The Experts below are selected from a list of 192 Experts worldwide ranked by ideXlab platform
Alexander Domoshnitsky - One of the best experts on this subject based on the ideXlab platform.
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Nonoscillation and Stability of the Second Order Ordinary Differential Equations with a Damping Term
arXiv: Dynamical Systems, 2008Co-Authors: Leonid Berezansky, Elena Braverman, Alexander DomoshnitskyAbstract:In this paper we consider the linear ordinary Equation of the second order $$ L x(t)\equiv \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=f(t), \eqno{(1)} $$ and the Corresponding Homogeneous Equation $$ \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=0. \eqno{(2)} $$ Note that $[\alpha ,\beta ]$ is called a nonoscillation interval if every nontrivial solution has at most one zero on this interval. Many investigations which seem to have no connection such as differential inequalities, the Polia-Mammana decomposition (i.e. representation of the operator $L$ in the form of products of the first order differential operators), unique solvability of the interpolation problems, kernels oscillation, separation of zeros, zones of Lyapunov's stability and some others have a certain common basis - nonoscillation. Presumably Sturm was the first to consider the two problems which naturally appear here: to develop corollaries of nonoscillation and to find methods to check nonoscillation. In this paper we obtain several tests for nonoscillation on the semiaxis and apply them to propose new results on asymptotic properties and the exponential stability of the second order Equation (2). Using the Floquet representations and upper and lower estimates of nonoscillation intervals of oscillatory solutions we deduce results on the exponential and Lyapunov's stability and instability of Equation (2).
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Nonoscillation and Stability of the Second Order Ordinary Differential Equations with a Damping Term
Functional differential equations, 2004Co-Authors: Leonid Berezansky, Elena Braverman, Alexander DomoshnitskyAbstract:In this paper we consider the linear ordinary Equation of the second order and the Corresponding Homogeneous Equation Note that [a, ,8] is called a nonoscillation interval if every nontrivial solution has at most one zero on this interval. Many investigations which seem to have no connection such as differential inequalities, the Polia.-Mammana decomposition (i.e. representation of the operator £in the form of products of the first order differential operators), unique solvability of the interpolation problems, kernels oscillation, separation of zeros, zones of Lyapunov's stability and some others have a certain common basis- nonoscillation. Presumably Sturm was the first to consider the two problems which naturally appear here: to develop corollaries of nonoscillation and to find methods to check nonoscillation. In this paper we obtain several tests for nonoscillation on the semiaxi.s and apply them to propose new results on asymptotic properties and the exponential stability of the second order Equation (0.2). Using the Floquet representations and upper and lower estimates of nonoscillation intervals of oscillatory solutions we deduce results on the exponential and Lyapunov's stability and instability of Equation (0.2).
Peter J. Park - One of the best experts on this subject based on the ideXlab platform.
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MULTISCALE NUMERICAL METHODS FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION EquationS
International Journal of Computational Methods, 2004Co-Authors: Peter J. Park, Thomas Y. HouAbstract:We present an efficient and robust approach in the finite element framework for numerical solutions that exhibit multiscale behavior, with applications to singularly perturbed convection-diffusion problems. The first type of Equation we study is the convectiondominated convection-diffusion Equation, with periodic or random coefficients; the second type of Equation is an elliptic Equation with singularities due to discontinuous coefficients and non-smooth boundaries. In both cases, standard methods for purely hyperbolic or elliptic problems perform poorly due to sharp boundary and internal layers in the solution. We propose a framework in which the finite element basis functions are designed to capture the local small-scale behavior correctly. When the structure of the layers can be determined locally, we apply the multiscale finite element method, in which we solve the Corresponding Homogeneous Equation on each element to capture the small scale features of the differential operator. We demonstrate the effectiveness of this method by computing the enhanced diffusivity scaling for a passive scalar in the cellular flow. We also carry out the asymptotic error analysis for its convergence rate and perform numerical experiments for verification. For a random flow with nonlocal layer structure, we use a variational principle to gain additional information in our attempt to design asymptotic basis functions. We also apply the same framework for elliptic Equations with discontinuous coefficients or non-smooth boundaries. In that case, we construct local basis function near singularities using infinite element method in order to resolve extreme singularity. Numerical results on problems with various singularities confirm the efficiency and accuracy of this approach.
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Multiscale numerical methods for the singularly perturbed convection-diffusion Equation
2000Co-Authors: Peter J. ParkAbstract:We develop efficient and robust numerical methods in the finite element framework for numerical solutions of the singularly perturbed convection-diffusion Equation and of a degenerate elliptic Equation. The standard methods for purely elliptic or hyperbolic problems perform poorly when there are sharp boundary and internal layers in the solution caused by the dominant convective effect. We offer a new approach in which we design the finite element basis functions that capture the local behavior correctly. When the structure of the layers can be determined locally, we apply the multiscale finite element method in which we solve the Corresponding Homogeneous Equation on each element to capture the small scale features of the differential operator. We demonstrate the effectiveness of this method by computing the enhanced diffusivity scaling for a passive scalar in the cellular flow. We carry out the asymptotic error analysis for its convergence rate and perform numerical experiments for verification. When the layer structure is nonlocal, we use a variational principle to gain additional information. For a random velocity field, this variational principle provides correct scaling results. This allows us to design asymptotic basis functions that can capture the global layers correctly. The same approach is also extended to elliptic problems with high contrast coefficients. When an asymptotic result is available, it is incorporated naturally into the finite element setting developed earlier. When there is a strong singularity due to a discontinuous coefficient, we construct the basis functions using the infinite element method. Our methods can handle singularities efficiently and are not sensitive to the large contrast.
M.i. Ramazanov - One of the best experts on this subject based on the ideXlab platform.
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Spectrum of Volterra integral operator of the second kind
2016Co-Authors: Meiramkul M. Amangaliyeva, Muvasharkhan T. Jenaliyev, Madi Ergaliev, M.i. RamazanovAbstract:The article addresses the singular Volterra integral Equation of the second kind, which has the ’incompressible’ kernel. It is shown that the Corresponding Homogeneous Equation on |λ| ≥ exp{|arg λ|}, arg λ ∈ [−π,π] has a continuous spectrum, and the multiplicity of the characteristic numbers grows with increasing |λ|. We use the Carleman-Vekua regularization method. We introduce the characteristic integral Equation. We prove that the initial integral Equation has eigen-functions, the multiplicity of which depends on the value of the spectral parameter λ. We prove the solvability theorem of the nonHomogeneous Equation in a case when the right-hand side of the Equation belongs to a certain class.
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Uniqueness and non-uniqueness of solutions of the boundary value problems of the heat Equation
2015Co-Authors: Meiramkul M. Amangaliyeva, Muvasharkhan T. Jenaliyev, Minzilya T. Kosmakova, M.i. RamazanovAbstract:The article addresses the singular Volterra integral Equation of the second kind which has the “incompressible” kernel. It is shown that the Corresponding Homogeneous Equation on |λ| > exp{|arg λ|}, arg λ ∈ [−π, π] has a continuous spectrum, and the multiplicity of the characteristic numbers grows with increasing |λ|. The Equation is reduced to Abel Equation by the regularization method. The eigenfunctions of the Equation are found in an explicit form. We prove the solvability theorem of the inHomogeneous Equation in a case when the right-hand side of the Equation belongs to a certain class.
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On the spectrum of Volterra integral Equation with the “incompressible” kernel
2014Co-Authors: Meiramkul M. Amangaliyeva, Muvasharkhan T. Jenaliyev, Minzilya T. Kosmakova, M.i. RamazanovAbstract:The article addresses the singular Volterra integral Equation of the second kind which has the “incompressible” kernel. It is shown that the Corresponding Homogeneous Equation for |λ| > 1 has a continuous spectrum, and the multiplicity of the characteristic numbers grows with increasing |λ|. The Equation is reduced to Abel Equation by using the regularization method. The eigenfunctions of the Equation are found in an explicit form. We prove the solvability theorem of the nonHomogeneous Equation in a case when the right-hand side of the Equation belongs to a certain class.
Fengshan Long - One of the best experts on this subject based on the ideXlab platform.
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application of a lie group admitted by a Homogeneous Equation for group classification of a Corresponding inHomogeneous Equation
Communications in Nonlinear Science and Numerical Simulation, 2017Co-Authors: Fengshan Long, Adisak Karnbanjong, A Suriyawichitseranee, Yurii N. Grigoriev, Sergey V MeleshkoAbstract:Abstract This paper proposes an algorithm for group classification of a nonHomogeneous Equation using the group analysis provided for the Corresponding Homogeneous Equation. The approach is illustrated by a partial differential Equation, an integro-differential Equation, and a delay partial differential Equation.
Elena Braverman - One of the best experts on this subject based on the ideXlab platform.
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Nonoscillation and Stability of the Second Order Ordinary Differential Equations with a Damping Term
arXiv: Dynamical Systems, 2008Co-Authors: Leonid Berezansky, Elena Braverman, Alexander DomoshnitskyAbstract:In this paper we consider the linear ordinary Equation of the second order $$ L x(t)\equiv \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=f(t), \eqno{(1)} $$ and the Corresponding Homogeneous Equation $$ \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=0. \eqno{(2)} $$ Note that $[\alpha ,\beta ]$ is called a nonoscillation interval if every nontrivial solution has at most one zero on this interval. Many investigations which seem to have no connection such as differential inequalities, the Polia-Mammana decomposition (i.e. representation of the operator $L$ in the form of products of the first order differential operators), unique solvability of the interpolation problems, kernels oscillation, separation of zeros, zones of Lyapunov's stability and some others have a certain common basis - nonoscillation. Presumably Sturm was the first to consider the two problems which naturally appear here: to develop corollaries of nonoscillation and to find methods to check nonoscillation. In this paper we obtain several tests for nonoscillation on the semiaxis and apply them to propose new results on asymptotic properties and the exponential stability of the second order Equation (2). Using the Floquet representations and upper and lower estimates of nonoscillation intervals of oscillatory solutions we deduce results on the exponential and Lyapunov's stability and instability of Equation (2).
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Nonoscillation and Stability of the Second Order Ordinary Differential Equations with a Damping Term
Functional differential equations, 2004Co-Authors: Leonid Berezansky, Elena Braverman, Alexander DomoshnitskyAbstract:In this paper we consider the linear ordinary Equation of the second order and the Corresponding Homogeneous Equation Note that [a, ,8] is called a nonoscillation interval if every nontrivial solution has at most one zero on this interval. Many investigations which seem to have no connection such as differential inequalities, the Polia.-Mammana decomposition (i.e. representation of the operator £in the form of products of the first order differential operators), unique solvability of the interpolation problems, kernels oscillation, separation of zeros, zones of Lyapunov's stability and some others have a certain common basis- nonoscillation. Presumably Sturm was the first to consider the two problems which naturally appear here: to develop corollaries of nonoscillation and to find methods to check nonoscillation. In this paper we obtain several tests for nonoscillation on the semiaxi.s and apply them to propose new results on asymptotic properties and the exponential stability of the second order Equation (0.2). Using the Floquet representations and upper and lower estimates of nonoscillation intervals of oscillatory solutions we deduce results on the exponential and Lyapunov's stability and instability of Equation (0.2).
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A Fast Spectral Subtractional Solver for Elliptic Equations
Journal of Scientific Computing, 2004Co-Authors: Elena Braverman, Boris Epstein, Moshe Israeli, Amir AverbuchAbstract:The paper presents a fast subtractional spectral algorithm for the solution of the Poisson Equation and the Helmholtz Equation which does not require an extension of the original domain. It takes O ( N ^2 log N ) operations, where N is the number of collocation points in each direction. The method is based on the eigenfunction expansion of the right hand side with integration and the successive solution of the Corresponding Homogeneous Equation using Modified Fourier Method. Both the right hand side and the boundary conditions are not assumed to have any periodicity properties. This algorithm is used as a preconditioner for the iterative solution of elliptic Equations with non-constant coefficients. The procedure enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points. We also apply the basic solver to the solution of the Poisson Equation in complex geometries.
