Parameter dependent equation
Parameter dependent equation. Aug 29, 2023 · We investigate the existence of multiple periodic solutions for a class of second order parameter-dependent equations of the form $ x''+f(t, x) = sp(t) $. It is assumed that the diffusion coefficient is a constant in each layer. Dependence on Parameters. Guerra*, M. org are unblocked. parameter-dependent equations Olivier ZAHMyand Anthony NOUYyz January 28, 2016 Abstract We propose a method for the construction of preconditioners of parameter-dependent matrices for the solution of large systems of parameter-dependent equations. . 4). More than one parameter can be employed when necessary. The interest is in Ambrosetti–Prodi type alternatives which provide the existence of zero Jul 1, 2024 · In this paper we investigate a parameter-dependent nonlocal differential equations with convolution coefficients. Garkun1, Suvendu K. The goal is to find approximate parameter-to-solution maps that have a small number of terms. Under additional growth conditions, we obtain upper and lower bounds for the parameter. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. we say that x is an independent variable: it can be freely set to any value (or any value within the given domain) and the value of the function is then computed. kasandbox. Jan 1, 2013 · Optimization of the solution of the parameter-dependent Sylvester equation. The stabilizing property of the solution to MARE is presented. The major contribution Jan 22, 2010 · Consider the following ODE with time-dependent parameters: y'(t) + f(t)y(t) = g(t) and the given initial condition: y(0) = 1 This is an example of an ODE with time Nov 18, 2013 · We discuss associative analogues of classical Yang-Baxter equation meromorphically dependent on parameters. We propose a classification of all solutions for one-dimensional associative Yang-Baxter equations. Consider the parameterized integral equation x(t;q) = x0(q)+ Z t t0 a(x(˝;q);˝;t;q)d˝; t 2 [t0;t1]; (3. First, by selecting a virtual reference point in Dec 17, 2013 · On the basis of existing results for Lyapunov equations and parameter-dependent linear systems, we prove that the tensor containing all solution samples typically allows for an excellent low multilinear rank approximation. August 24-29, 2014 Toward a Rational Matrix Approximation of the Parameter-Dependent Riccati Equation Solution J. 1) where f, u are elements of a general complex Banach space E, K is a linear operator on E, N is, in general, a nonlinear operator defined on certain domain in E, and A is a complex parameter. Ideas: We are interested in the dependence of differential equations on the initial data and any parameters present; in particular, we want open neighborhoods in “parameter space” for which the dependence is continuous. com/patrickjmt !! Solving a Dependent System Sep 18, 2024 · Abstract: In this paper we generalize Krylov's theory on parameter-dependent stochastic differential equations to the framework of rough stochastic differential equations (rough SDEs), as initially introduced by Friz, Hocquet and Lê. 2 Let x0: [t 0;tf] 7!R n be a solution of (3. Although the formulation is in the setting of model following adaptive control, the realization of the adaptive controll er does not Sep 18, 2019 · We study the periodic boundary value problem associated with the $$\\phi $$ ϕ -Laplacian equation of the form $$(\\phi (u'))'+f(u)u'+g(t,u)=s$$ ( ϕ ( u ′ ) ) ′ + f ( u ) u ′ + g ( t , u ) = s , where s is a real parameter, f and g are continuous functions, and g is T-periodic in the variable t. Based May 31, 2017 · Our aim is to numerically solve parameter-dependent Lyapunov equations using the reduced basis method. 5) has the form of (3. It is inspired by the multi-modes method and the ensemble method and extends those methods into a more general and unified framework. Van den Bergh. Parametric Equation – Explanation and Examples. Barik2, Aleksey K. Consider the parameterized integral equation t x(t,q) = ¯x0(q) + a(x( ,q), ,t,q)d , t ≤ [t0,t1], (3. We consider quasilinear elliptic equations, including the following Modified Nonlinear Schrödinger Equation as a special example: $ \begin{equation*} \left\{ \begin Aug 1, 2021 · The aim of this paper is to provide an efficient method for solving a family of parameter dependent, algebraic Lyapunov equations in an infinite dimensional setting. May 25, 2021 · Open x-simple regions naturally arise in the context of fundamental sets of solutions of linear ordinary differential equations depending on a parameter. The collinearly improved kernel along with the impact-parameter dependent BK equation has been used to demonstrate, that the previously established problem of Coulomb tails can be highly suppressed and the new solutions allow for a correct description of data, restoring thus the pre- Feb 14, 2024 · 6. This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The pro-posed method is an interpolation of the matrix inverse based on a projection of the Oct 1, 2008 · In this paper we present an online parameter identification method that is based on a non-linear parameter-to-output operator and, as opposed to methods available so far, works both for finite- and infinite-dimensional dynamical systems, e. The main idea of our approach is to construct local basis functions that encode the New spectral-parameter dependent solutions of the Yang-Baxter equation Alexander. Then, the uniqueness is proved for the almost stabilizing and Dec 15, 2023 · We propose a generalized multiscale finite element method combined with a balanced truncation to solve a parameter-dependent parabolic problem. Our analysis is based on previous work on reduced modeling and (weak) greedy algorithms for parameter dependent PDEs and abstract equations in Banach spaces. In mathematics, a parametric equation is explained as: “A form of the equation that has an independent variable in terms of which any other equation is defined, and dependent variables involved in such an equation are continuous functions of the independent parameter. Our work is In this article, we propose a modified Allen–Cahn (AC) equation with a space-dependent interfacial parameter. Therefore, a more relaxed stability Aug 7, 2023 · This paper deals with a novel direct state-dependent Riccati equation (SDRE) controller designed for trajectory tracking of underactuated autonomous underwater vehicles (AUVs) in the presence of parameter perturbation. When numerically solving the AC equation with a constant interfacial parameter over large domains, a substantial number of grid points are essential, which leads to significant computational costs. We discover that such equations enter in a description of a general class of parameter-dependent Poisson structures and double Lie and Poisson structures in sense of M. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n-width. The adaptive controller is intended to augment a nominal, fixed gain, observer based output feedback control law. Moreover, several properties of such equations depend on whether the underlying domain Ω of the independent variable x and the parameter t is x-simple and how the x-simplicity is combined with the connectedness, in particular, whether all the In addition to curves and surfaces, parametric equations can describe manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. Fedorov3,4,5 and Vladimir Gritsev2,3⋆ 1 Institute of Applied Physics of the National Academy of Sciences of Belarus, Belarus 2 Institute of Physics, University of Amsterdam, The Netherlands 3 Russian Quantum Center If none of the non-negative parameters α, β, γ, δ vanishes, three can be absorbed into the normalization of variables to leave only one parameter: since the first equation is homogeneous in x, and the second one in y, the parameters β/α and δ/γ are absorbable in the normalizations of y and x respectively, and γ into the normalization Thanks to all of you who support me on Patreon. Mar 26, 2021 · Stochastic differential equations (SDEs) are popular tools to analyse time series data in many areas, such as mathematical finance, physics, and biology. We consider the Sylvester equation (4) (A 0 − v C 1 C 2 T) X + X (B 0 − v D 1 D 2 T) = E, where A 0 is m × m, B 0 is n × n, C 1 and C 2 are m × r 1, D 1 and D 2 are n × r 2 matrices and the unknown matrix X as well as the right-hand side matrix E are m × n Apr 26, 2017 · In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). , both for ordinary differential equations and time-dependent partial differential equations. kastatic. g. 1. Mar 13, 2024 · Our aim is to numerically solve parameter-dependent Lyapunov equations using the reduced basis method. Such equations arise in parametric model order reduction. S. It is shown that in the case of two layers one can find a solution formula consisting of Mar 10, 2021 · Within the framework of parameter dependent PDEs, we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. Feb 5, 2017 · The problem is addressed in the context of the abstract system (1. Parametric equation, a type of equation that employs an independent variable called a parameter (often denoted by t) and in which dependent variables are defined as continuous functions of the parameter and are not dependent on another existing variable. The Jan 1, 2017 · PDF | Our aim is to numerically solve parameter-dependent Lyapunov equations using the reduced basis method. If you're seeing this message, it means we're having trouble loading external resources on our website. Despite the traditional SDRE regulator control, the proposed closed-loop SDRE controller design chiefly consists of two parts. Consider the differential equation \begin {equation} \dot x = f (t, x, \alpha),\qquad x (0) = x_0, \label {eq-ivp-with-parameter} \end {equation} where $\alpha$ is a parameter (i. Properties like the existence of fundamental sets of solutions or characterizations of such sets via nonvanishing Wronskians are sensitive to the topological properties of the underlying domain of the independent variable and the Jul 3, 2024 · Solution of parameter-dependent diffusion equation in layered media Antti Autio Antti Hannukainen∗ July 3, 2024 Abstract This work studies the parameter-dependent diffusion equation in a two-dimensional domain consisting of locally mirror symmetric layers. The main challenges are to accurately and efficiently approximate the POD Jan 1, 2022 · Typical examples are the heat equation where the system operator depends on the thermal diffusivity parameter and control is provided through an external heat source (cf. Such equations arise in parametric model | Find, read and cite all the research you Apr 1, 2024 · Then we show how to use SSMTool perform model reduction on parameter-dependent center and unstable manifolds in Sections 5 Reduction on parameter-dependent center manifold (PCM), 6 Reduction on parameter-dependent unstable manifold (PUM) respectively. One should think of a system of equations as being an implicit equation for its solution set, and of the parametric form as being the parameterized equation for the same set. May 25, 2021 · The well-known solution theory for (systems of) linear ordinary differential equations undergoes significant changes when introducing an additional real parameter. An example is also given to illustrate the main results. f (x) = 3x2 + 2x + 1. In this chapter we show how to build models of parameter-dependent systems and how to use these models in robust stability analysis and controller design. We propose a classi cation of all solutions for Jan 1, 2014 · Proceedings of the 19th World Congress The International Federation of Automatic Control Cape Town, South Africa. Yagoubi* and P. 5) t0 where q ≤ R is a parameter. Jan 1, 2018 · Request PDF | Sign-changing solutions for a parameter-dependent quasilinear equation | We consider quasilinear elliptic equations, including the following Modified Nonlinear Schrödinger Equation Aug 1, 2021 · The aim of this paper is to provide an efficient method for solving a family of parameter dependent, algebraic Lyapunov equations in an infinite dimensional setting. As an updated version of the standard multiscale method, the generalized multiscale method contains the necessary eigenvalue computation, in which the enriched multiscale basis functions are picked up from a snapshot space on users’ demand. 1a) y ˙ ζ = A ζ y ζ + B ζ u, with the parameter dependent initial condition (1. of ODE depend continuously on initial conditions and other parameters. . We discover that such equations enter in a description of a general class of parameter We discuss associative analogues of classical Yang-Baxter equation meromorphically dependent on parameters. We restrict ourselves to the systems The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the mass density, the vacuum energy, or the spatial curvature). The Jan 1, 2022 · Typical examples are the heat equation where the system operator depends on the thermal diffusivity parameter and control is provided through an external heat source (cf. Oct 2, 2020 · to study the existence of localized nodal solutions for parameter-dependent semiclassical quasilinear Schr odinger equations, under a certain parametric conditions. Independent and Dependent Variables. We restrict ourselves to the systems that affinely depend on the parameter, as our main strategy In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. Example 3. Aug 20, 2024 · parameter an independent variable that both \(x\) and \(y\) depend on in a parametric curve; usually represented by the variable \(t\) parametric curve the graph of the parametric equations \(x(t)\) and \(y(t)\) over an interval \(a≤t≤b\) combined with the equations parametric equations Nov 1, 1997 · INTRODUCTION AND MAIN OBJECTIVES Consider the general parameter-dependent Hammerstein-type equations of the form u = f + AKNu, (1. Let be a unital associative algebra. (2020). For every xed value of q integral equation (3. 5) where q 2 R is a parameter. In particular, we assume that the coefficients have both small scales and high contrast (where the high contrast refers to the large variations in the coefficients). The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in 1884 that the van 't Hoff equation for the temperature dependence of equilibrium constants suggests such a formula for the rates of both forward and x is the formal parameter (the parameter) of the defined function. 1: Arrhenius Equation Expand/collapse global location where temperature is the independent variable and the rate constant is the dependent variable. Sep 17, 2022 · It is an expression that produces all points of the line in terms of one parameter, \(z\). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle Mar 18, 2019 · For such case, we combine the proper orthogonal decomposition (POD)technique in time with greedy algorithm in parameter for producing optimal reduced basis (RB) space. When we write a function such as. Variables and Parameters. Third, a numerical simulation method for the class of path-dependent More specifically, in contrast to the usual linear heat equation with constant co-efficients, we are interested in a nonlinear heat equation with temperature-dependent material parameters. 3. Second, we construct maximum likelihood estimators of these parameters and then discuss their strong consistency. 2 Continuous Dependence On Parameters In this section our main objective is to establish sufficient conditions under which solutions of ODE depend continuously on initial conditions and other parameters. 6 below), or the wave equation, with parameter standing for a damping constant and/or velocity of propagation, and controlled by external disturbances. Evaluating the Hubble parameter at the present time yields Hubble's constant which is the proportionality constant of Hubble's law. 2. e. We compare the behavior of its solutions with suitable linear and piecewise linear equations near positive infinity and infinity. The main idea of the framework is to reformulate the underlying problem into another problem with parameter-independent Aug 8, 2011 · A parameter dependent Riccati equation approach is taken to analyze the stability properties of an output feedback adaptive control law design. You da real mvps! $1 per month helps!! :) https://www. These reductions construct parametric ROMs that enable efficient extraction of the self Mar 13, 2024 · As a particular application, we consider the approximation of solution manifolds of linear parameter-dependent partial differential equations with a probabilistic interpretation through the Feynman-Kac formula. 5) with q Sep 12, 2003 · 3. Numerical results for time-dependent convection–diffusion problem and Burgers’ equation illustrate the highly accurate and efficiency of the presented method. Introduction In this article, we study the existence of localized nodal solutions for the param-eter-dependent semiclassical quasilinear Schr odinger equation "2 XN i;j=1 D j(b Jan 5, 2022 · This paper develops and analyzes a general iterative framework for solving parameter-dependent and random convection–diffusion problems. We propose a flexible model In this paper, we propose a multiscale approach for solving the parameter-dependent elliptic equation with highly heterogeneous coefficients. Theorem 3. Using the Birkhoff–Kellogg type theorem, existence of positive solutions is established. Equation also appears in the theory of Heisenberg ferromagnets and magnons [6, 10, 26], in dissipative quantum mechanics [], and in condensed matter theory [], and has received considerable attention in mathematical analysis during the last 10 years. If you're behind a web filter, please make sure that the domains *. They provide a mechanistic description of the phenomenon of interest, and their parameters often have a clear interpretation. These advantages come at the cost of requiring a relatively simple model specification. [ 1 ] (2) Some parameter-dependent slack matrices that depend on the time-varying delay and its derivative are introduced in the affine integral inequality, zero equations, and S-procedure, providing extra freedom for optimizing the Lyapunov matrices compared with the existing using constant slack matrices. Existence, uniqueness and asymptotic behavior of initial boundary value problems under appropriate assumptions on the material parameters are established. a constant that appears in the differential equation). org and *. When the function is evaluated for a given value, as in f(3): or, y = f(3) = 3 + 2 = 5, 3 is the actual parameter (the argument) for evaluation by the defined function; it is a given value (actual value) that is substituted for the formal parameter of the defined Nov 7, 2016 · which has been involved in models of superfluid films in fluid mechanics and plasma physics [11, 12, 25]. First, we prove the existence and uniqueness of these equations under non-Lipschitz conditions. Nov 18, 2013 · We discuss associative analogues of classical Yang-Baxter equation meromorphically dependent on parameters. Chevrel* LUNAM Université, EMN (Ecole des Mines de Nantes), IRCCyN UMR CNRS 6597, FRANCE (e-mails: [email protected], [email protected] and Jul 2, 2024 · This work studies the parameter-dependent diffusion equation in a two-dimensional domain consisting of locally mirror symmetric layers. In its most general form, the parameter-dependent Yang–Baxter equation is an equation for (, ′), a parameter-dependent element of the tensor product (here, and ′ are the parameters, which usually range over the real numbers ℝ in the case of an additive parameter, or over positive real numbers ℝ + in the case of a multiplicative parameter). ” In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). Nonlinear Systems: Parameter Dependence. 1b) y ζ (0) = y ζ i, where ζ ∈ R is a random variable following a probability law η, A ζ is an operator on X, the state space, B ζ is a control operator, y ζ (t) ∈ X is the parameter Jul 15, 2024 · We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. We consider a stochastic equation of the form $$ dX_t^\zeta = b_t(\zeta,X_t^\zeta) \ dt + \sigma_t(\zeta,X_t May 15, 2015 · This technical note deals with a modified algebraic Riccati equation (MARE) and its corresponding inequality and difference equation, which arise in modified optimal control and filtering problems and are introduced into the cooperative control problems recently. patreon. The linear parameter-dependent systems, called also Linear Parameter-Varying Systems (LPV systems), arise in many applications, for instance in robotics and aerospace systems. Furthermore, in this context, the nonlinearity $ f $ does not satisfy the usual sign condition, and the global Apr 21, 2022 · This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters. lgpai lbpsa qgqupwi aogg axyvb bcvzj krzydko ufftztss mnbsz phsxem