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Ozgur Yildirim - One of the best experts on this subject based on the ideXlab platform.
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on stability of a third order of accuracy difference scheme for hyperbolic nonlocal bvp with self adjoint Operator
Abstract and Applied Analysis, 2013Co-Authors: Allaberen Ashyralyev, Ozgur YildirimAbstract:A third order of accuracy absolutely stable difference schemes is presented for nonlocal boundary value hyperbolic problem of the differential equations in a Hilbert space with self-adjoint positive Definite Operator . Stability estimates for solution of the difference scheme are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions is considered.
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second order of accuracy stable difference schemes for hyperbolic problems subject to nonlocal conditions with self adjoint Operator
NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics, 2011Co-Authors: Allaberen Ashyralyev, Ozgur YildirimAbstract:In the present paper, two new second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value problem {d2u(t)dt2+Au(t) = f(t) (0≤t≤1),u(0) = ∑ j = 1nαju(λj)+φ,ut(0) = ∑ j = 1nβjut(λj)+ψ,0<λ1<λ2<…<λn≤1 for differential equations in a Hilbert space H with the self‐adjoint positive Definite Operator A. The stability estimates for the solutions of these difference schemes are established. In practice, one‐dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of difference schemes for the nonlocal boundary value hyperbolic problems are obtained and the numerical results are presented to support our theoretical statements.
Allaberen Ashyralyev - One of the best experts on this subject based on the ideXlab platform.
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on stability of a third order of accuracy difference scheme for hyperbolic nonlocal bvp with self adjoint Operator
Abstract and Applied Analysis, 2013Co-Authors: Allaberen Ashyralyev, Ozgur YildirimAbstract:A third order of accuracy absolutely stable difference schemes is presented for nonlocal boundary value hyperbolic problem of the differential equations in a Hilbert space with self-adjoint positive Definite Operator . Stability estimates for solution of the difference scheme are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions is considered.
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second order of accuracy stable difference schemes for hyperbolic problems subject to nonlocal conditions with self adjoint Operator
NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics, 2011Co-Authors: Allaberen Ashyralyev, Ozgur YildirimAbstract:In the present paper, two new second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value problem {d2u(t)dt2+Au(t) = f(t) (0≤t≤1),u(0) = ∑ j = 1nαju(λj)+φ,ut(0) = ∑ j = 1nβjut(λj)+ψ,0<λ1<λ2<…<λn≤1 for differential equations in a Hilbert space H with the self‐adjoint positive Definite Operator A. The stability estimates for the solutions of these difference schemes are established. In practice, one‐dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of difference schemes for the nonlocal boundary value hyperbolic problems are obtained and the numerical results are presented to support our theoretical statements.
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on modified crank nicholson difference schemes for stochastic parabolic equation
Numerical Functional Analysis and Optimization, 2008Co-Authors: Allaberen AshyralyevAbstract:We consider the modified Crank–Nicholson difference schemes for the approximate solution of the initial value Cauchy problem for stochastic parabolic equation in a Hilbert space H with the self-adjoint positive Definite Operator A. Here: (1) w t is a standard Wiener process given on the probability space (Ω,F,P). (2) f(t) is an element of space that consists of H 1-value processes for which the condition is satisfied. The estimate of convergence for the solution of these difference schemes is obtained.
Habtu Zegeye - One of the best experts on this subject based on the ideXlab platform.
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implicit approximation scheme for the solution of k positive Definite Operator equation
Abstract and Applied Analysis, 2014Co-Authors: Naseer Shahzad, Arif Rafiq, Habtu ZegeyeAbstract:We construct an implicit sequence suitable for the approximation of solutions of K-positive Definite Operator equations in real Banach spaces. Furthermore, implicit error estimate is obtained and the convergence is shown to be faster in comparsion to the explicit error estimate obtained by Osilike and Udomene (2001).
Naseer Shahzad - One of the best experts on this subject based on the ideXlab platform.
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implicit approximation scheme for the solution of k positive Definite Operator equation
Abstract and Applied Analysis, 2014Co-Authors: Naseer Shahzad, Arif Rafiq, Habtu ZegeyeAbstract:We construct an implicit sequence suitable for the approximation of solutions of K-positive Definite Operator equations in real Banach spaces. Furthermore, implicit error estimate is obtained and the convergence is shown to be faster in comparsion to the explicit error estimate obtained by Osilike and Udomene (2001).
Zhuangyi Liu - One of the best experts on this subject based on the ideXlab platform.
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regularity analysis for an abstract thermoelastic system with inertial term
ESAIM: Control Optimisation and Calculus of Variations, 2021Co-Authors: Zhaobin Kuang, Zhuangyi Liu, Hugo Fernandez D SareAbstract:In this paper, we provide a complete regularity analysis for the following abstract thermoelastic system with inertial term where A is a self-adjoint, positive Definite Operator on a complex Hilbert space H and It is regarded as the second part of Fernandez Sare et al. [J. Diff. Eqs. 267 (2019) 7085–7134]. where the asymptotic stability of this model was investigated. We are able to decompose the region E into three parts where the associated semigroups are analytic, of Gevrey classes of specific order, and non-smoothing, respectively. Moreover, by a detailed spectral analysis, we will show that the orders of Gevrey class are sharp, under proper conditions. We also show that the orders of polynomial stability obtained in Fernandez Sare et al. [J. Diff. Eqs. 267 (2019) 7085–7134] are optimal.
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Regularity analysis for an abstract thermoelastic system with inertial term
'EDP Sciences', 2021Co-Authors: Zhaobin Kuang, Zhuangyi Liu, Hugo Fernandez D SareAbstract:In this paper, we provide a complete regularity analysis for the following abstract thermoelastic system with inertial term ρutt+lAγu tt+σAu-mAαθ =0, cθt+mAαu t+kAβθ =0, u(0)=u0,ut(0)=v0, θ(0)=θ0, where A is a self-adjoint, positive Definite Operator on a complex Hilbert space H and (α,β,γ)∈E=[0,β+12 ]×[0,1]×[0,1]. It is regarded as the second part of Fernández Sare et al. [J. Diff. Eqs. 267 (2019) 7085–7134]. where the asymptotic stability of this model was investigated. We are able to decompose the region E into three parts where the associated semigroups are analytic, of Gevrey classes of specific order, and non-smoothing, respectively. Moreover, by a detailed spectral analysis, we will show that the orders of Gevrey class are sharp, under proper conditions. We also show that the orders of polynomial stability obtained in Fernández Sare et al. [J. Diff. Eqs. 267 (2019) 7085–7134] are optimal
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regularity analysis for an abstract system of coupled hyperbolic and parabolic equations
Journal of Differential Equations, 2015Co-Authors: Jianghao Hao, Zhuangyi Liu, Jiongmin YongAbstract:Abstract In this paper, we provide a complete regularity analysis for the following abstract system of coupled hyperbolic and parabolic equations { u t t = − A u + γ A α w , w t = − γ A α u t − k A β w , u ( 0 ) = u 0 , u t ( 0 ) = v 0 , w ( 0 ) = w 0 , where A is a self-adjoint, positive Definite Operator on a complex Hilbert space H , and ( α , β ) ∈ [ 0 , 1 ] × [ 0 , 1 ] . We are able to decompose the unit square of the parameter ( α , β ) into three parts where the semigroup associated with the system is analytic, of specific order Gevrey classes, and non-smoothing, respectively. Moreover, we will show that the orders of Gevrey class is sharp, under proper conditions.
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stability of an abstract system of coupled hyperbolic and parabolic equations
Zeitschrift für Angewandte Mathematik und Physik, 2013Co-Authors: Jianghao Hao, Zhuangyi LiuAbstract:In this paper, we provide a complete stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$\begin{array}{ll}\;\;u_{tt} = -Au + \gamma A^{\alpha} \theta,\\ \quad \theta_t = -\gamma A^{\alpha}u_t - kA^{\beta}\theta,\\ u(0) = u_0, \quad u_t(0) = v_0, \quad \theta(0) = \theta_0\end{array}$$ where A is a self-adjoint, positive Definite Operator on a Hilbert space H. For \({(\alpha,\beta) \in [0,1] \times [0,1]}\) , the region of exponential stability had been identified in Ammar-Khodja et al. (ESAIM Control Optim Calc Var 4:577–593,1999). Our contribution is to show that the rest of the region can be classified as region of polynomial stability and region of instability. Moreover, we obtain the optimality of the order of polynomial stability.