesis, we consider parabolic partial differential equations such as the heat equation, the Fokker-Planck equation, and the porous media equation. Our aim is to develop methods that provide a priori estimates for solutions with singular initial data. These estimates are obtained by understanding the time decay of norms of solutions. First, we derive regularity results for the heat equation by estimating the decay of Lebesgue norms. Then, we apply similar methods to the Fokker-Planck equation with suitable assumptions on the advection and diffusion. Finally, we conclude by extending our techniques to the porous media equation. The sharpness of our results is confirmed by examining known solutions of these equations. The main contribution of this thesis is the use of functional inequalities to express decay of norms as differential inequalities. These are then combined with ODE methods to deduce estimates for the norms of solutions and their derivatives.
Almost periodic solutions to systems of parabolic equations
Directory of Open Access Journals (Sweden)
Janpou Nee
1994-01-01
Full Text Available In this paper we show that the second-order differential solution is 2-almost periodic, provided it is 2-bounded, and the growth of the components of a non-linear function of a system of parabolic equation is bounded by any pair of con-secutive eigenvalues of the associated Dirichlet boundary value problems.
Positive solutions with single and multi-peak for semilinear elliptic ...
Indian Academy of Sciences (India)
LI WANG
2018-04-24
Apr 24, 2018 ... [2] Bahri A and Lions P, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 14(3) (1997) 365–413. [3] Cao D, and Noussair E, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in RN , Ann. Inst. H.
Conditional stability in determination of initial data for stochastic parabolic equations
International Nuclear Information System (INIS)
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.
Numerical Schemes for Rough Parabolic Equations
Energy Technology Data Exchange (ETDEWEB)
Deya, Aurelien, E-mail: deya@iecn.u-nancy.fr [Universite de Nancy 1, Institut Elie Cartan Nancy (France)
2012-04-15
This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0,1) perturbed by a non-linear rough signal. It is the continuation of Deya (Electron. J. Probab. 16:1489-1518, 2011) and Deya et al. (Probab. Theory Relat. Fields, to appear), where the existence and uniqueness of a solution has been established. The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs. The results apply to a geometric 2-rough path, which covers the case of the multidimensional fractional Brownian motion with Hurst index H>1/3.
On the wave equation with semilinear porous acoustic boundary conditions
Graber, Philip Jameson; Said-Houari, Belkacem
2012-01-01
The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. © 2012 Elsevier Inc.
On the wave equation with semilinear porous acoustic boundary conditions
Graber, Philip Jameson
2012-05-01
The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. © 2012 Elsevier Inc.
On the Approximate Controllability of Some Semilinear Parabolic Boundary-Value Problems
International Nuclear Information System (INIS)
Diaz, J. I.; Henry, J.; Ramos, A. M.
1998-01-01
We prove the approximate controllability of several nonlinear parabolic boundary-value problems by means of two different methods: the first one can be called a Cancellation method and the second one uses the Kakutani fixed-point theorem
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.
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.
On the Schauder estimates of solutions to parabolic equations
International Nuclear Information System (INIS)
Han Qing
1998-01-01
This paper gives a priori estimates on asymptotic polynomials of solutions to parabolic differential equations at any points. This leads to a pointwise version of Schauder estimates. The result improves the classical Schauder estimates in a way that the estimates of solutions and their derivatives at one point depend on the coefficient and nonhomogeneous terms at this particular point
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.
Directory of Open Access Journals (Sweden)
Ling Zhang
2017-10-01
Full Text Available Abstract 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 1 2 $\\frac{1}{2}$ 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.
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.
GOSWAMI, DEEPJYOTI; PANI, AMIYA K.; YADAV, SANGITA
2014-01-01
AWe propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal L2-error estimate is derived for the semidiscrete approximation when the initial data is in L2. A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain. © 2014 Australian Mathematical Society.
Modeling mode interactions in boundary layer flows via the Parabolized Floquet Equations
Ran, Wei; Zare, Armin; Hack, M. J. Philipp; Jovanović, Mihailo R.
2017-01-01
In this paper, we develop a linear model to study interactions between different modes in slowly-growing boundary layer flows. Our method consists of two steps. First, we augment the Blasius boundary layer profile with a disturbance field resulting from the linear Parabolized Stability Equations (PSE) to obtain the modified base flow; and, second, we combine Floquet analysis with the linear PSE to capture the spatial evolution of flow fluctuations. This procedure yields the Parabolized Floque...
Nonlinear variational inequalities of semilinear parabolic type
Directory of Open Access Journals (Sweden)
Park Jong-Yeoul
2001-01-01
Full Text Available The existence of solutions for the nonlinear functional differential equation governed by the variational inequality is studied. The regularity and a variation of solutions of the equation are also given.
Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints
International Nuclear Information System (INIS)
Pham, H.
2002-01-01
This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove the existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature
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.
An accurate solution of parabolic equations by expansion in ultraspherical polynomials
International Nuclear Information System (INIS)
Doha, E.H.
1986-11-01
An ultraspherical expansion technique is applied to obtain numerically the solution of the third boundary value problem for linear parabolic partial differential equation in one-space variable. The differential equation with its boundary and initial conditions is reduced to a system of ordinary differential equations for the coefficients of the expansion. This system may be solved analytically or numerically in a step-by-step manner. The method in its present form may be considered as a generalization of that of Dew and Scraton. The extension of the method to the polar-type equations is also considered. (author). 12 refs, 1 tab
Ebert, Marcelo R
2018-01-01
This book provides an overview of different topics related to the theory of partial differential equations. Selected exercises are included at the end of each chapter to prepare readers for the “research project for beginners” proposed at the end of the book. It is a valuable resource for advanced graduates and undergraduate students who are interested in specializing in this area. The book is organized in five parts: In Part 1 the authors review the basics and the mathematical prerequisites, presenting two of the most fundamental results in the theory of partial differential equations: the Cauchy-Kovalevskaja theorem and Holmgren's uniqueness theorem in its classical and abstract form. It also introduces the method of characteristics in detail and applies this method to the study of Burger's equation. Part 2 focuses on qualitative properties of solutions to basic partial differential equations, explaining the usual properties of solutions to elliptic, parabolic and hyperbolic equations for the archetypes...
Scalable implicit methods for reaction-diffusion equations in two and three space dimensions
Energy Technology Data Exchange (ETDEWEB)
Veronese, S.V.; Othmer, H.G. [Univ. of Utah, Salt Lake City, UT (United States)
1996-12-31
This paper describes the implementation of a solver for systems of semi-linear parabolic partial differential equations in two and three space dimensions. The solver is based on a parallel implementation of a non-linear Alternating Direction Implicit (ADI) scheme which uses a Cartesian grid in space and an implicit time-stepping algorithm. Various reordering strategies for the linearized equations are used to reduce the stride and improve the overall effectiveness of the parallel implementation. We have successfully used this solver for large-scale reaction-diffusion problems in computational biology and medicine in which the desired solution is a traveling wave that may contain rapid transitions. A number of examples that illustrate the efficiency and accuracy of the method are given here; the theoretical analysis will be presented.
International Nuclear Information System (INIS)
Dehghan, Mehdi; Tatari, Mehdi
2008-01-01
In this research, the He's variational iteration technique is used for computing an unknown time-dependent parameter in an inverse quasilinear parabolic partial differential equation. Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and physics, as they appear in various engineering models. The He's variational iteration method is an analytical procedure for finding solutions of differential equations, is based on the use of Lagrange multipliers for identification of an optimal value of a parameter in a functional. To show the efficiency of the new approach, several test problems are presented for one-, two- and three-dimensional cases
Sound field computations in the Bay of Bengal using parabolic equation method
Digital Repository Service at National Institute of Oceanography (India)
Navelkar, G.S.; Somayajulu, Y.K.; Murty, C.S.
Effect of the cold core eddy in the Bay of Bengal on acoustic propagation was analysed by parabolic equation (PE) method. Source depth, frequency and propagation range considered respectively for the two numerical experiments are 150 m, 400 Hz, 650...
Implicational (Semilinear) Logics II: Additional Connectives and Characterizations of Semilinearity
Czech Academy of Sciences Publication Activity Database
Cintula, Petr; Noguera, Carles
2016-01-01
Roč. 55, č. 3 (2016), s. 353-372 ISSN 0933-5846 R&D Projects: GA ČR GA13-14654S EU Projects: European Commission(XE) 247584 - MATOMUVI Institutional support: RVO:67985807 ; RVO:67985556 Keywords : abstract algebraic logic * implicational logics * disjunctional logics * semilinear logics * non-classical logics * transfer theorems Subject RIV: BA - General Mathematics Impact factor: 0.394, year: 2016
On the curve of critical exponents for nonlinear elliptic problems in the case of a zero mass
Il'yasov, Ya. Sh.
2017-03-01
For semilinear elliptic equations -Δ u = λ| u| p-2 u-| u| q-2 u, boundary value problems in bounded and unbounded domains are considered. In the plane of exponents p × q, the so-called curves of critical exponents are defined that divide this plane into domains with qualitatively different properties of the boundary value problems and the corresponding parabolic equations. New solvability conditions for boundary value problems, conditions for the stability and instability of stationary solutions, and conditions for the existence of global solutions to parabolic equations are found.
New model reduction technique for a class of parabolic partial differential equations
Vajta, Miklos
1991-01-01
A model reduction (or lumping) technique for a class of parabolic-type partial differential equations is given, and its application is discussed. The frequency response of the temperature distribution in any multilayer solid is developed and given by a matrix expression. The distributed transfer
He, Zi; Chen, Ru-Shan
2016-03-01
An efficient three-dimensional time domain parabolic equation (TDPE) method is proposed to fast analyze the narrow-angle wideband EM scattering properties of electrically large targets. The finite difference (FD) of Crank-Nicolson (CN) scheme is used as the traditional tool to solve the time-domain parabolic equation. However, a huge computational resource is required when the meshes become dense. Therefore, the alternating direction implicit (ADI) scheme is introduced to discretize the time-domain parabolic equation. In this way, the reduced transient scattered fields can be calculated line by line in each transverse plane for any time step with unconditional stability. As a result, less computational resources are required for the proposed ADI-based TDPE method when compared with both the traditional CN-based TDPE method and the finite-different time-domain (FDTD) method. By employing the rotating TDPE method, the complete bistatic RCS can be obtained with encouraging accuracy for any observed angle. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method.
ON A PARABOLIC FREE BOUNDARY EQUATION MODELING PRICE FORMATION
MARKOWICH, P. A.
2009-10-01
We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results. © 2009 World Scientific Publishing Company.
ON A PARABOLIC FREE BOUNDARY EQUATION MODELING PRICE FORMATION
MARKOWICH, P. A.; MATEVOSYAN, N.; PIETSCHMANN, J.-F.; WOLFRAM, M.-T.
2009-01-01
We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results. © 2009 World Scientific Publishing Company.
Semilinear Schrödinger equations
Cazenave, Thierry
2003-01-01
The nonlinear Schrödinger equation has received a great deal of attention from mathematicians, in particular because of its applications to nonlinear optics. It is also a good model dispersive equation, since it is often technically simpler than other dispersive equations, such as the wave or Korteweg-de Vries equation. Particularly useful tools in studying the nonlinear Schrödinger equation are energy and Strichartz's estimates. This book presents various mathematical aspects of the nonlinear Schrödinger equation. It examines both problems of local nature (local existence of solutions, unique
International Nuclear Information System (INIS)
Hinestroza Gutierrez, D.
2006-08-01
In this work a new and promising algorithm based on the minimization of especial functional that depends on two regularization parameters is considered for the identification of the heat conduction coefficient in the parabolic equation. This algorithm uses the adjoint and sensibility equations. One of the regularization parameters is associated with the heat-coefficient (as in conventional Tikhonov algorithms) but the other is associated with the calculated solution. (author)
International Nuclear Information System (INIS)
Hinestroza Gutierrez, D.
2006-12-01
In this work a new and promising algorithm based in the minimization of especial functional that depends on two regularization parameters is considered for identification of the heat conduction coefficient in the parabolic equation. This algorithm uses the adjoint and sensibility equations. One of the regularization parameters is associated with the heat-coefficient (as in conventional Tikhonov algorithms) but the other is associated with the calculated solution. (author)
Inverse Problems for a Parabolic Integrodifferential Equation in a Convolutional Weak Form
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Kairi Kasemets
2013-01-01
Full Text Available We deduce formulas for the Fréchet derivatives of cost functionals of several inverse problems for a parabolic integrodifferential equation in a weak formulation. The method consists in the application of an integrated convolutional form of the weak problem and all computations are implemented in regular Sobolev spaces.
Directory of Open Access Journals (Sweden)
Fatima G. Khushtova
2016-03-01
Full Text Available In this paper Cauchy problem for a parabolic equation with Bessel operator and with Riemann–Liouville partial derivative is considered. The representation of the solution is obtained in terms of integral transform with Wright function in the kernel. It is shown that when this equation becomes the fractional diffusion equation, obtained solution becomes the solution of Cauchy problem for the corresponding equation. The uniqueness of the solution in the class of functions that satisfy the analogue of Tikhonov condition is proved.
Directory of Open Access Journals (Sweden)
M. G. Crandall
1999-07-01
Full Text Available We study existence of continuous weak (viscosity solutions of Dirichlet and Cauchy-Dirichlet problems for fully nonlinear uniformly elliptic and parabolic equations. Two types of results are obtained in contexts where uniqueness of solutions fails or is unknown. For equations with merely measurable coefficients we prove solvability of the problem, while in the continuous case we construct maximal and minimal solutions. Necessary barriers on external cones are also constructed.
Optimal Control Problems for Nonlinear Variational Evolution Inequalities
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Eun-Young Ju
2013-01-01
Full Text Available We deal with optimal control problems governed by semilinear parabolic type equations and in particular described by variational inequalities. We will also characterize the optimal controls by giving necessary conditions for optimality by proving the Gâteaux differentiability of solution mapping on control variables.
Implementation of compact finite-difference method to parabolized Navier-Stokes equations
International Nuclear Information System (INIS)
Esfahanian, V.; Hejranfar, K.; Darian, H.M.
2005-01-01
The numerical simulation of the Parabolized Navier-Stokes (PNS) equations for supersonic/hypersonic flow field is obtained by using the fourth-order compact finite-difference method. The PNS equations in the general curvilinear coordinates are solved by using the implicit finite-difference algorithm of Beam and Warming. A shock fitting procedure is utilized to obtain the accurate solution in the vicinity of the shock. The computations are performed for hypersonic axisymmetric flow over a blunt cone. The present results for the flow field along with those of the second-order method are presented and accuracy analysis is performed to insure the fourth-order accuracy of the method. (author)
Luchko, Yuri
2013-05-30
In this paper, we consider a reaction-diffusion problem with an unknown nonlinear source function that has to be determined from overposed data. The underlying model is in the form of a time-fractional reaction-diffusion equation and the work generalizes some known results for the inverse problems posed for PDEs of parabolic type. For the inverse problem under consideration, a uniqueness result is proved and a numerical algorithm with some theoretical qualification is presented in the one-dimensional case. The key both to the uniqueness result and to the numerical algorithm relies on the maximum principle which has recently been shown to hold for the fractional diffusion equation. In order to show the effectiveness of the proposed method, results of numerical simulations are presented. © 2013 IOP Publishing Ltd.
Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay
Czech Academy of Sciences Publication Activity Database
Chueshov, I.; Rezunenko, Oleksandr
2015-01-01
Roč. 14, č. 5 (2015), s. 1685-1704 ISSN 1534-0392 R&D Projects: GA ČR GAP103/12/2431 Institutional support: RVO:67985556 Keywords : Parabolic evolution equations * state-dependent delay * global attractor * finite-dimension * exponential attractor Subject RIV: BC - Control Systems Theory Impact factor: 0.926, year: 2015 http://library.utia.cas.cz/separaty/2015/AS/rezunenko-0444705.pdf
Radio wave propagation and parabolic equation modeling
Apaydin, Gokhan
2018-01-01
A thorough understanding of electromagnetic wave propagation is fundamental to the development of sophisticated communication and detection technologies. The powerful numerical methods described in this book represent a major step forward in our ability to accurately model electromagnetic wave propagation in order to establish and maintain reliable communication links, to detect targets in radar systems, and to maintain robust mobile phone and broadcasting networks. The first new book on guided wave propagation modeling and simulation to appear in nearly two decades, Radio Wave Propagation and Parabolic Equation Modeling addresses the fundamentals of electromagnetic wave propagation generally, with a specific focus on radio wave propagation through various media. The authors explore an array of new applications, and detail various v rtual electromagnetic tools for solving several frequent electromagnetic propagation problems. All of the methods described are presented within the context of real-world scenari...
Telescopic projective methods for parabolic differential equations
Gear, C W
2003-01-01
Projective methods were introduced in an earlier paper [C.W. Gear, I.G. Kevrekidis, Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum, NEC Research Institute Report 2001-029, available from http://www.neci.nj.nec.com/homepages/cwg/projective.pdf Abbreviated version to appear in SISC] as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum, one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active. In this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi-discretization of PDEs with parabolic components.
Telescopic projective methods for parabolic differential equations
International Nuclear Information System (INIS)
Gear, C.W.; Kevrekidis, Ioannis G.
2003-01-01
Projective methods were introduced in an earlier paper [C.W. Gear, I.G. Kevrekidis, Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum, NEC Research Institute Report 2001-029, available from http://www.neci.nj.nec.com/homepages/cwg/projective.pdf Abbreviated version to appear in SISC] as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum, one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active. In this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi-discretization of PDEs with parabolic components
Directory of Open Access Journals (Sweden)
Arnaldo Simal do Nascimento
1997-12-01
Full Text Available We use $Gamma$--convergence to prove existence of stable multiple--layer stationary solutions (stable patterns to the reaction--diffusion equation. $$ eqalign{ {partial v_varepsilon over partial t} =& varepsilon^2, hbox{div}, (k_1(xabla v_varepsilon + k_2(x(v_varepsilon -alpha(Beta-v_varepsilon (v_varepsilon -gamma_varepsilon(x,,hbox{ in }Omegaimes{Bbb R}^+ cr &v_varepsilon(x,0 = v_0 quad {partial v_varepsilon over partial widehat{n}} = 0,, quadhbox{ for } xin partialOmega,, t >0,.} $$ Given nested simple closed curves in ${Bbb R}^2$, we give sufficient conditions on their curvature so that the reaction--diffusion problem possesses a family of stable patterns. In particular, we extend to two-dimensional domains and to a spatially inhomogeneous source term, a previous result by Yanagida and Miyata.
A gradient estimate for solutions to parabolic equations with discontinuous coefficients
Fan, Jishan; Kim, Kyoungsun; Nagayasu, Sei; Nakamura, Gen
2011-01-01
Li-Vogelius and Li-Nirenberg gave a gradient estimate for solutions of strongly elliptic equations and systems of divergence forms with piecewise smooth coefficients, respectively. The discontinuities of the coefficients are assumed to be given by manifolds of codimension 1, which we called them emph{manifolds of discontinuities}. Their gradient estimate is independent of the distances between manifolds of discontinuities. In this paper, we gave a parabolic version of their results. T...
Approximation of entropy solutions to degenerate nonlinear parabolic equations
Abreu, Eduardo; Colombeau, Mathilde; Panov, Evgeny Yu
2017-12-01
We approximate the unique entropy solutions to general multidimensional degenerate parabolic equations with BV continuous flux and continuous nondecreasing diffusion function (including scalar conservation laws with BV continuous flux) in the periodic case. The approximation procedure reduces, by means of specific formulas, a system of PDEs to a family of systems of the same number of ODEs in the Banach space L^∞, whose solutions constitute a weak asymptotic solution of the original system of PDEs. We establish well posedness, monotonicity and L^1-stability. We prove that the sequence of approximate solutions is strongly L^1-precompact and that it converges to an entropy solution of the original equation in the sense of Carrillo. This result contributes to justify the use of this original method for the Cauchy problem to standard multidimensional systems of fluid dynamics for which a uniqueness result is lacking.
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.
A note on numerical solution of a parabolic-Schrödinger equation
Ozdemir, Yildirim; Alp, Mustafa
2016-08-01
In the present study, a nonlocal boundary value problem for a parabolic-Schrödinger equation is considered. The stability estimates for the solution of the given problem is established. The first and second order of difference schemes are presented for approximately solving a specific nonlocal boundary problem. The theoretical statements for the solution of these difference schemes are supported by the result of numerical examples.
Ashyralyyeva, Maral; Ashyraliyev, Maksat
2016-08-01
In the present paper, a second order of accuracy difference scheme for the approximate solution of a source identification problem for hyperbolic-parabolic equations is constructed. Theorem on stability estimates for the solution of this difference scheme and their first and second order difference derivatives is presented. In applications, this abstract result permits us to obtain the stability estimates for the solutions of difference schemes for approximate solutions of two source identification problems for hyperbolic-parabolic equations.
Recovering a coefficient in a parabolic equation using an iterative approach
Azhibekova, Aliya S.
2016-06-01
In this paper we are concerned with the problem of determining a coefficient in a parabolic equation using an iterative approach. We investigate an inverse coefficient problem in the difference form. To recover the coefficient, we minimize a residual functional between the observed and calculated values. This is done in a constructive way by fitting a finite-difference approximation to the inverse problem. We obtain some theoretical estimates for a direct and adjoint problem. Using these estimates we prove monotonicity of the objective functional and the convergence of iteration sequences.
Czech Academy of Sciences Publication Activity Database
Krisztin, T.; Rezunenko, Oleksandr
2016-01-01
Roč. 260, č. 5 (2016), s. 4454-4472 ISSN 0022-0396 R&D Projects: GA ČR GAP103/12/2431 Institutional support: RVO:67985556 Keywords : Parabolic partial differential equations * State dependent delay * Solution manifold Subject RIV: BC - Control Systems Theory Impact factor: 1.988, year: 2016 http://library.utia.cas.cz/separaty/2016/AS/rezunenko-0457879.pdf
Directory of Open Access Journals (Sweden)
Hatem Mejjaoli
2008-12-01
Full Text Available We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.
Application of the implicit MacCormack scheme to the parabolized Navier-Stokes equations
Lawrence, J. L.; Tannehill, J. C.; Chaussee, D. S.
1984-01-01
MacCormack's implicit finite-difference scheme was used to solve the two-dimensional parabolized Navier-Stokes (PNS) equations. This method for solving the PNS equations does not require the inversion of block tridiagonal systems of algebraic equations and permits the original explicit MacCormack scheme to be employed in those regions where implicit treatment is not needed. The advantages and disadvantages of the present adaptation are discussed in relation to those of the conventional Beam-Warming scheme for a flat plate boundary layer test case. Comparisons are made for accuracy, stability, computer time, computer storage, and ease of implementation. The present method was also applied to a second test case of hypersonic laminar flow over a 15% compression corner. The computed results compare favorably with experiment and a numerical solution of the complete Navier-Stokes equations.
The Initial and Neumann Boundary Value Problem for a Class Parabolic Monge-Ampère Equation
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Juan Wang
2013-01-01
Full Text Available We consider the existence, uniqueness, and asymptotic behavior of a classical solution to the initial and Neumann boundary value problem for a class nonlinear parabolic equation of Monge-Ampère type. We show that such solution exists for all times and is unique. It converges eventually to a solution that satisfies a Neumann type problem for nonlinear elliptic equation of Monge-Ampère type.
Mixed hyperbolic-second-order-parabolic formulations of general relativity
International Nuclear Information System (INIS)
Paschalidis, Vasileios
2008-01-01
Two new formulations of general relativity are introduced. The first one is a parabolization of the Arnowitt-Deser-Misner formulation and is derived by the addition of combinations of the constraints and their derivatives to the right-hand side of the Arnowitt-Deser-Misner evolution equations. The desirable property of this modification is that it turns the surface of constraints into a local attractor because the constraint propagation equations become second-order parabolic independently of the gauge conditions employed. This system may be classified as mixed hyperbolic--second-order parabolic. The second formulation is a parabolization of the Kidder-Scheel-Teukolsky formulation and is a manifestly mixed strongly hyperbolic--second-order-parabolic set of equations, bearing thus resemblance to the compressible Navier-Stokes equations. As a first test, a stability analysis of flat space is carried out and it is shown that the first modification exponentially damps and smoothes all constraint-violating modes. These systems provide a new basis for constructing schemes for long-term and stable numerical integration of the Einstein field equations.
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.
Analysis of nonlinear parabolic equations modeling plasma diffusion across a magnetic field
International Nuclear Information System (INIS)
Hyman, J.M.; Rosenau, P.
1984-01-01
We analyse the evolutionary behavior of the solution of a pair of coupled quasilinear parabolic equations modeling the diffusion of heat and mass of a magnetically confined plasma. The solutions's behavior, due to the nonlinear diffusion coefficients, exhibits many new phenomena. In short time, the solution converges into a highly organized symmetric pattern that is almost completely independent of initial data. The asymptotic dynamics then become very simple and take place in a finite dimensional space. These conclusions are backed by extensive numerical experimentation
Upwind algorithm for the parabolized Navier-Stokes equations
Lawrence, Scott L.; Tannehill, John C.; Chausee, Denny S.
1989-01-01
A new upwind algorithm based on Roe's scheme has been developed to solve the two-dimensional parabolized Navier-Stokes equations. This method does not require the addition of user-specified smoothing terms for the capture of discontinuities such as shock waves. Thus, the method is easy to use and can be applied without modification to a wide variety of supersonic flowfields. The advantages and disadvantages of this adaptation are discussed in relation to those of the conventional Beam-Warming (1978) scheme in terms of accuracy, stability, computer time and storage requirements, and programming effort. The new algorithm has been validated by applying it to three laminar test cases, including flat-plate boundary-layer flow, hypersonic flow past a 15-deg compression corner, and hypersonic flow into a converging inlet. The computed results compare well with experiment and show a dramatic improvement in the resolution of flowfield details when compared with results obtained using the conventional Beam-Warming algorithm.
First-order partial differential equations
Rhee, Hyun-Ku; Amundson, Neal R
2001-01-01
This first volume of a highly regarded two-volume text is fully usable on its own. After going over some of the preliminaries, the authors discuss mathematical models that yield first-order partial differential equations; motivations, classifications, and some methods of solution; linear and semilinear equations; chromatographic equations with finite rate expressions; homogeneous and nonhomogeneous quasilinear equations; formation and propagation of shocks; conservation equations, weak solutions, and shock layers; nonlinear equations; and variational problems. Exercises appear at the end of mo
A Note on the Asymptotic Behavior of Parabolic Monge-Ampère Equations on Riemannian Manifolds
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Qiang Ru
2013-01-01
Full Text Available We study the asymptotic behavior of the parabolic Monge-Ampère equation in , in , where is a compact complete Riemannian manifold, λ is a positive real parameter, and is a smooth function. We show a meaningful asymptotic result which is more general than those in Huisken, 1997.
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
Differential invariants of generic parabolic Monge–Ampère equations
International Nuclear Information System (INIS)
Ferraioli, D Catalano; Vinogradov, A M
2012-01-01
Some new results on the geometry of classical parabolic Monge–Ampère equations (PMAs) are presented. PMAs are either integrable, or non-integrable according to the integrability of its characteristic distribution. All integrable PMAs are locally equivalent to the equation u xx = 0. We study non-integrable PMAs by associating with each of them a one-dimensional distribution on the corresponding first-order jet manifold, called the directing distribution. According to some property of this distribution, non-integrable PMAs are subdivided into three classes, one generic and two special. Generic PMAs are completely characterized by their directing distributions, and we study canonical models of the latter, projective curve bundles (PCB). A PCB is a one-dimensional sub-bundle of the projectivized cotangent bundle of a four-dimensional manifold. Differential invariants of projective curves composing such a bundle are used to construct a series of contact differential invariants for corresponding PMAs. These give a solution of the equivalence problem for generic PMAs with respect to contact transformations. The introduced invariants measure the nonlinearity of PMAs in an exact manner. (paper)
Nonlinear scalar field equations. Pt. 1
International Nuclear Information System (INIS)
Berestycki, H.; Lions, P.L.
1983-01-01
This paper as well as a subsequent one is concerned with the existence of nontrivial solutions for some semi-linear elliptic equations in Rsup(N). Such problems are motivated in particular by the search for certain kinds of solitary waves (stationary states) in nonlinear equations of the Klein-Gordon or Schroedinger type. (orig./HSI)
Goswami, Deepjyoti; Pani, Amiya K.; Yadav, Sangita
2013-01-01
In the first part of this article, a new mixed method is proposed and analyzed for parabolic integro-differential equations (PIDE) with nonsmooth initial data. Compared to the standard mixed method for PIDE, the present method does not bank on a
Existence results for impulsive semilinear damped differential inclusions
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Mouffak Benchohra
2003-06-01
Full Text Available In this paper we investigate the existence of mild solutions for first and second order impulsive semilinear evolution inclusions in separable Banach spaces. By using suitable fixed point theorems, we study the case when the multivalued map has convex and nonconvex values.
Numerical approach to optimal portfolio in a power utility regime-switching model
Gyulov, Tihomir B.; Koleva, Miglena N.; Vulkov, Lubin G.
2017-12-01
We consider a system of weakly coupled degenerate semi-linear parabolic equations of optimal portfolio in a regime-switching with power utility function, derived by A.R. Valdez and T. Vargiolu [14]. First, we discuss some basic properties of the solution of this system. Then, we develop and analyze implicit-explicit, flux limited finite difference schemes for the differential problem. Numerical experiments are discussed.
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
Self-accelerating parabolic cylinder waves in 1-D
Energy Technology Data Exchange (ETDEWEB)
Yuce, C., E-mail: cyuce@anadolu.edu.tr
2016-11-25
Highlights: • We find a new class of self-accelerating waves. • We show that parabolic cylinder waves self-accelerates in a parabolic potential. • We discuss that truncated parabolic cylinder waves propagates large distance without almost being non-diffracted in free space. - Abstract: We introduce a new self-accelerating wave packet solution of the Schrodinger equation in one dimension. We obtain an exact analytical parabolic cylinder wave for the inverted harmonic potential. We show that truncated parabolic cylinder waves exhibits their accelerating feature.
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)
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.
An upwind algorithm for the parabolized Navier-Stokes equations
Lawrence, S. L.; Tannehill, J. C.; Chaussee, D. S.
1986-01-01
A new upwind algorithm based on Roe's scheme has been developed to solve the two-dimensional parabolized Navier-Stokes (PNS) equations. This method does not require the addition of user specified smoothing terms for the capture of discontinuities such as shock waves. Thus, the method is easy to use and can be applied without modification to a wide variety of supersonic flowfields. The advantages and disadvantages of this adaptation are discussed in relation to those of the conventional Beam-Warming scheme in terms of accuracy, stability, computer time and storage, and programming effort. The new algorithm has been validated by applying it to three laminar test cases including flat plate boundary-layer flow, hypersonic flow past a 15 deg compression corner, and hypersonic flow into a converging inlet. The computed results compare well with experiment and show a dramatic improvement in the resolution of flowfield details when compared with the results obtained using the conventional Beam-Warming algorithm.
Efficient solution of parabolic equations by Krylov approximation methods
Gallopoulos, E.; Saad, Y.
1990-01-01
Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms.
On the invariant measure for the nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Zhidkov, P.R.
1991-01-01
The invariant measure for the nonlinear Schroedinger equation is constructed. In fact, it is assumed that the nonlinearity in the equation is semilinear. The main aim of the paper is the explanation of the Fermi - Past - Ulam phenomenon. Poincare theorem gives the answer to this question. 17 refs
Nonlinear damped Schrodinger equation in two space dimensions
Directory of Open Access Journals (Sweden)
Tarek Saanouni
2015-04-01
Full Text Available In this article, we study the initial value problem for a semi-linear damped Schrodinger equation with exponential growth nonlinearity in two space dimensions. We show global well-posedness and exponential decay.
Coercive properties of elliptic-parabolic operator
International Nuclear Information System (INIS)
Duong Min Duc.
1987-06-01
Using a generalized Poincare inequality, we study the coercive properties of a class of elliptic-parabolic partial differential equations, which contains many degenerate elliptic equations considered by the other authors. (author). 16 refs
International Nuclear Information System (INIS)
Levenshtam, V B
2006-01-01
We justify the averaging method for abstract parabolic equations with stationary principal part that contain non-linearities (subordinate to the principal part) some of whose terms are rapidly oscillating in time with zero mean and are proportional to the square root of the frequency of oscillation. Our interest in the exponent 1/2 is motivated by the fact that terms proportional to lower powers of the frequency have no influence on the average. For linear equations of the same type, we justify an algorithm for the study of the stability of solutions in the case when the stationary averaged problem has eigenvalues on the imaginary axis (the critical case)
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
Identifying Initial Condition in Degenerate Parabolic Equation with Singular Potential
Directory of Open Access Journals (Sweden)
K. Atifi
2017-01-01
Full Text Available A hybrid algorithm and regularization method are proposed, for the first time, to solve the one-dimensional degenerate inverse heat conduction problem to estimate the initial temperature distribution from point measurements. The evolution of the heat is given by a degenerate parabolic equation with singular potential. This problem can be formulated in a least-squares framework, an iterative procedure which minimizes the difference between the given measurements and the value at sensor locations of a reconstructed field. The mathematical model leads to a nonconvex minimization problem. To solve it, we prove the existence of at least one solution of problem and we propose two approaches: the first is based on a Tikhonov regularization, while the second approach is based on a hybrid genetic algorithm (married genetic with descent method type gradient. Some numerical experiments are given.
Manning, Robert M.
2012-01-01
The method of moments is used to define and derive expressions for laser beam deflection and beam radius broadening for high-energy propagation through the Earth s atmosphere. These expressions are augmented with the integral invariants of the corresponding nonlinear parabolic equation that describes the electric field of high-energy laser beam to propagation to yield universal equations for the aforementioned quantities; the beam deflection is a linear function of the propagation distance whereas the beam broadening is a quadratic function of distance. The coefficients of these expressions are then derived from a thin screen approximation solution of the nonlinear parabolic equation to give corresponding analytical expressions for a target located outside the Earth s atmospheric layer. These equations, which are graphically presented for a host of propagation scenarios, as well as the thin screen model, are easily amenable to the phase expansions of the wave front for the specification and design of adaptive optics algorithms to correct for the inherent phase aberrations. This work finds application in, for example, the analysis of beamed energy propulsion for space-based vehicles.
Blow-Up Analysis for a Quasilinear Degenerate Parabolic Equation with Strongly Nonlinear Source
Directory of Open Access Journals (Sweden)
Pan Zheng
2012-01-01
Full Text Available We investigate the blow-up properties of the positive solution of the Cauchy problem for a quasilinear degenerate parabolic equation with strongly nonlinear source ut=div(|∇um|p−2∇ul+uq, (x,t∈RN×(0,T, where N≥1, p>2 , and m, l, q>1, and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and nonexistence of global solutions of the Cauchy problem. Moreover, under some suitable conditions we prove single-point blow-up for a large class of radial decreasing solutions.
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).
Well-posedness of nonlocal parabolic differential problems with dependent operators.
Ashyralyev, Allaberen; Hanalyev, Asker
2014-01-01
The nonlocal boundary value problem for the parabolic differential equation v'(t) + A(t)v(t) = f(t) (0 ≤ t ≤ T), v(0) = v(λ) + φ, 0 exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.
Chernoff's distribution and parabolic partial differential equations
P. Groeneboom; S.P. Lalley; N.M. Temme (Nico)
2013-01-01
textabstractWe give an alternative route to the derivation of the distribution of the maximum and the location of the maximum of one-sided and two-sided Brownian motion with a negative parabolic drift, using the Feynman-Kac formula with stopping times. The derivation also uses an interesting
Gradient-type methods in inverse parabolic problems
International Nuclear Information System (INIS)
Kabanikhin, Sergey; Penenko, Aleksey
2008-01-01
This article is devoted to gradient-based methods for inverse parabolic problems. In the first part, we present a priori convergence theorems based on the conditional stability estimates for linear inverse problems. These theorems are applied to backwards parabolic problem and sideways parabolic problem. The convergence conditions obtained coincide with sourcewise representability in the self-adjoint backwards parabolic case but they differ in the sideways case. In the second part, a variational approach is formulated for a coefficient identification problem. Using adjoint equations, a formal gradient of an objective functional is constructed. A numerical test illustrates the performance of conjugate gradient algorithm with the formal gradient.
Manning, Robert M.
2004-01-01
The extended wide-angle parabolic wave equation applied to electromagnetic wave propagation in random media is considered. A general operator equation is derived which gives the statistical moments of an electric field of a propagating wave. This expression is used to obtain the first and second order moments of the wave field and solutions are found that transcend those which incorporate the full paraxial approximation at the outset. Although these equations can be applied to any propagation scenario that satisfies the conditions of application of the extended parabolic wave equation, the example of propagation through atmospheric turbulence is used. It is shown that in the case of atmospheric wave propagation and under the Markov approximation (i.e., the delta-correlation of the fluctuations in the direction of propagation), the usual parabolic equation in the paraxial approximation is accurate even at millimeter wavelengths. The comprehensive operator solution also allows one to obtain expressions for the longitudinal (generalized) second order moment. This is also considered and the solution for the atmospheric case is obtained and discussed. The methodology developed here can be applied to any qualifying situation involving random propagation through turbid or plasma environments that can be represented by a spectral density of permittivity fluctuations.
A three-point backward finite-difference method has been derived for a system of mixed hyperbolic¯¯parabolic (convection¯¯diffusion) partial differential equations (mixed PDEs). The method resorts to the three-point backward differenci...
Directory of Open Access Journals (Sweden)
Sukjung Hwang
2015-11-01
Full Text Available Here we generalize quasilinear parabolic p-Laplacian type equations to obtain the prototype equation $$ u_t - \\hbox{div} \\Big(\\frac{g(|Du|}{|Du|} Du\\Big = 0, $$ where g is a nonnegative, increasing, and continuous function trapped in between two power functions $|Du|^{g_0 -1}$ and $|Du|^{g_1 -1}$ with $1
A parabolic model for dimple potentials
International Nuclear Information System (INIS)
Aydin, Melike Cibik; Uncu, Haydar; Deniz, Coskun
2013-01-01
We study the truncated parabolic function and demonstrate that it is a representation of the Dirac δ function. We also show that the truncated parabolic function, used as a potential in the Schrödinger equation, has the same bound state spectrum, tunneling and reflection amplitudes as the Dirac δ potential, as the width of the parabola approximates to zero. Dirac δ potential is used to model dimple potentials which are utilized to increase the phase-space density of a Bose–Einstein condensate in a harmonic trap. We show that a harmonic trap with a δ function at the origin is a limiting case of the harmonic trap with a symmetric truncated parabolic potential around the origin. Hence, the truncated parabolic is a better candidate for modeling the dimple potentials. (paper)
Implicational (semilinear) logics III: completeness properties
Czech Academy of Sciences Publication Activity Database
Cintula, Petr; Noguera, Carles
2018-01-01
Roč. 57, 3-4 (2018), s. 391-420 ISSN 0933-5846 R&D Projects: GA ČR GA13-14654S EU Projects: European Commission(XE) 689176 - SYSMICS Institutional support: RVO:67985807 ; RVO:67985556 Keywords : abstract algebraic logic * protoalgebraic logics * implicational logics * disjunctional logics * semilinear logics * non-classical logics * completeness theorems * rational completeness Subject RIV: BA - General Mathematics; BA - General Mathematics (UTIA-B) OBOR OECD: Computer science s, information science , bioinformathics (hardware development to be 2.2, social aspect to be 5.8) Impact factor: 0.394, year: 2016
Implicational (semilinear) logics III: completeness properties
Czech Academy of Sciences Publication Activity Database
Cintula, Petr; Noguera, Carles
2018-01-01
Roč. 57, 3-4 (2018), s. 391-420 ISSN 0933-5846 R&D Projects: GA ČR GA13-14654S EU Projects: European Commission(XE) 689176 - SYSMICS Institutional support: RVO:67985807 ; RVO:67985556 Keywords : abstract algebraic logic * protoalgebraic logics * implicational logics * disjunctional logics * semilinear logics * non-classical logics * completeness theorems * rational completeness Subject RIV: BA - General Mathematics; BA - General Mathematics (UTIA-B) OBOR OECD: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8) Impact factor: 0.394, year: 2016
Bounded and Periodic Solutions of Semilinear Impulsive Periodic System on Banach Spaces
Directory of Open Access Journals (Sweden)
Wei W
2008-01-01
Full Text Available Abstract A class of semilinear impulsive periodic system on Banach spaces is considered. First, we introduce the -periodic PC-mild solution of semilinear impulsive periodic system. By virtue of Gronwall lemma with impulse, the estimate on the PC-mild solutions is derived. The continuity and compactness of the new constructed Poincaré operator determined by impulsive evolution operator corresponding to homogenous linear impulsive periodic system are shown. This allows us to apply Horn's fixed-point theorem to prove the existence of -periodic PC-mild solutions when PC-mild solutions are ultimate bounded. This extends the study on periodic solutions of periodic system without impulse to periodic system with impulse on general Banach spaces. At last, an example is given for demonstration.
The matrix nonlinear Schrodinger equation in dimension 2
DEFF Research Database (Denmark)
Zuhan, L; Pedersen, Michael
2001-01-01
In this paper we study the existence of global solutions to the Cauchy problem for the matrix nonlinear Schrodinger equation (MNLS) in 2 space dimensions. A sharp condition for the global existence is obtained for this equation. This condition is in terms of an exact stationary solution...... of a semilinear elliptic equation. In the scalar case, the MNLS reduces to the well-known cubic nonlinear Schrodinger equation for which existence of solutions has been studied by many authors. (C) 2001 Academic Press....
Hasani, Mojtaba H; Gharibzadeh, Shahriar; Farjami, Yaghoub; Tavakkoli, Jahan
2013-09-01
Various numerical algorithms have been developed to solve the Khokhlov-Kuznetsov-Zabolotskaya (KZK) parabolic nonlinear wave equation. In this work, a generalized time-domain numerical algorithm is proposed to solve the diffraction term of the KZK equation. This algorithm solves the transverse Laplacian operator of the KZK equation in three-dimensional (3D) Cartesian coordinates using a finite-difference method based on the five-point implicit backward finite difference and the five-point Crank-Nicolson finite difference discretization techniques. This leads to a more uniform discretization of the Laplacian operator which in turn results in fewer calculation gridding nodes without compromising accuracy in the diffraction term. In addition, a new empirical algorithm based on the LU decomposition technique is proposed to solve the system of linear equations obtained from this discretization. The proposed empirical algorithm improves the calculation speed and memory usage, while the order of computational complexity remains linear in calculation of the diffraction term in the KZK equation. For evaluating the accuracy of the proposed algorithm, two previously published algorithms are used as comparison references: the conventional 2D Texas code and its generalization for 3D geometries. The results show that the accuracy/efficiency performance of the proposed algorithm is comparable with the established time-domain methods.
three solutions for a semilinear elliptic boundary value problem
Indian Academy of Sciences (India)
69
Keywords: The Laplacian operator, elliptic problem, Nehari man- ifold, three critical points, weak solution. 1. Introduction. Let Ω be a smooth bounded domain in RN , N ≥ 3 . In this work, we show the existence of at least three solutions for the semilinear elliptic boundary- value problem: (Pλ).. −∆u = f(x)|u(x)|p−2u(x) + ...
A Pseudo-Temporal Multi-Grid Relaxation Scheme for Solving the Parabolized Navier-Stokes Equations
White, J. A.; Morrison, J. H.
1999-01-01
A multi-grid, flux-difference-split, finite-volume code, VULCAN, is presented for solving the elliptic and parabolized form of the equations governing three-dimensional, turbulent, calorically perfect and non-equilibrium chemically reacting flows. The space marching algorithms developed to improve convergence rate and or reduce computational cost are emphasized. The algorithms presented are extensions to the class of implicit pseudo-time iterative, upwind space-marching schemes. A full approximate storage, full multi-grid scheme is also described which is used to accelerate the convergence of a Gauss-Seidel relaxation method. The multi-grid algorithm is shown to significantly improve convergence on high aspect ratio grids.
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.
The blow-up problem for a semilinear parabolic equation with a potential
Cortazar, Carmen; Elgueta, Manuel; Rossi, Julio D.
2007-11-01
Let [Omega] be a bounded smooth domain in . We consider the problem ut=[Delta]u+V(x)up in [Omega]×[0,T), with Dirichlet boundary conditions u=0 on [not partial differential][Omega]×[0,T) and initial datum u(x,0)=M[phi](x) where M[greater-or-equal, slanted]0, [phi] is positive and compatible with the boundary condition. We give estimates for the blow-up time of solutions for large values of M. As a consequence of these estimates we find that, for M large, the blow-up set concentrates near the points where [phi]p-1V attains its maximum.
The blow-up problem for a semilinear parabolic equation with a potential
Cortazar, C.; Elgueta, M.; Rossi, J. D.
2006-01-01
Let $\\Omega$ be a bounded smooth domain in $\\RR^N$. We consider the problem $u_t= \\Delta u + V(x) u^p$ in $\\Omega \\times [0,T)$, with Dirichlet boundary conditions $u=0$ on $\\partial \\Omega \\times [0,T)$ and initial datum $u(x,0)= M \\phi (x)$ where $M \\geq 0$, $\\phi$ is positive and compatible with the boundary condition. We give estimates for the blow up time of solutions for large values of $M$. As a consequence of these estimates we find that, for $M$ large, the blow up set concentrates ne...
Flux form Semi-Lagrangian methods for parabolic problems
Directory of Open Access Journals (Sweden)
Bonaventura Luca
2016-09-01
Full Text Available A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and stability analysis is proposed. Numerical examples validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection diffusion and nonlinear parabolic problems.
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.
Recovering the source and initial value simultaneously in a parabolic equation
International Nuclear Information System (INIS)
Zheng, Guang-Hui; Wei, Ting
2014-01-01
In this paper, we consider an inverse problem to simultaneously reconstruct the source term and initial data associated with a parabolic equation based on the additional temperature data at a terminal time t = T and the temperature data on an accessible part of a boundary. The conditional stability and uniqueness of the inverse problem are established. We apply a variational regularization method to recover the source and initial value. The existence, uniqueness and stability of the minimizer of the corresponding variational problem are obtained. Taking the minimizer as a regularized solution for the inverse problem, under an a priori and an a posteriori parameter choice rule, the convergence rates of the regularized solution under a source condition are also given. Furthermore, the source condition is characterized by an optimal control approach. Finally, we use a conjugate gradient method and a stopping criterion given by Morozov's discrepancy principle to solve the variational problem. Numerical experiments are provided to demonstrate the feasibility of the method. (papers)
Iterative Methods for Solving Nonlinear Parabolic Problem in Pension Saving Management
Koleva, M. N.
2011-11-01
In this work we consider a nonlinear parabolic equation, obtained from Riccati like transformation of the Hamilton-Jacobi-Bellman equation, arising in pension saving management. We discuss two numerical iterative methods for solving the model problem—fully implicit Picard method and mixed Picard-Newton method, which preserves the parabolic characteristics of the differential problem. Numerical experiments for comparison the accuracy and effectiveness of the algorithms are discussed. Finally, observations are given.
Discrete Weighted Pseudo Asymptotic Periodicity of Second Order Difference Equations
Directory of Open Access Journals (Sweden)
Zhinan Xia
2014-01-01
Full Text Available We define the concept of discrete weighted pseudo-S-asymptotically periodic function and prove some basic results including composition theorem. We investigate the existence, and uniqueness of discrete weighted pseudo-S-asymptotically periodic solution to nonautonomous semilinear difference equations. Furthermore, an application to scalar second order difference equations is given. The working tools are based on the exponential dichotomy theory and fixed point theorem.
Semilinear inclusions with nonlocal conditions without compactness in non-reflexive spaces
Czech Academy of Sciences Publication Activity Database
Benedetti, I.; Väth, Martin
2016-01-01
Roč. 48, č. 2 (2016), s. 613-636 ISSN 1230-3429 Institutional support: RVO:67985840 Keywords : weak measure of noncompactness * weakly condensing map * semilinear differential inclusion Subject RIV: BA - General Mathematics Impact factor: 0.667, year: 2016 http://www.apcz.pl/czasopisma/index.php/TMNA/article/view/TMNA.2016.061
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.
Studies with Parabolic Parabolic Linear Parabolic (PPLP) momentum function in the LHC
Solfaroli Camillocci, Matteo; Timko, Helga; Wenninger, Jorg; CERN. Geneva. ATS Department
2018-01-01
Measurements performed with a Parabolic Parabolic Linear Parabolic (PPLP) momentum function in the LHC. Three attempts have been performed with a pilot bunch and one with nominal bunch (1.1x1011 p/bunch).
Error Analysis of a Finite Element Method for the Space-Fractional Parabolic Equation
Jin, Bangti; Lazarov, Raytcho; Pasciak, Joseph; Zhou, Zhi
2014-01-01
© 2014 Society for Industrial and Applied Mathematics We consider an initial boundary value problem for a one-dimensional fractional-order parabolic equation with a space fractional derivative of Riemann-Liouville type and order α ∈ (1, 2). We study a spatial semidiscrete scheme using the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and the Crank-Nicolson method. Error estimates in the L2(D)- and Hα/2 (D)-norm are derived for the semidiscrete scheme and in the L2(D)-norm for the fully discrete schemes. These estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data. Extensive numerical results are presented to illustrate the theoretical results.
An hp-local Discontinuous Galerkin Method for Parabolic Integro-Differential Equations
Pani, Amiya K.
2010-06-06
In this article, a priori error bounds are derived for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that error estimates in L 2-norm of the gradient as well as of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps us to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains. © 2010 Springer Science+Business Media, LLC.
An hp-local Discontinuous Galerkin Method for Parabolic Integro-Differential Equations
Pani, Amiya K.; Yadav, Sangita
2010-01-01
In this article, a priori error bounds are derived for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that error estimates in L 2-norm of the gradient as well as of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps us to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains. © 2010 Springer Science+Business Media, LLC.
Optimal linear-quadratic control of coupled parabolic-hyperbolic PDEs
Aksikas, I.; Moghadam, A. Alizadeh; Forbes, J. F.
2017-10-01
This paper focuses on the optimal control design for a system of coupled parabolic-hypebolic partial differential equations by using the infinite-dimensional state-space description and the corresponding operator Riccati equation. Some dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the linear-quadratic (LQ)-optimal control problem. A state LQ-feedback operator is computed by solving the operator Riccati equation, which is converted into a set of algebraic and differential Riccati equations, thanks to the eigenvalues and the eigenvectors of the parabolic operator. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ-optimal controller designed in the early portion of the paper is implemented for the original nonlinear model. Numerical simulations are performed to show the controller performances.
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.
Numerical performance of the parabolized ADM (PADM) formulation of General Relativity
Paschalidis, Vasileios; Hansen, Jakob; Khokhlov, Alexei
2007-01-01
In a recent paper the first coauthor presented a new parabolic extension (PADM) of the standard 3+1 Arnowitt, Deser, Misner formulation of the equations of general relativity. By parabolizing first-order ADM in a certain way, the PADM formulation turns it into a mixed hyperbolic - second-order parabolic, well-posed system. The surface of constraints of PADM becomes a local attractor for all solutions and all possible well-posed gauge conditions. This paper describes a numerical implementation...
International Nuclear Information System (INIS)
Doering, C.R.
1985-01-01
Applications of nonlinear parabolic stochastic differential equations with additive colored noise in equilibrium and nonequilibrium statistical mechanics and quantum field theory are developed in detail, providing a new unified mathematical approach to many problems. The existence and uniqueness of solutions to these equations is established, and some of the properties of the solutions are investigated. In particular, asymptotic expansions for the correlation functions of the solutions are introduced and compared to rigorous nonperturbative bounds on the moments. It is found that the perturbative analysis is in qualitative disagreement with the exact result in models corresponding to cut-off self-interacting nonperturbatively renormalizable scalar quantum field theories. For these theories the nonlinearities cannot be considered as perturbations of the linearized theory
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.
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.
Directory of Open Access Journals (Sweden)
Meili Li
2015-01-01
Full Text Available The approximate controllability of semilinear neutral stochastic integrodifferential inclusions with infinite delay in an abstract space is studied. Sufficient conditions are established for the approximate controllability. The results are obtained by using the theory of analytic resolvent operator, the fractional power theory, and the theorem of nonlinear alternative for Kakutani maps. Finally, an example is provided to illustrate the theory.
Viscosity solutions of fully nonlinear functional parabolic PDE
Directory of Open Access Journals (Sweden)
Liu Wei-an
2005-01-01
Full Text Available By the technique of coupled solutions, the notion of viscosity solutions is extended to fully nonlinear retarded parabolic equations. Such equations involve many models arising from optimal control theory, economy and finance, biology, and so forth. The comparison principle is shown. Then the existence and uniqueness are established by the fixed point theory.
International Nuclear Information System (INIS)
Beauchard, K; Cannarsa, P; Yamamoto, M
2014-01-01
The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates, seems hard to apply to the case of Grushin-type operators of interest to this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse source problems for such operators, with locally distributed measurements in an arbitrary space dimension. For this purpose, we follow a mixed strategy which combines the approach due to Lebeau and Robbiano, relying on Fourier decomposition and Carleman inequalities for heat equations with non-smooth coefficients (solved by the Fourier modes). As a corollary, we obtain a direct proof of the observability of multidimensional Grushin-type parabolic equations, with locally distributed observations—which is equivalent to null controllability with locally distributed controls. (paper)
Monge-Ampere equations and tensorial functors
International Nuclear Information System (INIS)
Tunitsky, Dmitry V
2009-01-01
We consider differential-geometric structures associated with Monge-Ampere equations on manifolds and use them to study the contact linearization of such equations. We also consider the category of Monge-Ampere equations (the morphisms are contact diffeomorphisms) and a number of subcategories. We are chiefly interested in subcategories of Monge-Ampere equations whose objects are locally contact equivalent to equations linear in the second derivatives (semilinear equations), linear in derivatives, almost linear, linear in the second derivatives and independent of the first derivatives, linear, linear and independent of the first derivatives, equations with constant coefficients or evolution equations. We construct a number of functors from the category of Monge-Ampere equations and from some of its subcategories to the category of tensorial objects (that is, multi-valued sections of tensor bundles). In particular, we construct a pseudo-Riemannian metric for every generic Monge-Ampere equation. These functors enable us to establish effectively verifiable criteria for a Monge-Ampere equation to belong to the subcategories listed above.
Numerical performance of the parabolized ADM formulation of general relativity
International Nuclear Information System (INIS)
Paschalidis, Vasileios; Hansen, Jakob; Khokhlov, Alexei
2008-01-01
In a recent paper [Vasileios Paschalidis, Phys. Rev. D 78, 024002 (2008).], the first coauthor presented a new parabolic extension (PADM) of the standard 3+1 Arnowitt, Deser, Misner (ADM) formulation of the equations of general relativity. By parabolizing first-order ADM in a certain way, the PADM formulation turns it into a well-posed system which resembles the structure of mixed hyperbolic-second-order parabolic partial differential equations. The surface of constraints of PADM becomes a local attractor for all solutions and all possible well-posed gauge conditions. This paper describes a numerical implementation of PADM and studies its accuracy and stability in a series of standard numerical tests. Numerical properties of PADM are compared with those of standard ADM and its hyperbolic Kidder, Scheel, Teukolsky (KST) extension. The PADM scheme is numerically stable, convergent, and second-order accurate. The new formulation has better control of the constraint-violating modes than ADM and KST.
Beshtokov, M. Kh.
2016-10-01
A nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients is considered. For solving this problem, a priori estimates in the differential and difference forms are obtained. The a priori estimates imply the uniqueness and stability of the solution on a layer with respect to the initial data and the right-hand side and the convergence of the solution of the difference problem to the solution of the differential problem.
Strongly nonlinear parabolic variational inequalities.
Browder, F E; Brézis, H
1980-02-01
An existence and uniqueness result is established for a general class of variational inequalities for parabolic partial differential equations of the form partial differentialu/ partial differentialt + A(u) + g(u) = f with g nondecreasing but satisfying no growth condition. The proof is based upon a type of compactness result for solutions of variational inequalities that should find a variety of other applications.
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Chengdong Yang
2015-01-01
Full Text Available This paper addresses the exponential synchronization problem of a class of master-slave distributed parameter systems (DPSs with spatially variable coefficients and spatiotemporally variable nonlinear perturbation, modeled by a couple of semilinear parabolic partial differential equations (PDEs. With a locally Lipschitz constraint, the perturbation is a continuous function of time, space, and system state. Firstly, a sufficient condition for the robust exponential synchronization of the unforced semilinear master-slave PDE systems is investigated for all admissible nonlinear perturbations. Secondly, a robust distributed proportional-spatial derivative (P-sD state feedback controller is desired such that the closed-loop master-slave PDE systems achieve exponential synchronization. Using Lyapunov’s direct method and the technique of integration by parts, the main results of this paper are presented in terms of spatial differential linear matrix inequalities (SDLMIs. Finally, two numerical examples are provided to show the effectiveness of the proposed methods applied to the robust exponential synchronization problem of master-slave PDE systems with nonlinear perturbation.
An Explicit Upwind Algorithm for Solving the Parabolized Navier-Stokes Equations
Korte, John J.
1991-01-01
An explicit, upwind algorithm was developed for the direct (noniterative) integration of the 3-D Parabolized Navier-Stokes (PNS) equations in a generalized coordinate system. The new algorithm uses upwind approximations of the numerical fluxes for the pressure and convection terms obtained by combining flux difference splittings (FDS) formed from the solution of an approximate Riemann (RP). The approximate RP is solved using an extension of the method developed by Roe for steady supersonic flow of an ideal gas. Roe's method is extended for use with the 3-D PNS equations expressed in generalized coordinates and to include Vigneron's technique of splitting the streamwise pressure gradient. The difficulty associated with applying Roe's scheme in the subsonic region is overcome. The second-order upwind differencing of the flux derivatives are obtained by adding FDS to either an original forward or backward differencing of the flux derivative. This approach is used to modify an explicit MacCormack differencing scheme into an upwind differencing scheme. The second order upwind flux approximations, applied with flux limiters, provide a method for numerically capturing shocks without the need for additional artificial damping terms which require adjustment by the user. In addition, a cubic equation is derived for determining Vegneron's pressure splitting coefficient using the updated streamwise flux vector. Decoding the streamwise flux vector with the updated value of Vigneron's pressure splitting improves the stability of the scheme. The new algorithm is applied to 2-D and 3-D supersonic and hypersonic laminar flow test cases. Results are presented for the experimental studies of Holden and of Tracy. In addition, a flow field solution is presented for a generic hypersonic aircraft at a Mach number of 24.5 and angle of attack of 1 degree. The computed results compare well to both experimental data and numerical results from other algorithms. Computational times required
Distribution-valued weak solutions to a parabolic problem arising in financial mathematics
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Michael Eydenberg
2009-07-01
Full Text Available We study distribution-valued solutions to a parabolic problem that arises from a model of the Black-Scholes equation in option pricing. We give a minor generalization of known existence and uniqueness results for solutions in bounded domains $Omega subset mathbb{R}^{n+1}$ to give existence of solutions for certain classes of distributions $fin mathcal{D}'(Omega$. We also study growth conditions for smooth solutions of certain parabolic equations on $mathbb{R}^nimes (0,T$ that have initial values in the space of distributions.
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Wenwan Ding
2016-01-01
Full Text Available An improved fractal sea surface model, which can describe the capillary waves very well, is introduced to simulate the one-dimension rough sea surface. In this model, the propagation of electromagnetic waves (EWs is computed by the parabolic equation (PE method using the finite-difference (FD algorithm. The numerical simulation results of the introduced model are compared with those of the Miller-Brown model and the Elfouhaily spectrum inversion model. It has been shown that the effects of the fine structure of the sea surface on the EWs propagation in the introduced model are more apparent than those in the other two models.
Monte Carlo method for solving a parabolic problem
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Tian Yi
2016-01-01
Full Text Available In this paper, we present a numerical method based on random sampling for a parabolic problem. This method combines use of the Crank-Nicolson method and Monte Carlo method. In the numerical algorithm, we first discretize governing equations by Crank-Nicolson method, and obtain a large sparse system of linear algebraic equations, then use Monte Carlo method to solve the linear algebraic equations. To illustrate the usefulness of this technique, we apply it to some test problems.
Mutational analysis a joint framework for Cauchy problems in and beyond vector spaces
Lorenz, Thomas
2010-01-01
Ordinary differential equations play a central role in science and have been extended to evolution equations in Banach spaces. For many applications, however, it is difficult to specify a suitable normed vector space. Shapes without a priori restrictions, for example, do not have an obvious linear structure. This book generalizes ordinary differential equations beyond the borders of vector spaces with a focus on the well-posed Cauchy problem in finite time intervals. Here are some of the examples: - Feedback evolutions of compact subsets of the Euclidean space - Birth-and-growth processes of random sets (not necessarily convex) - Semilinear evolution equations - Nonlocal parabolic differential equations - Nonlinear transport equations for Radon measures - A structured population model - Stochastic differential equations with nonlocal sample dependence and how they can be coupled in systems immediately - due to the joint framework of Mutational Analysis. Finally, the book offers new tools for modelling.
Beshtokov, M. Kh.
2017-12-01
Boundary value problems for loaded third-order pseudo-parabolic equations with variable coefficients are considered. A priori estimates for the solutions of the problems in the differential and difference formulations are obtained. These a priori estimates imply the uniqueness and stability of the solution with respect to the initial data and the right-hand side on a layer, as well as the convergence of the solution of each difference problem to the solution of the corresponding differential problem.
Semilinear damped wave equation in locally uniform spaces
Czech Academy of Sciences Publication Activity Database
Michálek, Martin; Pražák, D.; Slavík, J.
2017-01-01
Roč. 16, č. 5 (2017), s. 1673-1695 ISSN 1534-0392 EU Projects: European Commission(XE) 320078 - MATHEF Institutional support: RVO:67985840 Keywords : damped wave equations * nonlinear damping * unbounded domains Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 0.801, year: 2016 http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=14110
Energy Technology Data Exchange (ETDEWEB)
Buerger, R.; Frid, H.; Karlsen, K.H.
2002-07-01
We consider a free boundary problem of a quasilinear strongly degenerate parabolic equation arising from a model of pressure filtration of flocculated suspensions. We provide definitions of generalized solutions of the free boundary problem in the framework of L2 divergence-measure fields. The formulation of boundary conditions is based on a Gauss-Green theorem for divergence-measure fields on bounded domains with Lipschitz deformable boundaries and avoids referring to traces of the solution. This allows to consider generalized solutions from a larger class than BV. Thus it is not necessary to derive the usual uniform estimates on spatial and time derivatives of the solutions of the corresponding regularized problem requires in the BV approach. We first prove existence and uniqueness of the solution of the regularized parabolic free boundary problem and then apply the vanishing viscosity method to prove existence of a generalized solution to the degenerate free boundary problem. (author)
Grinevich, P. G.; Santini, P. M.
2016-10-01
Written in the evolutionary form, the multidimensional integrable dispersionless equations, exactly like the soliton equations in 2+1 dimensions, become nonlocal. In particular, the Pavlov equation is brought to the form v t = v x v y - ∂ x -1 ∂ y [ v y + v x 2], where the formal integral ∂ x -1 becomes the asymmetric integral - int_x^∞ {dx'} . We show that this result could be guessed using an apparently new integral geometry lemma. It states that the integral of a sufficiently general smooth function f( X, Y) over a parabola in the plane ( X, Y) can be expressed in terms of the integrals of f( X, Y) over straight lines not intersecting the parabola. We expect that this result can have applications in two-dimensional linear tomography problems with an opaque parabolic obstacle.
Said-Houari, Belkacem
2012-03-01
In this paper, we consider a viscoelastic wave equation with an absorbing term and space-time dependent damping term. Based on the weighted energy method, and by assuming that the kernel decaying exponentially, we obtain the L2 decay rates of the solutions. More precisely, we show that the decay rates are the same as those obtained in Lin et al. (2010) [15] for the semilinear wave equation with absorption term. © 2011 Elsevier Inc.
Said-Houari, Belkacem
2012-01-01
In this paper, we consider a viscoelastic wave equation with an absorbing term and space-time dependent damping term. Based on the weighted energy method, and by assuming that the kernel decaying exponentially, we obtain the L2 decay rates of the solutions. More precisely, we show that the decay rates are the same as those obtained in Lin et al. (2010) [15] for the semilinear wave equation with absorption term. © 2011 Elsevier Inc.
Well-Posedness of Nonlocal Parabolic Differential Problems with Dependent Operators
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Allaberen Ashyralyev
2014-01-01
Full Text Available The nonlocal boundary value problem for the parabolic differential equation v'(t+A(tv(t=f(t (0≤t≤T, v(0=v(λ+φ, 0<λ≤T in an arbitrary Banach space E with the dependent linear positive operator A(t is investigated. The well-posedness of this problem is established in Banach spaces C0β,γ(Eα-β of all Eα-β-valued continuous functions φ(t on [0,T] satisfying a Hölder condition with a weight (t+τγ. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.
Oladeinde, Mobolaji Humphrey; Akpobi, John Ajokpaoghene
2011-10-01
The hydrodynamic and magnetohydrodynamic (MHD) lubrication problem of infinitely wide inclined and parabolic slider bearings is solved numerically using the finite element method. The bearing configurations are discretized into three-node isoparametric quadratic elements. Stiffness integrals obtained from the weak form of the governing equations are solved using Gauss quadrature to obtain a finite number of stiffness matrices. The global system of equations obtained from enforcing nodal continuity of pressure for the bearings are solved using the Gauss-Seidel iterative scheme with a convergence criterion of 10-10. Numerical computations reveal that, when compared for similar profile and couple stress parameters, greater pressure builds up in a parabolic slider compared to an inclined slider, indicating a greater wedge effect in the parabolic slider. The parabolic slider bearing is also shown to develop a greater load capacity when lubricated with magnetic fluids. The superior performance of parabolic slider bearing is more pronounced at greater Hartmann numbers for identical bearing structural parameters. It is also shown that when load carrying capacity is the yardstick for comparison, the parabolic slider bearings are superior to the inclined bearings when lubricated with couple stress or magnetic lubricants.
Existence and concentration of semiclassical states for nonlinear Schrodinger equations
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Shaowei Chen
2012-05-01
Full Text Available In this article, we study the semilinear Schrodinger equation $$ -epsilon^2Delta u+ u+ V(xu=f(u,quad uin H^1(mathbb{R}^N, $$ where $Ngeq 2$ and $epsilon>0$ is a small parameter. The function $V$ is bounded in $mathbb{R}^N$, $inf_{mathbb{R}^N}(1+V(x>0$ and it has a possibly degenerate isolated critical point. Under some conditions on f, we prove that as $epsilono 0$, this equation has a solution which concentrates at the critical point of V.
A compactness lemma of Aubin type and its application to degenerate parabolic equations
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Anvarbek Meirmanov
2014-10-01
Full Text Available Let $\\Omega\\subset \\mathbb{R}^{n}$ be a regular domain and $\\Phi(s\\in C_{\\rm loc}(\\mathbb{R}$ be a given function. If $\\mathfrak{M}\\subset L_2(0,T;W^1_2(\\Omega \\cap L_{\\infty}(\\Omega\\times (0,T$ is bounded and the set $\\{\\partial_t\\Phi(v|\\,v\\in \\mathfrak{M}\\}$ is bounded in $L_2(0,T;W^{-1}_2(\\Omega$, then there is a sequence $\\{v_k\\}\\in \\mathfrak{M}$ such that $v_k\\rightharpoonup v \\in L^2(0,T;W^1_2(\\Omega$, and $v_k\\to v$, $\\Phi(v_k\\to \\Phi(v$ a.e. in $\\Omega_T=\\Omega\\times (0,T$. This assertion is applied to prove solvability of the one-dimensional initial and boundary-value problem for a degenerate parabolic equation arising in the Buckley-Leverett model of two-phase filtration. We prove existence and uniqueness of a weak solution, establish the property of finite speed of propagation and construct a self-similar solution.
A short proof of increased parabolic regularity
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Stephen Pankavich
2015-08-01
Full Text Available We present a short proof of the increased regularity obtained by solutions to uniformly parabolic partial differential equations. Though this setting is fairly introductory, our new method of proof, which uses a priori estimates and an inductive method, can be extended to prove analogous results for problems with time-dependent coefficients, advection-diffusion or reaction diffusion equations, and nonlinear PDEs even when other tools, such as semigroup methods or the use of explicit fundamental solutions, are unavailable.
Controllability and Stabilization of Bilinear and Semilinear Partial Differential Equations
DEFF Research Database (Denmark)
Krishnaswamy, Vijayaraghavan
The topic of the thesis is the investigation of the question of controllability of weakly nonlinear partial differntial equations. The method is based on the Hilbert Uniqueness Method.......The topic of the thesis is the investigation of the question of controllability of weakly nonlinear partial differntial equations. The method is based on the Hilbert Uniqueness Method....
Analytic semigroups and optimal regularity in parabolic problems
Lunardi, Alessandra
2012-01-01
The book shows how the abstract methods of analytic semigroups and evolution equations in Banach spaces can be fruitfully applied to the study of parabolic problems. Particular attention is paid to optimal regularity results in linear equations. Furthermore, these results are used to study several other problems, especially fully nonlinear ones. Owing to the new unified approach chosen, known theorems are presented from a novel perspective and new results are derived. The book is self-contained. It is addressed to PhD students and researchers interested in abstract evolution equations and in p
Goswami, Deepjyoti
2013-05-01
In the first part of this article, a new mixed method is proposed and analyzed for parabolic integro-differential equations (PIDE) with nonsmooth initial data. Compared to the standard mixed method for PIDE, the present method does not bank on a reformulation using a resolvent operator. Based on energy arguments combined with a repeated use of an integral operator and without using parabolic type duality technique, optimal L2 L2-error estimates are derived for semidiscrete approximations, when the initial condition is in L2 L2. Due to the presence of the integral term, it is, further, observed that a negative norm estimate plays a crucial role in our error analysis. Moreover, the proposed analysis follows the spirit of the proof techniques used in deriving optimal error estimates for finite element approximations to PIDE with smooth data and therefore, it unifies both the theories, i.e., one for smooth data and other for nonsmooth data. Finally, we extend the proposed analysis to the standard mixed method for PIDE with rough initial data and provide an optimal error estimate in L2, L 2, which improves upon the results available in the literature. © 2013 Springer Science+Business Media New York.
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Yuan Wang
2015-01-01
Full Text Available Our work is devoted to a class of optimal control problems of parabolic partial differential equations. Because of the partial differential equations constraints, it is rather difficult to solve the optimization problem. The gradient of the cost function can be found by the adjoint problem approach. Based on the adjoint problem approach, the gradient of cost function is proved to be Lipschitz continuous. An improved conjugate method is applied to solve this optimization problem and this algorithm is proved to be convergent. This method is applied to set-point values in continuous cast secondary cooling zone. Based on the real data in a plant, the simulation experiments show that the method can ensure the steel billet quality. From these experiment results, it is concluded that the improved conjugate gradient algorithm is convergent and the method is effective in optimal control problem of partial differential equations.
On a modified form of navier-stokes equations for three-dimensional flows.
Venetis, J
2015-01-01
A rephrased form of Navier-Stokes equations is performed for incompressible, three-dimensional, unsteady flows according to Eulerian formalism for the fluid motion. In particular, we propose a geometrical method for the elimination of the nonlinear terms of these fundamental equations, which are expressed in true vector form, and finally arrive at an equivalent system of three semilinear first order PDEs, which hold for a three-dimensional rectangular Cartesian coordinate system. Next, we present the related variational formulation of these modified equations as well as a general type of weak solutions which mainly concern Sobolev spaces.
Schottky diode model for non-parabolic dispersion in narrow-gap semiconductor and few-layer graphene
Ang, Yee Sin; Ang, L. K.; Zubair, M.
Despite the fact that the energy dispersions are highly non-parabolic in many Schottky interfaces made up of 2D material, experimental results are often interpreted using the conventional Schottky diode equation which, contradictorily, assumes a parabolic energy dispersion. In this work, the Schottky diode equation is derived for narrow-gap semiconductor and few-layer graphene where the energy dispersions are highly non-parabolic. Based on Kane's non-parabolic band model, we obtained a more general Kane-Schottky scaling relation of J (T2 + γkBT3) which connects the contrasting J T2 in the conventional Schottky interface and the J T3 scaling in graphene-based Schottky interface via a non-parabolicity parameter, γ. For N-layer graphene of ABC -stacking and of ABA -stacking, the scaling relation follows J T 2 / N + 1 and J T3 respectively. Intriguingly, the Richardson constant extracted from the experimental data using an incorrect scaling can differ with the actual value by more than two orders of magnitude. Our results highlights the importance of using the correct scaling relation in order to accurately extract important physical properties, such as the Richardson constant and the Schottky barrier's height.
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Tomasz S. Zabawa
2005-01-01
Full Text Available The Dirichlet problem for an infinite weakly coupled system of semilinear differential-functional equations of elliptic type is considered. It is shown the existence of solutions to this problem. The result is based on Chaplygin's method of lower and upper functions.
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Reem A. Al-Omair
2009-03-01
Full Text Available In this paper we prove the existence of a mild solution for a semilinear evolution differential inclusion with nonlocal condition and governed by a family of linear operators, not necessarily bounded or closed, in a Banach space. No compactness assumption is assumed on the evolution operator generated by the family operators. Also, we prove that the set of mild solutions is compact.
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.
Rubin, S. G.
1982-01-01
Recent developments with finite-difference techniques are emphasized. The quotation marks reflect the fact that any finite discretization procedure can be included in this category. Many so-called finite element collocation and galerkin methods can be reproduced by appropriate forms of the differential equations and discretization formulas. Many of the difficulties encountered in early Navier-Stokes calculations were inherent not only in the choice of the different equations (accuracy), but also in the method of solution or choice of algorithm (convergence and stability, in the manner in which the dependent variables or discretized equations are related (coupling), in the manner that boundary conditions are applied, in the manner that the coordinate mesh is specified (grid generation), and finally, in recognizing that for many high Reynolds number flows not all contributions to the Navier-Stokes equations are necessarily of equal importance (parabolization, preferred direction, pressure interaction, asymptotic and mathematical character). It is these elements that are reviewed. Several Navier-Stokes and parabolized Navier-Stokes formulations are also presented.
Behavior of positive radial solutions of a quasilinear equation with a weighted Laplacian
Marta Garcia-Huidobro
2001-01-01
We obtain a classification result for positive radially symmetric solutions of the semilinear equation $$ -mathop{m div}(ilde a(|x|)abla u)=ilde b(|x|)|u|^{delta-1}u, $$ on a punctured ball. The weight functions $ilde a$ and $ilde b$ are $C^1$ on the punctured ball, are positive and measurable almost everywhere, and satisfy certain growth conditions near zero.
Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation
International Nuclear Information System (INIS)
Caraballo, Tomas; Kloeden, Peter E.; Schmalfuss, Bjoern
2004-01-01
We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of anon-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semilinear stochastic partial differential equations with Lipschitz continuous non-linearities
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.
Nobile, Fabio; Tempone, Raul
2009-01-01
We consider the problem of numerically approximating statistical moments of the solution of a time- dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen-Loève expansions driven by a finite number of uncorrelated random variables. After approxi- mating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. © 2009 John Wiley & Sons, Ltd.
Nobile, Fabio
2009-11-05
We consider the problem of numerically approximating statistical moments of the solution of a time- dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen-Loève expansions driven by a finite number of uncorrelated random variables. After approxi- mating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. © 2009 John Wiley & Sons, Ltd.
Behavior of positive radial solutions of a quasilinear equation with a weighted Laplacian
Directory of Open Access Journals (Sweden)
Marta Garcia-Huidobro
2001-01-01
Full Text Available We obtain a classification result for positive radially symmetric solutions of the semilinear equation $$ -mathop{m div}(ilde a(|x|abla u=ilde b(|x||u|^{delta-1}u, $$ on a punctured ball. The weight functions $ilde a$ and $ilde b$ are $C^1$ on the punctured ball, are positive and measurable almost everywhere, and satisfy certain growth conditions near zero.
Generalized Second Law of Thermodynamics in Parabolic LTB Inhomogeneous Cosmology
International Nuclear Information System (INIS)
Sheykhi, A.; Moradpour, H.; Sarab, K. Rezazadeh; Wang, B.
2015-01-01
We study thermodynamics of the parabolic Lemaitre–Tolman–Bondi (LTB) cosmology supported by a perfect fluid source. This model is the natural generalization of the flat Friedmann–Robertson–Walker (FRW) universe, and describes an inhomogeneous universe with spherical symmetry. After reviewing some basic equations in the parabolic LTB cosmology, we obtain a relation for the deceleration parameter in this model. We also obtain a condition for which the universe undergoes an accelerating phase at the present time. We use the first law of thermodynamics on the apparent horizon together with the Einstein field equations to get a relation for the apparent horizon entropy in LTB cosmology. We find out that in LTB model of cosmology, the apparent horizon's entropy could be feeded by a term, which incorporates the effects of the inhomogeneity. We consider this result and get a relation for the total entropy evolution, which is used to examine the generalized second law of thermodynamics for an accelerating universe. We also verify the validity of the second law and the generalized second law of thermodynamics for a universe filled with some kinds of matters bounded by the event horizon in the framework of the parabolic LTB model. (paper)
Nonlinear elliptic partial differential equations an introduction
Le Dret, Hervé
2018-01-01
This textbook presents the essential parts of the modern theory of nonlinear partial differential equations, including the calculus of variations. After a short review of results in real and functional analysis, the author introduces the main mathematical techniques for solving both semilinear and quasilinear elliptic PDEs, and the associated boundary value problems. Key topics include infinite dimensional fixed point methods, the Galerkin method, the maximum principle, elliptic regularity, and the calculus of variations. Aimed at graduate students and researchers, this textbook contains numerous examples and exercises and provides several comments and suggestions for further study.
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.
A parabolic singular perturbation problem with an internal layer
Grasman, J.; Shih, S.D.
2004-01-01
A method is presented to approximate with singular perturbation methods a parabolic differential equation for the quarter plane with a discontinuity at the corner. This discontinuity gives rise to an internal layer. It is necessary to match the local solution in this layer with the one in a corner
Monotone difference schemes for weakly coupled elliptic and parabolic systems
P. Matus (Piotr); F.J. Gaspar Lorenz (Franscisco); L. M. Hieu (Le Minh); V.T.K. Tuyen (Vo Thi Kim)
2017-01-01
textabstractThe present paper is devoted to the development of the theory of monotone difference schemes, approximating the so-called weakly coupled system of linear elliptic and quasilinear parabolic equations. Similarly to the scalar case, the canonical form of the vector-difference schemes is
A model reduction approach to numerical inversion for a parabolic partial differential equation
International Nuclear Information System (INIS)
Borcea, Liliana; Druskin, Vladimir; Zaslavsky, Mikhail; Mamonov, Alexander V
2014-01-01
We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magnetic field. The algorithm presented in this paper considers inversion in one and two dimensions. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the Gauss–Newton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated accurate simulations of the time-domain response. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments. (paper)
A model reduction approach to numerical inversion for a parabolic partial differential equation
Borcea, Liliana; Druskin, Vladimir; Mamonov, Alexander V.; Zaslavsky, Mikhail
2014-12-01
We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magnetic field. The algorithm presented in this paper considers inversion in one and two dimensions. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the Gauss-Newton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated accurate simulations of the time-domain response. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments.
Fast Time and Space Parallel Algorithms for Solution of Parabolic Partial Differential Equations
Fijany, Amir
1993-01-01
In this paper, fast time- and Space -Parallel agorithms for solution of linear parabolic PDEs are developed. It is shown that the seemingly strictly serial iterations of the time-stepping procedure for solution of the problem can be completed decoupled.
Energy decay for solutions to semilinear systems of elastic waves in exterior domains
Directory of Open Access Journals (Sweden)
Marcio V. Ferreira
2006-05-01
Full Text Available We consider the dynamical system of elasticity in the exterior of a bounded open domain in 3-D with smooth boundary. We prove that under the effect of "weak" dissipation, the total energy decays at a uniform rate as $t o +infty$, provided the initial data is "small" at infinity. No assumptions on the geometry of the obstacle are required. The results are then applied to a semilinear problem proving global existence and decay for small initial data.
Energy Technology Data Exchange (ETDEWEB)
Kirtley, K.R.
1988-10-01
A new coupled parabolic-marching method was developed to solve the three-dimensional incompressible Navier-Stokes equation for turbulent turbomachinery flows. Earlier space-marching methods were analyzed to determine their global stability during multiple passes of the computational domain. The methods were found to be unconditionally unstable even when an extra equation for the pressure, namely the Poisson equation for the pressure, was used between passes of the domain. Relaxation of one constraint during the solution process was found to be necessary for the successful calculation of a complex flow.Thus, the method of pseudocompressibility was introduced into the partially parabolized Navier-Stokes equation to relax the mass flow constraint during a forward-marching integration as well as globally stable during successive passes of the domain. With consistent discretization, the new method was found to be convergent.
Convergence of shock waves between conical and parabolic boundaries
Energy Technology Data Exchange (ETDEWEB)
Yanuka, D.; Zinowits, H. E.; Antonov, O.; Efimov, S.; Virozub, A.; Krasik, Ya. E. [Physics Department, Technion, Haifa 32000 (Israel)
2016-07-15
Convergence of shock waves, generated by underwater electrical explosions of cylindrical wire arrays, between either parabolic or conical bounding walls is investigated. A high-current pulse with a peak of ∼550 kA and rise time of ∼300 ns was applied for the wire array explosion. Strong self-emission from an optical fiber placed at the origin of the implosion was used for estimating the time of flight of the shock wave. 2D hydrodynamic simulations coupled with the equations of state of water and copper showed that the pressure obtained in the vicinity of the implosion is ∼7 times higher in the case of parabolic walls. However, comparison with a spherical wire array explosion showed that the pressure in the implosion vicinity in that case is higher than the pressure in the current experiment with parabolic bounding walls because of strong shock wave reflections from the walls. It is shown that this drawback of the bounding walls can be significantly minimized by optimization of the wire array geometry.
Zapata, M. A. Uh; Van Bang, D. Pham; Nguyen, K. D.
2016-05-01
This paper presents a parallel algorithm for the finite-volume discretisation of the Poisson equation on three-dimensional arbitrary geometries. The proposed method is formulated by using a 2D horizontal block domain decomposition and interprocessor data communication techniques with message passing interface. The horizontal unstructured-grid cells are reordered according to the neighbouring relations and decomposed into blocks using a load-balanced distribution to give all processors an equal amount of elements. In this algorithm, two parallel successive over-relaxation methods are presented: a multi-colour ordering technique for unstructured grids based on distributed memory and a block method using reordering index following similar ideas of the partitioning for structured grids. In all cases, the parallel algorithms are implemented with a combination of an acceleration iterative solver. This solver is based on a parabolic-diffusion equation introduced to obtain faster solutions of the linear systems arising from the discretisation. Numerical results are given to evaluate the performances of the methods showing speedups better than linear.
Time-optimal control of infinite order distributed parabolic systems involving time lags
Directory of Open Access Journals (Sweden)
G.M. Bahaa
2014-06-01
Full Text Available A time-optimal control problem for linear infinite order distributed parabolic systems involving constant time lags appear both in the state equation and in the boundary condition is presented. Some particular properties of the optimal control are discussed.
a numerical analysis of the energy behavior of a parabolic trough ...
African Journals Online (AJOL)
M. Ghodbane
A computer program was developed in Matlab after discretization equations. For the calculation of energy balance was asks these assumptions: The heat transfer fluid is incompressible;. The parabolic shape is symmetrical;. The ambient temperature around the concentrator is uniform;. The effect of the shadow of ...
O(2) Hopf bifurcation of viscous shock waves in a channel
Pogan, Alin; Yao, Jinghua; Zumbrun, Kevin
2015-07-01
Extending work of Texier and Zumbrun in the semilinear non-reflection symmetric case, we study O(2) transverse Hopf bifurcation, or "cellular instability", of viscous shock waves in a channel, for a class of quasilinear hyperbolic-parabolic systems including the equations of thermoviscoelasticity. The main difficulties are to (i) obtain Fréchet differentiability of the time- T solution operator by appropriate hyperbolic-parabolic energy estimates, and (ii) handle O(2) symmetry in the absence of either center manifold reduction (due to lack of spectral gap) or (due to nonstandard quasilinear hyperbolic-parabolic form) the requisite framework for treatment by spatial dynamics on the space of time-periodic functions, the two standard treatments for this problem. The latter issue is resolved by Lyapunov-Schmidt reduction of the time- T map, yielding a four-dimensional problem with O(2) plus approximate S1 symmetry, which we treat "by hand" using direct Implicit Function Theorem arguments. The former is treated by balancing information obtained in Lagrangian coordinates with that from associated constraints. Interestingly, this argument does not apply to gas dynamics or magnetohydrodynamics (MHD), due to the infinite-dimensional family of Lagrangian symmetries corresponding to invariance under arbitrary volume-preserving diffeomorphisms.
Cyclotron heating rate in a parabolic mirror
International Nuclear Information System (INIS)
Smith, P.K.
1984-01-01
Cyclotron resonance heating rates are found for a parabolic magnetic mirror. The equation of motion for perpendicular velocity is solved, including the radial magnetic field terms neglected in earlier papers. The expression for heating rate involves an infinite series of Anger's and Weber's functions, compared with a single term of the unrevised expression. The new results show an increase of heating rate compared with previous results. A simple expression is given for the ratio of the heating rates. (author)
Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in the annulus
Directory of Open Access Journals (Sweden)
Safa Dridi
2015-01-01
Full Text Available In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem: \\[-\\Delta u=q(xu^{\\sigma }\\;\\text{in}\\;\\Omega,\\quad u_{|\\partial\\Omega}=0.\\] Here \\(\\Omega\\ is an annulus in \\(\\mathbb{R}^{n}\\, \\(n\\geq 3\\, \\(\\sigma \\lt 1\\ and \\(q\\ is a positive function in \\(\\mathcal{C}_{loc}^{\\gamma }(\\Omega \\, \\(0\\lt\\gamma \\lt 1\\, satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub- and supersolutions with Karamata regular variation theory.
Energy Technology Data Exchange (ETDEWEB)
Johansson, Lisa
2003-07-01
Low-frequency, long-range sound propagation over a sea surface has been calculated using a wide-angel Cranck-Nicholson Parabolic Equation method. The model is developed to investigate noise from off-shore wind turbines. The calculations are made using normal meteorological conditions of the Baltic Sea. Special consideration has been made to a wind phenomenon called low level jet with strong winds on rather low altitude. The effects of water waves on sound propagation have been incorporated in the ground boundary condition using a boss model. This way of including roughness in sound propagation models is valid for water wave heights that are small compared to the wave length of the sound. Nevertheless, since only low frequency sound is considered, waves up to the mean wave height of the Baltic Sea can be included in this manner. The calculation model has been tested against benchmark cases and agrees well with measurements. The calculations show that channelling of sound occurs at downwind conditions and that the sound propagation tends towards cylindrical spreading. The effects of the water waves are found to be fairly small.
Aksikas, I.; Moghadam, A. Alizadeh; Forbes, J. F.
2018-04-01
This paper deals with the design of an optimal state-feedback linear-quadratic (LQ) controller for a system of coupled parabolic-hypebolic non-autonomous partial differential equations (PDEs). The infinite-dimensional state space representation and the corresponding operator Riccati differential equation are used to solve the control problem. Dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the LQ-optimal control problem and also to guarantee the exponential stability of the closed-loop system. Thanks to the eigenvalues and eigenfunctions of the parabolic operator and also the fact that the hyperbolic-associated operator Riccati differential equation can be converted to a scalar Riccati PDE, an algorithm to solve the LQ control problem has been presented. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ optimal controller designed in the early portion of the paper is implemented for the original non-linear model. Numerical simulations are performed to show the controller performances.
Radu, F.A.; Pop, I.S.; Knabner, P.; Bermúdez de Castro, A.; Gómez, D.; Quintela, P.; Salgado, P.
2006-01-01
In this paper we discuss some iterative approaches for solving the nonlinear algebraic systems encountered as fully discrete counterparts of some degenerate (fast diffusion) parabolic problems. After regularization, we combine a mixed finite element discretization with the Euler implicit scheme. For
International Nuclear Information System (INIS)
Caraballo, T.; Kloeden, P.E.
2006-01-01
Under a one-sided dissipative Lipschitz condition on its drift, a stochastic evolution equation with additive noise of the reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions. A similar situation holds for each Galerkin approximation and each implicit Euler scheme applied to these Galerkin approximations. Moreover, the stationary solution of the Euler scheme converges pathwise to that of the Galerkin system as the stepsize tends to zero and the stationary solutions of the Galerkin systems converge pathwise to that of the evolution equation as the dimension increases. The analysis is carried out on random partial and ordinary differential equations obtained from their stochastic counterparts by subtraction of appropriate Ornstein-Uhlenbeck stationary solutions
Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations
Lorz, Alexander
2011-01-17
Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses coexist? We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution. © Taylor & Francis Group, LLC.
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.
Shock wave convergence in water with parabolic wall boundaries
International Nuclear Information System (INIS)
Yanuka, D.; Shafer, D.; Krasik, Ya.
2015-01-01
The convergence of shock waves in water, where the cross section of the boundaries between which the shock wave propagates is either straight or parabolic, was studied. The shock wave was generated by underwater electrical explosions of planar Cu wire arrays using a high-current generator with a peak output current of ∼45 kA and rise time of ∼80 ns. The boundaries of the walls between which the shock wave propagates were symmetric along the z axis, which is defined by the direction of the exploding wires. It was shown that with walls having a parabolic cross section, the shock waves converge faster and the pressure in the vicinity of the line of convergence, calculated by two-dimensional hydrodynamic simulations coupled with the equations of state of water and copper, is also larger
Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations
Directory of Open Access Journals (Sweden)
I. Amirali
2014-01-01
Full Text Available Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.
Generalized heat-transport equations: parabolic and hyperbolic models
Rogolino, Patrizia; Kovács, Robert; Ván, Peter; Cimmelli, Vito Antonio
2018-03-01
We derive two different generalized heat-transport equations: the most general one, of the first order in time and second order in space, encompasses some well-known heat equations and describes the hyperbolic regime in the absence of nonlocal effects. Another, less general, of the second order in time and fourth order in space, is able to describe hyperbolic heat conduction also in the presence of nonlocal effects. We investigate the thermodynamic compatibility of both models by applying some generalizations of the classical Liu and Coleman-Noll procedures. In both cases, constitutive equations for the entropy and for the entropy flux are obtained. For the second model, we consider a heat-transport equation which includes nonlocal terms and study the resulting set of balance laws, proving that the corresponding thermal perturbations propagate with finite speed.
Cotté, B.
2018-05-01
This study proposes to couple a source model based on Amiet's theory and a parabolic equation code in order to model wind turbine noise emission and propagation in an inhomogeneous atmosphere. Two broadband noise generation mechanisms are considered, namely trailing edge noise and turbulent inflow noise. The effects of wind shear and atmospheric turbulence are taken into account using the Monin-Obukhov similarity theory. The coupling approach, based on the backpropagation method to preserve the directivity of the aeroacoustic sources, is validated by comparison with an analytical solution for the propagation over a finite impedance ground in a homogeneous atmosphere. The influence of refraction effects is then analyzed for different directions of propagation. The spectrum modification related to the ground effect and the presence of a shadow zone for upwind receivers are emphasized. The validity of the point source approximation that is often used in wind turbine noise propagation models is finally assessed. This approximation exaggerates the interference dips in the spectra, and is not able to correctly predict the amplitude modulation.
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.
Design and Realisation of a Parabolic Solar Cooker
International Nuclear Information System (INIS)
Ouannene, M; Chaouachi, B; Gabsi, S
2009-01-01
The sun s energy is really powerful. Solar energy is renewable and it s free. We can use it to make electricity, to heat buildings and to cook. The field of cooking consumes many fossil fuels such as gas and wood. Million people cannot find enough gas and/or wood to cook, so using solar cookers is a good idea. During this work, we designed, built and studied a parabolic solar cooker. The characteristic equations and the experimental results are given
Laser propagation and compton scattering in parabolic plasma channel
Dongguo, L; Yokoya, K; Hirose, T
2003-01-01
A Gaussian laser beam propagating in a parabolic plasma channel is discussed in this paper. For a weak laser, plasma density perturbation induced by interaction between the laser field and plasma is very small, the refractive index can be assumed to be constant with respect to time variable. For a parabolic plasma channel, through the static propagation equation, we obtain an analytical solution of the profile function of the Gaussian laser beam for an unmatched case and give the general condition for the matched case. As the laser intensity increases, an effect due to strong laser fields is included. We discuss how to design and select the distribution of plasma density for a certain experiment in which a plasma channel is utilized to guide a laser beam. The number of scattered photons (X-rays) generated through Compton backscattering in a plasma channel is discussed. (author)
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
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
Designing High-Efficiency Thin Silicon Solar Cells Using Parabolic-Pore Photonic Crystals
Bhattacharya, Sayak; John, Sajeev
2018-04-01
We demonstrate the efficacy of wave-interference-based light trapping and carrier transport in parabolic-pore photonic-crystal, thin-crystalline silicon (c -Si) solar cells to achieve above 29% power conversion efficiencies. Using a rigorous solution of Maxwell's equations through a standard finite-difference time domain scheme, we optimize the design of the vertical-parabolic-pore photonic crystal (PhC) on a 10 -μ m -thick c -Si solar cell to obtain a maximum achievable photocurrent density (MAPD) of 40.6 mA /cm2 beyond the ray-optical, Lambertian light-trapping limit. For a slanted-parabolic-pore PhC that breaks x -y symmetry, improved light trapping occurs due to better coupling into parallel-to-interface refraction modes. We achieve the optimum MAPD of 41.6 mA /cm2 for a tilt angle of 10° with respect to the vertical axis of the pores. This MAPD is further improved to 41.72 mA /cm2 by introducing a 75-nm SiO2 antireflective coating on top of the solar cell. We use this MAPD and the associated charge-carrier generation profile as input for a numerical solution of Poisson's equation coupled with semiconductor drift-diffusion equations using a Shockley-Read-Hall and Auger recombination model. Using experimentally achieved surface recombination velocities of 10 cm /s , we identify semiconductor doping profiles that yield power conversion efficiencies over 29%. Practical considerations of additional upper-contact losses suggest efficiencies close to 28%. This improvement beyond the current world record is largely due to an open-circuit voltage approaching 0.8 V enabled by reduced bulk recombination in our thin silicon architecture while maintaining a high short-circuit current through wave-interference-based light trapping.
Blow-up boundary regimes for general quasilinear parabolic equations in multidimensional domains
International Nuclear Information System (INIS)
Shishkov, A E; Shchelkov, A G
1999-01-01
A new approach (not based on the techniques of barriers) to the study of asymptotic properties of the generalized solutions of parabolic initial boundary-value problems with finite-time blow-up of the boundary values is proposed. Precise conditions on the blow-up pattern are found that guarantee uniform localization of the solution for an arbitrary compactly supported initial function. The main result of the paper consists in obtaining precise sufficient conditions for the singular (or blow-up) set of an arbitrary solution to remain within the boundary of the domain
Rohit Tripathi; Sumit Tiwari; G. N. Tiwari
2016-01-01
In present study, an approach is adopted where photovoltaic thermal flat plate collector is integrated with compound parabolic concentrator. Analytical expression of temperature dependent electrical efficiency of N number of partially covered Photovoltaic Thermal (PVT) - Compound Parabolic Concentrator (CPC) water collector connected in series has been derived with the help of basic thermal energy balance equations. Analysis has been carried for winter weather condition at Delhi location, Ind...
The absence of global positive solutions of systems of semilinear elliptic inequalities in cones
International Nuclear Information System (INIS)
Laptev, G G
2000-01-01
Let K be a cone in R N , N>=2. We establish conditions for the absence of global non-trivial non-negative solutions of semilinear elliptic inequalities and systems of inequalities of the form -div(|x| α Du)>=|x| β u q , u| ∂K =0. We find the critical exponent q* that divides the domains of existence of these solutions from those of their absence. We prove that in the limiting case q=q* there are no solutions. The method is to multiply the system by a special factor and integrate the inequalities thus obtained
Generators with interior degeneracy on spaces of L^2 type
Directory of Open Access Journals (Sweden)
Genni Fragnelli
2012-10-01
Full Text Available We consider operators in divergence and in nondivergence form with degeneracy at the interior of the space domain. Characterizing the domain of the operators, we prove that they generate positive analytic semigroups on spaces of L^2 type. Finally, some applications to linear and semilinear parabolic evolution problems and to linear hyperbolic ones are presented.
Directory of Open Access Journals (Sweden)
Nikita Agarwal
2017-07-01
Full Text Available In this article, we study the approximate controllability and homegenization results of a semi-linear elliptic problem with Robin boundary condition in a periodically perforated domain. We prove the existence of minimal norm control using Lions constructive approach, which is based on Fenchel-Rockafeller duality theory, and by means of Zuazua's fixed point arguments. Then, as the homogenization parameter goes to zero, we link the limit of the optimal controls (the limit of fixed point of the controllability problems with the optimal control of the corresponding homogenized problem.
Theoretical Study of the Compound Parabolic Trough Solar Collector
Dr. Subhi S. Mahammed; Dr. Hameed J. Khalaf; Tadahmun A. Yassen
2012-01-01
Theoretical design of compound parabolic trough solar collector (CPC) without tracking is presented in this work. The thermal efficiency is obtained by using FORTRAN 90 program. The thermal efficiency is between (60-67)% at mass flow rate between (0.02-0.03) kg/s at concentration ratio of (3.8) without need to tracking system.The total and diffused radiation is calculated for Tikrit city by using theoretical equations. Good agreement between present work and the previous work.
Soneson, Joshua E
2017-04-01
Wide-angle parabolic models are commonly used in geophysics and underwater acoustics but have seen little application in medical ultrasound. Here, a wide-angle model for continuous-wave high-intensity ultrasound beams is derived, which approximates the diffraction process more accurately than the commonly used Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation without increasing implementation complexity or computing time. A method for preventing the high spatial frequencies often present in source boundary conditions from corrupting the solution is presented. Simulations of shallowly focused axisymmetric beams using both the wide-angle and standard parabolic models are compared to assess the accuracy with which they model diffraction effects. The wide-angle model proposed here offers improved focusing accuracy and less error throughout the computational domain than the standard parabolic model, offering a facile method for extending the utility of existing KZK codes.
Fixed point of the parabolic renormalization operator
Lanford III, Oscar E
2014-01-01
This monograph grew out of the authors' efforts to provide a natural geometric description for the class of maps invariant under parabolic renormalization and for the Inou-Shishikura fixed point itself as well as to carry out a computer-assisted study of the parabolic renormalization operator. It introduces a renormalization-invariant class of analytic maps with a maximal domain of analyticity and rigid covering properties and presents a numerical scheme for computing parabolic renormalization of a germ, which is used to compute the Inou-Shishikura renormalization fixed point. Inside, readers will find a detailed introduction into the theory of parabolic bifurcation, Fatou coordinates, Écalle-Voronin conjugacy invariants of parabolic germs, and the definition and basic properties of parabolic renormalization. The systematic view of parabolic renormalization developed in the book and the numerical approach to its study will be interesting to both experts in the field as well as graduate students wishi...
Analytic method for solitary solutions of some partial differential equations
International Nuclear Information System (INIS)
Ugurlu, Yavuz; Kaya, Dogan
2007-01-01
In this Letter by considering an improved tanh function method, we found some exact solutions of the clannish random walker's parabolic equation, the modified Korteweg-de Vries (KdV) equation, and the Sharma-Tasso-Olver (STO) equation with its fission and fusion, the Jaulent-Miodek equation
Shishkin, G. I.
2015-11-01
An initial-boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation with a perturbation parameter ɛ (ɛ ∈ (0, 1]) multiplying the highest order derivative. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform mesh is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. The scheme does not converge ɛ-uniformly in the maximum norm as the number of its grid nodes is increased. When the solution of the difference scheme converges, which occurs if N -1 ≪ ɛ and N -1 0 ≪ 1, where N and N 0 are the numbers of grid intervals in x and t, respectively, the scheme is not ɛ-uniformly well conditioned or stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions on the "parameters" of the difference scheme and of the computer (namely, on ɛ, N, N 0, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions. Additionally, the conditions are obtained under which the perturbed numerical solution has the same order of convergence as the solution of the unperturbed standard difference scheme.
DEFF Research Database (Denmark)
Lomonaco, Luna; Petersen, Carsten Lunde; Shen, Weixiao
2017-01-01
We prove that any C1+BV degree d ≥ 2 circle covering h having all periodic orbits weakly expanding, is conjugate by a C1+BV diffeomorphism to a metrically expanding map. We use this to connect the space of parabolic external maps (coming from the theory of parabolic-like maps) to metrically expan...
Moving Least Squares Method for a One-Dimensional Parabolic Inverse Problem
Directory of Open Access Journals (Sweden)
Baiyu Wang
2014-01-01
Full Text Available This paper investigates the numerical solution of a class of one-dimensional inverse parabolic problems using the moving least squares approximation; the inverse problem is the determination of an unknown source term depending on time. The collocation method is used for solving the equation; some numerical experiments are presented and discussed to illustrate the stability and high efficiency of the method.
Annealed asymptotics for the parabolic Anderson model with a moving catalyst
Gärtner, J.; Heydenreich, M.O.
2006-01-01
This paper deals with the solution u to the parabolic Anderson equation ¿u/¿t=¿¿u+¿u on the lattice . We consider the case where the potential ¿ is time-dependent and has the form ¿(t,x)=d0(x-Yt) with Yt being a simple random walk with jump rate 2d. The solution u may be interpreted as the
Tracking local control of a parabolic trough collector
International Nuclear Information System (INIS)
Ajona, J.I.; Alberdi, J.; Gamero, E.; Blanco, J.
1992-01-01
In the local control, the sun position related to the trough collector is measured by two photo-resistors. The provided electronic signal is then compared with reference levels in order to get a set of B logical signals which form a byte. This byte and the commands issued by a programmable controller are connected to the inputs of o P.R.O.M. memory which is programmed with the logical equations of the control system. The memory output lines give the control command of the parabolic trough collector motor. (Author)
Theoretical Study of the Compound Parabolic Trough Solar Collector
Directory of Open Access Journals (Sweden)
Dr. Subhi S. Mahammed
2012-06-01
Full Text Available Theoretical design of compound parabolic trough solar collector (CPC without tracking is presented in this work. The thermal efficiency is obtained by using FORTRAN 90 program. The thermal efficiency is between (60-67% at mass flow rate between (0.02-0.03 kg/s at concentration ratio of (3.8 without need to tracking system.The total and diffused radiation is calculated for Tikrit city by using theoretical equations. Good agreement between present work and the previous work.
Analytic method for solitary solutions of some partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Ugurlu, Yavuz [Firat University, Department of Mathematics, 23119 Elazig (Turkey); Kaya, Dogan [Firat University, Department of Mathematics, 23119 Elazig (Turkey)], E-mail: dkaya@firat.edu.tr
2007-10-22
In this Letter by considering an improved tanh function method, we found some exact solutions of the clannish random walker's parabolic equation, the modified Korteweg-de Vries (KdV) equation, and the Sharma-Tasso-Olver (STO) equation with its fission and fusion, the Jaulent-Miodek equation.
The flow of an incompressible electroconductive fluid past a thin airfoil. The parabolic profile
Directory of Open Access Journals (Sweden)
Adrian CARABINEANU
2014-04-01
Full Text Available We study the two-dimensional steady flow of an ideal incompressible perfectly conducting fluid past an insulating thin parabolic airfoil. We consider the linearized Euler and Maxwell equations and Ohm's law. We use the integral representations for the velocity, magnetic induction and pressure and the boundary conditions to obtain an integral equation for the jump of the pressure across the airfoil. We give some graphic representations for the lift coefficient, velocity and magnetic induction.
Parabolic transformation cloaks for unbounded and bounded cloaking of matter waves
Chang, Yu-Hsuan; Lin, De-Hone
2014-01-01
Parabolic quantum cloaks with unbounded and bounded invisible regions are presented with the method of transformation design. The mass parameters of particles for perfect cloaking are shown to be constant along the parabolic coordinate axes of the cloaking shells. The invisibility performance of the cloaks is inspected from the viewpoints of waves and probability currents. The latter shows the controllable characteristic of a probability current by a quantum cloak. It also provides us with a simpler and more efficient way of exhibiting the performance of a quantum cloak without the solutions of the transformed wave equation. Through quantitative analysis of streamline structures in the cloaking shell, one defines the efficiency of the presented quantum cloak in the situation of oblique incidence. The cloaking models presented here give us more choices for testing and applying quantum cloaking.
Optimal control for parabolic-hyperbolic system with time delay
International Nuclear Information System (INIS)
Kowalewski, A.
1985-07-01
In this paper we consider an optimal control problem for a system described by a linear partial differential equation of the parabolic-hyperbolic type with time delay in the state. The right-hand side of this equation and the initial conditions are not continuous functions usually, but they are measurable functions belonging to L 2 or Lsup(infinity) spaces. Therefore, the solution of this equation is given by a certain Sobolev space. The time delay in the state is constant, but it can be also a function of time. The control time T is fixed in our problem. Making use of the Milutin-Dubovicki theorem, necessary and sufficient conditions of optimality with the quadratic performance functional and constrained control are derived for the Dirichlet problem. The flow chart of the algorithm which can be used in the numerical solving of certain optimization problems for distributed systems is also presented. (author)
A note on Chudnovskyʼs Fuchsian equations
Brezhnev, Yurii V.
We show that four exceptional Fuchsian equations, each determined by the four parabolic singularities, known as the Chudnovsky equations, are transformed into each other by algebraic transformations. We describe equivalence of these equations and their counterparts on tori. The latters are the Fuchsian equations on elliptic curves and their equivalence is characterized by transcendental transformations which are represented explicitly in terms of elliptic and theta functions.
Directory of Open Access Journals (Sweden)
Shirmayil Bagirov
2018-01-01
Full Text Available In the domain $Q_{R}'= \\{ x:| x| >R\\}\\times( 0,+\\infty$ we consider the problem $$\\displaylines{ \\frac{\\partial u_1}{\\partial t}+\\Delta^2 u_1-\\frac{C_1}{|x| ^4}u_1 =| x| ^{\\sigma _1}| u_2| ^{q_1}, \\quad u_1| _{t=0}=u_{10}( x\\geq0, \\cr \\frac{\\partial u_2}{\\partial t}+\\Delta^2 u_2-\\frac{C_2}{| x| ^4}u_2=| x| ^{\\sigma _2}| u_1| ^{q_2},\\quad u_2| _{t=0}=u_{20}( x\\geq0, \\cr \\int_0^\\infty \\int_{\\partial B_{R}} u_i\\,ds\\,dt\\geq 0, \\quad \\int_0^\\infty \\int_{\\partial B_{R}}\\Delta u_i\\,ds\\,dt\\leq 0, }$$ where $\\sigma_i\\in \\mathbb{R} $, $ q_i>1 $, $ 0\\leq C_i<( \\frac{n( n-4 }{4} ^2$, $ i=1,2 $. Sufficient condition for the nonexistence of global solutions is obtained.The proof is based on the method of test functions.
Stable estimation of two coefficients in a nonlinear Fisher–KPP equation
International Nuclear Information System (INIS)
Cristofol, Michel; Roques, Lionel
2013-01-01
We consider the inverse problem of determining two non-constant coefficients in a nonlinear parabolic equation of the Fisher–Kolmogorov–Petrovsky–Piskunov type. For the equation u t = DΔu + μ(x) u − γ(x)u 2 in (0, T) × Ω, which corresponds to a classical model of population dynamics in a bounded heterogeneous environment, our results give a stability inequality between the couple of coefficients (μ, γ) and some observations of the solution u. These observations consist in measurements of u: in the whole domain Ω at two fixed times, in a subset ω⊂⊂Ω during a finite time interval and on the boundary of Ω at all times t ∈ (0, T). The proof relies on parabolic estimates together with the parabolic maximum principle and Hopf’s lemma which enable us to use a Carleman inequality. This work extends previous studies on the stable determination of non-constant coefficients in parabolic equations, as it deals with two coefficients and with a nonlinear term. A consequence of our results is the uniqueness of the couple of coefficients (μ, γ), given the observation of u. This uniqueness result was obtained in a previous paper but in the one-dimensional case only. (paper)
An evolution infinity Laplace equation modelling dynamic elasto-plastic torsion
Messelmi, Farid
2017-12-01
We consider in this paper a parabolic partial differential equation involving the infinity Laplace operator and a Leray-Lions operator with no coercitive assumption. We prove the existence and uniqueness of the corresponding approached problem and we show that at the limit the solution solves the parabolic variational inequality arising in the elasto-plastic torsion problem.
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...
Fragnelli, Genni
2016-01-01
The authors consider a parabolic problem with degeneracy in the interior of the spatial domain, and they focus on observability results through Carleman estimates for the associated adjoint problem. The novelties of the present paper are two. First, the coefficient of the leading operator only belongs to a Sobolev space. Second, the degeneracy point is allowed to lie even in the interior of the control region, so that no previous result can be adapted to this situation; however, different cases can be handled, and new controllability results are established as a consequence.
Weak self-adjoint differential equations
International Nuclear Information System (INIS)
Gandarias, M L
2011-01-01
The concepts of self-adjoint and quasi self-adjoint equations were introduced by Ibragimov (2006 J. Math. Anal. Appl. 318 742-57; 2007 Arch. ALGA 4 55-60). In Ibragimov (2007 J. Math. Anal. Appl. 333 311-28), a general theorem on conservation laws was proved. In this paper, we generalize the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. We find a class of weak self-adjoint quasi-linear parabolic equations. The property of a differential equation to be weak self-adjoint is important for constructing conservation laws associated with symmetries of the differential equation. (fast track communication)
Botta, E.F.F.; Dijkstra, D.; Veldman, A.E.P.
1972-01-01
The numerical method of solution for the semi-infinite flat plate has been extended to the case of the parabolic cylinder. Results are presented for the skin friction, the friction drag, the pressure and the pressure drag. The drag coefficients have been checked by means of an application of the
A global numerical solution of the radial Schroedinger equation by second-order perturbation theory
International Nuclear Information System (INIS)
Adam, G.
1979-01-01
A global numerical method, which uses second-order perturbation theory, is described for the solution of the radial Schroedinger equation. The perturbative numerical (PN) solution is derived in two stages: first, the original potential is approximated by a piecewise continuous parabolic function, and second, the resulting Schroedinger equation is solved on each integration step by second-order perturbation theory, starting with a step function reference approximation for the parabolic potential. We get a manageable PN algorithm, which shows an order of accuracy equal to six in the solution of the original Schroedinger equation, and is very stable against round off errors. (author)
Numerical Methods for Partial Differential Equations
Guo, Ben-yu
1987-01-01
These Proceedings of the first Chinese Conference on Numerical Methods for Partial Differential Equations covers topics such as difference methods, finite element methods, spectral methods, splitting methods, parallel algorithm etc., their theoretical foundation and applications to engineering. Numerical methods both for boundary value problems of elliptic equations and for initial-boundary value problems of evolution equations, such as hyperbolic systems and parabolic equations, are involved. The 16 papers of this volume present recent or new unpublished results and provide a good overview of current research being done in this field in China.
International Nuclear Information System (INIS)
Bakhos, Tania; Saibaba, Arvind K.; Kitanidis, Peter K.
2015-01-01
We consider the problem of estimating parameters in large-scale weakly nonlinear inverse problems for which the underlying governing equations is a linear, time-dependent, parabolic partial differential equation. A major challenge in solving these inverse problems using Newton-type methods is the computational cost associated with solving the forward problem and with repeated construction of the Jacobian, which represents the sensitivity of the measurements to the unknown parameters. Forming the Jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time-dependent parabolic partial differential equations corresponding to multiple sources and receivers. We propose an efficient method based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach to solve the resulting shifted systems of equations efficiently. Our proposed solver speeds up the computation of the forward and adjoint problems, thus yielding significant speedup in total inversion time. We consider an application from Transient Hydraulic Tomography (THT), which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests. The algorithms discussed are applied to a synthetic example taken from THT to demonstrate the resulting computational gains of this proposed method
Energy Technology Data Exchange (ETDEWEB)
Bakhos, Tania, E-mail: taniab@stanford.edu [Institute for Computational and Mathematical Engineering, Stanford University (United States); Saibaba, Arvind K. [Department of Electrical and Computer Engineering, Tufts University (United States); Kitanidis, Peter K. [Institute for Computational and Mathematical Engineering, Stanford University (United States); Department of Civil and Environmental Engineering, Stanford University (United States)
2015-10-15
We consider the problem of estimating parameters in large-scale weakly nonlinear inverse problems for which the underlying governing equations is a linear, time-dependent, parabolic partial differential equation. A major challenge in solving these inverse problems using Newton-type methods is the computational cost associated with solving the forward problem and with repeated construction of the Jacobian, which represents the sensitivity of the measurements to the unknown parameters. Forming the Jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time-dependent parabolic partial differential equations corresponding to multiple sources and receivers. We propose an efficient method based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach to solve the resulting shifted systems of equations efficiently. Our proposed solver speeds up the computation of the forward and adjoint problems, thus yielding significant speedup in total inversion time. We consider an application from Transient Hydraulic Tomography (THT), which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests. The algorithms discussed are applied to a synthetic example taken from THT to demonstrate the resulting computational gains of this proposed method.
Local Properties of Solutions to Non-Autonomous Parabolic PDEs with State-Dependent Delays
Czech Academy of Sciences Publication Activity Database
Rezunenko, Oleksandr
2012-01-01
Roč. 2, č. 2 (2012), s. 56-71 ISSN 2158-611X R&D Projects: GA ČR(CZ) GAP103/12/2431 Institutional support: RVO:67985556 Keywords : partial differential equations * state-dependent delay * invariance principle Subject RIV: BC - Control Systems Theory http://library.utia.cas.cz/separaty/2012/AS/rezunenko- local properties of solutions to non-autonomous parabolic PDEs with state-dependent delay s.pdf
Test results, Industrial Solar Technology parabolic trough solar collector
Energy Technology Data Exchange (ETDEWEB)
Dudley, V.E. [EG and G MSI, Albuquerque, NM (United States); Evans, L.R.; Matthews, C.W. [Sandia National Labs., Albuquerque, NM (United States)
1995-11-01
Sandia National Laboratories and Industrial Solar Technology are cost-sharing development of advanced parabolic trough technology. As part of this effort, several configurations of an IST solar collector were tested to determine the collector efficiency and thermal losses with black chrome and black nickel receiver selective coatings, combined with aluminized film and silver film reflectors, using standard Pyrex{reg_sign} and anti-reflective coated Pyrex{reg_sign} glass receiver envelopes. The development effort has been successful, producing an advanced collector with 77% optical efficiency, using silver-film reflectors, a black nickel receiver coating, and a solgel anti-reflective glass receiver envelope. For each receiver configuration, performance equations were empirically derived relating collector efficiency and thermal losses to the operating temperature. Finally, equations were derived showing collector performance as a function of input insolation value, incident angle, and operating temperature.
Bayesian Inference for Linear Parabolic PDEs with Noisy Boundary Conditions
Ruggeri, Fabrizio; Sawlan, Zaid A; Scavino, Marco; Tempone, Raul
2016-01-01
In this work we develop a hierarchical Bayesian setting to infer unknown parameters in initial-boundary value problems (IBVPs) for one-dimensional linear parabolic partial differential equations. Noisy boundary data and known initial condition are assumed. We derive the likelihood function associated with the forward problem, given some measurements of the solution field subject to Gaussian noise. Such function is then analytically marginalized using the linearity of the equation. Gaussian priors have been assumed for the time-dependent Dirichlet boundary values. Our approach is applied to synthetic data for the one-dimensional heat equation model, where the thermal diffusivity is the unknown parameter. We show how to infer the thermal diffusivity parameter when its prior distribution is lognormal or modeled by means of a space-dependent stationary lognormal random field. We use the Laplace method to provide approximated Gaussian posterior distributions for the thermal diffusivity. Expected information gains and predictive posterior densities for observable quantities are numerically estimated for different experimental setups.
Bayesian Inference for Linear Parabolic PDEs with Noisy Boundary Conditions
Ruggeri, Fabrizio
2015-01-07
In this work we develop a hierarchical Bayesian setting to infer unknown parameters in initial-boundary value problems (IBVPs) for one-dimensional linear parabolic partial differential equations. Noisy boundary data and known initial condition are assumed. We derive the likelihood function associated with the forward problem, given some measurements of the solution field subject to Gaussian noise. Such function is then analytically marginalized using the linearity of the equation. Gaussian priors have been assumed for the time-dependent Dirichlet boundary values. Our approach is applied to synthetic data for the one-dimensional heat equation model, where the thermal diffusivity is the unknown parameter. We show how to infer the thermal diffusivity parameter when its prior distribution is lognormal or modeled by means of a space-dependent stationary lognormal random field. We use the Laplace method to provide approximated Gaussian posterior distributions for the thermal diffusivity. Expected information gains and predictive posterior densities for observable quantities are numerically estimated for different experimental setups.
Bayesian Inference for Linear Parabolic PDEs with Noisy Boundary Conditions
Ruggeri, Fabrizio
2016-01-06
In this work we develop a hierarchical Bayesian setting to infer unknown parameters in initial-boundary value problems (IBVPs) for one-dimensional linear parabolic partial differential equations. Noisy boundary data and known initial condition are assumed. We derive the likelihood function associated with the forward problem, given some measurements of the solution field subject to Gaussian noise. Such function is then analytically marginalized using the linearity of the equation. Gaussian priors have been assumed for the time-dependent Dirichlet boundary values. Our approach is applied to synthetic data for the one-dimensional heat equation model, where the thermal diffusivity is the unknown parameter. We show how to infer the thermal diffusivity parameter when its prior distribution is lognormal or modeled by means of a space-dependent stationary lognormal random field. We use the Laplace method to provide approximated Gaussian posterior distributions for the thermal diffusivity. Expected information gains and predictive posterior densities for observable quantities are numerically estimated for different experimental setups.
Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations
Lorz, Alexander; Mirrahimi, Sepideh; Perthame, Benoî t
2011-01-01
simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution. © Taylor & Francis Group, LLC.
Stability analysis of impulsive parabolic complex networks
Energy Technology Data Exchange (ETDEWEB)
Wang Jinliang, E-mail: wangjinliang1984@yahoo.com.cn [Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, XueYuan Road, No. 37, HaiDian District, Beijing 100191 (China); Wu Huaining [Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, XueYuan Road, No. 37, HaiDian District, Beijing 100191 (China)
2011-11-15
Highlights: > Two impulsive parabolic complex network models are proposed. > The global exponential stability of impulsive parabolic complex networks are considered. > The robust global exponential stability of impulsive parabolic complex networks are considered. - Abstract: In the present paper, two kinds of impulsive parabolic complex networks (IPCNs) are considered. In the first one, all nodes have the same time-varying delay. In the second one, different nodes have different time-varying delays. Using the Lyapunov functional method combined with the inequality techniques, some global exponential stability criteria are derived for the IPCNs. Furthermore, several robust global exponential stability conditions are proposed to take uncertainties in the parameters of the IPCNs into account. Finally, numerical simulations are presented to illustrate the effectiveness of the results obtained here.
Stability analysis of impulsive parabolic complex networks
International Nuclear Information System (INIS)
Wang Jinliang; Wu Huaining
2011-01-01
Highlights: → Two impulsive parabolic complex network models are proposed. → The global exponential stability of impulsive parabolic complex networks are considered. → The robust global exponential stability of impulsive parabolic complex networks are considered. - Abstract: In the present paper, two kinds of impulsive parabolic complex networks (IPCNs) are considered. In the first one, all nodes have the same time-varying delay. In the second one, different nodes have different time-varying delays. Using the Lyapunov functional method combined with the inequality techniques, some global exponential stability criteria are derived for the IPCNs. Furthermore, several robust global exponential stability conditions are proposed to take uncertainties in the parameters of the IPCNs into account. Finally, numerical simulations are presented to illustrate the effectiveness of the results obtained here.
Institute of Scientific and Technical Information of China (English)
张铁; 李长军
2001-01-01
The object of this paper is to investigate the superconvergence properties of finite element approximations to parabolic and hyperbolic integro-differential equations. The quasi projection technique introduced earlier by Douglas et al. is developed to derive the O(h2r) order knot superconvergence in the case of a single space variable, and to show the optimal order negative norm estimates in the case of several space variables.
Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations
Barles, Guy; Chasseigne, Emmanuel; Ciomaga, Adina; Imbert, Cyril
2011-01-01
We establish new Hoelder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hoelder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the loc...
Asymptotic problems for stochastic partial differential equations
Salins, Michael
Stochastic partial differential equations (SPDEs) can be used to model systems in a wide variety of fields including physics, chemistry, and engineering. The main SPDEs of interest in this dissertation are the semilinear stochastic wave equations which model the movement of a material with constant mass density that is exposed to both determinstic and random forcing. Cerrai and Freidlin have shown that on fixed time intervals, as the mass density of the material approaches zero, the solutions of the stochastic wave equation converge uniformly to the solutions of a stochastic heat equation, in probability. This is called the Smoluchowski-Kramers approximation. In Chapter 2, we investigate some of the multi-scale behaviors that these wave equations exhibit. In particular, we show that the Freidlin-Wentzell exit place and exit time asymptotics for the stochastic wave equation in the small noise regime can be approximated by the exit place and exit time asymptotics for the stochastic heat equation. We prove that the exit time and exit place asymptotics are characterized by quantities called quasipotentials and we prove that the quasipotentials converge. We then investigate the special case where the equation has a gradient structure and show that we can explicitly solve for the quasipotentials, and that the quasipotentials for the heat equation and wave equation are equal. In Chapter 3, we study the Smoluchowski-Kramers approximation in the case where the material is electrically charged and exposed to a magnetic field. Interestingly, if the system is frictionless, then the Smoluchowski-Kramers approximation does not hold. We prove that the Smoluchowski-Kramers approximation is valid for systems exposed to both a magnetic field and friction. Notably, we prove that the solutions to the second-order equations converge to the solutions of the first-order equation in an Lp sense. This strengthens previous results where convergence was proved in probability.
International Nuclear Information System (INIS)
Pierantozzi, T.; Vazquez, L.
2005-01-01
Through fractional calculus and following the method used by Dirac to obtain his well-known equation from the Klein-Gordon equation, we analyze a possible interpolation between the Dirac and the diffusion equations in one space dimension. We study the transition between the hyperbolic and parabolic behaviors by means of the generalization of the D'Alembert formula for the classical wave equation and the invariance under space and time inversions of the interpolating fractional evolution equations Dirac like. Such invariance depends on the values of the fractional index and is related to the nonlocal property of the time fractional differential operator. For this system of fractional evolution equations, we also find an associated conserved quantity analogous to the Hamiltonian for the classical Dirac case
Solar parabolic dish technology evaluation report
Lucas, J. W.
1984-01-01
The activities of the JPL Solar Thermal Power Systems Parabolic Dish Project for FY 1983 are summarized. Included are discussions on designs of module development including concentrator, receiver, and power conversion subsystems together with a separate discussion of field tests, Small Community Experiment system development, and tests at the Parabolic Dish Test Site.
F John's stability conditions versus A Carasso's SECB constraint for backward parabolic problems
International Nuclear Information System (INIS)
Lee, Jinwoo; Sheen, Dongwoo
2009-01-01
In order to solve backward parabolic problems John (1960 Commun. Pure. Appl. Math.13 551–85) introduced the two constraints ||u(T)|| ≤ M and ||u(0) − g|| ≤ δ where u(t) satisfies the backward heat equation for t in (0, T) with the initial data u(0). The slow evolution from the continuation boundary (SECB) constraint was introduced by Carasso (1994 SIAM J. Numer. Anal. 31 1535–57) to attain continuous dependence on data for backward parabolic problems even at the continuation boundary t = T. The additional 'SECB constraint' guarantees a significant improvement in stability up to t = T. In this paper, we prove that the same type of stability can be obtained by using only two constraints among the three. More precisely, we show that the a priori boundedness condition ||u(T)|| ≤ M is redundant. This implies that Carasso's SECB condition can be used to replace the a priori boundedness condition of John with an improved stability estimate. Also, a new class of regularized solutions is introduced for backward parabolic problems with an SECB constraint. The new regularized solutions are optimally stable and we also provide a constructive scheme to compute. Finally, numerical examples are provided
Numerical simulation of solar parabolic trough collector performance in the Algeria Saharan region
International Nuclear Information System (INIS)
Marif, Yacine; Benmoussa, Hocine; Bouguettaia, Hamza; Belhadj, Mohamed M.; Zerrouki, Moussa
2014-01-01
Highlights: • The parabolic trough collector performance is examined. • The finite difference method is proposed and validated. • Two fluids are considered water and TherminolVP-1™. - Abstract: In order to determine the optical and thermal performance of a solar parabolic trough collector under the climate conditions of Algerian Sahara, a computer program based on one dimensional implicit finite difference method with energy balance approach has been developed. The absorber pipe, glass envelope and fluid were divided into several segments and the partial derivation in the differential equations was replaced by the backward finite difference terms in each segment. Two fluids were considered, liquid water and TherminolVP-1™ synthetic oil. Furthermore, the intensity of the direct solar radiation was estimated by monthly average values of the atmospheric Linke turbidity factor for different tracking systems. According to the simulation findings, the one axis polar East–West and horizontal East–West tracking systems were most desirable for a parabolic trough collector throughout the whole year. In addition, it is found that the thermal efficiency was about 69.73–72.24%, which decreases with the high synthetic oil fluid temperatures and increases in the lower water temperature by 2%
From ordinary to partial differential equations
Esposito, Giampiero
2017-01-01
This book is addressed to mathematics and physics students who want to develop an interdisciplinary view of mathematics, from the age of Riemann, Poincaré and Darboux to basic tools of modern mathematics. It enables them to acquire the sensibility necessary for the formulation and solution of difficult problems, with an emphasis on concepts, rigour and creativity. It consists of eight self-contained parts: ordinary differential equations; linear elliptic equations; calculus of variations; linear and non-linear hyperbolic equations; parabolic equations; Fuchsian functions and non-linear equations; the functional equations of number theory; pseudo-differential operators and pseudo-differential equations. The author leads readers through the original papers and introduces new concepts, with a selection of topics and examples that are of high pedagogical value.
Parabolic features and the erosion rate on Venus
Strom, Robert G.
1993-01-01
The impact cratering record on Venus consists of 919 craters covering 98 percent of the surface. These craters are remarkably well preserved, and most show pristine structures including fresh ejecta blankets. Only 35 craters (3.8 percent) have had their ejecta blankets embayed by lava and most of these occur in the Atla-Beta Regio region; an area thought to be recently active. parabolic features are associated with 66 of the 919 craters. These craters range in size from 6 to 105 km diameter. The parabolic features are thought to be the result of the deposition of fine-grained ejecta by winds in the dense venusian atmosphere. The deposits cover about 9 percent of the surface and none appear to be embayed by younger volcanic materials. However, there appears to be a paucity of these deposits in the Atla-Beta Regio region, and this may be due to the more recent volcanism in this area of Venus. Since parabolic features are probably fine-grain, wind-deposited ejecta, then all impact craters on Venus probably had these deposits at some time in the past. The older deposits have probably been either eroded or buried by eolian processes. Therefore, the present population of these features is probably associated with the most recent impact craters on the planet. Furthermore, the size/frequency distribution of craters with parabolic features is virtually identical to that of the total crater population. This suggests that there has been little loss of small parabolic features compared to large ones, otherwise there should be a significant and systematic paucity of craters with parabolic features with decreasing size compared to the total crater population. Whatever is erasing the parabolic features apparently does so uniformly regardless of the areal extent of the deposit. The lifetime of parabolic features and the eolian erosion rate on Venus can be estimated from the average age of the surface and the present population of parabolic features.
Fujita Exponent for a Nonlinear Degenerate Parabolic Equation with Localized Source
Directory of Open Access Journals (Sweden)
Yulan Wang
2014-01-01
Full Text Available This paper is devoted to understand the blow-up properties of reaction-diffusion equations which combine a localized reaction term with nonlinear diffusion. In particular, we study the critical exponent of a p-Laplacian equation with a localized reaction. We obtain the Fujita exponent qc of the equation.
Elliptic equation for random walks. Application to transport in microporous media
DEFF Research Database (Denmark)
Shapiro, Alexander
2007-01-01
We consider a process of random walks with arbitrary residence time distribution. We show that in many cases this process may not be described by the classical (Fick) parabolic diffusion equation, but an elliptic equation. An additional term proportional to the second time derivative takes into a...
Thermal behaviour of solar air heater with compound parabolic concentrator
International Nuclear Information System (INIS)
Tchinda, Rene
2008-01-01
A mathematical model for computing the thermal performance of an air heater with a truncated compound parabolic concentrator having a flat one-sided absorber is presented. A computer code that employs an iterative solution procedure is constructed to solve the governing energy equations and to estimate the performance parameters of the collector. The effects of the air mass flow rate, the wind speed and the collector length on the thermal performance of the present air heater are investigated. Predictions for the performance of the solar heater also exhibit reasonable agreement, with experimental data with an average error of 7%
Complex energy eigenvalues of a linear potential with a parabolical barrier
International Nuclear Information System (INIS)
Malherbe, J.B.
1978-01-01
The physical meaning and restrictions of complex energy eigenvalues are briefly discussed. It is indicated that a quasi-stationary phase describes an idealised disintegration system. Approximate resonance-eigenvalues of the one dimensional Schrodinger equation with a linear potential and parabolic barrier are calculated by means of Connor's semiclassical method. This method is based on the generalized WKB-method of Miller and Good. The results obtained confirm the correctness of a model representation which explains the unusual distribution of eigenvalues by certain other linear potentials in a complex energy level [af
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...
Improved stochastic approximation methods for discretized parabolic partial differential equations
Guiaş, Flavius
2016-12-01
We present improvements of the stochastic direct simulation method, a known numerical scheme based on Markov jump processes which is used for approximating solutions of ordinary differential equations. This scheme is suited especially for spatial discretizations of evolution partial differential equations (PDEs). By exploiting the full path simulation of the stochastic method, we use this first approximation as a predictor and construct improved approximations by Picard iterations, Runge-Kutta steps, or a combination. This has as consequence an increased order of convergence. We illustrate the features of the improved method at a standard benchmark problem, a reaction-diffusion equation modeling a combustion process in one space dimension (1D) and two space dimensions (2D).
Manufacturing parabolic mirrors
CERN PhotoLab
1975-01-01
The photo shows the construction of a vertical centrifuge mounted on an air cushion, with a precision of 1/10000 during rotation, used for the manufacture of very high=precision parabolic mirrors. (See Annual Report 1974.)
Nonlinear stochastic heat equations with cubic nonlinearities and additive Q-regular noise in R^1
Directory of Open Access Journals (Sweden)
Henri Schurz
2010-09-01
Full Text Available Semilinear stochastic heat equations perturbed by cubic-type nonlinearities and additive space-time noise with homogeneous boundary conditions are discussed in R^1. The space-time noise is supposed to be Gaussian in time and possesses a Fourier expansion in space along the eigenfunctions of underlying Lapace operators. We follow the concept of approximate strong (classical Fourier solutions. The existence of unique continuous L^2-bounded solutions is proved. Furthermore, we present a procedure for its numerical approximation based on nonstandard methods (linear-implicit and justify their stability and consistency. The behavior of related total energy functional turns out to be crucial in the presented analysis.
Partial differential equations in action complements and exercises
Salsa, Sandro
2015-01-01
This textbook presents problems and exercises at various levels of difficulty in the following areas: Classical Methods in PDEs (diffusion, waves, transport, potential equations); Basic Functional Analysis and Distribution Theory; Variational Formulation of Elliptic Problems; and Weak Formulation for Parabolic Problems and for the Wave Equation. Thanks to the broad variety of exercises with complete solutions, it can be used in all basic and advanced PDE courses.
A New Algorithm for System of Integral Equations
Directory of Open Access Journals (Sweden)
Abdujabar Rasulov
2014-01-01
Full Text Available We develop a new algorithm to solve the system of integral equations. In this new method no need to use matrix weights. Beacause of it, we reduce computational complexity considerable. Using the new algorithm it is also possible to solve an initial boundary value problem for system of parabolic equations. To verify the efficiency, the results of computational experiments are given.
Photovoltaic applications of Compound Parabolic Concentrator (CPC)
Winston, R.
1975-01-01
The use of a compound parabolic concentrator as field collector, in conjunction with a primary focusing concentrator for photovoltaic applications is studied. The primary focusing concentrator can be a parabolic reflector, an array of Fresnel mirrors, a Fresnel lens or some other lens. Silicon solar cell grid structures are proposed that increase efficiency with concentration up to 10 suns. A ray tracing program has been developed to determine energy distribution at the exit of a compound parabolic concentrator. Projected total cost of a CPC/solar cell system will be between 4 and 5 times lower than for flat plate silicon cell arrays.
Use of fast Fourier transforms for solving partial differential equations in physics
Le Bail, R C
1972-01-01
The use of fast Fourier techniques for the direct solution of an important class of elliptic, parabolic, and hyperbolic partial differential equations in two dimensions is described. Extensions to higher-order and higher-dimension equations as well as to integrodifferential equations are presented, and several numerical examples with their resulting precision and timing are reported. (12 refs).
Motsa, S S; Magagula, V M; Sibanda, P
2014-01-01
This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.
Directory of Open Access Journals (Sweden)
S. S. Motsa
2014-01-01
Full Text Available This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs. The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.
Directory of Open Access Journals (Sweden)
Bruno de Andrade
2009-01-01
Full Text Available We study the existence and uniqueness of almost automorphic (resp., pseudo-almost automorphic solutions to a first-order differential equation with linear part dominated by a Hille-Yosida type operator with nondense domain.
International Nuclear Information System (INIS)
Guidi, Leonardo F.; Marchetti, D.H.U.
2003-01-01
We establish a comparison between Rakib-Sivashinsky and Michelson-Sivashinsky quasilinear parabolic differential equations governing the weak thermal limit of flame front propagating in channels. For the former equation, we give a complete description of all steady solutions and present their local and global stability analysis. For the latter, bi-coalescent and interpolating unstable steady solutions are introduced and shown to be more numerous than the previous known coalescent solutions. These facts are argued to be responsible for the disagreement between the observed dynamics in numerical experiments and the exact (linear) stability analysis and give ingredients to construct quasi-stable solutions describing parabolic steadily propagating flame with centered tip
Thin-Layer Solutions of the Helmholtz and Related Equations
Ockendon, J. R.
2012-01-01
This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations-the Helmholtz, modified Helmholtz, and convection-diffusion equations, and the heat conduction equation in the frequency domain-and the connections between these equations for this particular class of solutions.S pecifically, we consider thin-layer solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero.F or the wellstudied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail.E xamples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes.It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation.W e also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others.In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions. © 2012 Society for Industrial and Applied Mathematics.
New finite volume methods for approximating partial differential equations on arbitrary meshes
International Nuclear Information System (INIS)
Hermeline, F.
2008-12-01
This dissertation presents some new methods of finite volume type for approximating partial differential equations on arbitrary meshes. The main idea lies in solving twice the problem to be dealt with. One addresses the elliptic equations with variable (anisotropic, antisymmetric, discontinuous) coefficients, the parabolic linear or non linear equations (heat equation, radiative diffusion, magnetic diffusion with Hall effect), the wave type equations (Maxwell, acoustics), the elasticity and Stokes'equations. Numerous numerical experiments show the good behaviour of this type of method. (author)
Integration of differential equations by the pseudo-linear (PL) approximation
International Nuclear Information System (INIS)
Bonalumi, Riccardo A.
1998-01-01
A new method of integrating differential equations was originated with the technique of approximately calculating the integrals called the pseudo-linear (PL) procedure: this method is A-stable. This article contains the following examples: 1st order ordinary differential equations (ODEs), 2nd order linear ODEs, stiff system of ODEs (neutron kinetics), one-dimensional parabolic (diffusion) partial differential equations. In this latter case, this PL method coincides with the Crank-Nicholson method
Controllability of partial differential equations governed by multiplicative controls
Khapalov, Alexander Y
2010-01-01
The goal of this monograph is to address the issue of the global controllability of partial differential equations in the context of multiplicative (or bilinear) controls, which enter the model equations as coefficients. The mathematical models we examine include the linear and nonlinear parabolic and hyperbolic PDE's, the Schrödinger equation, and coupled hybrid nonlinear distributed parameter systems modeling the swimming phenomenon. The book offers a new, high-quality and intrinsically nonlinear methodology to approach the aforementioned highly nonlinear controllability problems.
Ó, João; Tomei, Carlos
2014-01-01
This volume is a collection of articles presented at the Workshop for Nonlinear Analysis held in João Pessoa, Brazil, in September 2012. The influence of Bernhard Ruf, to whom this volume is dedicated on the occasion of his 60th birthday, is perceptible throughout the collection by the choice of themes and techniques. The many contributors consider modern topics in the calculus of variations, topological methods and regularity analysis, together with novel applications of partial differential equations. In keeping with the tradition of the workshop, emphasis is given to elliptic operators inserted in different contexts, both theoretical and applied. Topics include semi-linear and fully nonlinear equations and systems with different nonlinearities, at sub- and supercritical exponents, with spectral interactions of Ambrosetti-Prodi type. Also treated are analytic aspects as well as applications such as diffusion problems in mathematical genetics and finance and evolution equations related to electromechanical ...
Federal technology alert. Parabolic-trough solar water heating
Energy Technology Data Exchange (ETDEWEB)
NONE
1998-04-01
Parabolic-trough solar water heating is a well-proven renewable energy technology with considerable potential for application at Federal facilities. For the US, parabolic-trough water-heating systems are most cost effective in the Southwest where direct solar radiation is high. Jails, hospitals, barracks, and other facilities that consistently use large volumes of hot water are particularly good candidates, as are facilities with central plants for district heating. As with any renewable energy or energy efficiency technology requiring significant initial capital investment, the primary condition that will make a parabolic-trough system economically viable is if it is replacing expensive conventional water heating. In combination with absorption cooling systems, parabolic-trough collectors can also be used for air-conditioning. Industrial Solar Technology (IST) of Golden, Colorado, is the sole current manufacturer of parabolic-trough solar water heating systems. IST has an Indefinite Delivery/Indefinite Quantity (IDIQ) contract with the Federal Energy Management Program (FEMP) of the US Department of Energy (DOE) to finance and install parabolic-trough solar water heating on an Energy Savings Performance Contract (ESPC) basis for any Federal facility that requests it and for which it proves viable. For an ESPC project, the facility does not pay for design, capital equipment, or installation. Instead, it pays only for guaranteed energy savings. Preparing and implementing delivery or task orders against the IDIQ is much simpler than the standard procurement process. This Federal Technology Alert (FTA) of the New Technology Demonstration Program is one of a series of guides to renewable energy and new energy-efficient technologies.
Handbook of Nonlinear Partial Differential Equations
Polyanin, Andrei D
2011-01-01
New to the Second Edition More than 1,000 pages with over 1,500 new first-, second-, third-, fourth-, and higher-order nonlinear equations with solutions Parabolic, hyperbolic, elliptic, and other systems of equations with solutions Some exact methods and transformations Symbolic and numerical methods for solving nonlinear PDEs with Maple(t), Mathematica(R), and MATLAB(R) Many new illustrative examples and tables A large list of references consisting of over 1,300 sources To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology. They
Finite-dimensional attractor for a composite system of wave/plate equations with localized damping
International Nuclear Information System (INIS)
Bucci, Francesca; Toundykov, Daniel
2010-01-01
The long-term behaviour of solutions to a model for acoustic–structure interactions is addressed; the system consists of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are the existence of a global attractor for the dynamics generated by this composite system as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dimension—established in a previous work by Bucci et al (2007 Commun. Pure Appl. Anal. 6 113–40) only in the presence of full-interior acoustic damping—holds even in the case of localized dissipation. This nontrivial generalization is inspired by, and consistent with, the recent advances in the study of wave equations with nonlinear localized damping
Global existence of solutions for semilinear damped wave equation in 2-D exterior domain
Ikehata, Ryo
We consider a mixed problem of a damped wave equation utt-Δ u+ ut=| u| p in the two dimensional exterior domain case. Small global in time solutions can be constructed in the case when the power p on the nonlinear term | u| p satisfies p ∗=2Japon. 55 (2002) 33) plays an effective role.
Nearly Interactive Parabolized Navier-Stokes Solver for High Speed Forebody and Inlet Flows
Benson, Thomas J.; Liou, May-Fun; Jones, William H.; Trefny, Charles J.
2009-01-01
A system of computer programs is being developed for the preliminary design of high speed inlets and forebodies. The system comprises four functions: geometry definition, flow grid generation, flow solver, and graphics post-processor. The system runs on a dedicated personal computer using the Windows operating system and is controlled by graphical user interfaces written in MATLAB (The Mathworks, Inc.). The flow solver uses the Parabolized Navier-Stokes equations to compute millions of mesh points in several minutes. Sample two-dimensional and three-dimensional calculations are demonstrated in the paper.
Adaptive distributed parameter and input estimation in linear parabolic PDEs
Mechhoud, Sarra
2016-01-01
In this paper, we discuss the on-line estimation of distributed source term, diffusion, and reaction coefficients of a linear parabolic partial differential equation using both distributed and interior-point measurements. First, new sufficient identifiability conditions of the input and the parameter simultaneous estimation are stated. Then, by means of Lyapunov-based design, an adaptive estimator is derived in the infinite-dimensional framework. It consists of a state observer and gradient-based parameter and input adaptation laws. The parameter convergence depends on the plant signal richness assumption, whereas the state convergence is established using a Lyapunov approach. The results of the paper are illustrated by simulation on tokamak plasma heat transport model using simulated data.
Thermal behaviour of a solar air heater with a compound parabolic concentrator
International Nuclear Information System (INIS)
Tchinda, R.
2005-11-01
A mathematical model for computing the thermal performance of an air heater with a truncated compound parabolic concentrator having a flat one-sided absorber is presented. A computed code that employs an iterative solution procedure is constructed to solve the governing energy equations and to estimate the performance parameters of the collector. The effects of the air mass flow rate, the wind speed and the collector length on the thermal performance of the present air heater are investigated. Prediction for the performance of the solar heater also exhibits reasonable agreement with experimental data with an average error of 7%. (author)
Moduli of Parabolic Higgs Bundles and Atiyah Algebroids
DEFF Research Database (Denmark)
Logares, Marina; Martens, Johan
2010-01-01
In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one on the dual of an Atiyah algebroid over the moduli space of parabolic vector bundle...
Pseudodifferential equations over non-Archimedean spaces
Zúñiga-Galindo, W A
2016-01-01
Focusing on p-adic and adelic analogues of pseudodifferential equations, this monograph presents a very general theory of parabolic-type equations and their Markov processes motivated by their connection with models of complex hierarchic systems. The Gelfand-Shilov method for constructing fundamental solutions using local zeta functions is developed in a p-adic setting and several particular equations are studied, such as the p-adic analogues of the Klein-Gordon equation. Pseudodifferential equations for complex-valued functions on non-Archimedean local fields are central to contemporary harmonic analysis and mathematical physics and their theory reveals a deep connection with probability and number theory. The results of this book extend and complement the material presented by Vladimirov, Volovich and Zelenov (1994) and Kochubei (2001), which emphasize spectral theory and evolution equations in a single variable, and Albeverio, Khrennikov and Shelkovich (2010), which deals mainly with the theory and applica...
Numerical approximations of difference functional equations and applications
Directory of Open Access Journals (Sweden)
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.
Directory of Open Access Journals (Sweden)
Wen-ku Shi
2016-01-01
Full Text Available The composite stiffness of parabolic leaf springs with variable stiffness is difficult to calculate using traditional integral equations. Numerical integration or FEA may be used but will require computer-aided software and long calculation times. An efficient method for calculating the composite stiffness of parabolic leaf springs with variable stiffness is developed and evaluated to reduce the complexity of calculation and shorten the calculation time. A simplified model for double-leaf springs with variable stiffness is built, and a composite stiffness calculation method for the model is derived using displacement superposition and material deformation continuity. The proposed method can be applied on triple-leaf and multileaf springs. The accuracy of the calculation method is verified by the rig test and FEA analysis. Finally, several parameters that should be considered during the design process of springs are discussed. The rig test and FEA analytical results indicate that the calculated results are acceptable. The proposed method can provide guidance for the design and production of parabolic leaf springs with variable stiffness. The composite stiffness of the leaf spring can be calculated quickly and accurately when the basic parameters of the leaf spring are known.
Jiang, Daijun; Li, Zhiyuan; Liu, Yikan; Yamamoto, Masahiro
2017-05-01
In this paper, we first establish a weak unique continuation property for time-fractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic equations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the uniqueness for an inverse problem on determining the spatial component in the source term by interior measurements. Numerically, we reformulate our inverse source problem as an optimization problem, and propose an iterative thresholding algorithm. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm.
Moduli space of Parabolic vector bundles over hyperelliptic curves
Indian Academy of Sciences (India)
27
This has been generalized for higher dimensional varieties by Maruyama ... Key words and phrases. Parabolic structure .... Let E be a vector bundle of rank r on X. Recall that a parabolic ..... Let us understand this picture geometrically. Let ω1 ...
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.
Existence of extremal periodic solutions for quasilinear parabolic equations
Directory of Open Access Journals (Sweden)
Siegfried Carl
1997-01-01
bounded domain under periodic Dirichlet boundary conditions. Our main goal is to prove the existence of extremal solutions among all solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, comparison and truncation techniques and suitably constructed special testfunction.
A compact representation of drawing movements with sequences of parabolic primitives.
Directory of Open Access Journals (Sweden)
Felix Polyakov
2009-07-01
Full Text Available Some studies suggest that complex arm movements in humans and monkeys may optimize several objective functions, while others claim that arm movements satisfy geometric constraints and are composed of elementary components. However, the ability to unify different constraints has remained an open question. The criterion for a maximally smooth (minimizing jerk motion is satisfied for parabolic trajectories having constant equi-affine speed, which thus comply with the geometric constraint known as the two-thirds power law. Here we empirically test the hypothesis that parabolic segments provide a compact representation of spontaneous drawing movements. Monkey scribblings performed during a period of practice were recorded. Practiced hand paths could be approximated well by relatively long parabolic segments. Following practice, the orientations and spatial locations of the fitted parabolic segments could be drawn from only 2-4 clusters, and there was less discrepancy between the fitted parabolic segments and the executed paths. This enabled us to show that well-practiced spontaneous scribbling movements can be represented as sequences ("words" of a small number of elementary parabolic primitives ("letters". A movement primitive can be defined as a movement entity that cannot be intentionally stopped before its completion. We found that in a well-trained monkey a movement was usually decelerated after receiving a reward, but it stopped only after the completion of a sequence composed of several parabolic segments. Piece-wise parabolic segments can be generated by applying affine geometric transformations to a single parabolic template. Thus, complex movements might be constructed by applying sequences of suitable geometric transformations to a few templates. Our findings therefore suggest that the motor system aims at achieving more parsimonious internal representations through practice, that parabolas serve as geometric primitives and that non
Arumugam, S.; Ramakrishna, P.; Sangavi, S.
2018-02-01
Improvements in heating technology with solar energy is gaining focus, especially solar parabolic collectors. Solar heating in conventional parabolic collectors is done with the help of radiation concentration on receiver tubes. Conventional receiver tubes are open to atmosphere and loose heat by ambient air currents. In order to reduce the convection losses and also to improve the aperture area, we designed a tube with cavity. This study is a comparative performance behaviour of conventional tube and cavity model tube. The performance formulae were derived for the cavity model based on conventional model. Reduction in overall heat loss coefficient was observed for cavity model, though collector heat removal factor and collector efficiency were nearly same for both models. Improvement in efficiency was also observed in the cavity model’s performance. The approach towards the design of a cavity model tube as the receiver tube in solar parabolic collectors gave improved results and proved as a good consideration.
A One-Dimensional Wave Equation with White Noise Boundary Condition
International Nuclear Information System (INIS)
Kim, Jong Uhn
2006-01-01
We discuss the Cauchy problem for a one-dimensional wave equation with white noise boundary condition. We also establish the existence of an invariant measure when the noise is additive. Similar problems for parabolic equations were discussed by several authors. To our knowledge, there is only one work which investigated the initial-boundary value problem for a wave equation with random noise at the boundary. We handle a more general case by a different method. Our result on the existence of an invariant measure relies on the author's recent work on a certain class of stochastic evolution equations
Yan, Na; Baas, Andreas
2015-04-01
Parabolic dunes are one of a few common aeolian landforms which are highly controlled by eco-geomorphic interactions. Parabolic dunes, on the one hand, can be developed from highly mobile dune landforms, barchans for instance, in an ameliorated vegetation condition; or on the other hand, they can be reactivated and transformed back into mobile dunes due to vegetation deterioration. The fundamental mechanisms and eco-geomorphic interactions controlling both dune transformations remain poorly understood. To bridge the gap between complex processes involved in dune transformations on a relatively long temporal scale and real world monitoring records on a very limited temporal scale, this research has extended the DECAL model to incorporate 'dynamic' growth functions and the different 'growth' of perennial shrubs between growing and non-growing seasons, informed by field measurements and remote sensing analysis, to explore environmental controls and eco-geomorphic interactions of both types of dune transformation. A non-dimensional 'dune stabilising index' is proposed to capture the interactions between environmental controls (i.e. the capabilities of vegetation to withstand wind erosion and sand burial, the sandy substratum thickness, the height of the initial dune, and the sand transport potential), and establish the linkage between these controls and the geometry of a stabilising dune. An example demonstrates how to use the power-law relationship between the dune stabilising index and the normalised migration distance to assist in extrapolating the historical trajectories of transforming dunes. The modelling results also show that a slight increase in vegetation cover of an initial parabolic dune can significantly increase the reactivation threshold of climatic impact (both drought stress and wind strength) required to reactivate a stabilising parabolic dune into a barchan. Four eco-geomorphic interaction zones that govern a barchan-to-parabolic dune transformation
Finite-time blow-up for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
Hashira, Takahiro; Ishida, Sachiko; Yokota, Tomomi
2018-05-01
This paper deals with the quasilinear degenerate Keller-Segel systems of parabolic-parabolic type in a ball of RN (N ≥ 2). In the case of non-degenerate diffusion, Cieślak-Stinner [3,4] proved that if q > m + 2/N, where m denotes the intensity of diffusion and q denotes the nonlinearity, then there exist initial data such that the corresponding solution blows up in finite time. As to the case of degenerate diffusion, it is known that a solution blows up if q > m + 2/N (see Ishida-Yokota [13]); however, whether the blow-up time is finite or infinite has been unknown. This paper gives an answer to the unsolved problem. Indeed, the finite-time blow-up of energy solutions is established when q > m + 2/N.
Projection scheme for a reflected stochastic heat equation with additive noise
Higa, Arturo Kohatsu; Pettersson, Roger
2005-02-01
We consider a projection scheme as a numerical solution of a reflected stochastic heat equation driven by a space-time white noise. Convergence is obtained via a discrete contraction principle and known convergence results for numerical solutions of parabolic variational inequalities.
Semigroup methods for evolution equations on networks
Mugnolo, Delio
2014-01-01
This concise text is based on a series of lectures held only a few years ago and originally intended as an introduction to known results on linear hyperbolic and parabolic equations. Yet the topic of differential equations on graphs, ramified spaces, and more general network-like objects has recently gained significant momentum and, well beyond the confines of mathematics, there is a lively interdisciplinary discourse on all aspects of so-called complex networks. Such network-like structures can be found in virtually all branches of science, engineering and the humanities, and future research thus calls for solid theoretical foundations. This book is specifically devoted to the study of evolution equations – i.e., of time-dependent differential equations such as the heat equation, the wave equation, or the Schrödinger equation (quantum graphs) – bearing in mind that the majority of the literature in the last ten years on the subject of differential equations of graphs has been devoted to ellip...
Perturbation theory for continuous stochastic equations
International Nuclear Information System (INIS)
Chechetkin, V.R.; Lutovinov, V.S.
1987-01-01
The various general perturbational schemes for continuous stochastic equations are considered. These schemes have many analogous features with the iterational solution of Schwinger equation for S-matrix. The following problems are discussed: continuous stochastic evolution equations for probability distribution functionals, evolution equations for equal time correlators, perturbation theory for Gaussian and Poissonian additive noise, perturbation theory for birth and death processes, stochastic properties of systems with multiplicative noise. The general results are illustrated by diffusion-controlled reactions, fluctuations in closed systems with chemical processes, propagation of waves in random media in parabolic equation approximation, and non-equilibrium phase transitions in systems with Poissonian breeding centers. The rate of irreversible reaction X + X → A (Smoluchowski process) is calculated with the use of general theory based on continuous stochastic equations for birth and death processes. The threshold criterion and range of fluctuational region for synergetic phase transition in system with Poissonian breeding centers are also considered. (author)
Directory of Open Access Journals (Sweden)
Khaled Zaki
2016-12-01
Full Text Available We establish the existence of solutions for the nonlinear parabolic problem with Dirichlet homogeneous boundary conditions, $$ \\frac{\\partial u}{\\partial t} - \\sum_{i=1}^N\\frac{\\partial}{\\partial x_i} \\Big( d_i(u\\frac{\\partial u}{\\partial x_i} \\Big =\\mu,\\quad u(t=0=u_0, $$ in a bounded domain. The coefficients $d_i(s$ are continuous on an interval $]-\\infty,m[$, there exists an index p such that $d_p(u$ blows up at a finite value m of the unknown u, and $\\mu$ is a diffuse measure.
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.)
Energy Technology Data Exchange (ETDEWEB)
Sternberg, K
2007-02-08
Molten carbonate fuel cells (MCFCs) allow an efficient and environmentally friendly energy production by converting the chemical energy contained in the fuel gas in virtue of electro-chemical reactions. In order to predict the effect of the electro-chemical reactions and to control the dynamical behavior of the fuel cell a mathematical model has to be found. The molten carbonate fuel cell (MCFC) can indeed be described by a highly complex,large scale, semi-linear system of partial differential algebraic equations. This system includes a reaction-diffusion-equation of parabolic type, several reaction-transport-equations of hyperbolic type, several ordinary differential equations and finally a system of integro-differential algebraic equations which describes the nonlinear non-standard boundary conditions for the entire partial differential algebraic equation system (PDAE-system). The existence of an analytical or the computability of a numerical solution for this high-dimensional PDAE-system depends on the kind of the differential equations and their special characteristics. Apart from theoretical investigations, the real process has to be controlled, more precisely optimally controlled. Hence, on the basis of the PDAE-system an optimal control problem is set up, whose analytical and numerical solvability is closely linked to the solvability of the PDAE-system. Moreover the solution of that optimal control problem is made more difficult by inaccuracies in the underlying database, which does not supply sufficiently accurate values for the model parameters. Therefore the optimal control problem must also be investigated with respect to small disturbances of model parameters. The aim of this work is to analyze the relevant dynamic behavior of MCFCs and to develop concepts for their optimal process control. Therefore this work is concerned with the simulation, the optimal control and the sensitivity analysis of a mathematical model for MCDCs, which can be characterized
Temperature waves and the Boltzmann kinetic equation for phonons
International Nuclear Information System (INIS)
Urushev, D.; Borisov, M.; Vavrek, A.
1988-01-01
The ordinary parabolic equation for thermal conduction based on the Fourier empiric law as well as the generalized thermal conduction equation based on the Maxwell law have been derived from the Boltzmann equation for the phonons within the relaxation time approximation. The temperature waves of the so-called second sound in crystals at low temperatures are transformed into Fourier waves at low frequencies with respect to the characteristic frequency of the U-processes. These waves are transformed into temperature waves similar to the second sound waves in He II at frequences higher than the U-processes characteristic. 1 fig., 19 refs
Boundary Control of Linear Uncertain 1-D Parabolic PDE Using Approximate Dynamic Programming.
Talaei, Behzad; Jagannathan, Sarangapani; Singler, John
2018-04-01
This paper develops a near optimal boundary control method for distributed parameter systems governed by uncertain linear 1-D parabolic partial differential equations (PDE) by using approximate dynamic programming. A quadratic surface integral is proposed to express the optimal cost functional for the infinite-dimensional state space. Accordingly, the Hamilton-Jacobi-Bellman (HJB) equation is formulated in the infinite-dimensional domain without using any model reduction. Subsequently, a neural network identifier is developed to estimate the unknown spatially varying coefficient in PDE dynamics. Novel tuning law is proposed to guarantee the boundedness of identifier approximation error in the PDE domain. A radial basis network (RBN) is subsequently proposed to generate an approximate solution for the optimal surface kernel function online. The tuning law for near optimal RBN weights is created, such that the HJB equation error is minimized while the dynamics are identified and closed-loop system remains stable. Ultimate boundedness (UB) of the closed-loop system is verified by using the Lyapunov theory. The performance of the proposed controller is successfully confirmed by simulation on an unstable diffusion-reaction process.
International Nuclear Information System (INIS)
David Kearney; Hank Price
1999-01-01
Technology roadmapping is a needs-driven technology planning process to help identify, select, and develop technology alternatives to satisfy a set of market needs. The DOE's Office of Power Technologies' Concentrating Solar Power (CSP) Program recently sponsored a technology roadmapping workshop for parabolic trough technology. The workshop was attended by an impressive cross section of industry and research experts. The goals of the workshop were to evaluate the market potential for trough power projects, develop a better understanding of the current state of the technology, and to develop a conceptual plan for advancing the state of parabolic trough technology. This report documents and extends the roadmap that was conceptually developed during the workshop
Directory of Open Access Journals (Sweden)
M. Tshipa
2017-12-01
Full Text Available A theoretical investigation of the effects of spatial variation of confining electric potential on photoionization cross section (PCS in a spherical quantum dot is presented. The potential profiles considered here are the shifted parabolic potential and the inverse lateral shifted parabolic potential compared with the well-studied parabolic potential. The primary findings are that parabolic potential and the inverse lateral shifted parabolic potential blue shift the peaks of the PCS while the shifted parabolic potential causes a red shift.
Hermitian-Einstein metrics on parabolic stable bundles
International Nuclear Information System (INIS)
Li Jiayu; Narasimhan, M.S.
1995-12-01
Let M-bar be a compact complex manifold of complex dimension two with a smooth Kaehler metric and D a smooth divisor on M-bar. If E is a rank 2 holomorphic vector bundle on M-bar with a stable parabolic structure along D, we prove the existence of a metric on E' = E module MbarD (compatible with the parabolic structure) which is Hermitian-Einstein with respect to the restriction of Kaehler metric of M-barD. A converse is also proved. (author). 24 refs
Explosive solutions of elliptic equations with absorption and non ...
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
control theory and have been first studied by Lasry and Lions [8]. The corresponding parabolic equation was considered in Quittner [12]. In terms of the dynamic programming approach, an explosive solution of (1) corresponds to a value function (or Bellman function) associated to an infinite exit cost (see [8]). Bandle and ...
Diffusive instabilities in hyperbolic reaction-diffusion equations
Zemskov, Evgeny P.; Horsthemke, Werner
2016-03-01
We investigate two-variable reaction-diffusion systems of the hyperbolic type. A linear stability analysis is performed, and the conditions for diffusion-driven instabilities are derived. Two basic types of eigenvalues, real and complex, are described. Dispersion curves for both types of eigenvalues are plotted and their behavior is analyzed. The real case is related to the Turing instability, and the complex one corresponds to the wave instability. We emphasize the interesting feature that the wave instability in the hyperbolic equations occurs in two-variable systems, whereas in the parabolic case one needs three reaction-diffusion equations.
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.
Gerbi, Sté phane; Said-Houari, Belkacem
2011-01-01
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.
A semilinear parabolic–elliptic chemotaxis system with critical mass in any space dimension
International Nuclear Information System (INIS)
Montaru, Alexandre
2013-01-01
We study radial solutions in a ball of R N of a semilinear, parabolic–elliptic Patlak–Keller–Segel system with a nonlinear sensitivity involving a critical power. For N = 2, the latter reduces to the classical ‘linear’ model, well known for its critical mass 8π. We show that a critical mass phenomenon also occurs for N ⩾ 3, but with a strongly different qualitative behaviour. More precisely, if the total mass of cells is smaller or equal to the critical mass M-bar , then the cell density converges to a regular steady state that is supported strictly inside the ball as time goes to infinity. In the case of the critical mass, this result is nontrivial since there exists a continuum of stationary solutions and is moreover in sharp contrast with the case N = 2 where infinite-time blow-up occurs. If the total mass of cells is larger than M-bar , then all radial solutions blow up in finite time. This actually follows from the existence (unlike for N = 2) of a family of self-similar, blowing-up solutions that are supported strictly inside the ball. (paper)
Describing Quadratic Cremer Point Polynomials by Parabolic Perturbations
DEFF Research Database (Denmark)
Sørensen, Dan Erik Krarup
1996-01-01
We describe two infinite order parabolic perturbation proceduresyielding quadratic polynomials having a Cremer fixed point. The main ideais to obtain the polynomial as the limit of repeated parabolic perturbations.The basic tool at each step is to control the behaviour of certain externalrays.......Polynomials of the Cremer type correspond to parameters at the boundary of ahyperbolic component of the Mandelbrot set. In this paper we concentrate onthe main cardioid component. We investigate the differences between two-sided(i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove...... the existence of polynomials having an explicitlygiven external ray accumulating both at the Cremer point and at its non-periodicpreimage. We think of the Julia set as containing a "topologists double comb".In the one-sided case we prove a weaker result: the existence of polynomials havingan explicitly given...
Parabolized Navier-Stokes solutions of separation and trailing-edge flows
Brown, J. L.
1983-01-01
A robust, iterative solution procedure is presented for the parabolized Navier-Stokes or higher order boundary layer equations as applied to subsonic viscous-inviscid interaction flows. The robustness of the present procedure is due, in part, to an improved algorithmic formulation. The present formulation is based on a reinterpretation of stability requirements for this class of algorithms and requires only second order accurate backward or central differences for all streamwise derivatives. Upstream influence is provided for through the algorithmic formulation and iterative sweeps in x. The primary contribution to robustness, however, is the boundary condition treatment, which imposes global constraints to control the convergence path. Discussed are successful calculations of subsonic, strong viscous-inviscid interactions, including separation. These results are consistent with Navier-Stokes solutions and triple deck theory.
Mathematical analysis of partial differential equations modeling electrostatic MEMS
Esposito, Pierpaolo; Guo, Yujin
2010-01-01
Micro- and nanoelectromechanical systems (MEMS and NEMS), which combine electronics with miniature-size mechanical devices, are essential components of modern technology. It is the mathematical model describing "electrostatically actuated" MEMS that is addressed in this monograph. Even the simplified models that the authors deal with still lead to very interesting second- and fourth-order nonlinear elliptic equations (in the stationary case) and to nonlinear parabolic equations (in the dynamic case). While nonlinear eigenvalue problems-where the stationary MEMS models fit-are a well-developed
Mechatronic Prototype of Parabolic Solar Tracker.
Morón, Carlos; Díaz, Jorge Pablo; Ferrández, Daniel; Ramos, Mari Paz
2016-06-15
In the last 30 years numerous attempts have been made to improve the efficiency of the parabolic collectors in the electric power production, although most of the studies have focused on the industrial production of thermoelectric power. This research focuses on the application of this concentrating solar thermal power in the unexplored field of building construction. To that end, a mechatronic prototype of a hybrid paraboloidal and cylindrical-parabolic tracker based on the Arduido technology has been designed. The prototype is able to measure meteorological data autonomously in order to quantify the energy potential of any location. In this way, it is possible to reliably model real commercial equipment behavior before its deployment in buildings and single family houses.
Weyl states and Fermi arcs in parabolic bands
Doria, Mauro M.; Perali, Andrea
2017-07-01
Weyl fermions are shown to exist inside a parabolic band in a single electronic layer, where the kinetic energy of carriers is given by the non-relativistic Schroedinger equation. There are Fermi arcs as a direct consequence of the folding of a ring-shaped Fermi surface inside the first Brillouin zone. Our results stem from the decomposition of the kinetic energy into the sum of the square of the Weyl state, the coupling to the local magnetic field and the Rashba interaction. The Weyl fermions break the space and time reflection symmetries present in the kinetic energy, thus allowing for the onset of a weak three-dimensional magnetic field around the layer. This field brings topological stability to the current-carrying states through a Chern number. In the special limit for which the Weyl state becomes gapless, this magnetic interaction is shown to be purely attractive, thus suggesting the onset of a superconducting condensate of zero helicity states.
Interaction Potential between Parabolic Rotator and an Outside Particle
Directory of Open Access Journals (Sweden)
Dan Wang
2014-01-01
Full Text Available At micro/nanoscale, the interaction potential between parabolic rotator and a particle located outside the rotator is studied on the basis of the negative exponential pair potential 1/Rn between particles. Similar to two-dimensional curved surfaces, we confirm that the potential of the three-dimensional parabolic rotator and outside particle can also be expressed as a unified form of curvatures; that is, it can be written as the function of curvatures. Furthermore, we verify that the driving forces acting on the particle may be induced by the highly curved micro/nano-parabolic rotator. Curvatures and the gradient of curvatures are the essential elements forming the driving forces. Through the idealized numerical experiments, the accuracy of the curvature-based potential is preliminarily proved.
Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation
Energy Technology Data Exchange (ETDEWEB)
Cortazar, C.; Elgueta, M. [Universidad Catolica, Santiago (Chile); Felmer, P. [Universidad de Chile, Santiago (Chile)
1996-12-31
We will consider in this paper a semilinear elliptic equation {triangle}u + f(u) = 0 in {Omega}, (1.5) where the function f is locally Lipschitz in (0,{infinity}) and continuous in (0,{infinity}). We study symmetry properties of nonnegative solutions of this equation in two different situations: first we assume {Omega} = IR{sup N}, and second we consider {Omega} {ne} IR{sup N} and we provide (1.5) with overdetermined boundary conditions. Next we describe our results in the first case, that is, when {Omega} = IR{sup N}. We will consider the following hypothesis on the nonlinear function f (F) f(0) {le} 0, f continuous in (0,+{infinity}), locally Lipschitz in (0,+{infinity}) and there exists {alpha} > 0 so that f is strictly decreasing in [0,{alpha}]. We note that the support of a solution of (1.5) is not known a priori and so we have in fact a free boundary involved. Our goal is to determine the shape of this support and the symmetry properties of the solution.
Dinar, N.
1978-01-01
Several aspects of multigrid methods are briefly described. The main subjects include the development of very efficient multigrid algorithms for systems of elliptic equations (Cauchy-Riemann, Stokes, Navier-Stokes), as well as the development of control and prediction tools (based on local mode Fourier analysis), used to analyze, check and improve these algorithms. Preliminary research on multigrid algorithms for time dependent parabolic equations is also described. Improvements in existing multigrid processes and algorithms for elliptic equations were studied.
An air-based corrugated cavity-receiver for solar parabolic trough concentrators
International Nuclear Information System (INIS)
Bader, Roman; Pedretti, Andrea; Barbato, Maurizio; Steinfeld, Aldo
2015-01-01
Highlights: • We analyze a novel tubular cavity-receiver for solar parabolic trough collectors. • Four-fold solar concentration ratio is reached compared to conventional receivers. • Efficient operation at up to 500 °C is possible. • The pumping power requirement is found to be acceptably low. - Abstract: A tubular cavity-receiver that uses air as the heat transfer fluid is evaluated numerically using a validated heat transfer model. The receiver is designed for use on a large-span (9 m net concentrator aperture width) solar parabolic trough concentrator. Through the combination of a parabolic primary concentrator with a nonimaging secondary concentrator, the collector reaches a solar concentration ratio of 97.5. Four different receiver configurations are considered, with smooth or V-corrugated absorber tube and single- or double-glazed aperture window. The collector’s performance is characterized by its optical efficiency and heat loss. The optical efficiency is determined with the Monte Carlo ray-tracing method. Radiative heat exchange inside the receiver is calculated with the net radiation method. The 2D steady-state energy equation, which couples conductive, convective, and radiative heat transfer, is solved for the solid domains of the receiver cross-section, using finite-volume techniques. Simulations for Sevilla/Spain at the summer solstice at solar noon (direct normal solar irradiance: 847 W m −2 , solar incidence angle: 13.9°) yield collector efficiencies between 60% and 65% at a heat transfer fluid temperature of 125 °C and between 37% and 42% at 500 °C, depending on the receiver configuration. The optical losses amount to more than 30% of the incident solar radiation and constitute the largest source of energy loss. For a 200 m long collector module operated between 300 and 500 °C, the isentropic pumping power required to pump the HTF through the receiver is between 11 and 17 kW
Mechatronic Prototype of Parabolic Solar Tracker
Directory of Open Access Journals (Sweden)
Carlos Morón
2016-06-01
Full Text Available In the last 30 years numerous attempts have been made to improve the efficiency of the parabolic collectors in the electric power production, although most of the studies have focused on the industrial production of thermoelectric power. This research focuses on the application of this concentrating solar thermal power in the unexplored field of building construction. To that end, a mechatronic prototype of a hybrid paraboloidal and cylindrical-parabolic tracker based on the Arduido technology has been designed. The prototype is able to measure meteorological data autonomously in order to quantify the energy potential of any location. In this way, it is possible to reliably model real commercial equipment behavior before its deployment in buildings and single family houses.
Nanofocusing parabolic refractive x-ray lenses
International Nuclear Information System (INIS)
Schroer, C.G.; Kuhlmann, M.; Hunger, U.T.; Guenzler, T.F.; Kurapova, O.; Feste, S.; Frehse, F.; Lengeler, B.; Drakopoulos, M.; Somogyi, A.; Simionovici, A.S.; Snigirev, A.; Snigireva, I.; Schug, C.; Schroeder, W.H.
2003-01-01
Parabolic refractive x-ray lenses with short focal distance can generate intensive hard x-ray microbeams with lateral extensions in the 100 nm range even at a short distance from a synchrotron radiation source. We have fabricated planar parabolic lenses made of silicon that have a focal distance in the range of a few millimeters at hard x-ray energies. In a crossed geometry, two lenses were used to generate a microbeam with a lateral size of 380 nm by 210 nm at 25 keV in a distance of 42 m from the synchrotron radiation source. Using diamond as the lens material, microbeams with a lateral size down to 20 nm and below are conceivable in the energy range from 10 to 100 keV
Energy Technology Data Exchange (ETDEWEB)
Ajona, J I; Alberdi, J; Gamero, E; Blanco, J
1992-07-01
In the local control, the sun position related to the trough collector is measured by two photo-resistors. The provided electronic signal is then compared with reference levels in order to get a set of B logical signals which form a byte. This byte and the commands issued by a programmable controller are connected to the inputs of o P.R.O.M. memory which is programmed with the logical equations of the control system. The memory output lines give the control command of the parabolic trough collector motor. (Author)
Computer aided FEA simulation of EN45A parabolic leaf spring
Directory of Open Access Journals (Sweden)
Krishan Kumar
2013-04-01
Full Text Available This paper describes computer aided finite element analysis of parabolic leaf spring. The present work is an improvement in design of EN45A parabolic leaf spring used by a light commercial automotive vehicle. Development of a leaf spring is a long process which requires lots of test to validate the design and manufacturing variables. A three-layer parabolic leaf spring of EN45A has been taken for this work. The thickness of leaves varies from center to the outer side following a parabolic pattern. These leaf springs are designed to become lighter, but also provide a much improved ride to the vehicle through a reduction on interleaf friction. The CAD modeling of parabolic leaf spring has been done in CATIA V5 and for analysis the model is imported in ANSYS-11 workbench. The finite element analysis (FEA of the leaf spring has been carried out by initially discretizing the model into finite number of elements and nodes and then applying the necessary boundary conditions. Maximum displacement, directional displacement, equivalent stress and weight of the assembly are the output targets of this analysis for comparison & validation of the work.
A comparative study of the parabolized Navier-Stokes code using various grid-generation techniques
Kaul, U. K.; Chaussee, D. S.
1985-01-01
The parabolized Navier-Stokes (PNS) equations are used to calculate the flow-field characteristics about the hypersonic research aircraft X-24C. A comparison of the results obtained using elliptic, hyperbolic and algebraic grid generators is presented. The outer bow shock is treated as a sharp discontinuity, and the discontinuities within the shock layer are captured. Surface pressures and heat-transfer results at angles of attack of 6 deg and 20 deg, obtained using the three grid generators, are compared. The PNS equations are marched downstream over the body in both Cartesian and cylindrical base coordinate systems, and the results are compared. A robust marching procedure is demonstrated by successfully using large marching-step sizes with the implicit shock fitting procedure. A correlation is found between the marching-step size, Reynolds number and the angle of attack at fixed values of smoothing and stability coefficients for the marching scheme.
International Nuclear Information System (INIS)
LaChapelle, J.
2004-01-01
A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schroedinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette
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...
Czech Academy of Sciences Publication Activity Database
Escudero, C.; Gazzola, F.; Hakl, Robert; Torres, P.J.
2015-01-01
Roč. 140, č. 4 (2015), s. 385-393 ISSN 0862-7959 Institutional support: RVO:67985840 Keywords : higher order parabolic equation * existence of solution * blow-up in finite time Subject RIV: BA - General Mathematics http://hdl.handle.net/10338.dmlcz/144457
Amitai, Dganit; Averbuch, Amir; Itzikowitz, Samuel; Turkel, Eli
1991-01-01
A major problem in achieving significant speed-up on parallel machines is the overhead involved with synchronizing the concurrent process. Removing the synchronization constraint has the potential of speeding up the computation. The authors present asynchronous (AS) and corrected-asynchronous (CA) finite difference schemes for the multi-dimensional heat equation. Although the discussion concentrates on the Euler scheme for the solution of the heat equation, it has the potential for being extended to other schemes and other parabolic partial differential equations (PDEs). These schemes are analyzed and implemented on the shared memory multi-user Sequent Balance machine. Numerical results for one and two dimensional problems are presented. It is shown experimentally that the synchronization penalty can be about 50 percent of run time: in most cases, the asynchronous scheme runs twice as fast as the parallel synchronous scheme. In general, the efficiency of the parallel schemes increases with processor load, with the time level, and with the problem dimension. The efficiency of the AS may reach 90 percent and over, but it provides accurate results only for steady-state values. The CA, on the other hand, is less efficient, but provides more accurate results for intermediate (non steady-state) values.
MFE revisited : part 1: adaptive grid-generation using the heat equation
Zegeling, P.A.
1996-01-01
In this paper the moving-nite-element method (MFE) is used to solve the heat equation, with an articial time component, to give a non-uniform (steady-state) grid that is adapted to a given prole. It is known from theory and experiments that MFE, applied to parabolic PDEs, gives adaptive grids which
Parabolic Trough Solar Power for Competitive U.S. Markets
International Nuclear Information System (INIS)
Price, Henry W.
1998-01-01
Nine parabolic trough power plants located in the California Mojave Desert represent the only commercial development of large-scale solar power plants to date. Although all nine plants continue to operate today, no new solar power plants have been completed since 1990. Over the last several years, the parabolic trough industry has focused much of its efforts on international market opportunities. Although the power market in developing countries appears to offer a number of opportunities for parabolic trough technologies due to high growth and the availability of special financial incentives for renewables, these markets are also plagued with many difficulties for developers. In recent years, there has been some renewed interest in the U.S. domestic power market as a result of an emerging green market and green pricing incentives. Unfortunately, many of these market opportunities and incentives focus on smaller, more modular technologies (such as photovoltaics or wind power), and as a result they tend to exclude or are of minimum long-term benefit to large-scale concentrating solar power technologies. This paper looks at what is necessary for large-scale parabolic trough solar power plants to compete with state-of-the-art fossil power technology in a competitive U.S. power market
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.
Oscillatory bifurcation for semilinear ordinary differential equations
Directory of Open Access Journals (Sweden)
Tetsutaro Shibata
2016-06-01
\\] where $f(u = u + (1/2\\sin^k u$ ($k \\ge 2$ and $\\lambda > 0$ is a bifurcation parameter. It is known that $\\lambda$ is parameterized by the maximum norm $\\alpha = \\Vert u_\\lambda\\Vert_\\infty$ of the solution $u_\\lambda$ associated with $\\lambda$ and is written as $\\lambda = \\lambda(k,\\alpha$. When we focus on the asymptotic behavior of $\\lambda(k,\\alpha$ as $\\alpha \\to \\infty$, it is natural to expect that $\\lambda(k, \\alpha \\to \\pi^2/4$, and its convergence rate is common to $k$. Contrary to this expectation, we show that $\\lambda(2n_1+1,\\alpha$ tends to $\\pi^2/4$ faster than $\\lambda(2n_2,\\alpha$ as $\\alpha \\to \\infty$, where $n_1\\ge 1,\\ n_2 \\ge 1$ are arbitrary given integers.
Modeling Flow Rate to Estimate Hydraulic Conductivity in a Parabolic Ceramic Water Filter
Directory of Open Access Journals (Sweden)
Ileana Wald
2012-01-01
Full Text Available In this project we model volumetric flow rate through a parabolic ceramic water filter (CWF to determine how quickly it can process water while still improving its quality. The volumetric flow rate is dependent upon the pore size of the filter, the surface area, and the height of water in the filter (hydraulic head. We derive differential equations governing this flow from the conservation of mass principle and Darcy's Law and find the flow rate with respect to time. We then use methods of calculus to find optimal specifications for the filter. This work is related to the research conducted in Dr. James R. Mihelcic's Civil and Environmental Engineering Lab at USF.
Use of a Parabolic Microphone to Detect Hidden Subjects in Search and Rescue.
Bowditch, Nathaniel L; Searing, Stanley K; Thomas, Jeffrey A; Thompson, Peggy K; Tubis, Jacqueline N; Bowditch, Sylvia P
2018-03-01
This study compares a parabolic microphone to unaided hearing in detecting and comprehending hidden callers at ranges of 322 to 2510 m. Eight subjects were placed 322 to 2510 m away from a central listening point. The subjects were concealed, and their calling volume was calibrated. In random order, subjects were asked to call the name of a state for 5 minutes. Listeners with parabolic microphones and others with unaided hearing recorded the direction of the call (detection) and name of the state (comprehension). The parabolic microphone was superior to unaided hearing in both detecting subjects and comprehending their calls, with an effect size (Cohen's d) of 1.58 for detection and 1.55 for comprehension. For each of the 8 hidden subjects, there were 24 detection attempts with the parabolic microphone and 54 to 60 attempts by unaided listeners. At the longer distances (1529-2510 m), the parabolic microphone was better at detecting callers (83% vs 51%; P<0.00001 by χ 2 ) and comprehension (57% vs 12%; P<0.00001). At the shorter distances (322-1190 m), the parabolic microphone offered advantages in detection (100% vs 83%; P=0.000023) and comprehension (86% vs 51%; P<0.00001), although not as pronounced as at the longer distances. Use of a 66-cm (26-inch) parabolic microphone significantly improved detection and comprehension of hidden calling subjects at distances between 322 and 2510 m when compared with unaided hearing. This study supports the use of a parabolic microphone in search and rescue to locate responsive subjects in favorable weather and terrain. Copyright © 2017 The Authors. Published by Elsevier Inc. All rights reserved.
Sasakian and Parabolic Higgs Bundles
Biswas, Indranil; Mj, Mahan
2018-03-01
Let M be a quasi-regular compact connected Sasakian manifold, and let N = M/ S 1 be the base projective variety. We establish an equivalence between the class of Sasakian G-Higgs bundles over M and the class of parabolic (or equivalently, ramified) G-Higgs bundles over the base N.
Blow-up of solutions for the sixth-order thin film equation with ...
Indian Academy of Sciences (India)
College of Mathematics and Statistics, Nanjing University of Information ... By using the improved energy estimate method and by constructing second-order ... In the last 20 years, higher-order nonlinear parabolic partial differential ... in [11] to deal with the second-order p-Laplacian equation (see also [12–14] for further.
Analysis of an upstream weighted collocation approximation to the transport equation
International Nuclear Information System (INIS)
Shapiro, A.; Pinder, G.F.
1981-01-01
The numerical behavior of a modified orthogonal collocation method, as applied to the transport equations, can be examined through the use of a Fourier series analysis. The necessity of such a study becomes apparent in the analysis of several techniques which emulate classical upstream weighting schemes. These techniques are employed in orthogonal collocation and other numerical methods as a means of handling parabolic partial differential equations with significant first-order terms. Divergent behavior can be shown to exist in one upstream weighting method applied to orthogonal collocation
Nanofocusing Parabolic Refractive X-Ray Lenses
International Nuclear Information System (INIS)
Schroer, C.G.; Kuhlmann, M.; Hunger, U.T.; Guenzler, T.F.; Kurapova, O.; Feste, S.; Lengeler, B.; Drakopoulos, M.; Somogyi, A.; Simionovici, A. S.; Snigirev, A.; Snigireva, I.
2004-01-01
Parabolic refractive x-ray lenses with short focal distance can generate intensive hard x-ray microbeams with lateral extensions in the 100nm range even at short distance from a synchrotron radiation source. We have fabricated planar parabolic lenses made of silicon that have a focal distance in the range of a few millimeters at hard x-ray energies. In a crossed geometry, two lenses were used to generate a microbeam with a lateral size of 330nm by 110nm at 25keV in a distance of 41.8m from the synchrotron radiation source. First microdiffraction and fluorescence microtomography experiments were carried out with these lenses. Using diamond as lens material, microbeams with lateral size down to 20nm and below are conceivable in the energy range from 10 to 100keV
Quasilinear parabolic variational inequalities with multi-valued lower-order terms
Carl, Siegfried; Le, Vy K.
2014-10-01
In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain : Find and an such that where is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function is assumed to be upper semicontinuous only, so that Clarke's generalized gradient is included as a special case. Thus, parabolic variational-hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational-hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.
Explicit nonlinear finite element geometric analysis of parabolic leaf springs under various loads.
Kong, Y S; Omar, M Z; Chua, L B; Abdullah, S
2013-01-01
This study describes the effects of bounce, brake, and roll behavior of a bus toward its leaf spring suspension systems. Parabolic leaf springs are designed based on vertical deflection and stress; however, loads are practically derived from various modes especially under harsh road drives or emergency braking. Parabolic leaf springs must sustain these loads without failing to ensure bus and passenger safety. In this study, the explicit nonlinear dynamic finite element (FE) method is implemented because of the complexity of experimental testing A series of load cases; namely, vertical push, wind-up, and suspension roll are introduced for the simulations. The vertical stiffness of the parabolic leaf springs is related to the vehicle load-carrying capability, whereas the wind-up stiffness is associated with vehicle braking. The roll stiffness of the parabolic leaf springs is correlated with the vehicle roll stability. To obtain a better bus performance, two new parabolic leaf spring designs are proposed and simulated. The stress level during the loadings is observed and compared with its design limit. Results indicate that the newly designed high vertical stiffness parabolic spring provides the bus a greater roll stability and a lower stress value compared with the original design. Bus safety and stability is promoted, as well as the load carrying capability.
Explicit Nonlinear Finite Element Geometric Analysis of Parabolic Leaf Springs under Various Loads
Directory of Open Access Journals (Sweden)
Y. S. Kong
2013-01-01
Full Text Available This study describes the effects of bounce, brake, and roll behavior of a bus toward its leaf spring suspension systems. Parabolic leaf springs are designed based on vertical deflection and stress; however, loads are practically derived from various modes especially under harsh road drives or emergency braking. Parabolic leaf springs must sustain these loads without failing to ensure bus and passenger safety. In this study, the explicit nonlinear dynamic finite element (FE method is implemented because of the complexity of experimental testing A series of load cases; namely, vertical push, wind-up, and suspension roll are introduced for the simulations. The vertical stiffness of the parabolic leaf springs is related to the vehicle load-carrying capability, whereas the wind-up stiffness is associated with vehicle braking. The roll stiffness of the parabolic leaf springs is correlated with the vehicle roll stability. To obtain a better bus performance, two new parabolic leaf spring designs are proposed and simulated. The stress level during the loadings is observed and compared with its design limit. Results indicate that the newly designed high vertical stiffness parabolic spring provides the bus a greater roll stability and a lower stress value compared with the original design. Bus safety and stability is promoted, as well as the load carrying capability.
An energy-stable convex splitting for the phase-field crystal equation
Vignal, P.; Dalcin, L.; Brown, D. L.; Collier, N.; Calo, V. M.
2015-01-01
Abstract The phase-field crystal equation, a parabolic, sixth-order and nonlinear partial differential equation, has generated considerable interest as a possible solution to problems arising in molecular dynamics. Nonetheless, solving this equation is not a trivial task, as energy dissipation and mass conservation need to be verified for the numerical solution to be valid. This work addresses these issues, and proposes a novel algorithm that guarantees mass conservation, unconditional energy stability and second-order accuracy in time. Numerical results validating our proofs are presented, and two and three dimensional simulations involving crystal growth are shown, highlighting the robustness of the method. © 2015 Elsevier Ltd.
An energy-stable convex splitting for the phase-field crystal equation
Vignal, P.
2015-10-01
Abstract The phase-field crystal equation, a parabolic, sixth-order and nonlinear partial differential equation, has generated considerable interest as a possible solution to problems arising in molecular dynamics. Nonetheless, solving this equation is not a trivial task, as energy dissipation and mass conservation need to be verified for the numerical solution to be valid. This work addresses these issues, and proposes a novel algorithm that guarantees mass conservation, unconditional energy stability and second-order accuracy in time. Numerical results validating our proofs are presented, and two and three dimensional simulations involving crystal growth are shown, highlighting the robustness of the method. © 2015 Elsevier Ltd.
Kashchenko, Sergey A.
2016-12-01
The dynamics of second-order equations with nonlinear delayed feedback and a large coefficient of a delayed equation is investigated using asymptotic methods. Based on special methods of quasi-normal forms, a new construction is elaborated for obtaining the main terms of asymptotic expansions of asymptotic residual solutions. It is shown that the dynamical properties of the above equations are determined mostly by the behavior of the solutions of some special families of parabolic boundary value problems. A comparative analysis of the dynamics of equations with the delayed feedback of three types is carried out.
Nonimaging secondary concentrators for large rim angle parabolic troughs with tubular absorbers.
Ries, H; Spirkl, W
1996-05-01
For parabolic trough solar collectors with tubular absorbers, we design new tailored secondary concentrators. The design is applicable for any rim angle of a parabolic reflector. With the secondary, the concentration can be increased by a factor of more than 2 with a compact secondary reflector consisting of a single piece, even for the important case of a rim angle of 90 deg. The parabolic reflector can be used without changes; the reduced absorber is still tubular but smaller than the original absorber and slightly displaced toward the primary.
Canonical generators of the cohomology of moduli of parabolic bundles on curves
International Nuclear Information System (INIS)
Biswas, I.; Raghavendra, N.
1994-11-01
We determine generators of the rational cohomology algebras of moduli spaces of parabolic vector bundles on a curve, under some 'primality' conditions on the parabolic datum. These generators are canonical in a precise sense. Our results are new even for usual vector bundles (i.e., vector bundles without parabolic structure) whose rank is greater than 2 and is coprime to the degree; in this case, they are generalizations of a theorem of Newstead on the moduli of vector bundles of rank 2 and odd degree. (author). 11 refs
International Nuclear Information System (INIS)
Liang, Z.X.; Zhang, Z.D.; Liu, W.M.
2005-01-01
We present a family of exact solutions of the one-dimensional nonlinear Schroedinger equation which describes the dynamics of a bright soliton in Bose-Einstein condensates with the time-dependent interatomic interaction in an expulsive parabolic potential. Our results show that, under a safe range of parameters, the bright soliton can be compressed into very high local matter densities by increasing the absolute value of the atomic scattering length, which can provide an experimental tool for investigating the range of validity of the one-dimensional Gross-Pitaevskii equation. We also find that the number of atoms in the bright soliton keeps dynamic stability: a time-periodic atomic exchange is formed between the bright soliton and the background
Spectral methods for time dependent partial differential equations
Gottlieb, D.; Turkel, E.
1983-01-01
The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.
Stochastic Differential Equations and Kondratiev Spaces
Energy Technology Data Exchange (ETDEWEB)
Vaage, G.
1995-05-01
The purpose of this mathematical thesis was to improve the understanding of physical processes such as fluid flow in porous media. An example is oil flowing in a reservoir. In the first of five included papers, Hilbert space methods for elliptic boundary value problems are used to prove the existence and uniqueness of a large family of elliptic differential equations with additive noise without using the Hermite transform. The ideas are then extended to the multidimensional case and used to prove existence and uniqueness of solution of the Stokes equations with additive noise. The second paper uses functional analytic methods for partial differential equations and presents a general framework for proving existence and uniqueness of solutions to stochastic partial differential equations with multiplicative noise, for a large family of noises. The methods are applied to equations of elliptic, parabolic as well as hyperbolic type. The framework presented can be extended to the multidimensional case. The third paper shows how the ideas from the second paper can be extended to study the moving boundary value problem associated with the stochastic pressure equation. The fourth paper discusses a set of stochastic differential equations. The fifth paper studies the relationship between the two families of Kondratiev spaces used in the thesis. 102 refs.
Artificial neural networks approach on solar parabolic dish cooker
International Nuclear Information System (INIS)
Lokeswaran, S.; Eswaramoorthy, M.
2011-01-01
This paper presents heat transfer analysis of solar parabolic dish cooker using Artificial Neural Network (ANN). The objective of this study to envisage thermal performance parameters such as receiver plate and pot water temperatures of the solar parabolic dish cooker by using the ANN for experimental data. An experiment is conducted under two cases (1) cooker with plain receiver and (2) cooker with porous receiver. The Back Propagation (BP) algorithm is used to train and test networks and ANN predictions are compared with experimental results. Different network configurations are studied by the aid of searching a relatively better network for prediction. The results showed a good regression analysis with the correlation coefficients in the range of 0.9968-0.9992 and mean relative errors (MREs) in the range of 1.2586-4.0346% for the test data set. Thus ANN model can successfully be used for the prediction of the thermal performance parameters of parabolic dish cooker with reasonable degree of accuracy. (authors)
Parabolic equations in biology growth, reaction, movement and diffusion
Perthame, Benoît
2015-01-01
This book presents several fundamental questions in mathematical biology such as Turing instability, pattern formation, reaction-diffusion systems, invasion waves and Fokker-Planck equations. These are classical modeling tools for mathematical biology with applications to ecology and population dynamics, the neurosciences, enzymatic reactions, chemotaxis, invasion waves etc. The book presents these aspects from a mathematical perspective, with the aim of identifying those qualitative properties of the models that are relevant for biological applications. To do so, it uncovers the mechanisms at work behind Turing instability, pattern formation and invasion waves. This involves several mathematical tools, such as stability and instability analysis, blow-up in finite time, asymptotic methods and relative entropy properties. Given the content presented, the book is well suited as a textbook for master-level coursework.
Peng, Qiujin; Qiao, Zhonghua; Sun, Shuyu
2017-01-01
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.
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.
Master equations for degenerate systems: electron radiative cascade in a Coulomb potential
International Nuclear Information System (INIS)
Uskov, D B; Pratt, R H
2004-01-01
We examine the effects of degeneracy and its lifting for the problem of electron radiative cascade, described by master equations of the Lindblad form (quantum optical master equations). A weak external field approximation is used to study the resulting gradual transformation of cascade dynamics between degenerate and non-degenerate forms. Exploiting the spherical symmetry properties of the system we demonstrate significant difference between perturbations commuting with angular momentum and perturbations breaking the spherical symmetry, such as a homogeneous external field. We discuss the possibility and the general approach for reduction of the Lindblad master equations in the case of spectral degeneracy to the Pauli balance equations. This determines the appropriate choice of basis as, for example, spherical or parabolic
Parabolic dish collectors - A solar option
Truscello, V. C.
1981-05-01
A description is given of several parabolic-dish high temperature solar thermal systems currently undergoing performance trials. A single parabolic dish has the potential for generating 20 to 30 kW of electricity with fluid temperatures from 300 to 1650 C. Each dish is a complete power-producing unit, and may function either independently or as part of a group of linked modules. The two dish designs under consideration are of 11 and 12 meter diameters, yielding receiver operating temperatures of 925 and 815 C, respectively. The receiver designs described include (1) an organic working fluid (toluene) Rankine cycle engine; (2) a Brayton open cycle unit incorporating a hybrid combustion chamber and nozzle and a shaft-coupled permanent magnet alternator; and (3) a modified Stirling cycle device originally designed for automotive use. Also considered are thermal buffer energy storage and thermochemical transport and storage.
Spheroidal corrections to the spherical and parabolic bases of the hydrogen atom
International Nuclear Information System (INIS)
Mardyan, L.G.; Pogosyan, G.S.; Sisakyan, A.N.
1986-01-01
This paper introduces the bases of the hydrogen atom and obtains recursion relations that determine the expansion of the spheroidal basis with respect to its parabolic basis. The leading spheroidal corrections to the spherical and parabolic bases are calculated by perturbation theory
Some integral representations and limits for (products of) the parabolic cylinder function
Veestraeten, D.
2016-01-01
Recently, [Veestraeten D. On the inverse transform of Laplace transforms that contain (products of) the parabolic cylinder function. Integr Transf Spec F 2015;26:859-871] derived inverse Laplace transforms for Laplace transforms that contain products of two parabolic cylinder functions by exploiting
Output feedback control of heat transport mechanisms in parabolic distributed solar collectors
Elmetennani, Shahrazed
2016-08-05
This paper presents an output feedback control for distributed parabolic solar collectors. The controller aims at forcing the outlet temperature to track a desired reference in order to manage the produced heat despite the external disturbances. The proposed control strategy is derived using the distributed physical model of the system to avoid the loss of information due to model approximation schemes. The system dynamics are driven to follow reference dynamics defined by a transport equation with a constant velocity, which allows to control the transient behavior and the response time of the closed loop. The designed controller depends only on the accessible measured variables which makes it easy for real time implementation and useful for industrial plants. Simulation results show the efficiency of the reference tracking closed loop under different working conditions.
Class of unconditionally stable second-order implicit schemes for hyperbolic and parabolic equations
International Nuclear Information System (INIS)
Lui, H.C.
The linearized Burgers equation is considered as a model u/sub t/ tau/sub x/ = bu/sub xx/, where the subscripts t and x denote the derivatives of the function u with respect to time t and space x; a and b are constants (b greater than or equal to 0). Numerical schemes for solving the equation are described that are second-order accurate, unconditionally stable, and dissipative of higher order. (U.S.)
4th Italian-Japanese workshop on Geometric Properties for Parabolic and Elliptic PDE’s
Ishige, Kazuhiro; Nitsch, Carlo; Salani, Paolo
2016-01-01
This book collects recent research papers by respected specialists in the field. It presents advances in the field of geometric properties for parabolic and elliptic partial differential equations, an area that has always attracted great attention. It settles the basic issues (existence, uniqueness, stability and regularity of solutions of initial/boundary value problems) before focusing on the topological and/or geometric aspects. These topics interact with many other areas of research and rely on a wide range of mathematical tools and techniques, both analytic and geometric. The Italian and Japanese mathematical schools have a long history of research on PDEs and have numerous active groups collaborating in the study of the geometric properties of their solutions. .
Maximum principles for boundary-degenerate linear parabolic differential operators
Feehan, Paul M. N.
2013-01-01
We develop weak and strong maximum principles for boundary-degenerate, linear, parabolic, second-order partial differential operators, $Lu := -u_t-\\tr(aD^2u)-\\langle b, Du\\rangle + cu$, with \\emph{partial} Dirichlet boundary conditions. The coefficient, $a(t,x)$, is assumed to vanish along a non-empty open subset, $\\mydirac_0!\\sQ$, called the \\emph{degenerate boundary portion}, of the parabolic boundary, $\\mydirac!\\sQ$, of the domain $\\sQ\\subset\\RR^{d+1}$, while $a(t,x)$ may be non-zero at po...
Irreversible thermodynamics, parabolic law and self-similar state in grain growth
International Nuclear Information System (INIS)
Rios, P.R.
2004-01-01
The formalism of the thermodynamic theory of irreversible processes is applied to grain growth to investigate the nature of the self-similar state and its corresponding parabolic law. Grain growth does not reach a steady state in the sense that the entropy production remains constant. However, the entropy production can be written as a product of two factors: a scale factor that tends to zero for long times and a scaled entropy production. It is suggested that the parabolic law and the self-similar state may be associated with the minimum of this scaled entropy production. This result implies that the parabolic law and the self-similar state have a sound irreversible thermodynamical basis
Effects of parabolic motion on an isothermal vertical plate with constant mass flux
Directory of Open Access Journals (Sweden)
R. Muthucumaraswamy
2014-12-01
Full Text Available An analytical study of free convection flow near a parabolic started infinite vertical plate with isothermal in the presence of uniform mass flux was considered. The mathematical model is reduced to a system of linear partial differential equations for the velocity, the concentration and the temperature; the closed form exact solutions were obtained by the Laplace transform technique. The velocity, temperature and concentration profiles for the different parameters as thermal Grashof number Gr, mass Grashof number Gc, Prandtl number Pr, Schmidt number Sc and time t were graphed and the numerical values for the skin friction were as tabulated. It is observed that the velocity is enhanced as the time increased and the velocity is decreased as the Prandtl number increased.
Optimal investment models with stochastic volatility: the time ...
African Journals Online (AJOL)
Therefore, a transform is primordial to express the value function in terms of a semilinear PDE with quadratic growth on the derivative term. Some proofs for the existence of smooth solution to this equation have been provided for this equation by Pham [11]. In that paper they illustrated some common stochastic volatility ...
Development of Vector Parabolic Equation Technique for Propagation in Urban and Tunnel Environments
2010-09-01
that of the former is geared towards determining the transport amplitude, having found the eikonal by some other means. Among the principal...FOR MODELING RADIO TRANSMISSION LOSS 1761 We can then use the following asymptotic ansatz (10) where (11) and is the tunnel width [26]. The eikonal is a...equation and equating terms of the same order of , we can define the eikonal and find the vector PE [4] for the straight waveguide (12) where is the
The dynamics of parabolic flight: Flight characteristics and passenger percepts
Karmali, Faisal; Shelhamer, Mark
2008-09-01
Flying a parabolic trajectory in an aircraft is one of the few ways to create freefall on Earth, which is important for astronaut training and scientific research. Here we review the physics underlying parabolic flight, explain the resulting flight dynamics, and describe several counterintuitive findings, which we corroborate using experimental data. Typically, the aircraft flies parabolic arcs that produce approximately 25 s of freefall (0 g) followed by 40 s of enhanced force (1.8 g), repeated 30-60 times. Although passengers perceive gravity to be zero, in actuality acceleration, and not gravity, has changed, and thus we caution against the terms "microgravity" and "zero gravity." Despite the aircraft trajectory including large (45°) pitch-up and pitch-down attitudes, the occupants experience a net force perpendicular to the floor of the aircraft. This is because the aircraft generates appropriate lift and thrust to produce the desired vertical and longitudinal accelerations, respectively, although we measured moderate (0.2 g) aft-ward accelerations during certain parts of these trajectories. Aircraft pitch rotation (average 3°/s) is barely detectable by the vestibular system, but could influence some physics experiments. Investigators should consider such details in the planning, analysis, and interpretation of parabolic-flight experiments.
Renardy, M.
A semigroup approach to differential-delay equations is developed which reduces such equations to ordinary differential equations on a Banach space of histories and seems more suitable for certain partial integro-differential equations than the standard theory. The method is applied to prove a local-time existence theorem for equations of the form utt = g( uxt, uxt) x, where {∂g}/{∂u xt} > 0 . On a formal level, it is demonstrated that the stretching of filaments of viscoelastic liquids can be described by an equation of this form.
Numerical Analysis of Partial Differential Equations
Lions, Jacques-Louis
2011-01-01
S. Albertoni: Alcuni metodi di calcolo nella teoria della diffusione dei neutroni.- I. Babuska: Optimization and numerical stability in computations.- J.H. Bramble: Error estimates in elliptic boundary value problems.- G. Capriz: The numerical approach to hydrodynamic problems.- A. Dou: Energy inequalities in an elastic cylinder.- T. Doupont: On the existence of an iterative method for the solution of elliptic difference equation with an improved work estimate.- J. Douglas, J.R. Cannon: The approximation of harmonic and parabolic functions of half-spaces from interior data.- B.E. Hubbard: Erro
A note on Hermitian-Einstein metrics on parabolic stable bundles
International Nuclear Information System (INIS)
Li Jiayu; Narasimhan, M.S.
2000-01-01
Let M-bar be a compact complex manifold of complex dimension two with a smooth Kaehler metric and D a smooth divisor on M-bar. If E is a rank 2 holomorphic vector bundle on M-bar with a stable parabolic structure along D, we prove that there exists a Hermitian-Einstein metric on E' = E-vertical bar M-barbackslashD compatible with the parabolic structure, and whose curvature is square integrable. (author)
Cagnetti, Filippo; Gomes, Diogo A.; Mitake, Hiroyoshi; Tran, Hung V.
2015-01-01
We investigate large-time asymptotics for viscous Hamilton-Jacobi equations with possibly degenerate diffusion terms. We establish new results on the convergence, which are the first general ones concerning equations which are neither uniformly parabolic nor first order. Our method is based on the nonlinear adjoint method and the derivation of new estimates on long time averaging effects. It also extends to the case of weakly coupled systems.
Elliptic random-walk equation for suspension and tracer transport in porous media
DEFF Research Database (Denmark)
Shapiro, Alexander; Bedrikovetsky, P. G.
2008-01-01
. The new theory predicts delay of the maximum of the tracer, compared to the velocity of the flow, while its forward "tail" contains much more particles than in the solution of the classical parabolic (advection-dispersion) equation. This is in agreement with the experimental observations and predictions......We propose a new approach to transport of the suspensions and tracers in porous media. The approach is based on a modified version of the continuous time random walk (CTRW) theory. In the framework of this theory we derive an elliptic transport equation. The new equation contains the time...... of the CTRW theory. (C) 2008 Elsevier B.V. All rights reserved....
Domain Decomposition Solvers for Frequency-Domain Finite Element Equations
Copeland, Dylan
2010-10-05
The paper is devoted to fast iterative solvers for frequency-domain finite element equations approximating linear and nonlinear parabolic initial boundary value problems with time-harmonic excitations. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple linear elliptic system for the amplitudes belonging to the sine- and to the cosine-excitation or a large nonlinear elliptic system for the Fourier coefficients in the linear and nonlinear case, respectively. The fast solution of the corresponding linear and nonlinear system of finite element equations is crucial for the competitiveness of this method. © 2011 Springer-Verlag Berlin Heidelberg.
Domain Decomposition Solvers for Frequency-Domain Finite Element Equations
Copeland, Dylan; Kolmbauer, Michael; Langer, Ulrich
2010-01-01
The paper is devoted to fast iterative solvers for frequency-domain finite element equations approximating linear and nonlinear parabolic initial boundary value problems with time-harmonic excitations. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple linear elliptic system for the amplitudes belonging to the sine- and to the cosine-excitation or a large nonlinear elliptic system for the Fourier coefficients in the linear and nonlinear case, respectively. The fast solution of the corresponding linear and nonlinear system of finite element equations is crucial for the competitiveness of this method. © 2011 Springer-Verlag Berlin Heidelberg.
Krylov, N. V.; Priola, E.
2017-09-01
We show, among other things, how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on the time variable with the same constants as in the case of the one-dimensional heat equation. The method is quite general and is based on using the Poisson stochastic process. It also applies to equations involving non-local operators. It looks like no other methods are available at this time and it is a very challenging problem to find a purely analytical approach to proving such results.
On Critical Spaces for the Navier-Stokes Equations
Prüss, Jan; Wilke, Mathias
2017-10-01
The abstract theory of critical spaces developed in Prüss and Wilke (J Evol Equ, 2017. doi: 10.1007/s00028-017-0382-6), Prüss et al. (Critical spaces for quasilinear parabolic evolution equations and applications, 2017) is applied to the Navier-Stokes equations in bounded domains with Navier boundary conditions as well as no-slip conditions. Our approach unifies, simplifies and extends existing work in the L_p -L_q setting, considerably. As an essential step, it is shown that the strong and weak Stokes operators with Navier conditions admit an H^∞-calculus with H^∞-angle 0, and the real and complex interpolation spaces of these operators are identified.
Solutions to variational inequalities of parabolic type
Zhu, Yuanguo
2006-09-01
The existence of strong solutions to a kind of variational inequality of parabolic type is investigated by the theory of semigroups of linear operators. As an application, an abstract semi permeable media problem is studied.
Directory of Open Access Journals (Sweden)
Bixiang Wang
2013-08-01
Full Text Available We prove the existence and uniqueness of random attractors for the p-Laplace equation driven simultaneously by non-autonomous deterministic and stochastic forcing. The nonlinearity of the equation is allowed to have a polynomial growth rate of any order which may be greater than p. We further establish the upper semicontinuity of random attractors as the intensity of noise approaches zero. In addition, we show the pathwise periodicity of random attractors when all non-autonomous deterministic forcing terms are time periodic.
Stability for semilinear parabolic problems in L_2 and W^{1,2}
Czech Academy of Sciences Publication Activity Database
Gurevich, P.; Väth, Martin
2016-01-01
Roč. 35, č. 3 (2016), s. 333-357 ISSN 0232-2064 Institutional support: RVO:67985840 Keywords : asymptotic stability * existence * uniqueness Subject RIV: BA - General Mathematics Impact factor: 0.643, year: 2016 https://www.ems-ph.org/journals/show_abstract.php?issn=0232-2064&vol=35&iss=3&rank=5
Integrated parabolic nanolenses on MicroLED color pixels
Demory, Brandon; Chung, Kunook; Katcher, Adam; Sui, Jingyang; Deng, Hui; Ku, Pei-Cheng
2018-04-01
A parabolic nanolens array coupled to the emission of a nanopillar micro-light emitting diode (LED) color pixel is shown to reduce the far field divergence. For a blue wavelength LED, the total emission is 95% collimated within a 0.5 numerical aperture zone, a 3.5x improvement over the same LED without a lens structure. This corresponds to a half-width at half-maximum (HWHM) line width reduction of 2.85 times. Using a resist reflow and etchback procedure, the nanolens array dimensions and parabolic shape are formed. Experimental measurement of the far field emission shows a HWHM linewidth reduction by a factor of 2x, reducing the divergence over the original LED.
Global Carleman estimates for degenerate parabolic operators with applications
Cannarsa, P; Vancostenoble, J
2016-01-01
Degenerate parabolic operators have received increasing attention in recent years because they are associated with both important theoretical analysis, such as stochastic diffusion processes, and interesting applications to engineering, physics, biology, and economics. This manuscript has been conceived to introduce the reader to global Carleman estimates for a class of parabolic operators which may degenerate at the boundary of the space domain, in the normal direction to the boundary. Such a kind of degeneracy is relevant to study the invariance of a domain with respect to a given stochastic diffusion flow, and appears naturally in climatology models.
Study of a new solar adsorption refrigerator powered by a parabolic trough collector
International Nuclear Information System (INIS)
El Fadar, A.; Mimet, A.; Azzabakh, A.; Perez-Garcia, M.; Castaing, J.
2009-01-01
This paper presents the study of solar adsorption cooling machine, where the reactor is heated by a parabolic trough collector (PTC) and is coupled with a heat pipe (HP). This reactor contains a porous medium constituted of activated carbon, reacting by adsorption with ammonia. We have developed a model, based on the equilibrium equations of the refrigerant, adsorption isotherms, heat and mass transfer within the adsorbent bed and energy balance in the hybrid system components. From real climatic data, the model computes the performances of the machine. In comparison with other systems powered by flat plate or evacuated tube collectors, the predicted results, have illustrated the ability of the proposed system to achieve a high performance due to high efficiency of PTC, and high flux density of heat pipe
Computation of rational solutions for a first-order nonlinear differential equation
Directory of Open Access Journals (Sweden)
Djilali Behloul
2011-09-01
Full Text Available In this article, we study differential equations of the form $y'=sum A_i(xy^i/sum B_i(xy^i$ which can be elliptic, hyperbolic, parabolic, Riccati, or quasi-linear. We show how rational solutions can be computed in a systematic manner. Such results are most likely to find applications in the theory of limit cycles as indicated by Gine et al [4].
First Middle East Aircraft Parabolic Flights for ISU Participant Experiments
Pletser, Vladimir; Frischauf, Norbert; Cohen, Dan; Foster, Matthew; Spannagel, Ruven; Szeszko, Adam; Laufer, Rene
2017-06-01
Aircraft parabolic flights are widely used throughout the world to create microgravity environment for scientific and technology research, experiment rehearsal for space missions, and for astronaut training before space flights. As part of the Space Studies Program 2016 of the International Space University summer session at the Technion - Israel Institute of Technology, Haifa, Israel, a series of aircraft parabolic flights were organized with a glider in support of departmental activities on `Artificial and Micro-gravity' within the Space Sciences Department. Five flights were organized with manoeuvres including several parabolas with 5 to 6 s of weightlessness, bank turns with acceleration up to 2 g and disorientation inducing manoeuvres. Four demonstration experiments and two experiments proposed by SSP16 participants were performed during the flights by on board operators. This paper reports on the microgravity experiments conducted during these parabolic flights, the first conducted in the Middle East for science and pedagogical experiments.
Temperature Performance Evaluation of Parabolic Dishes Covered ...
African Journals Online (AJOL)
Aweda
The parabolic dish with glass material gave the highest temperature of .... 3: Second day variation temperature and time using different materials. 8. 10 .... the sun rays at that particular time. ... especially between 11:00 am and 3:00 pm when.
Partial differential equations with numerical methods
Larsson, Stig
2003-01-01
The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. As preparation, the two-point boundary value problem and the initial-value problem for ODEs are discussed in separate chapters. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. Some background on linear functional analysis and Sobolev spaces, and also on numerical linear algebra, is reviewed in two appendices.
Numerical solution of a reaction-diffusion equation
International Nuclear Information System (INIS)
Moyano, Edgardo A.; Scarpettini, Alberto F.
2000-01-01
The purpose of the present work to continue the observations and the numerical experiences on a reaction-diffusion model, that is a simplified form of the neutronic flux equation. The model is parabolic, nonlinear, with Dirichlet boundary conditions. The purpose is to approximate non trivial solutions, asymptotically stables for t → ∞, that is solutions that tend to the elliptic problem, in the Lyapunov sense. It belongs to the so-called reaction-diffusion equations of semi linear kind, that is, linear equations in the heat operator and they have a nonlinear reaction function, in this case f (u, a, b) = u (a - b u), being u concentration, a and b parameters. The study of the incidence of these parameters take an interest to the neutronic flux physics. So that we search non trivial, positive and bounded solutions. The used algorithm is based on the concept of monotone and ordered sequences, and on the existence theorem of Amann and Sattinger. (author)
Spike-adding in parabolic bursters: The role of folded-saddle canards
Desroches, Mathieu; Krupa, Martin; Rodrigues, Serafim
2016-09-01
The present work develops a new approach to studying parabolic bursting, and also proposes a novel four-dimensional canonical and polynomial-based parabolic burster. In addition to this new polynomial system, we also consider the conductance-based model of the Aplysia R15 neuron known as the Plant model, and a reduction of this prototypical biophysical parabolic burster to three variables, including one phase variable, namely the Baer-Rinzel-Carillo (BRC) phase model. Revisiting these models from the perspective of slow-fast dynamics reveals that the number of spikes per burst may vary upon parameter changes, however the spike-adding process occurs in an explosive fashion that involves special solutions called canards. This spike-adding canard explosion phenomenon is analysed by using tools from geometric singular perturbation theory in tandem with numerical bifurcation techniques. We find that the bifurcation structure persists across all considered systems, that is, spikes within the burst are incremented via the crossing of an excitability threshold given by a particular type of canard orbit, namely the true canard of a folded-saddle singularity. However there can be a difference in the spike-adding transitions in parameter space from one case to another, according to whether the process is continuous or discontinuous, which depends upon the geometry of the folded-saddle canard. Using these findings, we construct a new polynomial approximation of the Plant model, which retains all the key elements for parabolic bursting, including the spike-adding transitions mediated by folded-saddle canards. Finally, we briefly investigate the presence of spike-adding via canards in planar phase models of parabolic bursting, namely the theta model by Ermentrout and Kopell.
Kazeykina, Anna; Muñoz, Claudio
2018-04-01
We continue our study on the Cauchy problem for the two-dimensional Novikov-Veselov (NV) equation, integrable via the inverse scattering transform for the two dimensional Schrödinger operator at a fixed energy parameter. This work is concerned with the more involved case of a positive energy parameter. For the solution of the linearized equation we derive smoothing and Strichartz estimates by combining new estimates for two different frequency regimes, extending our previous results for the negative energy case [18]. The low frequency regime, which our previous result was not able to treat, is studied in detail. At non-low frequencies we also derive improved smoothing estimates with gain of almost one derivative. Then we combine the linear estimates with a Fourier decomposition method and Xs,b spaces to obtain local well-posedness of NV at positive energy in Hs, s > 1/2. Our result implies, in particular, that at least for s > 1/2, NV does not change its behavior from semilinear to quasilinear as energy changes sign, in contrast to the closely related Kadomtsev-Petviashvili equations. As a complement to our LWP results, we also provide some new explicit solutions of NV at zero energy, generalizations of the lumps solutions, which exhibit new and nonstandard long time behavior. In particular, these solutions blow up in infinite time in L2.
Unconditionally stable difference methods for delay partial differential equations
Huang, Chengming; Vandewalle, Stefan
2012-01-01
This paper is concerned with the numerical solution of parabolic partial differential equations with time-delay. We focus in particular on the delay dependent stability analysis of difference methods that use a non-constrained mesh, i.e., the time step-size is not required to be a submultiple of the delay. We prove that the fully discrete system unconditionally preserves the delay dependent asymptotic stability of the linear test problem under consideration, when the following discretizati...
A parabolic mirror x-ray collimator
Franks, A.; Jackson, K.; Yacoot, A.
2000-05-01
A robust and stable x-ray collimator has been developed to produce a parallel beam of x-rays by total external reflection from a parabolic mirror. The width of the gold-coated silica mirror varies along its length, which allows it to be bent from a plane surface into a parabolic form by application of unequal bending forces at its ends. A family of parabolas of near constant focal length can be formed by changing the screw-applied bending force, thus allowing the collimator to cater for a range of wavelengths by the turning of a screw. Even with radiation with a wavelength as short as that as Mo Kicons/Journals/Common/alpha" ALT="alpha" ALIGN="TOP"/> 1 (icons/Journals/Common/lambda" ALT="lambda" ALIGN="TOP"/> = 0.07 nm), a gain in flux by a factor of 5.5 was achieved. The potential gain increases with wavelength, e.g. for Cu Kicons/Journals/Common/alpha" ALT="alpha" ALIGN="TOP"/> 1 radiation this amounts to over a factor of ten.
Transitional inertialess instabilities in driven multilayer channel flows
Papaefthymiou, Evangelos; Papageorgiou, Demetrios
2016-11-01
We study the nonlinear stability of viscous, immiscible multilayer flows in channels driven both by a pressure gradient and/or gravity in a slightly inclined channel. Three fluid phases are present with two internal interfaces. Novel weakly nonlinear models of coupled evolution equations are derived and we concentrate on inertialess flows with stably stratified fluids, with and without surface tension. These are 2 × 2 systems of second-order semilinear parabolic PDEs that can exhibit inertialess instabilities due to resonances between the interfaces - mathematically this is manifested by a transition from hyperbolic to elliptic behavior of the nonlinear flux functions. We consider flows that are linearly stable (i.e the nonlinear fluxes are hyperbolic initially) and use the theory of nonlinear systems of conservation laws to obtain a criterion (which can be verified easily) that can predict nonlinear stability or instability (i.e. nonlinear fluxes encounter ellipticity as they evolve spatiotemporally) at large times. In the former case the solution decays asymptotically to its base state, and in the latter nonlinear traveling waves emerge. EPSRC Grant Numbers EP/K041134 and EP/L020564.
Wind load design methods for ground-based heliostats and parabolic dish collectors
Energy Technology Data Exchange (ETDEWEB)
Peterka, J A; Derickson, R G [Colorado State Univ., Fort Collins, CO (United States). Fluid Dynamics and Diffusion Lab.
1992-09-01
The purpose of this design method is to define wind loads on flat heliostat and parabolic dish collectors in a simplified form. Wind loads are defined for both mean and peak loads accounting for the protective influence of upwind collectors, wind protective fences, or other wind-blockage elements. The method used to define wind loads was to generalize wind load data obtained during tests on model collectors, heliostats or parabolic dishes, placed in a modeled atmospheric wind in a boundary-layer wind-tunnel at Colorado State University. For both heliostats and parabolic dishes, loads are reported for solitary collectors and for collectors as elements of a field. All collectors were solid with negligible porosity; thus the effects of porosity in the collectors is not addressed.
Gas Turbine/Solar Parabolic Trough Hybrid Designs: Preprint
Energy Technology Data Exchange (ETDEWEB)
Turchi, C. S.; Ma, Z.; Erbes, M.
2011-03-01
A strength of parabolic trough concentrating solar power (CSP) plants is the ability to provide reliable power by incorporating either thermal energy storage or backup heat from fossil fuels. Yet these benefits have not been fully realized because thermal energy storage remains expensive at trough operating temperatures and gas usage in CSP plants is less efficient than in dedicated combined cycle plants. For example, while a modern combined cycle plant can achieve an overall efficiency in excess of 55%; auxiliary heaters in a parabolic trough plant convert gas to electricity at below 40%. Thus, one can argue the more effective use of natural gas is in a combined cycle plant, not as backup to a CSP plant. Integrated solar combined cycle (ISCC) systems avoid this pitfall by injecting solar steam into the fossil power cycle; however, these designs are limited to about 10% total solar enhancement. Without reliable, cost-effective energy storage or backup power, renewable sources will struggle to achieve a high penetration in the electric grid. This paper describes a novel gas turbine / parabolic trough hybrid design that combines solar contribution of 57% and higher with gas heat rates that rival that for combined cycle natural gas plants. The design integrates proven solar and fossil technologies, thereby offering high reliability and low financial risk while promoting deployment of solar thermal power.
Heat transfer analysis of parabolic trough solar receiver
International Nuclear Information System (INIS)
Padilla, Ricardo Vasquez; Demirkaya, Gokmen; Goswami, D. Yogi; Stefanakos, Elias; Rahman, Muhammad M.
2011-01-01
Highlights: → In this paper a detailed one dimensional numerical heat transfer analysis of a PTC is performed. → The receiver and envelope were divided into several segments and mass and energy balance were applied in each segment. → Improvements either in the heat transfer correlations or radiative heat transfer analysis are presented. → The proposed heat transfer model was validated with experimental data obtained from Sandia National Laboratory. → Our results showed a better agreement with experimental data compared to other models. -- Abstract: Solar Parabolic Trough Collectors (PTCs) are currently used for the production of electricity and applications with relatively higher temperatures. A heat transfer fluid circulates through a metal tube (receiver) with an external selective surface that absorbs solar radiation reflected from the mirror surfaces of the PTC. In order to reduce the heat losses, the receiver is covered by an envelope and the enclosure is usually kept under vacuum pressure. The heat transfer and optical analysis of the PTC is essential to optimize and understand its performance under different operating conditions. In this paper a detailed one dimensional numerical heat transfer analysis of a PTC is performed. The receiver and envelope were divided into several segments and mass and energy balance were applied in each segment. Improvements either in the heat transfer correlations or radiative heat transfer analysis are presented as well. The partial differential equations were discretized and the nonlinear algebraic equations were solved simultaneously. Finally, to validate the numerical results, the model was compared with experimental data obtained from Sandia National Laboratory (SNL) and other one dimensional heat transfer models. Our results showed a better agreement with experimental data compared to other models.
Improvement Design of Parabolic Trough
Ihsan, S. I.; Safian, M. A. I. M.; Taufek, M. A. M.; Mohiuddin, A. K. M.
2017-03-01
The performance of parabolic trough solar collector (PTSC) has been evaluated using different heat transfer working fluids; namely water and SAE20 W50 engine oil. New and slightly improved PTSC was developed to run the experimental study. Under the meteorological conditions of Malaysia, authors found that PTSC can operate at a higher temperature than water collector but the performance efficiency of collector using engine oil is much lower than the water collector.
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
Maximum principles and sharp constants for solutions of elliptic and parabolic systems
Kresin, Gershon
2012-01-01
The main goal of this book is to present results pertaining to various versions of the maximum principle for elliptic and parabolic systems of arbitrary order. In particular, the authors present necessary and sufficient conditions for validity of the classical maximum modulus principles for systems of second order and obtain sharp constants in inequalities of Miranda-Agmon type and in many other inequalities of a similar nature. Somewhat related to this topic are explicit formulas for the norms and the essential norms of boundary integral operators. The proofs are based on a unified approach using, on one hand, representations of the norms of matrix-valued integral operators whose target spaces are linear and finite dimensional, and, on the other hand, on solving certain finite dimensional optimization problems. This book reflects results obtained by the authors, and can be useful to research mathematicians and graduate students interested in partial differential equations.
Integrability of the Gross-Pitaevskii equation with Feshbach resonance management
International Nuclear Information System (INIS)
Zhao Dun; Luo Honggang; Chai Huayue
2008-01-01
In this Letter we study the integrability of a class of Gross-Pitaevskii equations managed by Feshbach resonance in an expulsive parabolic external potential. By using WTC test, we find a condition under which the Gross-Pitaevskii equation is completely integrable. Under the present model, this integrability condition is completely consistent with that proposed by Serkin, Hasegawa, and Belyaeva [V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102]. Furthermore, this integrability can also be explicitly shown by a transformation, which can convert the Gross-Pitaevskii equation into the well-known standard nonlinear Schroedinger equation. By this transformation, each exact solution of the standard nonlinear Schroedinger equation can be converted into that of the Gross-Pitaevskii equation, which builds a systematical connection between the canonical solitons and the so-called nonautonomous ones. The finding of this transformation has a significant contribution to understanding the essential properties of the nonautonomous solitons and the dynamics of the Bose-Einstein condensates by using the Feshbach resonance technique
Blowing-up semilinear wave equation with exponential nonlinearity ...
Indian Academy of Sciences (India)
H1-norm. Hence, it is legitimate to consider an exponential nonlinearity. Moreover, the choice of an exponential nonlinearity emerges from a possible control of solutions via a. Moser–Trudinger type inequality [1, 16, 19]. In fact, Nakamura and Ozawa [17] proved global well-posedness and scattering for small Cauchy data in ...
Analysis of the stress-deformed condition of the disassembly parabolic antenna
Odinets, M. N.; Kaygorodtseva, N. V.; Krysova, I. V.
2018-01-01
Active development of satellite communications and computer-aided design systems raises the problem of designing parabolic antennas on a new round of development. The aim of the work was to investigate the influence of the design of the mirror of a parabolic antenna on its endurance under wind load. The research task was an automated analysis of the stress-deformed condition of various designs of computer models of a paraboloid mirror (segmented or holistic) at modeling the exploitation conditions. The peculiarity of the research was that the assembly model of the antenna’s mirror was subjected to rigid connections on the contacting surfaces of the segments and only then the finite element grid was generated. The analysis showed the advantage of the design of the demountable antenna, which consists of cyclic segments, in front of the construction of the holistic antenna. Calculation of the stress-deformed condition of the antennas allows us to conclude that dividing the design of the antenna’s mirror on parabolic and cyclic segments increases it strength and rigidity. In the future, this can be used to minimize the mass of antenna and the dimensions of the disassembled antenna. The presented way of modeling a mirror of a parabolic antenna using to the method of the finite-element analysis can be used in the production of antennas.
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.
Polarization properties of linearly polarized parabolic scaling Bessel beams
Energy Technology Data Exchange (ETDEWEB)
Guo, Mengwen; Zhao, Daomu, E-mail: zhaodaomu@yahoo.com
2016-10-07
The intensity profiles for the dominant polarization, cross polarization, and longitudinal components of modified parabolic scaling Bessel beams with linear polarization are investigated theoretically. The transverse intensity distributions of the three electric components are intimately connected to the topological charge. In particular, the intensity patterns of the cross polarization and longitudinal components near the apodization plane reflect the sign of the topological charge. - Highlights: • We investigated the polarization properties of modified parabolic scaling Bessel beams with linear polarization. • We studied the evolution of transverse intensity profiles for the three components of these beams. • The intensity patterns of the cross polarization and longitudinal components can reflect the sign of the topological charge.
Parabolic approximation method for fast magnetosonic wave propagation in tokamaks
International Nuclear Information System (INIS)
Phillips, C.K.; Perkins, F.W.; Hwang, D.Q.
1985-07-01
Fast magnetosonic wave propagation in a cylindrical tokamak model is studied using a parabolic approximation method in which poloidal variations of the wave field are considered weak in comparison to the radial variations. Diffraction effects, which are ignored by ray tracing mthods, are included self-consistently using the parabolic method since continuous representations for the wave electromagnetic fields are computed directly. Numerical results are presented which illustrate the cylindrical convergence of the launched waves into a diffraction-limited focal spot on the cyclotron absorption layer near the magnetic axis for a wide range of plasma confinement parameters
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.
Temperature Performance Evaluation of Parabolic Dishes Covered ...
African Journals Online (AJOL)
Solar radiation reaching the earth is considered to be affected by some parameters like diffusion. This radiation is reflected or scattered by air molecules, cloud and aerosols (dust). Parabolic dishes made of different materials (glass, foil and painted surface) were used to concentrate energy on a copper calorimeter filled with ...
Banks, H. T.; Kunisch, K.
1982-01-01
Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.
Guan, Chao; Hasi, Eerdun; Zhang, Ping; Tao, Binbin; Liu, Dan; Zhou, Yanguang
2017-10-01
Since the 1970s, parabolic dunes at the southern fringe of the Hobq Desert, Inner Mongolia, China have exhibited many different shapes (V-shaped, U-shaped, and palmate) each with a unique mode of development. In the study area, parabolic dunes are mainly distributed in Regions A, B, and C with an intermittent river running from the south to the north. We used high-resolution remote-sensing images from 1970 to 2014 and RTK-GPS measurements to study the development modes of different dune shapes; the modes are characterized by the relationship between the intermittent river and dunes, formation of the incipient dune patterns, the predominant source supply of dunes, and the primary formation of different shapes (V-shaped, U-shaped, and palmate). Most parabolic dunes in Region A are V-shaped and closer to the bank of the river. The original barchans in this region exhibit "disconnected arms" behavior. With the sand blown out of the riverbed through gullies, the nebkhas on the disconnected arms acquire the external sand source through the "fertile island effect", thereby developing into triangular sand patches and further developing into V-shaped parabolic dunes. Most parabolic dunes in Regions B and C are palmate. The residual dunes cut by the re-channelization of river from transverse dune fields on the west bank are the main sand source of Region B. The parabolic dunes in Region C are the original barchans having then been transformed. The stoss slopes of V-shaped parabolic dunes along the riverbank are gradual and the dunes are flat in shape. The dune crest of V-shaped parabolic dune is the deposition area, which forms the "arc-shaped sand ridge". Their two arms are non-parallel; the lateral airflow of the arms jointly transport sand to the middle part of dunes, resulting in a narrower triangle that gradually becomes V-shaped. Palmate parabolic dunes have a steeper stoss slope and height. The dune crest of the palmate parabolic dune is the erosion area, which forms
Gradient remediability in linear distributed parabolic systems ...
African Journals Online (AJOL)
The aim of this paper is the introduction of a new concept that concerned the analysis of a large class of distributed parabolic systems. It is the general concept of gradient remediability. More precisely, we study with respect to the gradient observation, the existence of an input operator (gradient efficient actuators) ensuring ...
Ruggeri, Fabrizio
2016-05-12
In this work we develop a Bayesian setting to infer unknown parameters in initial-boundary value problems related to linear parabolic partial differential equations. We realistically assume that the boundary data are noisy, for a given prescribed initial condition. We show how to derive the joint likelihood function for the forward problem, given some measurements of the solution field subject to Gaussian noise. Given Gaussian priors for the time-dependent Dirichlet boundary values, we analytically marginalize the joint likelihood using the linearity of the equation. Our hierarchical Bayesian approach is fully implemented in an example that involves the heat equation. In this example, the thermal diffusivity is the unknown parameter. We assume that the thermal diffusivity parameter can be modeled a priori through a lognormal random variable or by means of a space-dependent stationary lognormal random field. Synthetic data are used to test the inference. We exploit the behavior of the non-normalized log posterior distribution of the thermal diffusivity. Then, we use the Laplace method to obtain an approximated Gaussian posterior and therefore avoid costly Markov Chain Monte Carlo computations. Expected information gains and predictive posterior densities for observable quantities are numerically estimated using Laplace approximation for different experimental setups.
Institute of Scientific and Technical Information of China (English)
高夫征
2005-01-01
A finite volume element predictor-correetor method for a class of nonlinear parabolic system of equations is presented and analyzed. Suboptimal L2 error estimate for the finite volume element predictor-corrector method is derived. A numerical experiment shows that the numerical results are consistent with theoretical analysis.
Survey of the status of finite element methods for partial differential equations
Temam, Roger
1986-01-01
The finite element methods (FEM) have proved to be a powerful technique for the solution of boundary value problems associated with partial differential equations of either elliptic, parabolic, or hyperbolic type. They also have a good potential for utilization on parallel computers particularly in relation to the concept of domain decomposition. This report is intended as an introduction to the FEM for the nonspecialist. It contains a survey which is totally nonexhaustive, and it also contains as an illustration, a report on some new results concerning two specific applications, namely a free boundary fluid-structure interaction problem and the Euler equations for inviscid flows.
Directory of Open Access Journals (Sweden)
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.
Yu, Fajun
2015-03-01
We present the nonautonomous discrete bright soliton solutions and their interactions in the discrete Ablowitz-Ladik (DAL) equation with variable coefficients, which possesses complicated wave propagation in time and differs from the usual bright soliton waves. The differential-difference similarity transformation allows us to relate the discrete bright soliton solutions of the inhomogeneous DAL equation to the solutions of the homogeneous DAL equation. Propagation and interaction behaviors of the nonautonomous discrete solitons are analyzed through the one- and two-soliton solutions. We study the discrete snaking behaviors, parabolic behaviors, and interaction behaviors of the discrete solitons. In addition, the interaction management with free functions and dynamic behaviors of these solutions is investigated analytically, which have certain applications in electrical and optical systems.
Mobile point sensors and actuators in the controllability theory of partial differential equations
Khapalov, Alexander Y
2017-01-01
This book presents a concise study of controllability theory of partial differential equations when they are equipped with actuators and/or sensors that are finite dimensional at every moment of time. Based on the author’s extensive research in the area of controllability theory, this monograph specifically focuses on the issues of controllability, observability, and stabilizability for parabolic and hyperbolic partial differential equations. The topics in this book also cover related applied questions such as the problem of localization of unknown pollution sources based on information obtained from point sensors that arise in environmental monitoring. Researchers and graduate students interested in controllability theory of partial differential equations and its applications will find this book to be an invaluable resource to their studies.
Packing of equal discs on a parabolic spiral lattice
International Nuclear Information System (INIS)
Xudong, F.; Bursill, L.A.; Julin, P.
1989-01-01
A contact disc model is investigated to determine the most closely-packed parabolic spiral lattice. The most space-efficient packings have divergence angles in agreement with the priority ranking of natural spiral structures
International Nuclear Information System (INIS)
Gómez-Varela, A I; Bao-Varela, C; Flores-Arias, M T
2014-01-01
The present paper considers two gain GRIN media, characterized by a complex parabolic and hyperbolic secant refractive index profile, for the design of uniform beam shaper systems. A general condition for beam shaping is obtained from the equation describing the evolution of the half-width of a plane Gaussian beam in the GRIN media. The simulation of the irradiance evolution of an input plane Gaussian beam—operating at 575 nm and beam waist radius of 0.45 mm—in each material is shown, in order to examine the beam shaping quality in terms of thickness of the active GRIN media and input beam wavelength. (paper)
Modeling, Simulation and Performance Evaluation of Parabolic Trough
African Journals Online (AJOL)
Mekuannint
Mekuannint Mesfin and Abebayehu Assefa. Department of Mechanical Engineering. Addis Ababa University ... off design weather conditions as well. Keywords: Parabolic Trough Collector (PTC);. Heat Transfer ... of a conventional Rankine cycle power plant with solar fields that are used to increase the temperature of heat ...
Modeling, Simulation and Performance Evaluation of Parabolic Trough
African Journals Online (AJOL)
Mekuannint
Heat Transfer Fluid (HTF); TRNSYS power plant model; STEC library; Solar Advisor Model (SAM);. TRNSYS solar field model; Solar Electric. Generation System (SEGS). INTRODUCTION. Parabolic troughs are currently most used means of power generation option of solar sources. Solar electric generation systems (SEGs) ...
Transport equations, Level Set and Eulerian mechanics. Application to fluid-structure coupling
International Nuclear Information System (INIS)
Maitre, E.
2008-11-01
My works were devoted to numerical analysis of non-linear elliptic-parabolic equations, to neutron transport equation and to the simulation of fabrics draping. More recently I developed an Eulerian method based on a level set formulation of the immersed boundary method to deal with fluid-structure coupling problems arising in bio-mechanics. Some of the more efficient algorithms to solve the neutron transport equation make use of the splitting of the transport operator taking into account its characteristics. In the present work we introduced a new algorithm based on this splitting and an adaptation of minimal residual methods to infinite dimensional case. We present the case where the velocity space is of dimension 1 (slab geometry) and 2 (plane geometry) because the splitting is simpler in the former
Weidemaier, P.
2005-06-01
The trace problem on the hypersurface y_n=0 is investigated for a function u=u(y,t) \\in L_q(0,T;W_{\\underline p}^{\\underline m}(\\mathbb R_+^n)) with \\partial_t u \\in L_q(0,T; L_{\\underline p}(\\mathbb R_+^n)), that is, Sobolev spaces with mixed Lebesgue norm L_{\\underline p,q}(\\mathbb R^n_+\\times(0,T))=L_q(0,T;L_{\\underline p}(\\mathbb R_+^n)) are considered; here \\underline p=(p_1,\\dots,p_n) is a vector and \\mathbb R^n_+=\\mathbb R^{n-1} \\times (0,\\infty). Such function spaces are useful in the context of parabolic equations. They allow, in particular, different exponents of summability in space and time. It is shown that the sharp regularity of the trace in the time variable is characterized by the Lizorkin-Triebel space F_{q,p_n}^{1-1/(p_nm_n)}(0,T;L_{\\widetilde{\\underline p}}(\\mathbb R^{n-1})), \\underline p=(\\widetilde{\\underline p},p_n). A similar result is established for first order spatial derivatives of u. These results allow one to determine the exact spaces for the data in the inhomogeneous Dirichlet and Neumann problems for parabolic equations of the second order if the solution is in the space L_q(0,T; W_p^2(\\Omega)) \\cap W_q^1(0,T;L_p(\\Omega)) with p \\le q.
Physiologic Pressure and Flow Changes During Parabolic Flight (Pilot Study)
Pantalos, George; Sharp, M. Keith; Mathias, John R.; Hargens, Alan R.; Watenpaugh, Donald E.; Buckey, Jay C.
1999-01-01
The objective of this study was to obtain measurement of cutaneous tissue perfusion central and peripheral venous pressure, and esophageal and abdominal pressure in human test subjects during parabolic flight. Hemodynamic data recorded during SLS-I and SLS-2 missions have resulted in the paradoxical finding of increased cardiac stroke volume in the presence of a decreased central venous pressure (CVP) following entry in weightlessness. The investigators have proposed that in the absence of gravity, acceleration-induced peripheral vascular compression is relieved, increasing peripheral vascular capacity and flow while reducing central and peripheral venous pressure, This pilot study seeks to measure blood pressure and flow in human test subjects during parabolic flight for different postures.
A parabolic-hyperbolic system modelling a moving cell
Directory of Open Access Journals (Sweden)
Fabiana Cardetti
2009-08-01
Full Text Available In this article, we study the existence and uniqueness of local solutions for a moving boundary problem governed by a coupled parabolic-hyperbolic system. The results can be applied to cell movement, extending a result obtained by Choi, Groulx, and Lui in 2005.
International Nuclear Information System (INIS)
Tchinda, R.
2005-11-01
Equations describing the heat transfer in symmetric, compound parabolic concentrating solar collectors (CPCs) with and without envelope have been established. The model takes into account the non linear behavior of these two systems. A theoretical numerical model has been developed to outline the effect of the envelope on the thermal and optical performance of CPCs. The effects of the flow rate, the plate length, the selective coating, etc. are studied. The over-all thermal loss coefficient and the enclosure absorption factor for both types are defined. It is found that the efficient configuration has an envelope. Theoretical computed values are in good agreement with the experimental values published in the literature. (author)
International Nuclear Information System (INIS)
Soria-Verdugo, Antonio; Goos, Elke; Arrieta-Sanagustín, Jorge; García-Hernando, Nestor
2016-01-01
Highlights: • Pyrolysis of biomass under parabolic and exponential temperature profiles is modeled. • The model is based on a simplified Distributed Activation Energy Model. • 4 biomasses are analyzed in TGA with parabolic and exponential temperature increases. • Deviations between the model prediction and TGA measurements are under 5 °C. - Abstract: A modification of the simplified Distributed Activation Energy Model is proposed to simulate the pyrolysis of biomass under parabolic and exponential temperature increases. The pyrolysis of pine wood, olive kernel, thistle flower and corncob was experimentally studied in a TGA Q500 thermogravimetric analyzer. The results of the measurements of nine different parabolic and exponential temperature increases for each sample were employed to validate the models proposed. The deviation between the experimental TGA measurements and the estimation of the reacted fraction during the pyrolysis of the four samples under parabolic and exponential temperature increases was lower than 5 °C for all the cases studied. The models derived in this work to describe the pyrolysis of biomass with parabolic and exponential temperature increases were found to be in good agreement with the experiments conducted in a thermogravimetric analyzer.
International Nuclear Information System (INIS)
Zhu, Yanqing; Shi, Jifu; Li, Yujian; Wang, Leilei; Huang, Qizhang; Xu, Gang
2016-01-01
Highlights: • A parabolic primary mirror field is designed to reduce the gap between adjacent mirrors. • The movable receiver can reduce the end losses. • The thermal efficiency of 66% is achieved at Guangzhou in winter. - Abstract: This paper proposes a stretched parabolic linear Fresnel reflector (SPLFR) collecting system. The primary optical mirror field of the SPLFR collecting system and the second-stage concentrator of compound parabolic collector are designed. The mirrors located at the parabolic line are close to each other, which effectively reduce the gap between the adjacent mirrors. The end losses of the receiver are very important, especially in a small-scale collecting system. A movable receiver is introduced for the reduction of the end losses. Moreover, a stretched structure of SPLFR is designed for wind resistance. Finally, the thermal performance of the SPLFR collecting system with fixed and movable receiver located in Guangzhou is tested. The maximum thermal efficiency obtained by this collecting system with movable receiver is 66% which avoid the end losses effectively, and the solar collector thermal loss coefficient is 1.32 W/m"2 °C. The results show that the SPLFR collecting system has excellent thermal performance and a promising application future. Meanwhile, this system will provide a valuable reference for concentrating solar power technology.
A moving mesh finite difference method for equilibrium radiation diffusion equations
Energy Technology Data Exchange (ETDEWEB)
Yang, Xiaobo, E-mail: xwindyb@126.com [Department of Mathematics, College of Science, China University of Mining and Technology, Xuzhou, Jiangsu 221116 (China); Huang, Weizhang, E-mail: whuang@ku.edu [Department of Mathematics, University of Kansas, Lawrence, KS 66045 (United States); Qiu, Jianxian, E-mail: jxqiu@xmu.edu.cn [School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, Fujian 361005 (China)
2015-10-01
An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation.
A moving mesh finite difference method for equilibrium radiation diffusion equations
International Nuclear Information System (INIS)
Yang, Xiaobo; Huang, Weizhang; Qiu, Jianxian
2015-01-01
An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation
International Nuclear Information System (INIS)
Budantsev, M. V.; Lavrov, R. A.; Pogosov, A. G.; Zhdanov, E. Yu.; Pokhabov, D. A.
2011-01-01
Extraordinary piecewise parabolic behavior of the magnetoresistance has been experimentally detected in the two-dimensional electron gas with a dense triangular lattice of antidots, where commensurability magnetoresistance oscillations are suppressed. The magnetic field range of 0–0.6 T can be divided into three wide regions, in each of which the magnetoresistance is described by parabolic dependences with high accuracy (comparable to the experimental accuracy) and the transition regions between adjacent regions are much narrower than the regions themselves. In the region corresponding to the weakest magnetic fields, the parabolic behavior becomes almost linear. The observed behavior is reproducible as the electron gas density changes, which results in a change in the resistance by more than an order of magnitude. Possible physical mechanisms responsible for the observed behavior, including so-called “memory effects,” are discussed.
International Nuclear Information System (INIS)
Bouri, C.; Selles, P.; Malegat, L.; Kwato Njock, M. G.
2006-01-01
Spherical and parabolic partial cross sections and asymmetry parameters, defined in the ejected electron frame, are presented for photoionization excitation of the helium atom at 0.1 eV above its double ionization threshold. A quantitative law giving the dominant spherical partial wave l dom for each excitation level n is obtained. The parabolic partial cross sections are shown to satisfy the same approximate selection rules as the related Rydberg series of doubly excited states (K,T) n A . The analysis of radial and angular correlations reveals the close relationship between double excitation, ionization excitation, and double ionization. Opposite to a widespread belief, the observed value of the asymmetry parameter is shown to result from the interplay of radial correlations and symmetry constraints, irrespective of angular correlations. Finally, the measurement of parabolic partial cross sections is proposed as a challenge to experimentalists
Finding the radical of an algebra of linear transformations
Cohen, A.M.; Ivanyos, G.; Wales, D.B.
1997-01-01
We present a method that reduces the problem of computing the radical of a matrix algebra over an arbitrary field to solving systems of semilinear equations. The complexity of the algorithm, measured in the number of arithmetic operations and the total number of the coefficients passed to an oracle
International Nuclear Information System (INIS)
Karabulut, Halit; Ipci, Duygu; Cinar, Can
2016-01-01
Highlights: • A numerical method has been developed for fully developed flows with constant wall temperature. • The governing equations were transformed to boundary fitted coordinates. • The Nusselt number of parabolic duct has been investigated. • Validation of the numerical method has been made by comparing published data. - Abstract: In motor-vehicles the use of more compact radiators have several advantages such as; improving the aerodynamic form of cars, reducing the weight and volume of the cars, reducing the material consumption and environmental pollutions, and enabling faster increase of the engine coolant temperature after starting to run and thereby improving the thermal efficiency. For the design of efficient and compact radiators, the robust determination of the heat transfer coefficient becomes imperative. In this study the external heat transfer coefficient of the radiator has been investigated for hydrodynamically and thermally fully developed flows in channels with constant wall temperature. In such situation the numerical treatment of the problem results in a trivial solution. To find a non-trivial solution the problem is treated either as an eigenvalue problem or as a thermally developing flow problem. In this study a numerical solution procedure has been developed and the heat transfer coefficients of the fully developed flow in triangular and parabolic air channels were investigated. The governing equations were transformed to boundary fitted coordinates and numerically solved. The non-trivial solution was obtained by means of guessing the temperature of any grid point within the solution domain. The correction of the guessed temperature was performed via smoothing the temperature profile on a line passing through the mentioned grid point. Results were compared with literature data and found to be consistent.
Two-Phase Fluid Simulation Using a Diffuse Interface Model with Peng--Robinson Equation of State
Qiao, Zhonghua
2014-01-01
In this paper, two-phase fluid systems are simulated using a diffusive interface model with the Peng-Robinson equation of state (EOS), a widely used realistic EOS for hydrocarbon fluid in the petroleum industry. We first utilize the gradient theory of thermodynamics and variational calculus to derive a generalized chemical equilibrium equation, which is mathematically a second-order elliptic partial differential equation (PDE) in molar density with a strongly nonlinear source term. To solve this PDE, we convert it to a time-dependent parabolic PDE with the main interest in its final steady state solution. A Lagrange multiplier is used to enforce mass conservation. The parabolic PDE is then solved by mixed finite element methods with a semi-implicit time marching scheme. Convex splitting of the energy functional is proposed to construct this time marching scheme, where the volume exclusion effect of an EOS is treated implicitly while the pairwise attraction effect of EOS is calculated explicitly. This scheme is proved to be unconditionally energy stable. Our proposed algorithm is able to solve successfully the spatially heterogeneous two-phase systems with the Peng-Robinson EOS in multiple spatial dimensions, the first time in the literature. Numerical examples are provided with realistic hydrocarbon components to illustrate the theory. Furthermore, our computational results are compared with laboratory experimental data and verified with the Young-Laplace equation with good agreement. This work sets the stage for a broad extension of efficient convex-splitting semi-implicit schemes for numerical simulation of phase field models with a realistic EOS in complex geometries of multiple spatial dimensions.
International Nuclear Information System (INIS)
Brain, P; Strimenopoulou, F; Ivarsson, M; Wilson, F J; Diukova, A; Wise, R G; Berry, E; Jolly, A; Hall, J E
2014-01-01
Conventional analysis of clinical resting electroencephalography (EEG) recordings typically involves assessment of spectral power in pre-defined frequency bands at specific electrodes. EEG is a potentially useful technique in drug development for measuring the pharmacodynamic (PD) effects of a centrally acting compound and hence to assess the likelihood of success of a novel drug based on pharmacokinetic–pharmacodynamic (PK–PD) principles. However, the need to define the electrodes and spectral bands to be analysed a priori is limiting where the nature of the drug-induced EEG effects is initially not known. We describe the extension to human EEG data of a generalised semi-linear canonical correlation analysis (GSLCCA), developed for small animal data. GSLCCA uses data from the whole spectrum, the entire recording duration and multiple electrodes. It provides interpretable information on the mechanism of drug action and a PD measure suitable for use in PK–PD modelling. Data from a study with low (analgesic) doses of the μ-opioid agonist, remifentanil, in 12 healthy subjects were analysed using conventional spectral edge analysis and GSLCCA. At this low dose, the conventional analysis was unsuccessful but plausible results consistent with previous observations were obtained using GSLCCA, confirming that GSLCCA can be successfully applied to clinical EEG data. (paper)
DEFF Research Database (Denmark)
Gammelmark, Søren; Eckardt, André
2013-01-01
felt by the two species. Using numerical simulations we predict that a finite parabolic potential can assist the adiabatic preparation of the antiferromagnet. The optimal strength of the parabolic inhomogeneity depends sensitively on the number imbalance between the two species. We also find...
International Nuclear Information System (INIS)
Sivakami, A.; Mahendran, M.
2010-01-01
The binding energy of a shallow hydrogenic impurity in a spherical quantum dot under hydrostatic pressure with square well potential is calculated using a variational approach within the effective mass approximation. The effect of conduction band non-parabolicity on these energies is also estimated. The binding energy is computed for GaAs spherical quantum dot as a function of dot size, hydrostatic pressure both in the presence and absence of the band non-parabolicity effect. Our results show that (i) the hydrostatic pressure increases the impurity binding energy when dot radius increases for a given pressure, (ii) the hydrostatic pressure with the band non-parabolicity effect effectively increases the binding energy such that the variation is large for smaller dots and (iii) the maximum contribution by the non-parabolicity effect is about 15% for narrow dots. Our results are in good agreement with Perez-Merchancano et al. [J. Phys. Condens. Matter 19 (2007) 026225] who have not considered the conduction band non-parabolicity effect.
International Nuclear Information System (INIS)
Zhao Jia-Sheng; Li Pan; Chen Xiao-Dong; Feng Su-Juan; Mao Qing-He
2012-01-01
The evolutions of the pulses propagating in decreasing and increasing gain distributed fiber amplifiers with finite gain bandwidths are investigated by simulations with the nonlinear Schrödinger equation. The results show that the parabolic pulse propagations in both the decreasing and the increasing gain amplifiers are restricted by the finite gain bandwidth. For a given input pulse, by choosing a small initial gain coefficient and gain variation rate, the whole gain for the pulse amplification limited by the gain bandwidth may be higher, which is helpful for the enhancement of the output linearly chirped pulse energy. Compared to the decreasing gain distributed fiber amplifier, the increasing gain distributed amplifier may be more conducive to suppress the pulse spectral broadening and increase the critical amplifier length for achieving a larger output linearly chirped pulse energy
On a Mixed Nonlinear One Point Boundary Value Problem for an Integrodifferential Equation
Directory of Open Access Journals (Sweden)
Mesloub Said
2008-01-01
Full Text Available This paper is devoted to the study of a mixed problem for a nonlinear parabolic integro-differential equation which mainly arise from a one dimensional quasistatic contact problem. We prove the existence and uniqueness of solutions in a weighted Sobolev space. Proofs are based on some a priori estimates and on the Schauder fixed point theorem. we also give a result which helps to establish the regularity of a solution.