The Experts below are selected from a list of 13023 Experts worldwide ranked by ideXlab platform
Sergey Zelik - One of the best experts on this subject based on the ideXlab platform.
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upper and lower bounds for the kolmogorov entropy of the attractor for the rde in an Unbounded Domain
Journal of Dynamics and Differential Equations, 2002Co-Authors: Messoud Efendiev, Sergey ZelikAbstract:The long-time behaviour of bounded solutions of a reaction-diffusion system in an Unbounded Domain Ω ⊂ ℝn, for which the nonlinearity f(u, ∇xu) explicitly depends on ∇xu is studied. We prove the existence of a global attractor, fractal dimension of which is infinite, and give upper and lower bounds for the Kolmogorov entropy of the attractor and analyze the sharpness of these bounds. © 2002 Plenum Publishing Corporation.
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the attractor for a nonlinear reaction diffusion system in an Unbounded Domain
Communications on Pure and Applied Mathematics, 2001Co-Authors: Messoud Efendiev, Sergey ZelikAbstract:In this paper the quasi-linear second-order parabolic systems of reaction-diffusion type in an Unbounded Domain are considered. Our aim is to study the long-time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered. © 2001 John Wiley & Sons, Inc.
Messoud Efendiev - One of the best experts on this subject based on the ideXlab platform.
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upper and lower bounds for the kolmogorov entropy of the attractor for the rde in an Unbounded Domain
Journal of Dynamics and Differential Equations, 2002Co-Authors: Messoud Efendiev, Sergey ZelikAbstract:The long-time behaviour of bounded solutions of a reaction-diffusion system in an Unbounded Domain Ω ⊂ ℝn, for which the nonlinearity f(u, ∇xu) explicitly depends on ∇xu is studied. We prove the existence of a global attractor, fractal dimension of which is infinite, and give upper and lower bounds for the Kolmogorov entropy of the attractor and analyze the sharpness of these bounds. © 2002 Plenum Publishing Corporation.
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the attractor for a nonlinear reaction diffusion system in an Unbounded Domain
Communications on Pure and Applied Mathematics, 2001Co-Authors: Messoud Efendiev, Sergey ZelikAbstract:In this paper the quasi-linear second-order parabolic systems of reaction-diffusion type in an Unbounded Domain are considered. Our aim is to study the long-time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered. © 2001 John Wiley & Sons, Inc.
Jiwei Zhang - One of the best experts on this subject based on the ideXlab platform.
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numerical solution of the time fractional sub diffusion equation on an Unbounded Domain in two dimensional space
East Asian Journal on Applied Mathematics, 2017Co-Authors: Jiwei ZhangAbstract:The numerical solution of the time-fractional sub-diffusion equation on an Unbounded Domain in two-dimensional space is considered, where a circular artificial boundary is introduced to divide the Unbounded Domain into a bounded computational Domain and an Unbounded exterior Domain. The local artificial boundary conditions for the fractional sub-diffusion equation are designed on the circular artificial boundary by a joint Laplace transform and Fourier series expansion, and some auxiliary variables are introduced to circumvent high-order derivatives in the artificial boundary conditions. The original problem defined on the Unbounded Domain is thus reduced to an initial boundary value problem on a bounded computational Domain. A finite difference and L1 approximation are applied for the space variables and the Caputo time-fractional derivative, respectively. Two numerical examples demonstrate the performance of the proposed method.
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artificial boundary conditions for nonlocal heat equations on Unbounded Domain
Communications in Computational Physics, 2017Co-Authors: Wei Zhang, Jiang Yang, Jiwei ZhangAbstract:This paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite Domain. Two classes of artificial boundary conditions (ABCs) are designed, namely, nonlocal analog Dirichlet-to-Neumann-type ABCs (global in time) and high-order Pade approximate ABCs (local in time). These ABCs reformulate the original problem into an initial-boundary-value (IBV) problem on a bounded Domain. For the global ABCs, we adopt a fast evolution to enhance computational efficiency and reduce memory storage. High order fully discrete schemes, both second-order in time and space, are given to discretize two reduced problems. Extensive numerical experiments are carried out to show the accuracy and efficiency of the proposed methods.
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computation of the schrodinger equation in the semiclassical regime on an Unbounded Domain
SIAM Journal on Numerical Analysis, 2014Co-Authors: Xu Yang, Jiwei ZhangAbstract:The study of this paper is twofold: On the one hand, we generalize the high-order local absorbing boundary conditions (LABCs) proposed in [J. Zhang et al., Comm. Comput. Phys., 10 (2011), pp. 742--766] to compute the Schrodinger equation in the semiclassical regime on an Unbounded Domain. We analyze the stability of the equation with LABCs and the convergence of the Crank--Nicolson scheme that discretizes it and we conclude that when the rescaled Planck constant $\varepsilon$ gets small, the accuracy deteriorates and the requirements on time step and mesh size get tough. This leads to the second part of our study. We propose an asymptotic method based on the frozen Gaussian approximation. The absorbing boundary condition is dealt with by a simple strategy that all the effects of the Gaussian functions which contribute to the outgoing waves will be eliminated by stopping the Hamiltonian flow of their centers when they get out of the Domain of interest. We present numerical examples in both one and two dime...
Vahagn Nersesyan - One of the best experts on this subject based on the ideXlab platform.
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Ergodicity for the randomly forced Navier-Stokes system in a two-dimensional Unbounded Domain
arXiv: Analysis of PDEs, 2019Co-Authors: Vahagn NersesyanAbstract:The ergodic properties of the randomly forced Navier-Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded Domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier-Stokes system in an Unbounded Domain satisfying the Poincar\'e inequality. Assuming that the system is perturbed by a bounded non-degenerate noise, we establish uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is carried out by developing the controllability approach of the papers arXiv:1803.01893 and arXiv:1802.03250 and using the asymptotic compactness of the dynamics.
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Ergodicity for the randomly forced Navier--Stokes system in a two-dimensional Unbounded Domain
2019Co-Authors: Vahagn NersesyanAbstract:The ergodic properties of the randomly forced Navier-Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded Domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier-Stokes system in an Unbounded Domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise, we establish uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is carried out by developing the controllability approach of the papers [Shi19, KNS18] and using the asymptotic compactness of the dynamics.
Zhizhong Sun - One of the best experts on this subject based on the ideXlab platform.
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the finite difference approximation for a class of fractional sub diffusion equations on a space Unbounded Domain
Journal of Computational Physics, 2013Co-Authors: Guanghua Gao, Zhizhong SunAbstract:In this paper, for a class of fractional sub-diffusion equations on a space Unbounded Domain, firstly, exact artificial boundary conditions, which involve the time-fractional derivatives, are derived using the Laplace transform technique. Then the original problem on the space Unbounded Domain is reduced to the initial-boundary value problem on a space bounded Domain. Secondly, an efficient finite difference approximation for the reduced initial-boundary problem on the space bounded Domain is constructed. Different from the method of order reduction used in [37] for the fractional sub-diffusion equations on a space half-infinite Domain, the presented difference scheme, which is more simple than that in the previous work, is developed using the direct discretization method, i.e. the approximate method of considering the governing equations at mesh points directly. The stability and convergence of the scheme with numerical accuracy O(@t^2^-^@c+h^2) are proved by means of discrete energy method and Sobolev imbedding inequality, where @c is the order of time-fractional derivative in the governing equation, @t and h are the temporal stepsize and spatial stepsize, respectively. Thirdly, a compact difference scheme for the case of @c=<2/3 is derived with the truncation errors of fourth-order accuracy for interior points and third-order accuracy for boundary points, respectively. Then the global convergence order O(@t^2^-^@c+h^4) of the compact difference scheme is proved. Finally, numerical experiments are used to verify the numerical accuracy and the efficiency of the obtained schemes.
